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Complexity measures

Note

Be sure you have gone through the Tutorial before going through the API here to have a good idea of the terminology used in ComplexityMeasures.jl.

Complexity measures API

The complexity measure API is defined by the complexity function, which may take as an input an ComplexityEstimator. The function complexity_normalized is also useful.

Approximate entropy

ComplexityMeasures.ApproximateEntropyType
ApproximateEntropy <: ComplexityEstimator
+Complexity measures · ComplexityMeasures.jl

Complexity measures

Note

Be sure you have gone through the Tutorial before going through the API here to have a good idea of the terminology used in ComplexityMeasures.jl.

Complexity measures API

The complexity measure API is defined by the complexity function, which may take as an input an ComplexityEstimator. The function complexity_normalized is also useful.

Approximate entropy

ComplexityMeasures.ApproximateEntropyType
ApproximateEntropy <: ComplexityEstimator
 ApproximateEntropy([x]; r = 0.2std(x), kwargs...)

An estimator for the approximate entropy (Pincus, 1991) complexity measure, used with complexity.

The keyword argument r is mandatory if an input timeseries x is not provided.

Keyword arguments

  • r::Real: The radius used when querying for nearest neighbors around points. Its value should be determined from the input data, for example as some proportion of the standard deviation of the data.
  • m::Int = 2: The embedding dimension.
  • τ::Int = 1: The embedding lag.
  • base::Real = MathConstants.e: The base to use for the logarithm. Pincus (1991) uses the natural logarithm.

Description

Approximate entropy (ApEn) is defined as

\[ApEn(m ,r) = \lim_{N \to \infty} \left[ \phi(x, m, r) - \phi(x, m + 1, r) \right].\]

Approximate entropy is estimated for a timeseries x, by first embedding x using embedding dimension m and embedding lag τ, then searching for similar vectors within tolerance radius r, using the estimator described below, with logarithms to the given base (natural logarithm is used in Pincus, 1991).

Specifically, for a finite-length timeseries x, an estimator for $ApEn(m ,r)$ is

\[ApEn(m, r, N) = \phi(x, m, r, N) - \phi(x, m + 1, r, N),\]

where N = length(x) and

\[\phi(x, k, r, N) = \dfrac{1}{N-(k-1)\tau} \sum_{i=1}^{N - (k-1)\tau} \log{\left( \sum_{j = 1}^{N-(k-1)\tau} \dfrac{\theta(d({\bf x}_i^m, {\bf x}_j^m) \leq r)}{N-(k-1)\tau} - \right)}.\]

Here, $\theta(\cdot)$ returns 1 if the argument is true and 0 otherwise, $d({\bf x}_i, {\bf x}_j)$ returns the Chebyshev distance between vectors ${\bf x}_i$ and ${\bf x}_j$, and the k-dimensional embedding vectors are constructed from the input timeseries $x(t)$ as

\[{\bf x}_i^k = (x(i), x(i+τ), x(i+2τ), \ldots, x(i+(k-1)\tau)).\]

Flexible embedding lag

In the original paper, they fix τ = 1. In our implementation, the normalization constant is modified to account for embeddings with τ != 1.

source

Sample entropy

ComplexityMeasures.SampleEntropyType
SampleEntropy([x]; r = 0.2std(x), kwargs...) <: ComplexityEstimator

An estimator for the sample entropy complexity measure (Richman and Moorman, 2000), used with complexity and complexity_normalized.

The keyword argument r is mandatory if an input timeseries x is not provided.

Keyword arguments

  • r::Real: The radius used when querying for nearest neighbors around points. Its value should be determined from the input data, for example as some proportion of the standard deviation of the data.
  • m::Int = 2: The embedding dimension.
  • τ::Int = 1: The embedding lag.

Description

An estimator for sample entropy using radius r, embedding dimension m, and embedding lag τ is

\[SampEn(m,r, N) = -\ln{\dfrac{A(r, N)}{B(r, N)}}.\]

Here,

\[\begin{aligned} + \right)}.\]

Here, $\theta(\cdot)$ returns 1 if the argument is true and 0 otherwise, $d({\bf x}_i, {\bf x}_j)$ returns the Chebyshev distance between vectors ${\bf x}_i$ and ${\bf x}_j$, and the k-dimensional embedding vectors are constructed from the input timeseries $x(t)$ as

\[{\bf x}_i^k = (x(i), x(i+τ), x(i+2τ), \ldots, x(i+(k-1)\tau)).\]

Flexible embedding lag

In the original paper, they fix τ = 1. In our implementation, the normalization constant is modified to account for embeddings with τ != 1.

source

Sample entropy

ComplexityMeasures.SampleEntropyType
SampleEntropy([x]; r = 0.2std(x), kwargs...) <: ComplexityEstimator

An estimator for the sample entropy complexity measure (Richman and Moorman, 2000), used with complexity and complexity_normalized.

The keyword argument r is mandatory if an input timeseries x is not provided.

Keyword arguments

  • r::Real: The radius used when querying for nearest neighbors around points. Its value should be determined from the input data, for example as some proportion of the standard deviation of the data.
  • m::Int = 2: The embedding dimension.
  • τ::Int = 1: The embedding lag.

Description

An estimator for sample entropy using radius r, embedding dimension m, and embedding lag τ is

\[SampEn(m,r, N) = -\ln{\dfrac{A(r, N)}{B(r, N)}}.\]

Here,

\[\begin{aligned} B(r, m, N) = \sum_{i = 1}^{N-m\tau} \sum_{j = 1, j \neq i}^{N-m\tau} \theta(d({\bf x}_i^m, {\bf x}_j^m) \leq r) \\ A(r, m, N) = \sum_{i = 1}^{N-m\tau} \sum_{j = 1, j \neq i}^{N-m\tau} \theta(d({\bf x}_i^{m+1}, {\bf x}_j^{m+1}) \leq r) \\ -\end{aligned},\]

where $\theta(\cdot)$ returns 1 if the argument is true and 0 otherwise, and $d(x, y)$ computes the Chebyshev distance between $x$ and $y$, and ${\bf x}_i^{m}$ and ${\bf x}_i^{m+1}$ are m-dimensional and m+1-dimensional embedding vectors, where k-dimensional embedding vectors are constructed from the input timeseries $x(t)$ as

\[{\bf x}_i^k = (x(i), x(i+τ), x(i+2τ), \ldots, x(i+(k-1)\tau)).\]

Quoting Richman & Moorman (2002): "SampEn(m,r,N) will be defined except when B = 0, in which case no regularity has been detected, or when A = 0, which corresponds to a conditional probability of 0 and an infinite value of SampEn(m,r,N)". In these cases, NaN is returned.

If computing the normalized measure, then the resulting sample entropy is on [0, 1].

Flexible embedding lag

The original algorithm fixes τ = 1. All formulas here are modified to account for any τ.

See also: entropy_sample.

source

Missing dispersion patterns

ComplexityMeasures.MissingDispersionPatternsType
MissingDispersionPatterns <: ComplexityEstimator
-MissingDispersionPatterns(o = Dispersion()) → mdp

An estimator for the number of missing dispersion patterns (MDP), a complexity measure which can be used to detect nonlinearity in time series (Zhou et al., 2023).

Used with complexity or complexity_normalized.

Description

When used with complexity, complexity(mdp) is syntactically equivalent with just missing_outcomes(o). When used with complexity_normalized, the normalization is simply missing_outcomes(o)/total_outcomes(o).

Encoding

Dispersion's linear mapping from CDFs to integers is based on equidistant partitioning of the interval [0, 1]. This is slightly different from Zhou et al. (2023), which uses the linear mapping $s_i := \text{round}(y + 0.5)$.

Usage

In Zhou et al. (2023), MissingDispersionPatterns is used to detect nonlinearity in time series by comparing the MDP for a time series x to values for an ensemble of surrogates of x, as per the standard analysis of TimeseriesSurrogates.jl If the MDP value of $x$ is significantly larger than some high quantile of the surrogate distribution, then it is taken as evidence for nonlinearity.

See also: Dispersion, ReverseDispersion, total_outcomes.

source

Reverse dispersion entropy

ComplexityMeasures.ReverseDispersionType
ReverseDispersion <: ComplexityEstimator
+\end{aligned},\]

where $\theta(\cdot)$ returns 1 if the argument is true and 0 otherwise, and $d(x, y)$ computes the Chebyshev distance between $x$ and $y$, and ${\bf x}_i^{m}$ and ${\bf x}_i^{m+1}$ are m-dimensional and m+1-dimensional embedding vectors, where k-dimensional embedding vectors are constructed from the input timeseries $x(t)$ as

\[{\bf x}_i^k = (x(i), x(i+τ), x(i+2τ), \ldots, x(i+(k-1)\tau)).\]

Quoting Richman & Moorman (2002): "SampEn(m,r,N) will be defined except when B = 0, in which case no regularity has been detected, or when A = 0, which corresponds to a conditional probability of 0 and an infinite value of SampEn(m,r,N)". In these cases, NaN is returned.

If computing the normalized measure, then the resulting sample entropy is on [0, 1].

Flexible embedding lag

The original algorithm fixes τ = 1. All formulas here are modified to account for any τ.

See also: entropy_sample.

source

Missing dispersion patterns

ComplexityMeasures.MissingDispersionPatternsType
MissingDispersionPatterns <: ComplexityEstimator
+MissingDispersionPatterns(o = Dispersion()) → mdp

An estimator for the number of missing dispersion patterns (MDP), a complexity measure which can be used to detect nonlinearity in time series (Zhou et al., 2023).

Used with complexity or complexity_normalized.

Description

When used with complexity, complexity(mdp) is syntactically equivalent with just missing_outcomes(o). When used with complexity_normalized, the normalization is simply missing_outcomes(o)/total_outcomes(o).

Encoding

Dispersion's linear mapping from CDFs to integers is based on equidistant partitioning of the interval [0, 1]. This is slightly different from Zhou et al. (2023), which uses the linear mapping $s_i := \text{round}(y + 0.5)$.

Usage

In Zhou et al. (2023), MissingDispersionPatterns is used to detect nonlinearity in time series by comparing the MDP for a time series x to values for an ensemble of surrogates of x, as per the standard analysis of TimeseriesSurrogates.jl If the MDP value of $x$ is significantly larger than some high quantile of the surrogate distribution, then it is taken as evidence for nonlinearity.

See also: Dispersion, ReverseDispersion, total_outcomes.

source

Reverse dispersion entropy

ComplexityMeasures.ReverseDispersionType
ReverseDispersion <: ComplexityEstimator
 ReverseDispersion(; c = 3, m = 2, τ = 1, check_unique = true)

Estimator for the reverse dispersion entropy complexity measure (Li et al., 2019).

Description

Li et al. (2019) defines the reverse dispersion entropy as

\[H_{rde} = \sum_{i = 1}^{c^m} \left(p_i - \dfrac{1}{{c^m}} \right)^2 = -\left( \sum_{i=1}^{c^m} p_i^2 \right) - \dfrac{1}{c^{m}}\]

where the probabilities $p_i$ are obtained precisely as for the Dispersion probability estimator. Relative frequencies of dispersion patterns are computed using the given encoding scheme , which defaults to encoding using the normal cumulative distribution function (NCDF), as implemented by GaussianCDFEncoding, using embedding dimension m and embedding delay τ. Recommended parameter values(Li et al., 2019) are m ∈ [2, 3], τ = 1 for the embedding, and c ∈ [3, 4, …, 8] categories for the Gaussian mapping.

If normalizing, then the reverse dispersion entropy is normalized to [0, 1].

The minimum value of $H_{rde}$ is zero and occurs precisely when the dispersion pattern distribution is flat, which occurs when all $p_i$s are equal to $1/c^m$. Because $H_{rde} \geq 0$, $H_{rde}$ can therefore be said to be a measure of how far the dispersion pattern probability distribution is from white noise.

Data requirements

The input must have more than one unique element for the default GaussianCDFEncoding to be well-defined. Li et al. (2019) recommends that x has at least 1000 data points.

If check_unique == true (default), then it is checked that the input has more than one unique value. If check_unique == false and the input only has one unique element, then a InexactError is thrown when trying to compute probabilities.

source

Statistical complexity

ComplexityMeasures.StatisticalComplexityType
StatisticalComplexity <: ComplexityEstimator
+\left( \sum_{i=1}^{c^m} p_i^2 \right) - \dfrac{1}{c^{m}}\]

where the probabilities $p_i$ are obtained precisely as for the Dispersion probability estimator. Relative frequencies of dispersion patterns are computed using the given encoding scheme , which defaults to encoding using the normal cumulative distribution function (NCDF), as implemented by GaussianCDFEncoding, using embedding dimension m and embedding delay τ. Recommended parameter values(Li et al., 2019) are m ∈ [2, 3], τ = 1 for the embedding, and c ∈ [3, 4, …, 8] categories for the Gaussian mapping.

If normalizing, then the reverse dispersion entropy is normalized to [0, 1].

The minimum value of $H_{rde}$ is zero and occurs precisely when the dispersion pattern distribution is flat, which occurs when all $p_i$s are equal to $1/c^m$. Because $H_{rde} \geq 0$, $H_{rde}$ can therefore be said to be a measure of how far the dispersion pattern probability distribution is from white noise.

Data requirements

The input must have more than one unique element for the default GaussianCDFEncoding to be well-defined. Li et al. (2019) recommends that x has at least 1000 data points.

If check_unique == true (default), then it is checked that the input has more than one unique value. If check_unique == false and the input only has one unique element, then a InexactError is thrown when trying to compute probabilities.

source

Statistical complexity

ComplexityMeasures.StatisticalComplexityType
StatisticalComplexity <: ComplexityEstimator
 StatisticalComplexity(; kwargs...)

An estimator for the statistical complexity and entropy, originally by (Rosso et al., 2007) and generalized by Rosso et al. (2013).

Our implementation extends the generalization to any valid distance metric, any OutcomeSpace with a priori known total_outcomes, any ProbabilitiesEstimator, and any normalizable discrete InformationMeasure.

Used with complexity.

Keyword arguments

  • o::OutcomeSpace = OrdinalPatterns{3}(). The OutcomeSpace, which controls how the input data are discretized.
  • pest::ProbabilitiesEstimator = RelativeAmount(): The ProbabilitiesEstimator used to estimate probabilities over the discretized input data.
  • hest = Renyi(): A DiscreteInfoEstimator or an InformationMeasure. Any information measure that defines information_maximum is valid here including extropies. The measure will be estimated using the PlugIn estimator if not given an estimator.
  • dist <: SemiMetric = JSDivergence(): The distance measure (from Distances.jl) to use for estimating the distance between the estimated probability distribution and a uniform distribution with the same maximal number of outcomes.

Description

Statistical complexity is defined as

\[C_q[P] = \mathcal{H}_q\cdot \mathcal{Q}_q[P],\]

where $Q_q$ is a "disequilibrium" obtained from a distance-measure and $H_q$ a disorder measure. In the original paper(Rosso et al., 2007), this complexity measure was defined via an ordinal pattern-based probability distribution (see OrdinalPatterns), using Shannon entropy as the information measure, and the Jensen-Shannon divergence as a distance measure.

Our implementation is a further generalization of the complexity measure developed in Rosso et al. (2013). We let $H_q$be any normalizable InformationMeasure, e.g. Shannon, Renyi or Tsallis entropy, and we let $Q_q$ be either on the Euclidean, Wooters, Kullback, q-Kullback, Jensen or q-Jensen distance as

\[Q_q[P] = Q_q^0\cdot D[P, P_e],\]

where $D[P, P_e]$ is the distance between the obtained distribution $P$ and a uniform distribution with the same maximum number of bins, measured by the distance measure dist.

Usage

The statistical complexity is exclusively used in combination with the chosen information measure (typically an entropy). The estimated value of the information measure can be accessed as a Ref value of the struct as

x = randn(100)
 c = StatisticalComplexity()
 compl = complexity(c, x)
-entr = first(entropy_complexity(c, x)) # both complexity and entropy value

complexity(c::StatisticalComplexity, x) returns only the statistical complexity. To obtain both the value of the entropy (or other information measure) and the statistical complexity together as a Tuple, use the wrapper entropy_complexity.

See also: entropy_complexity_curves.

source
ComplexityMeasures.entropy_complexity_curvesFunction
entropy_complexity_curves(c::StatisticalComplexity;
-    num_max=1, num_min=1000) -> (min_entropy_complexity, max_entropy_complexity)

Calculate the maximum complexity-entropy curve for the statistical complexity according to Rosso et al. (2007) for num_max * total_outcomes(c.o) different values of the normalized information measure of choice (in case of the maximum complexity curves) and num_min different values of the normalized information measure of choice (in case of the minimum complexity curve).

This function can also be used to compute the maximum "complexity-extropy curve" if c.hest is e.g. ShannonExtropy, which is the equivalent of the complexity-entropy curves, but using extropy instead of entropy.

Description

The way the statistical complexity is designed, there is a minimum and maximum possible complexity for data with a given value of an information measure. The calculation time of the maximum complexity curve grows as O(total_outcomes(c.o)^2), and thus takes very long for high numbers of outcomes. This function is inspired by S. Sippels implementation in statcomp (Sippel et al., 2016).

This function will work with any ProbabilitiesEstimator where total_outcomes is known a priori.

source

Lempel-Ziv complexity

ComplexityMeasures.LempelZiv76Type
LempelZiv76 <: ComplexityEstimator
-LempelZiv76()

The Lempel-Ziv, or LempelZiv76, complexity measure (Lempel and Ziv, 1976), which is used with complexity and complexity_normalized.

For results to be comparable across sequences with different length, use the normalized version. Normalized LempelZiv76-complexity is implemented as given in Amigó et al. (2004). The normalized measure is close to zero for very regular signals, while for random sequences, it is close to 1 with high probability[Amigó2004]. Note: the normalized LempelZiv76 complexity can be higher than 1[Amigó2004].

The LempelZiv76 measure applies only to binary sequences, i.e. sequences with a two-element alphabet (precisely two distinct outcomes). For performance optimization, we do not check the number of unique elements in the input. If your input sequence is not binary, you must encode it first using one of the implemented Encoding schemes (or encode your data manually).

source

Bubble entropy

ComplexityMeasures.BubbleEntropyType
BubbleEntropy <: ComplexityEstimator
+entr = first(entropy_complexity(c, x)) # both complexity and entropy value

complexity(c::StatisticalComplexity, x) returns only the statistical complexity. To obtain both the value of the entropy (or other information measure) and the statistical complexity together as a Tuple, use the wrapper entropy_complexity.

See also: entropy_complexity_curves.

source
ComplexityMeasures.entropy_complexity_curvesFunction
entropy_complexity_curves(c::StatisticalComplexity;
+    num_max=1, num_min=1000) -> (min_entropy_complexity, max_entropy_complexity)

Calculate the maximum complexity-entropy curve for the statistical complexity according to Rosso et al. (2007) for num_max * total_outcomes(c.o) different values of the normalized information measure of choice (in case of the maximum complexity curves) and num_min different values of the normalized information measure of choice (in case of the minimum complexity curve).

This function can also be used to compute the maximum "complexity-extropy curve" if c.hest is e.g. ShannonExtropy, which is the equivalent of the complexity-entropy curves, but using extropy instead of entropy.

Description

The way the statistical complexity is designed, there is a minimum and maximum possible complexity for data with a given value of an information measure. The calculation time of the maximum complexity curve grows as O(total_outcomes(c.o)^2), and thus takes very long for high numbers of outcomes. This function is inspired by S. Sippels implementation in statcomp (Sippel et al., 2016).

This function will work with any ProbabilitiesEstimator where total_outcomes is known a priori.

source

Lempel-Ziv complexity

ComplexityMeasures.LempelZiv76Type
LempelZiv76 <: ComplexityEstimator
+LempelZiv76()

The Lempel-Ziv, or LempelZiv76, complexity measure (Lempel and Ziv, 1976), which is used with complexity and complexity_normalized.

For results to be comparable across sequences with different length, use the normalized version. Normalized LempelZiv76-complexity is implemented as given in Amigó et al. (2004). The normalized measure is close to zero for very regular signals, while for random sequences, it is close to 1 with high probability[Amigó2004]. Note: the normalized LempelZiv76 complexity can be higher than 1[Amigó2004].

The LempelZiv76 measure applies only to binary sequences, i.e. sequences with a two-element alphabet (precisely two distinct outcomes). For performance optimization, we do not check the number of unique elements in the input. If your input sequence is not binary, you must encode it first using one of the implemented Encoding schemes (or encode your data manually).

source

Bubble entropy

ComplexityMeasures.BubbleEntropyType
BubbleEntropy <: ComplexityEstimator
 BubbleEntropy(; m = 3, τ = 1, definition = Renyi(q = 2))

The BubbleEntropy complexity estimator (Manis et al., 2017) is just a difference between two entropies, each computed with the BubbleSortSwaps outcome space, for embedding dimensions m + 1 and m, respectively.

Manis et al. (2017) use the Renyi entropy of order q = 2 as the information measure definition, but here you can use any InformationMeasure. Manis et al. (2017) formulates the "bubble entropy" as the normalized measure below, while here you can also compute the unnormalized measure.

Definition

For input data x, the "bubble entropy" is computed by first embedding the input data using embedding dimension m and embedding delay τ (call the embedded pts y), and then computing the difference between the two entropies:

\[BubbleEn_T(τ) = H_T(y, m + 1) - H_T(y, m)\]

where $H_T(y, m)$ and $H_T(y, m + 1)$ are entropies of type $T$ (e.g. Renyi) computed with the input data x embedded to dimension $m$ and $m+1$, respectively. Use complexity to compute this non-normalized version. Use complexity_normalized to compute the normalized difference of entropies:

\[BubbleEn_H(τ)^{norm} = \dfrac{H_T(x, m + 1) - H_T(x, m)}{max(H_T(x, m + 1)) - max(H_T(x, m))},\]

where the maximum of the entropies for dimensions m and m + 1 are computed using information_maximum.

Example

using ComplexityMeasures
 x = rand(1000)
 est = BubbleEntropy(m = 5, τ = 3)
-complexity(est, x)
source
+complexity(est, x)
source
diff --git a/previews/PR430/convenience/index.html b/previews/PR430/convenience/index.html index 28cb84e0..9dcd0951 100644 --- a/previews/PR430/convenience/index.html +++ b/previews/PR430/convenience/index.html @@ -1,7 +1,7 @@ Convenience functions · ComplexityMeasures.jl

Convenience functions

Common entropy-based measures

We provide a few convenience functions for widely used names for entropy or "entropy-like" quantities. Other arbitrary specialized convenience functions can easily be defined in a couple lines of code.

We emphasize that these functions really aren't anything more than 2-lines-of-code wrappers that call information with the appropriate OutcomeSpace and InformationMeasure.

ComplexityMeasures.entropy_waveletFunction
entropy_wavelet(x; wavelet = Wavelets.WT.Daubechies{12}(), base = 2)

Compute the wavelet entropy. This function is just a convenience call to:

est = WaveletOverlap(wavelet)
-information(Shannon(base), est, x)

See WaveletOverlap for more info.

source
ComplexityMeasures.entropy_dispersionFunction
entropy_dispersion(x; base = 2, kwargs...)

Compute the dispersion entropy. This function is just a convenience call to:

est = Dispersion(kwargs...)
-information(Shannon(base), est, x)

See Dispersion for more info.

source
ComplexityMeasures.entropy_approxFunction
entropy_approx(x; m = 2, τ = 1, r = 0.2 * Statistics.std(x), base = MathConstants.e)

Convenience syntax for computing the approximate entropy (Pincus, 1991) for timeseries x.

This is just a wrapper for complexity(ApproximateEntropy(; m, τ, r, base), x) (see also ApproximateEntropy).

source
ComplexityMeasures.entropy_distributionFunction
entropy_distribution(x; τ = 1, m = 3, n = 3, base = 2)

Compute the distribution entropy (Li et al., 2015) of x using embedding dimension m with delay/lag τ, using the Chebyshev distance metric, and using an n-element equally-spaced binning over the distribution of distances to estimate probabilities.

This function is just a convenience call to:

x = rand(1000000)
+information(Shannon(base), est, x)

See OrdinalPatterns for more info. Similarly, one can use WeightedOrdinalPatterns or AmplitudeAwareOrdinalPatterns for the weighted/amplitude-aware versions.

source
ComplexityMeasures.entropy_waveletFunction
entropy_wavelet(x; wavelet = Wavelets.WT.Daubechies{12}(), base = 2)

Compute the wavelet entropy. This function is just a convenience call to:

est = WaveletOverlap(wavelet)
+information(Shannon(base), est, x)

See WaveletOverlap for more info.

source
ComplexityMeasures.entropy_dispersionFunction
entropy_dispersion(x; base = 2, kwargs...)

Compute the dispersion entropy. This function is just a convenience call to:

est = Dispersion(kwargs...)
+information(Shannon(base), est, x)

See Dispersion for more info.

source
ComplexityMeasures.entropy_approxFunction
entropy_approx(x; m = 2, τ = 1, r = 0.2 * Statistics.std(x), base = MathConstants.e)

Convenience syntax for computing the approximate entropy (Pincus, 1991) for timeseries x.

This is just a wrapper for complexity(ApproximateEntropy(; m, τ, r, base), x) (see also ApproximateEntropy).

source
ComplexityMeasures.entropy_distributionFunction
entropy_distribution(x; τ = 1, m = 3, n = 3, base = 2)

Compute the distribution entropy (Li et al., 2015) of x using embedding dimension m with delay/lag τ, using the Chebyshev distance metric, and using an n-element equally-spaced binning over the distribution of distances to estimate probabilities.

This function is just a convenience call to:

x = rand(1000000)
 o = SequentialPairDistances(x, n, m, τ, metric = Chebyshev())
-h = information(Shannon(base = 2), o, x)

See SequentialPairDistances for more info.

source
+h = information(Shannon(base = 2), o, x)

See SequentialPairDistances for more info.

source diff --git a/previews/PR430/devdocs/index.html b/previews/PR430/devdocs/index.html index c821af77..f0bf94ac 100644 --- a/previews/PR430/devdocs/index.html +++ b/previews/PR430/devdocs/index.html @@ -2,4 +2,4 @@ ComplexityMeasures.jl Dev Docs · ComplexityMeasures.jl

ComplexityMeasures.jl Dev Docs

Good practices in developing a code base apply in every Pull Request. The Good Scientific Code Workshop is worth checking out for this.

All PRs contributing new functionality must be well tested and well documented. You only need to add tests for methods that you explicitly extended.

Adding a new OutcomeSpace

Mandatory steps

  1. Decide on the outcome space and how the estimator will map probabilities to outcomes.
  2. Define your type and make it subtype OutcomeSpace.
  3. Add a docstring to your type following the style of the docstrings of other estimators.
  4. If suitable, the estimator may be able to operate based on Encodings. If so, it is preferred to implement an Encoding subtype and extend the methods encode and decode. This will allow your outcome space to be used with a larger span of entropy and complexity methods without additional effort. Have a look at the file defining OrdinalPatterns for an idea of how this works.

If your new outcome space is counting-based, then

  1. Implement dispatch for counts_and_outcomes for your OutcomeSpace type. If the outcomes do not come for free, then instead you can extend counts and then explicitly add another method for counts_and_outcomes that calls counts first and then decodes the outcomes. Follow existing implementations for guidelines (see for example source code for Dispersion).
  2. Implement dispatch for codify. This will ensure that the outcome space also works automatically with any discrete estimators in the downstream CausalityTools.jl.

If your new outcome space is not counting-based, then

  1. Implement dispatch for probabilities_and_outcomes for your OutcomeSpace type. If the outcomes do not come for free, then instead you can extend probabilities and then explicitly add another method for probabilities_and_outcomes that calls probabilities first and then decodes the outcomes. Follow existing implementations for guidelines (see for example source code for NaiveKernel).

Finally,

  1. Implement dispatch for outcome_space and your OutcomeSpace type. The return value of outcome_space must be sorted (as in the default behavior of sort, in ascending order).
  2. Add your outcome space type to the table list in the documentation string of OutcomeSpace. If you made an encoding, also add it to corresponding table in the encodings section.

Optional steps

The following methods may be extended for your OutcomeSpace if doing so leads to performance benefits.

  1. total_outcomes. By default it returns the length of outcome_space. This is the function that most typically has performance benefits if implemented explicitly, so most existing estimators extend it by default.

Adding a new ProbabilitiesEstimator

Mandatory steps

  1. Define your type and make it subtype ProbabilitiesEstimator.
  2. Add a docstring to your type following the style of the docstrings of other ProbabilitiesEstimators.
  3. Implement dispatch for probabilities for your ProbabilitiesEstimator type. You'll then get probabilities_and_outcomes for free.
  4. Implement dispatch for allprobabilities_and_outcomes for your ProbabilitiesEstimator type.
  5. Add your new ProbabilitiesEstimator type to the list of probabilities estimators in the probabilities estimators documentation section.

Adding a new InformationMeasureEstimator

The type implementation should follow the declared API of InformationMeasureEstimator. If the type is a discrete measure, then extend information(e::YourType, p::Probabilities). If it is a differential measure, then extend information(e::YourType, x::InputData).

ComplexityMeasures.InformationMeasureEstimatorType
InformationMeasureEstimator{I <: InformationMeasure}

The supertype of all information measure estimators. Its direct subtypes are DiscreteInfoEstimator and DifferentialInfoEstimator.

Since all estimators must reference a measure definition in some way, we made the following interface decisions:

  1. all estimators have as first type parameter I <: InformationMeasure
  2. all estimators reference the information measure in a definition field
  3. all estimators are defined using Base.@kwdef so that they may be initialized with the syntax Estimator(; definition = Shannon()) (or any other).

Any concrete subtypes must follow the above, e.g.:

Base.@kwdef struct MyEstimator{I <: InformationMeasure, X} <: DiscreteInfoEstimator{I}
     definition::I
     x::X
-end
Why separate the *definition* of a measure from *estimators* of a measure?

In real applications, we generally don't have access to the underlying probability mass functions or densities required to compute the various entropy or extropy definitons. Therefore, these information measures must be estimated from finite data. Estimating a particular measure (e.g. Shannon entropy) can be done in many ways, each with its own own pros and cons. We aim to provide a complete library of literature estimators of the various information measures (PRs are welcome!).

source

Adding a new InformationMeasure

This amounts to adding a new definition of an information measure, not an estimator. It de-facto means adding a method for the discrete Plug-In estimator.

Mandatory steps

  1. Define your information measure definition type and make it subtype InformationMeasure.
  2. Implement dispatch for information(def::YourType, p::Probabilities). This is the Plug-In estimator for the discrete measure.
  3. Add a docstring to your type following the style of the docstrings of other information measure definitions, and should include the mathematical definition of the measure.
  4. Add your information measure definition type to the list of definitions in the docs/src/information_measures.md documentation page.
  5. Add a reference to your information measure definition in the docstring for InformationMeasure.

Optional steps

  1. If the maximum value of your information measure type is analytically computable for a probability distribution with a known number of elements, implementing dispatch for information_maximum automatically enables information_normalized for your type.

Adding a new MultiScaleAlgorithm

A new MultiScaleAlgorithm is simply a new way of coarse-graining input time series across multiple scales.

Mandatory steps

  1. Define a new type YourNewMultiScaleType <: MultiScaleAlgorithm. This type will define how coarse graining is performed.
  2. Implement dispatch for downsample, which transforms the original time series into a vector of coarse-grained time series, one per scale (may be nested if needed).
  3. Implement dispatch for the internal apply_multiscale function.
  4. Add an entry for your new type in the multiscale.md file.
  5. Add tests for your new type. You specifically need to implement analytical tests that verify that downsample is correctly implemented. For API tests, simply copy the tests from e.g. tests/multiscale/Composite.jl, and replace the multiscale coarse-graining algorithm with an instance of your algorithm.
  6. Hooray! You're new coarse-graining procedure is integrated with the entire ComplexityMeasures.jl ecosystem!
+end
Why separate the *definition* of a measure from *estimators* of a measure?

In real applications, we generally don't have access to the underlying probability mass functions or densities required to compute the various entropy or extropy definitons. Therefore, these information measures must be estimated from finite data. Estimating a particular measure (e.g. Shannon entropy) can be done in many ways, each with its own own pros and cons. We aim to provide a complete library of literature estimators of the various information measures (PRs are welcome!).

source

Adding a new InformationMeasure

This amounts to adding a new definition of an information measure, not an estimator. It de-facto means adding a method for the discrete Plug-In estimator.

Mandatory steps

  1. Define your information measure definition type and make it subtype InformationMeasure.
  2. Implement dispatch for information(def::YourType, p::Probabilities). This is the Plug-In estimator for the discrete measure.
  3. Add a docstring to your type following the style of the docstrings of other information measure definitions, and should include the mathematical definition of the measure.
  4. Add your information measure definition type to the list of definitions in the docs/src/information_measures.md documentation page.
  5. Add a reference to your information measure definition in the docstring for InformationMeasure.

Optional steps

  1. If the maximum value of your information measure type is analytically computable for a probability distribution with a known number of elements, implementing dispatch for information_maximum automatically enables information_normalized for your type.

Adding a new MultiScaleAlgorithm

A new MultiScaleAlgorithm is simply a new way of coarse-graining input time series across multiple scales.

Mandatory steps

  1. Define a new type YourNewMultiScaleType <: MultiScaleAlgorithm. This type will define how coarse graining is performed.
  2. Implement dispatch for downsample, which transforms the original time series into a vector of coarse-grained time series, one per scale (may be nested if needed).
  3. Implement dispatch for the internal apply_multiscale function.
  4. Add an entry for your new type in the multiscale.md file.
  5. Add tests for your new type. You specifically need to implement analytical tests that verify that downsample is correctly implemented. For API tests, simply copy the tests from e.g. tests/multiscale/Composite.jl, and replace the multiscale coarse-graining algorithm with an instance of your algorithm.
  6. Hooray! You're new coarse-graining procedure is integrated with the entire ComplexityMeasures.jl ecosystem!
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    Probabilities: KL-divergence of histograms

    In this example we show how simple it is to compute the KL-divergence (or any other distance function for probability distributions) using ComplexityMeasures.jl. For simplicity, we will compute the KL-divergence between the ValueBinnings of two timeseries.

    Note that it is crucial to use allprobabilities_and_outcomes instead of probabilities_and_outcomes.

    using ComplexityMeasures
    +fig
    Example block output

    Probabilities: KL-divergence of histograms

    In this example we show how simple it is to compute the KL-divergence (or any other distance function for probability distributions) using ComplexityMeasures.jl. For simplicity, we will compute the KL-divergence between the ValueBinnings of two timeseries.

    Note that it is crucial to use allprobabilities_and_outcomes instead of probabilities_and_outcomes.

    using ComplexityMeasures
     
     N = 1000
     t = range(0, 20π; length=N)
    @@ -122,7 +122,7 @@
         axislegend()
     end
     
    -fig
    Example block output

    All estimators approach the true differential entropy, but those based on order statistics are negatively biased for small sample sizes.

    Rényi entropy

    Here, we see how the LeonenkoProzantoSavani estimator approaches the known target Renyi entropy of a multivariate normal distribution for increasing time series length. We'll consider the Rényi entropy with q = 2.

    using ComplexityMeasures
    +fig
    Example block output

    All estimators approach the true differential entropy, but those based on order statistics are negatively biased for small sample sizes.

    Rényi entropy

    Here, we see how the LeonenkoProzantoSavani estimator approaches the known target Renyi entropy of a multivariate normal distribution for increasing time series length. We'll consider the Rényi entropy with q = 2.

    using ComplexityMeasures
     import ComplexityMeasures: information # we're overriding this function in the example
     using CairoMakie, Statistics
     using Distributions: MvNormal
    @@ -282,7 +282,7 @@
     lines!(az, t, z; color = Cycled(3), label = "h=$(h=round(h_z, sigdigits = 5))");
     for a in (ax, ay, az); axislegend(a); end
     for a in (ax, ay); hidexdecorations!(a; grid=false); end
    -fig
    Example block output

    Discrete entropies: properties

    Here, we show the sensitivity of the various entropies to variations in their parameters.

    Curado entropy

    Here, we reproduce Figure 2 from Curado and Nobre (2004), showing how the Curado entropy changes as function of the parameter a for a range of two-element probability distributions given by Probabilities([p, 1 - p] for p in 1:0.0:0.01:1.0).

    using ComplexityMeasures, CairoMakie
    +fig
    Example block output

    Discrete entropies: properties

    Here, we show the sensitivity of the various entropies to variations in their parameters.

    Curado entropy

    Here, we reproduce Figure 2 from Curado and Nobre (2004), showing how the Curado entropy changes as function of the parameter a for a range of two-element probability distributions given by Probabilities([p, 1 - p] for p in 1:0.0:0.01:1.0).

    using ComplexityMeasures, CairoMakie
     bs = [1.0, 1.5, 2.0, 3.0, 4.0, 10.0]
     ps = [Probabilities([p, 1 - p]) for p = 0.0:0.01:1.0]
     hs = [[information(Curado(; b = b), p) for p in ps] for b in bs]
    @@ -359,7 +359,7 @@
     
     axislegend(position = :rc)
     ylims!(0, max(maximum(pes), 1))
    -fig
    Example block output

    Discrete entropy: normalized entropy for comparing different signals

    When comparing different signals or signals that have different length, it is best to normalize entropies so that the "complexity" or "disorder" quantification is directly comparable between signals. Here is an example based on the wavelet entropy example where we use the spectral entropy instead of the wavelet entropy:

    using ComplexityMeasures
    +fig
    Example block output

    Discrete entropy: normalized entropy for comparing different signals

    When comparing different signals or signals that have different length, it is best to normalize entropies so that the "complexity" or "disorder" quantification is directly comparable between signals. Here is an example based on the wavelet entropy example where we use the spectral entropy instead of the wavelet entropy:

    using ComplexityMeasures
     N1, N2, a = 101, 10001, 10
     
     for N in (N1, N2)
    @@ -375,28 +375,28 @@
         end
     end
    entropy: 0.3131510976800182, normalized: 0.05520585619046334.
     entropy: 1.2651873361559447, normalized: 0.22304169026158024.
    -entropy: 3.0462934077109933, normalized: 0.5370354344147666.
    +entropy: 2.1999011914159317, normalized: 0.3878237365485253.
     entropy: 7.57800737722389e-5, normalized: 6.1669977445812415e-6.
     entropy: 0.8397257036159657, normalized: 0.06833704775520644.
    -entropy: 5.325523428890741, normalized: 0.43339217474759445.

    You see that while the direct entropy values of noisy signal changes strongly with N but they are almost the same for the normalized version. For the regular signals, the entropy decreases nevertheless because the noise contribution of the Fourier computation becomes less significant.

    Spatiotemporal permutation entropy

    Usage of a SpatialOrdinalPatterns estimator is straightforward. Here we get the spatial permutation entropy of a 2D array (e.g., an image):

    using ComplexityMeasures
    +entropy: 5.376111081921851, normalized: 0.437509008192283.

    You see that while the direct entropy values of noisy signal changes strongly with N but they are almost the same for the normalized version. For the regular signals, the entropy decreases nevertheless because the noise contribution of the Fourier computation becomes less significant.

    Spatiotemporal permutation entropy

    Usage of a SpatialOrdinalPatterns estimator is straightforward. Here we get the spatial permutation entropy of a 2D array (e.g., an image):

    using ComplexityMeasures
     x = rand(50, 50) # some image
     stencil = [1 1; 0 1] # or one of the other ways of specifying stencils
     est = SpatialOrdinalPatterns(stencil, x)
    -h = information(est, x)
    2.5837530071016235

    To apply this to timeseries of spatial data, simply loop over the call, e.g.:

    data = [rand(50, 50) for i in 1:10] # e.g., evolution of a 2D field of a PDE
    +h = information(est, x)
    2.5842567359792734

    To apply this to timeseries of spatial data, simply loop over the call, e.g.:

    data = [rand(50, 50) for i in 1:10] # e.g., evolution of a 2D field of a PDE
     est = SpatialOrdinalPatterns(stencil, first(data))
     h_vs_t = map(d -> information(est, d), data)
    10-element Vector{Float64}:
    - 2.582915918592923
    - 2.5832961331213165
    - 2.58352706962532
    - 2.583258828148142
    - 2.5827883551696167
    - 2.584051519443532
    - 2.5842431310732934
    - 2.584246956724435
    - 2.5838667924713214
    - 2.584661671877444

    Computing any other generalized spatiotemporal permutation entropy is trivial, e.g. with Renyi:

    x = reshape(repeat(1:5, 500) .+ 0.1*rand(500*5), 50, 50)
    + 2.5840003478401865
    + 2.584410546404469
    + 2.584483117976614
    + 2.5841448457304694
    + 2.584342871259514
    + 2.5838442746199277
    + 2.584813938228458
    + 2.5839104964168023
    + 2.5844029192100404
    + 2.5845805694416546

    Computing any other generalized spatiotemporal permutation entropy is trivial, e.g. with Renyi:

    x = reshape(repeat(1:5, 500) .+ 0.1*rand(500*5), 50, 50)
     est = SpatialOrdinalPatterns(stencil, x)
    -information(Renyi(q = 2), est, x)
    1.5563594031580732

    Spatial discrete entropy: Fabio

    Let's see how the normalized permutation and dispersion entropies increase for an image that gets progressively more noise added to it.

    using ComplexityMeasures
    +information(Renyi(q = 2), est, x)
    1.5558910386484375

    Spatial discrete entropy: Fabio

    Let's see how the normalized permutation and dispersion entropies increase for an image that gets progressively more noise added to it.

    using ComplexityMeasures
     using Distributions: Uniform
     using CairoMakie
     using Statistics
    @@ -433,7 +433,7 @@
         hidedecorations!(ax_i)  # hides ticks, grid and lables
         hidespines!(ax_i)  # hide the frame
     end
    -fig
    Example block output

    While the normalized SpatialOrdinalPatterns entropy quickly approaches its maximum value, the normalized SpatialDispersion entropy much better resolves the increase in entropy as the image gets noiser. This can probably be explained by the fact that the number of possible states (or total_outcomes) for any given stencil is larger for SpatialDispersion than for SpatialOrdinalPatterns, so the dispersion approach is much less sensitive to noise addition (i.e. noise saturation over the possible states is slower for SpatialDispersion).

    Complexity: reverse dispersion entropy

    Here, we compare regular dispersion entropy (Rostaghi and Azami, 2016), and reverse dispersion entropy (Li et al., 2019) for a time series consisting of normally distributed noise with a single spike in the middle of the signal. We compute the entropies over a range subsets of the data, using a sliding window consisting of 70 data points, stepping the window 10 time steps at a time. This example reproduces parts of figure 3 in (Li et al., 2019), but results here are not exactly the same as in the original paper, because their examples are based on randomly generated numbers and do not provide code that specify random number seeds.

    using ComplexityMeasures
    +fig
    Example block output

    While the normalized SpatialOrdinalPatterns entropy quickly approaches its maximum value, the normalized SpatialDispersion entropy much better resolves the increase in entropy as the image gets noiser. This can probably be explained by the fact that the number of possible states (or total_outcomes) for any given stencil is larger for SpatialDispersion than for SpatialOrdinalPatterns, so the dispersion approach is much less sensitive to noise addition (i.e. noise saturation over the possible states is slower for SpatialDispersion).

    Complexity: reverse dispersion entropy

    Here, we compare regular dispersion entropy (Rostaghi and Azami, 2016), and reverse dispersion entropy (Li et al., 2019) for a time series consisting of normally distributed noise with a single spike in the middle of the signal. We compute the entropies over a range subsets of the data, using a sliding window consisting of 70 data points, stepping the window 10 time steps at a time. This example reproduces parts of figure 3 in (Li et al., 2019), but results here are not exactly the same as in the original paper, because their examples are based on randomly generated numbers and do not provide code that specify random number seeds.

    using ComplexityMeasures
     using Random
     using CairoMakie
     using Distributions: Normal
    @@ -478,7 +478,7 @@
     
     axislegend(position = :rc)
     ylims!(0, max(maximum(pes), 1))
    -fig
    Example block output

    Complexity: missing dispersion patterns

    using ComplexityMeasures
    +fig
    Example block output

    Complexity: missing dispersion patterns

    using ComplexityMeasures
     using CairoMakie
     using DynamicalSystemsBase
     using TimeseriesSurrogates
    @@ -531,7 +531,7 @@
     axislegend(position = :rc)
     ylims!(0, 1.1)
     
    -fig
    Example block output

    We don't need to actually to compute the quantiles here to see that for the logistic map, across all time series lengths, the $N_{MDP}$ values are above the extremal values of the $N_{MDP}$ values for the surrogate ensembles. Thus, we conclude that the logistic map time series has nonlinearity (well, of course).

    For the univariate noise time series, there is considerable overlap between $N_{MDP}$ for the surrogate distributions and the original signal, so we can't claim nonlinearity for this signal.

    Of course, to robustly reject the null hypothesis, we'd need to generate a sufficient number of surrogate realizations, and actually compute quantiles to compare with.

    Complexity: approximate entropy

    Here, we reproduce the Henon map example with $R=0.8$ from Pincus (1991), comparing our values with relevant values from table 1 in Pincus (1991).

    We use DiscreteDynamicalSystem from DynamicalSystemsBase to represent the map, and use the trajectory function from the same package to iterate the map for different initial conditions, for multiple time series lengths.

    Finally, we summarize our results in box plots and compare the values to those obtained by Pincus (1991).

    using ComplexityMeasures
    +fig
    Example block output

    We don't need to actually to compute the quantiles here to see that for the logistic map, across all time series lengths, the $N_{MDP}$ values are above the extremal values of the $N_{MDP}$ values for the surrogate ensembles. Thus, we conclude that the logistic map time series has nonlinearity (well, of course).

    For the univariate noise time series, there is considerable overlap between $N_{MDP}$ for the surrogate distributions and the original signal, so we can't claim nonlinearity for this signal.

    Of course, to robustly reject the null hypothesis, we'd need to generate a sufficient number of surrogate realizations, and actually compute quantiles to compare with.

    Complexity: approximate entropy

    Here, we reproduce the Henon map example with $R=0.8$ from Pincus (1991), comparing our values with relevant values from table 1 in Pincus (1991).

    We use DiscreteDynamicalSystem from DynamicalSystemsBase to represent the map, and use the trajectory function from the same package to iterate the map for different initial conditions, for multiple time series lengths.

    Finally, we summarize our results in box plots and compare the values to those obtained by Pincus (1991).

    using ComplexityMeasures
     using DynamicalSystemsBase
     using DelayEmbeddings
     using CairoMakie
    @@ -596,7 +596,7 @@
     
     scatter!(a2, ts_lengths, [0.337, 0.385, NaN, 0.394];
         label = "Pincus (1991)", color = :black)
    -fig
    Example block output

    Complexity: sample entropy

    Completely regular signals should have sample entropy approaching zero, while less regular signals should have higher sample entropy.

    using ComplexityMeasures
    +fig
    Example block output

    Complexity: sample entropy

    Completely regular signals should have sample entropy approaching zero, while less regular signals should have higher sample entropy.

    using ComplexityMeasures
     using CairoMakie
     N, a = 2000, 10
     t = LinRange(0, 2*a*π, N)
    @@ -616,7 +616,7 @@
     lines!(az, t, z; color = Cycled(3), label = "h=$(h=round(h_z, sigdigits = 5))");
     for a in (ax, ay, az); axislegend(a); end
     for a in (ax, ay); hidexdecorations!(a; grid=false); end
    -fig
    Example block output

    Next, we compare the sample entropy obtained for different values of the radius r for uniform noise, normally distributed noise, and a periodic signal.

    using ComplexityMeasures
    +fig
    Example block output

    Next, we compare the sample entropy obtained for different values of the radius r for uniform noise, normally distributed noise, and a periodic signal.

    using ComplexityMeasures
     using CairoMakie
     using Statistics
     using Distributions: Normal
    @@ -643,7 +643,7 @@
     lines!(a1, rs, hs_N, label = "Gaussian noise, N(0, 1)")
     lines!(a1, rs, hs_periodic, label = "Periodic signal")
     axislegend()
    -fig
    Example block output

    Statistical complexity of iterated maps

    In this example, we reproduce parts of Fig. 1 in Rosso et al. (2007): We compute the statistical complexity of the Henon, logistic and Schuster map, as well as that of k-noise.

    using ComplexityMeasures
    +fig
    Example block output

    Statistical complexity of iterated maps

    In this example, we reproduce parts of Fig. 1 in Rosso et al. (2007): We compute the statistical complexity of the Henon, logistic and Schuster map, as well as that of k-noise.

    using ComplexityMeasures
     using Distances
     using DynamicalSystemsBase
     using CairoMakie
    @@ -739,7 +739,7 @@
     lines!(ax, min_curve; color=:black)
     lines!(ax, max_curve; color=:black)
     axislegend(; position=:lt)
    -fig
    Example block output

    Complexity: multiscale

    Let's use multiscale analysis to investigate the SampleEntropy of a signal across coarse-graining scales.

    using ComplexityMeasures
    +fig
    Example block output

    Complexity: multiscale

    Let's use multiscale analysis to investigate the SampleEntropy of a signal across coarse-graining scales.

    using ComplexityMeasures
     using CairoMakie
     
     N, a = 2000, 20
    @@ -755,4 +755,4 @@
     lines!(ax1, t, y; color = Cycled(1));
     ax2 = Axis(fig[2, 1]; ylabel = "Sample entropy (h)", xlabel = "Scale")
     scatterlines!(ax2, scales |> collect, hs; color = Cycled(1));
    -fig
    Example block output +figExample block output diff --git a/previews/PR430/index.html b/previews/PR430/index.html index 2f81844b..f39c9619 100644 --- a/previews/PR430/index.html +++ b/previews/PR430/index.html @@ -1,2 +1,2 @@ -ComplexityMeasures.jl · ComplexityMeasures.jl

    ComplexityMeasures.jl

    ComplexityMeasuresModule

    ComplexityMeasures.jl

    docsdev docsstable CI codecov Package Downloads Package Downloads publication

    ComplexityMeasures.jl is a Julia-based software for calculating 1000s of various kinds of probabilities, entropies, and other so-called complexity measures from a single-variable input datasets. For relational measures across many input datasets see its extension CausalityTools.jl. If you are a user of other programming languages (Python, R, MATLAB, ...), you can still use ComplexityMeasures.jl due to Julia's interoperability. For example, for Python use juliacall.

    A careful comparison with alternative widely used software shows that ComplexityMeasures.jl outclasses the alternatives in several objective aspects of comparison, such as computational performance, overall amount of measures, reliability, and extendability. See the associated publication for more details.

    The key features that it provides can be summarized as:

    • A rigorous framework for extracting probabilities from data, based on the mathematical formulation of probability spaces.
    • Several (12+) outcome spaces, i.e., ways to discretize data into probabilities.
    • Several estimators for estimating probabilities given an outcome space, which correct theoretically known estimation biases.
    • Several definitions of information measures, such as various flavours of entropies (Shannon, Tsallis, Curado...), extropies, and other complexity measures, that are used in the context of nonlinear dynamics, nonlinear timeseries analysis, and complex systems.
    • Several discrete and continuous (differential) estimators for entropies, which correct theoretically known estimation biases.
    • An extendable interface and well thought out API accompanied by dedicated developer documentation. This makes it trivial to define new outcome spaces, or new estimators for probabilities, information measures, or complexity measures and integrate them with everything else in the software without boilerplate code.

    ComplexityMeasures.jl can be used as a standalone package, or as part of other projects in the JuliaDynamics organization, such as DynamicalSystems.jl or CausalityTools.jl.

    To install it, run import Pkg; Pkg.add("ComplexityMeasures").

    All further information is provided in the documentation, which you can either find online or build locally by running the docs/make.jl file.

    Previously, this package was called Entropies.jl.

    source

    Latest news

    ComplexityMeasures.jl has been updated to v3!

    The software has been massively improved and its core principles were redesigned to be extendable, accessible, and more closely based on the rigorous mathematics of probabilities and entropies.

    For more details of this new release, please see our announcement post on discourse or the central Tutorial of the v3 documentation.

    In this v3 many concepts were renamed, but there is no formally breaking change. Everything that changed has been deprecated and is backwards compatible. You can see the CHANGELOG.md for more details!

    Documentation contents

    • Before anything else, we recommend users to go through our overarching Tutorial, which teaches not only central API functions, but also terminology and crucial core concepts:
    • Probabilities lists all outcome spaces and probabilities estimators.
    • Information measures lists all implemented information measure definitions and estimators (both discrete and differential).
    • Complexity measures lists all implemented complexity measures that are not functionals of probabilities (unlike information measures).
    • The Examples page lists dozens of runnable example code snippets along with their outputs.

    Input data for ComplexityMeasures.jl

    The input data type typically depend on the outcome space chosen. In general though, the standard DynamicalSystems.jl approach is taken and as such we have three types of input data:

    • Timeseries, which are AbstractVector{<:Real}, used in e.g. with WaveletOverlap.
    • Multi-variate timeseries, or datasets, or state space sets, which are StateSpaceSets, used e.g. with NaiveKernel. The short syntax SSSet may be used instead of StateSpaceSet.
    • Spatial data, which are higher dimensional standard Arrays, used e.g. with SpatialOrdinalPatterns.
    StateSpaceSets.StateSpaceSetType
    StateSpaceSet{D, T, V} <: AbstractVector{V}

    A dedicated interface for sets in a state space. It is an ordered container of equally-sized points of length D, with element type T, represented by a vector of type V. Typically V is SVector{D,T} or Vector{T} and the data are always stored internally as Vector{V}. SSSet is an alias for StateSpaceSet.

    The underlying Vector{V} can be obtained by vec(ssset), although this is almost never necessary because StateSpaceSet subtypes AbstractVector and extends its interface. StateSpaceSet also supports almost all sensible vector operations like append!, push!, hcat, eachrow, among others. When iterated over, it iterates over its contained points.

    Construction

    Constructing a StateSpaceSet is done in three ways:

    1. By giving in each individual columns of the state space set as Vector{<:Real}: StateSpaceSet(x, y, z, ...).
    2. By giving in a matrix whose rows are the state space points: StateSpaceSet(m).
    3. By giving in directly a vector of vectors (state space points): StateSpaceSet(v_of_v).

    All constructors allow for the keyword container which sets the type of V (the type of inner vectors). At the moment options are only SVector, MVector, or Vector, and by default SVector is used.

    Description of indexing

    When indexed with 1 index, StateSpaceSet behaves exactly like its encapsulated vector. i.e., a vector of vectors (state space points). When indexed with 2 indices it behaves like a matrix where each row is a point.

    In the following let i, j be integers, typeof(X) <: AbstractStateSpaceSet and v1, v2 be <: AbstractVector{Int} (v1, v2 could also be ranges, and for performance benefits make v2 an SVector{Int}).

    • X[i] == X[i, :] gives the ith point (returns an SVector)
    • X[v1] == X[v1, :], returns a StateSpaceSet with the points in those indices.
    • X[:, j] gives the jth variable timeseries (or collection), as Vector
    • X[v1, v2], X[:, v2] returns a StateSpaceSet with the appropriate entries (first indices being "time"/point index, while second being variables)
    • X[i, j] value of the jth variable, at the ith timepoint

    Use Matrix(ssset) or StateSpaceSet(matrix) to convert. It is assumed that each column of the matrix is one variable. If you have various timeseries vectors x, y, z, ... pass them like StateSpaceSet(x, y, z, ...). You can use columns(dataset) to obtain the reverse, i.e. all columns of the dataset in a tuple.

    source

    Total entropy/information/complexity measures

    ComplexityMeasures.jl offers thousands of measures computable right out of the box. To see an exact number of how many, see this calculation page.

    +ComplexityMeasures.jl · ComplexityMeasures.jl

    ComplexityMeasures.jl

    ComplexityMeasuresModule

    ComplexityMeasures.jl

    docsdev docsstable CI codecov Package Downloads Package Downloads publication

    ComplexityMeasures.jl is a Julia-based software for calculating 1000s of various kinds of probabilities, entropies, and other so-called complexity measures from a single-variable input datasets. For relational measures across many input datasets see its extension CausalityTools.jl. If you are a user of other programming languages (Python, R, MATLAB, ...), you can still use ComplexityMeasures.jl due to Julia's interoperability. For example, for Python use juliacall.

    A careful comparison with alternative widely used software shows that ComplexityMeasures.jl outclasses the alternatives in several objective aspects of comparison, such as computational performance, overall amount of measures, reliability, and extendability. See the associated publication for more details.

    The key features that it provides can be summarized as:

    • A rigorous framework for extracting probabilities from data, based on the mathematical formulation of probability spaces.
    • Several (12+) outcome spaces, i.e., ways to discretize data into probabilities.
    • Several estimators for estimating probabilities given an outcome space, which correct theoretically known estimation biases.
    • Several definitions of information measures, such as various flavours of entropies (Shannon, Tsallis, Curado...), extropies, and other complexity measures, that are used in the context of nonlinear dynamics, nonlinear timeseries analysis, and complex systems.
    • Several discrete and continuous (differential) estimators for entropies, which correct theoretically known estimation biases.
    • An extendable interface and well thought out API accompanied by dedicated developer documentation. This makes it trivial to define new outcome spaces, or new estimators for probabilities, information measures, or complexity measures and integrate them with everything else in the software without boilerplate code.

    ComplexityMeasures.jl can be used as a standalone package, or as part of other projects in the JuliaDynamics organization, such as DynamicalSystems.jl or CausalityTools.jl.

    To install it, run import Pkg; Pkg.add("ComplexityMeasures").

    All further information is provided in the documentation, which you can either find online or build locally by running the docs/make.jl file.

    Previously, this package was called Entropies.jl.

    source

    Latest news

    ComplexityMeasures.jl has been updated to v3!

    The software has been massively improved and its core principles were redesigned to be extendable, accessible, and more closely based on the rigorous mathematics of probabilities and entropies.

    For more details of this new release, please see our announcement post on discourse or the central Tutorial of the v3 documentation.

    In this v3 many concepts were renamed, but there is no formally breaking change. Everything that changed has been deprecated and is backwards compatible. You can see the CHANGELOG.md for more details!

    Documentation contents

    • Before anything else, we recommend users to go through our overarching Tutorial, which teaches not only central API functions, but also terminology and crucial core concepts:
    • Probabilities lists all outcome spaces and probabilities estimators.
    • Information measures lists all implemented information measure definitions and estimators (both discrete and differential).
    • Complexity measures lists all implemented complexity measures that are not functionals of probabilities (unlike information measures).
    • The Examples page lists dozens of runnable example code snippets along with their outputs.

    Input data for ComplexityMeasures.jl

    The input data type typically depend on the outcome space chosen. In general though, the standard DynamicalSystems.jl approach is taken and as such we have three types of input data:

    • Timeseries, which are AbstractVector{<:Real}, used in e.g. with WaveletOverlap.
    • Multi-variate timeseries, or datasets, or state space sets, which are StateSpaceSets, used e.g. with NaiveKernel. The short syntax SSSet may be used instead of StateSpaceSet.
    • Spatial data, which are higher dimensional standard Arrays, used e.g. with SpatialOrdinalPatterns.
    StateSpaceSets.StateSpaceSetType
    StateSpaceSet{D, T, V} <: AbstractVector{V}

    A dedicated interface for sets in a state space. It is an ordered container of equally-sized points of length D, with element type T, represented by a vector of type V. Typically V is SVector{D,T} or Vector{T} and the data are always stored internally as Vector{V}. SSSet is an alias for StateSpaceSet.

    The underlying Vector{V} can be obtained by vec(ssset), although this is almost never necessary because StateSpaceSet subtypes AbstractVector and extends its interface. StateSpaceSet also supports almost all sensible vector operations like append!, push!, hcat, eachrow, among others. When iterated over, it iterates over its contained points.

    Construction

    Constructing a StateSpaceSet is done in three ways:

    1. By giving in each individual columns of the state space set as Vector{<:Real}: StateSpaceSet(x, y, z, ...).
    2. By giving in a matrix whose rows are the state space points: StateSpaceSet(m).
    3. By giving in directly a vector of vectors (state space points): StateSpaceSet(v_of_v).

    All constructors allow for the keyword container which sets the type of V (the type of inner vectors). At the moment options are only SVector, MVector, or Vector, and by default SVector is used.

    Description of indexing

    When indexed with 1 index, StateSpaceSet behaves exactly like its encapsulated vector. i.e., a vector of vectors (state space points). When indexed with 2 indices it behaves like a matrix where each row is a point.

    In the following let i, j be integers, typeof(X) <: AbstractStateSpaceSet and v1, v2 be <: AbstractVector{Int} (v1, v2 could also be ranges, and for performance benefits make v2 an SVector{Int}).

    • X[i] == X[i, :] gives the ith point (returns an SVector)
    • X[v1] == X[v1, :], returns a StateSpaceSet with the points in those indices.
    • X[:, j] gives the jth variable timeseries (or collection), as Vector
    • X[v1, v2], X[:, v2] returns a StateSpaceSet with the appropriate entries (first indices being "time"/point index, while second being variables)
    • X[i, j] value of the jth variable, at the ith timepoint

    Use Matrix(ssset) or StateSpaceSet(matrix) to convert. It is assumed that each column of the matrix is one variable. If you have various timeseries vectors x, y, z, ... pass them like StateSpaceSet(x, y, z, ...). You can use columns(dataset) to obtain the reverse, i.e. all columns of the dataset in a tuple.

    source

    Total entropy/information/complexity measures

    ComplexityMeasures.jl offers thousands of measures computable right out of the box. To see an exact number of how many, see this calculation page.

    diff --git a/previews/PR430/information_measures/index.html b/previews/PR430/information_measures/index.html index 4c55dab5..ae409095 100644 --- a/previews/PR430/information_measures/index.html +++ b/previews/PR430/information_measures/index.html @@ -1,5 +1,5 @@ -Information measures (entropies and co.) · ComplexityMeasures.jl

    Information measures (entropies and co.)

    Note

    Be sure you have gone through the Tutorial before going through the API here to have a good idea of the terminology used in ComplexityMeasures.jl.

    Information measures API

    The information measure API is defined by the information function, which takes as an input an InformationMeasure, or some specialized DiscreteInfoEstimator or DifferentialInfoEstimator for estimating the discrete or differential variant of the measure. The functions information_maximum and information_normalized are also useful.

    ComplexityMeasures.InformationMeasureType
    InformationMeasure

    InformationMeasure is the supertype of all information measure definitions.

    In this package, we define "information measures" as functionals of probability mass functions ("discrete" measures), or of probability density functions ("differential" measures). Examples are (generalized) entropies such as Shannon or Renyi, or extropies like ShannonExtropy. Amigó et al. (2018) provides a useful review of generalized entropies.

    Used with

    Any of the information measures listed below can be used with

    • information, to compute a numerical value for the measure, given some input data.
    • information_maximum, to compute the maximum possible value for the measure.
    • information_normalized, to compute the normalized form of the measure (divided by the maximum possible value).

    The information_maximum/information_normalized functions only works with the discrete version of the measure. See docstrings for the above functions for usage examples.

    Implementations

    Estimators

    A particular information measure may have both a discrete and a continuous/differential definition, which are estimated using a DifferentialInfoEstimator or a DifferentialInfoEstimator, respectively.

    source
    ComplexityMeasures.informationMethod
    information([die::DiscreteInfoEstimator,] [est::ProbabilitiesEstimator,] o::OutcomeSpace, x) → h::Real
    +Information measures (entropies and co.) · ComplexityMeasures.jl

    Information measures (entropies and co.)

    Note

    Be sure you have gone through the Tutorial before going through the API here to have a good idea of the terminology used in ComplexityMeasures.jl.

    Information measures API

    The information measure API is defined by the information function, which takes as an input an InformationMeasure, or some specialized DiscreteInfoEstimator or DifferentialInfoEstimator for estimating the discrete or differential variant of the measure. The functions information_maximum and information_normalized are also useful.

    ComplexityMeasures.InformationMeasureType
    InformationMeasure

    InformationMeasure is the supertype of all information measure definitions.

    In this package, we define "information measures" as functionals of probability mass functions ("discrete" measures), or of probability density functions ("differential" measures). Examples are (generalized) entropies such as Shannon or Renyi, or extropies like ShannonExtropy. Amigó et al. (2018) provides a useful review of generalized entropies.

    Used with

    Any of the information measures listed below can be used with

    • information, to compute a numerical value for the measure, given some input data.
    • information_maximum, to compute the maximum possible value for the measure.
    • information_normalized, to compute the normalized form of the measure (divided by the maximum possible value).

    The information_maximum/information_normalized functions only works with the discrete version of the measure. See docstrings for the above functions for usage examples.

    Implementations

    Estimators

    A particular information measure may have both a discrete and a continuous/differential definition, which are estimated using a DifferentialInfoEstimator or a DifferentialInfoEstimator, respectively.

    source
    ComplexityMeasures.informationMethod
    information([die::DiscreteInfoEstimator,] [est::ProbabilitiesEstimator,] o::OutcomeSpace, x) → h::Real
     information(o::OutcomeSpace, x) → h::Real

    Estimate a discrete information measure from input data x using the provided DiscreteInfoEstimator and ProbabilitiesEstimator over the given OutcomeSpace.

    As an alternative, you can provide an InformationMeasure for the first argument (die) which will default to PlugIn estimation) for the information estimation. You may also skip the first argument (die), in which case Shannon() will be used. You may also skip the second argument (est), which will default to the RelativeAmount probabilities estimator. Note that some information measure estimators (e.g., GeneralizedSchuermann) operate directly on counts and hence ignore est.

    information([e::DiscreteInfoEstimator,] p::Probabilities) → h::Real
     information([e::DiscreteInfoEstimator,] c::Counts) → h::Real

    Like above, but estimate the information measure from the pre-computed Probabilities p or Counts. Counts are converted into probabilities using RelativeAmount, unless the estimator e uses counts directly.

    See also: information_maximum, information_normalized for a normalized version.

    Examples (naive estimation)

    The simplest way to estimate a discrete measure is to provide the InformationMeasure directly in combination with an OutcomeSpace. This will use the "naive" PlugIn estimator for the measure, and the "naive" RelativeAmount estimator for the probabilities.

    x = randn(100) # some input data
     o = ValueBinning(RectangularBinning(5)) # a 5-bin histogram outcome space
    @@ -20,46 +20,46 @@
     # Estimate information measures using the generic `Jackknife` estimator
     h_r = information(Jackknife(Renyi()), Shrinkage(), o, x)
     j_t = information(Jackknife(TsallisExtropy()), BayesianRegularization(), o, x)
    -j_r = information(Jackknife(RenyiExtropy()), RelativeAmount(), o, x)
    source
    ComplexityMeasures.informationMethod
    information(est::DifferentialInfoEstimator, x) → h::Real

    Estimate a differential information measure using the provided DifferentialInfoEstimator and input data x.

    Description

    The overwhelming majority of differential estimators estimate the Shannon entropy. If the same estimator can estimate different information measures (e.g. it can estimate both Shannon and Tsallis), then the information measure is provided as an argument to the estimator itself.

    See the table of differential information measure estimators in the docs for all differential information measure estimators.

    Currently, unlike for the discrete information measures, this method doesn't involve explicitly first computing a probability density function and then passing this density to an information measure definition. But in the future, we want to establish a density API similar to the probabilities API.

    Examples

    To compute the differential version of a measure, give it as the first argument to a DifferentialInfoEstimator and pass it to information.

    x = randn(1000)
    +j_r = information(Jackknife(RenyiExtropy()), RelativeAmount(), o, x)
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    ComplexityMeasures.informationMethod
    information(est::DifferentialInfoEstimator, x) → h::Real

    Estimate a differential information measure using the provided DifferentialInfoEstimator and input data x.

    Description

    The overwhelming majority of differential estimators estimate the Shannon entropy. If the same estimator can estimate different information measures (e.g. it can estimate both Shannon and Tsallis), then the information measure is provided as an argument to the estimator itself.

    See the table of differential information measure estimators in the docs for all differential information measure estimators.

    Currently, unlike for the discrete information measures, this method doesn't involve explicitly first computing a probability density function and then passing this density to an information measure definition. But in the future, we want to establish a density API similar to the probabilities API.

    Examples

    To compute the differential version of a measure, give it as the first argument to a DifferentialInfoEstimator and pass it to information.

    x = randn(1000)
     h_sh = information(Kraskov(Shannon()), x)
     h_vc = information(Vasicek(Shannon()), x)

    A normal distribution has a base-e Shannon differential entropy of 0.5*log(2π) + 0.5 nats.

    est = Kraskov(k = 5, base = ℯ) # Base `ℯ` for nats.
     h = information(est, randn(2_000_000))
    -abs(h - 0.5*log(2π) - 0.5) # ≈ 0.0001
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    ComplexityMeasures.information_maximumFunction
    information_maximum(e::InformationMeasure, o::OutcomeSpace [, x])

    Return the maximum value of the given information measure can have, given input data x and the given outcome space (the OutcomeSpace may also be specified by a ProbabilitiesEstimator).

    Like in outcome_space, for some outcome spaces, the possible outcomes are known without knowledge of input x, in which case the function dispatches to information_maximum(e, o).

    information_maximum(e::InformationMeasure, L::Int)

    The same as above, but computed directly from the number of total outcomes L.

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    ComplexityMeasures.information_normalizedFunction
    information_normalized([e::DiscreteInfoEstimator,] [est::ProbabilitiesEstimator,] o::OutcomeSpace, x) → h::Real

    Estimate the normalized version of the given discrete information measure, This is just the value of information divided its maximum possible value given o.

    The same convenience syntaxes as in information can be used here.

    Notice that there is no method information_normalized(e::DiscreteInfoEstimator, probs::Probabilities), because there is no way to know the number of possible outcomes (i.e., the total_outcomes) from probs.

    Normalized values

    For the PlugIn estimator, it is guaranteed that h̃ ∈ [0, 1]. For any other estimator, we can't guarantee this, since the estimator might over-correct. You should know what you're doing if using anything but PlugIn to estimate normalized values.

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    Entropies

    ComplexityMeasures.ShannonType
    Shannon <: InformationMeasure
    -Shannon(; base = 2)

    The Shannon (Shannon, 1948) entropy, used with information to compute:

    \[H(p) = - \sum_i p[i] \log(p[i])\]

    with the $\log$ at the given base.

    The maximum value of the Shannon entropy is $\log_{base}(L)$, which is the entropy of the uniform distribution with $L$ the total_outcomes.

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    ComplexityMeasures.information_maximumFunction
    information_maximum(e::InformationMeasure, o::OutcomeSpace [, x])

    Return the maximum value of the given information measure can have, given input data x and the given outcome space (the OutcomeSpace may also be specified by a ProbabilitiesEstimator).

    Like in outcome_space, for some outcome spaces, the possible outcomes are known without knowledge of input x, in which case the function dispatches to information_maximum(e, o).

    information_maximum(e::InformationMeasure, L::Int)

    The same as above, but computed directly from the number of total outcomes L.

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    ComplexityMeasures.information_normalizedFunction
    information_normalized([e::DiscreteInfoEstimator,] [est::ProbabilitiesEstimator,] o::OutcomeSpace, x) → h::Real

    Estimate the normalized version of the given discrete information measure, This is just the value of information divided its maximum possible value given o.

    The same convenience syntaxes as in information can be used here.

    Notice that there is no method information_normalized(e::DiscreteInfoEstimator, probs::Probabilities), because there is no way to know the number of possible outcomes (i.e., the total_outcomes) from probs.

    Normalized values

    For the PlugIn estimator, it is guaranteed that h̃ ∈ [0, 1]. For any other estimator, we can't guarantee this, since the estimator might over-correct. You should know what you're doing if using anything but PlugIn to estimate normalized values.

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    Entropies

    ComplexityMeasures.ShannonType
    Shannon <: InformationMeasure
    +Shannon(; base = 2)

    The Shannon (Shannon, 1948) entropy, used with information to compute:

    \[H(p) = - \sum_i p[i] \log(p[i])\]

    with the $\log$ at the given base.

    The maximum value of the Shannon entropy is $\log_{base}(L)$, which is the entropy of the uniform distribution with $L$ the total_outcomes.

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    ComplexityMeasures.RenyiType
    Renyi <: InformationMeasure
     Renyi(q, base = 2)
    -Renyi(; q = 1.0, base = 2)

    The Rényi generalized order-q entropy (Rényi, 1961), used with information to compute an entropy with units given by base (typically 2 or MathConstants.e).

    Description

    Let $p$ be an array of probabilities (summing to 1). Then the Rényi generalized entropy is

    \[H_q(p) = \frac{1}{1-q} \log \left(\sum_i p[i]^q\right)\]

    and generalizes other known entropies, like e.g. the information entropy ($q = 1$, see Shannon (1948)), the maximum entropy ($q=0$, also known as Hartley entropy), or the correlation entropy ($q = 2$, also known as collision entropy).

    The maximum value of the Rényi entropy is $\log_{base}(L)$, which is the entropy of the uniform distribution with $L$ the total_outcomes.

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    ComplexityMeasures.TsallisType
    Tsallis <: InformationMeasure
    +Renyi(; q = 1.0, base = 2)

    The Rényi generalized order-q entropy (Rényi, 1961), used with information to compute an entropy with units given by base (typically 2 or MathConstants.e).

    Description

    Let $p$ be an array of probabilities (summing to 1). Then the Rényi generalized entropy is

    \[H_q(p) = \frac{1}{1-q} \log \left(\sum_i p[i]^q\right)\]

    and generalizes other known entropies, like e.g. the information entropy ($q = 1$, see Shannon (1948)), the maximum entropy ($q=0$, also known as Hartley entropy), or the correlation entropy ($q = 2$, also known as collision entropy).

    The maximum value of the Rényi entropy is $\log_{base}(L)$, which is the entropy of the uniform distribution with $L$ the total_outcomes.

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    ComplexityMeasures.TsallisType
    Tsallis <: InformationMeasure
     Tsallis(q; k = 1.0, base = 2)
    -Tsallis(; q = 1.0, k = 1.0, base = 2)

    The Tsallis generalized order-q entropy (Tsallis, 1988), used with information to compute an entropy.

    base only applies in the limiting case q == 1, in which the Tsallis entropy reduces to Shannon entropy.

    Description

    The Tsallis entropy is a generalization of the Boltzmann-Gibbs entropy, with k standing for the Boltzmann constant. It is defined as

    \[S_q(p) = \frac{k}{q - 1}\left(1 - \sum_{i} p[i]^q\right)\]

    The maximum value of the Tsallis entropy is $k(L^{1 - q} - 1)/(1 - q)$, with $L$ the total_outcomes.

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    ComplexityMeasures.KaniadakisType
    Kaniadakis <: InformationMeasure
    -Kaniadakis(; κ = 1.0, base = 2.0)

    The Kaniadakis entropy (Tsallis, 2009), used with information to compute

    \[H_K(p) = -\sum_{i=1}^N p_i f_\kappa(p_i),\]

    \[f_\kappa (x) = \dfrac{x^\kappa - x^{-\kappa}}{2\kappa},\]

    where if $\kappa = 0$, regular logarithm to the given base is used, and 0 probabilities are skipped.

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    ComplexityMeasures.CuradoType
    Curado <: InformationMeasure
    -Curado(; b = 1.0)

    The Curado entropy (Curado and Nobre, 2004), used with information to compute

    \[H_C(p) = \left( \sum_{i=1}^N e^{-b p_i} \right) + e^{-b} - 1,\]

    with b ∈ ℛ, b > 0, and the terms outside the sum ensures that $H_C(0) = H_C(1) = 0$.

    The maximum entropy for Curado is $L(1 - \exp(-b/L)) + \exp(-b) - 1$ with $L$ the total_outcomes.

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    ComplexityMeasures.StretchedExponentialType
    StretchedExponential <: InformationMeasure
    +Tsallis(; q = 1.0, k = 1.0, base = 2)

    The Tsallis generalized order-q entropy (Tsallis, 1988), used with information to compute an entropy.

    base only applies in the limiting case q == 1, in which the Tsallis entropy reduces to Shannon entropy.

    Description

    The Tsallis entropy is a generalization of the Boltzmann-Gibbs entropy, with k standing for the Boltzmann constant. It is defined as

    \[S_q(p) = \frac{k}{q - 1}\left(1 - \sum_{i} p[i]^q\right)\]

    The maximum value of the Tsallis entropy is $k(L^{1 - q} - 1)/(1 - q)$, with $L$ the total_outcomes.

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    ComplexityMeasures.KaniadakisType
    Kaniadakis <: InformationMeasure
    +Kaniadakis(; κ = 1.0, base = 2.0)

    The Kaniadakis entropy (Tsallis, 2009), used with information to compute

    \[H_K(p) = -\sum_{i=1}^N p_i f_\kappa(p_i),\]

    \[f_\kappa (x) = \dfrac{x^\kappa - x^{-\kappa}}{2\kappa},\]

    where if $\kappa = 0$, regular logarithm to the given base is used, and 0 probabilities are skipped.

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    ComplexityMeasures.CuradoType
    Curado <: InformationMeasure
    +Curado(; b = 1.0)

    The Curado entropy (Curado and Nobre, 2004), used with information to compute

    \[H_C(p) = \left( \sum_{i=1}^N e^{-b p_i} \right) + e^{-b} - 1,\]

    with b ∈ ℛ, b > 0, and the terms outside the sum ensures that $H_C(0) = H_C(1) = 0$.

    The maximum entropy for Curado is $L(1 - \exp(-b/L)) + \exp(-b) - 1$ with $L$ the total_outcomes.

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    ComplexityMeasures.StretchedExponentialType
    StretchedExponential <: InformationMeasure
     StretchedExponential(; η = 2.0, base = 2)

    The stretched exponential, or Anteneodo-Plastino, entropy (Anteneodo and Plastino, 1999), used with information to compute

    \[S_{\eta}(p) = \sum_{i = 1}^N \Gamma \left( \dfrac{\eta + 1}{\eta}, - \log_{base}(p_i) \right) - -p_i \Gamma \left( \dfrac{\eta + 1}{\eta} \right),\]

    where $\eta \geq 0$, $\Gamma(\cdot, \cdot)$ is the upper incomplete Gamma function, and $\Gamma(\cdot) = \Gamma(\cdot, 0)$ is the Gamma function. Reduces to Shannon entropy for η = 1.0.

    The maximum entropy for StrechedExponential is a rather complicated expression involving incomplete Gamma functions (see source code).

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    Other information measures

    ComplexityMeasures.ShannonExtropyType
    ShannonExtropy <: InformationMeasure
    -ShannonExtropy(; base = 2)

    The Shannon extropy (Lad et al., 2015), used with information to compute

    \[J(x) = -\sum_{i=1}^N (1 - p[i]) \log{(1 - p[i])},\]

    for a probability distribution $P = \{p_1, p_2, \ldots, p_N\}$, with the $\log$ at the given base.

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    ComplexityMeasures.RenyiExtropyType
    RenyiExtropy <: InformationMeasure
    -RenyiExtropy(; q = 1.0, base = 2)

    The Rényi extropy (Liu and Xiao, 2023).

    Description

    RenyiExtropy is used with information to compute

    \[J_R(P) = \dfrac{-(n - 1) \log{(n - 1)} + (n - 1) \log{ \left( \sum_{i=1}^N {(1 - p[i])}^q \right)} }{q - 1}\]

    for a probability distribution $P = \{p_1, p_2, \ldots, p_N\}$, with the $\log$ at the given base. Alternatively, RenyiExtropy can be used with information_normalized, which ensures that the computed extropy is on the interval $[0, 1]$ by normalizing to to the maximal Rényi extropy, given by

    \[J_R(P) = (N - 1)\log \left( \dfrac{n}{n-1} \right) .\]

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    ComplexityMeasures.TsallisExtropyType
    TsallisExtropy <: InformationMeasure
    -TsallisExtropy(; base = 2)

    The Tsallis extropy (Xue and Deng, 2023).

    Description

    TsallisExtropy is used with information to compute

    \[J_T(P) = k \dfrac{N - 1 - \sum_{i=1}^N ( 1 - p[i])^q}{q - 1}\]

    for a probability distribution $P = \{p_1, p_2, \ldots, p_N\}$, with the $\log$ at the given base. Alternatively, TsallisExtropy can be used with information_normalized, which ensures that the computed extropy is on the interval $[0, 1]$ by normalizing to to the maximal Tsallis extropy, given by

    \[J_T(P) = \dfrac{(N - 1)N^{q - 1} - (N - 1)^q}{(q - 1)N^{q - 1}}\]

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    ComplexityMeasures.FluctuationComplexityType
    FluctuationComplexity <: InformationMeasure
    -FluctuationComplexity(; definition = Shannon(; base = 2), base = 2)

    The "fluctuation complexity" quantifies the standard deviation of the information content of the states $\omega_i$ around some summary statistic (InformationMeasure) of a PMF. Specifically, given some outcome space $\Omega$ with outcomes $\omega_i \in \Omega$ and a probability mass function $p(\Omega) = \{ p(\omega_i) \}_{i=1}^N$, it is defined as

    \[\sigma_I(p) := \sqrt{\sum_{i=1}^N p_i(I_i - H_*)^2}\]

    where $I_i = -\log_{base}(p_i)$ is the information content of the i-th outcome. The type of information measure $*$ is controlled by definition.

    The base controls the base of the logarithm that goes into the information content terms. Make sure that you pick a base that is consistent with the base chosen for the definition (relevant for e.g. Shannon).

    Properties

    If definition is the Shannon entropy, then we recover the Shannon-type information fluctuation complexity from (Bates and Shepard, 1993). Then the fluctuation complexity is zero for PMFs with only a single non-zero element, or for the uniform distribution.

    If definition is not Shannon entropy, then the properties of the measure varies, and does not necessarily share the properties (Bates and Shepard, 1993).

    Potential for new research

    As far as we know, using other information measures besides Shannon entropy for the fluctuation complexity hasn't been explored in the literature yet. Our implementation, however, allows for it. Please inform us if you try some new combinations!

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    Discrete information estimators

    ComplexityMeasures.DiscreteInfoEstimatorType
    DiscreteInfoEstimator

    The supertype of all discrete information measure estimators, which are used in combination with a ProbabilitiesEstimator as input to information or related functions.

    The first argument to a discrete estimator is always an InformationMeasure (defaults to Shannon).

    Description

    A discrete InformationMeasure is a functional of a probability mass function. To estimate such a measure from data, we must first estimate a probability mass function using a ProbabilitiesEstimator from the (encoded/discretized) input data, and then apply the estimator to the estimated probabilities. For example, the Shannon entropy is typically computed using the RelativeAmount estimator to compute probabilities, which are then given to the PlugIn estimator. Many other estimators exist, not only for Shannon entropy, but other information measures as well.

    We provide a library of both generic estimators such as PlugIn or Jackknife (which can be applied to any measure), as well as dedicated estimators such as MillerMadow, which computes Shannon entropy using the Miller-Madow bias correction. The list below gives a complete overview.

    Implementations

    The following estimators are generic and can compute any InformationMeasure.

    • PlugIn. The default, generic plug-in estimator of any information measure. It computes the measure exactly as stated in the definition, using the computed probability mass function.
    • Jackknife. Uses the a combination of the plug-in estimator and the jackknife principle to estimate the information measure.

    Shannon entropy estimators

    The following estimators are dedicated Shannon entropy estimators, which provide improvements over the naive PlugIn estimator.

    Info

    Any of the implemented DiscreteInfoEstimators can be used in combination with any ProbabilitiesEstimator as input to information. What this means is that every estimator actually comes in many different variants - one for each ProbabilitiesEstimator. For example, the MillerMadow estimator of Shannon entropy is typically calculated with RelativeAmount probabilities. But here, you can use for example the BayesianRegularization or the Shrinkage probabilities estimators instead, i.e. information(MillerMadow(), RelativeAmount(outcome_space), x) and information(MillerMadow(), BayesianRegularization(outcomes_space), x) are distinct estimators. This holds for all DiscreteInfoEstimators. Many of these estimators haven't been explored in the literature before, so feel free to explore, and please cite this software if you use it to explore some new estimator combination!

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    ComplexityMeasures.PlugInType
    PlugIn(e::InformationMeasure) <: DiscreteInfoEstimatorGeneric

    The PlugIn estimator is also called the empirical/naive/"maximum likelihood" estimator, and is used with information to any discrete InformationMeasure.

    It computes any quantity exactly as given by its formula. When computing an information measure, which here is defined as a probabilities functional, it computes the quantity directly from a probability mass function, which is derived from maximum-likelihood (RelativeAmount estimates of the probabilities.

    Bias of plug-in estimates

    The plugin-estimator of Shannon entropy underestimates the true entropy, with a bias that grows with the number of distinct outcomes (Arora et al., 2022)(Arora et al., 2022),

    \[bias(H_S^{plugin}) = -\dfrac{K-1}{2N} + o(N^-1).\]

    where K is the number of distinct outcomes, and N is the sample size. Many authors have tried to remedy this by proposing alternative Shannon entropy estimators. For example, the MillerMadow estimator is a simple correction to the plug-in estimator that adds back the bias term above. Many other estimators exist; see DiscreteInfoEstimators for an overview.

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    ComplexityMeasures.MillerMadowType
    MillerMadow <: DiscreteInfoEstimatorShannon
    -MillerMadow(measure::Shannon = Shannon())

    The MillerMadow estimator is used with information to compute the discrete Shannon entropy according to Miller (1955).

    Description

    The Miller-Madow estimator of Shannon entropy is given by

    \[H_S^{MM} = H_S^{plugin} + \dfrac{m - 1}{2N},\]

    where $H_S^{plugin}$ is the Shannon entropy estimated using the PlugIn estimator, m is the number of bins with nonzero probability (as defined in Paninski (2003)), and N is the number of observations.

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    ComplexityMeasures.GeneralizedSchuermannType
    GeneralizedSchuermann <: DiscreteInfoEstimatorShannon
    -GeneralizedSchuermann(definition = Shannon(); a = 1.0)

    The GeneralizedSchuermann estimator is used with information to compute the discrete Shannon entropy with the bias-corrected estimator given in Grassberger (2022).

    The "generalized" part of the name, as opposed to the Schürmann (2004) estimator (Schuermann), is due to the possibility of picking difference parameters $a_i$ for different outcomes. If different parameters are assigned to the different outcomes, a must be a vector of parameters of length length(outcomes), where the outcomes are obtained using outcomes. See Grassberger (2022) for more information. If a is a real number, then $a_i = a \forall i$, and the estimator reduces to the Schuermann estimator.

    Description

    For a set of $N$ observations over $M$ outcomes, the estimator is given by

    \[H_S^{opt} = \varphi(N) - \dfrac{1}{N} \sum_{i=1}^M n_i G_{n_i}(a_i),\]

    where $n_i$ is the observed frequency of the i-th outcome,

    \[G_n(a) = \varphi(n) + (-1)^n \int_0^a \dfrac{x^{n - 1}}{x + 1} dx,\]

    $G_n(1) = G_n$ and $G_n(0) = \varphi(n)$, and

    \[G_n = \varphi(n) + (-1)^n \int_0^1 \dfrac{x^{n - 1}}{x + 1} dx.\]

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    ComplexityMeasures.JackknifeType
    Jackknife <: DiscreteInfoEstimatorGeneric
    -Jackknife(definition::InformationMeasure = Shannon())

    The Jackknife estimator is used with information to compute any discrete InformationMeasure.

    The Jackknife estimator uses the generic jackknife principle to reduce bias. Zahl (1977) was the first to apply the jaccknife technique in the context of Shannon entropy estimation. Here, we've generalized his estimator to work with any InformationMeasure.

    Description

    As an example of the jackknife technique, here is the formula for a jackknife estimate of Shannon entropy

    \[H_S^{J} = N H_S^{plugin} - \dfrac{N-1}{N} \sum_{i=1}^N {H_S^{plugin}}^{-\{i\}},\]

    where $N$ is the sample size, $H_S^{plugin}$ is the plugin estimate of Shannon entropy, and ${H_S^{plugin}}^{-\{i\}}$ is the plugin estimate, but computed with the $i$-th sample left out.

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    ComplexityMeasures.HorvitzThompsonType
    HorvitzThompson <: DiscreteInfoEstimatorShannon
    -HorvitzThompson(measure::Shannon = Shannon())

    The HorvitzThompson estimator is used with information to compute the discrete Shannon entropy according to Horvitz and Thompson (1952).

    Description

    The Horvitz-Thompson estimator of Shannon entropy is given by

    \[H_S^{HT} = -\sum_{i=1}^M \dfrac{p_i \log(p_i) }{1 - (1 - p_i)^N},\]

    where $N$ is the sample size and $M$ is the number of outcomes. Given the true probability $p_i$ of the $i$-th outcome, $1 - (1 - p_i)^N$ is the probability that the outcome appears at least once in a sample of size $N$ (Arora et al., 2022). Dividing by this inclusion probability is a form of weighting, and compensates for situations where certain outcomes have so low probabilities that they are not often observed in a sample, for example in power-law distributions.

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    ComplexityMeasures.ChaoShenType
    ChaoShen <: DiscreteInfoEstimatorShannon
    -ChaoShen(definition::Shannon = Shannon())

    The ChaoShen estimator is used with information to compute the discrete Shannon entropy according to Chao and Shen (2003).

    Description

    This estimator is a modification of the HorvitzThompson estimator that multiplies each plugin probability estimate by an estimate of sample coverage. If $f_1$ is the number of singletons (outcomes that occur only once) in a sample of length $N$, then the sample coverage is $C = 1 - \dfrac{f_1}{N}$. The Chao-Shen estimator of Shannon entropy is then

    \[H_S^{CS} = -\sum_{i=1}^M \left( \dfrac{C p_i \log(C p_i)}{1 - (1 - C p_i)^N} \right),\]

    where $N$ is the sample size and $M$ is the number of outcomes. If $f_1 = N$, then $f_1$ is set to $f_1 = N - 1$ to ensure positive entropy (Arora et al., 2022).

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    Differential information estimators

    ComplexityMeasures.DifferentialInfoEstimatorType
    DifferentialInfoEstimator

    The supertype of all differential information measure estimators. These estimators compute an information measure in various ways that do not involve explicitly estimating a probability distribution.

    Each DifferentialInfoEstimators uses a specialized technique to approximate relevant densities/integrals, and is often tailored to one or a few types of information measures. For example, Kraskov estimates the Shannon entropy.

    See information for usage.

    Implementations

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    ComplexityMeasures.KraskovType
    Kraskov <: DifferentialInfoEstimator
    -Kraskov(definition = Shannon(); k::Int = 1, w::Int = 0)

    The Kraskov estimator computes the Shannon differential information of a multi-dimensional StateSpaceSet using the k-th nearest neighbor searches method from Kraskov et al. (2004), with logarithms to the base specified in definition.

    w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. Kraskov estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

    See also: information, KozachenkoLeonenko, DifferentialInfoEstimator.

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    ComplexityMeasures.KozachenkoLeonenkoType
    KozachenkoLeonenko <: DifferentialInfoEstimator
    -KozachenkoLeonenko(definition = Shannon(); w::Int = 0)

    The KozachenkoLeonenko estimator (Kozachenko and Leonenko, 1987) computes the Shannon differential information of a multi-dimensional StateSpaceSet, with logarithms to the base specified in definition.

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. KozachenkoLeonenko estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))]\]

    using the nearest neighbor method from Kozachenko and Leonenko (1987), as described in Charzyńska and Gambin (2016).

    w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    In contrast to Kraskov, this estimator uses only the closest neighbor.

    See also: information, Kraskov, DifferentialInfoEstimator.

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    ComplexityMeasures.ZhuType
    Zhu <: DifferentialInfoEstimator
    -Zhu(; definition = Shannon(), k = 1, w = 0)

    The Zhu estimator (Zhu et al., 2015) is an extension to KozachenkoLeonenko, and computes the Shannon differential information of a multi-dimensional StateSpaceSet, with logarithms to the base specified in definition.

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. Zhu estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))]\]

    by approximating densities within hyperrectangles surrounding each point xᵢ ∈ x using using k nearest neighbor searches. w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    See also: information, KozachenkoLeonenko, DifferentialInfoEstimator.

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    ComplexityMeasures.ZhuSinghType
    ZhuSingh <: DifferentialInfoEstimator
    -ZhuSingh(definition = Shannon(); k = 1, w = 0)

    The ZhuSingh estimator (Zhu et al., 2015) computes the Shannon differential information of a multi-dimensional StateSpaceSet, with logarithms to the base specified in definition.

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. ZhuSingh estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

    Like Zhu, this estimator approximates probabilities within hyperrectangles surrounding each point xᵢ ∈ x using using k nearest neighbor searches. However, it also considers the number of neighbors falling on the borders of these hyperrectangles. This estimator is an extension to the entropy estimator in Singh et al. (2003).

    w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    See also: information, DifferentialInfoEstimator.

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    ComplexityMeasures.GaoType
    Gao <: DifferentialInfoEstimator
    -Gao(definition = Shannon(); k = 1, w = 0, corrected = true)

    The Gao estimator (Gao et al., 09–12 May 2015) computes the Shannon differential information, using a k-th nearest-neighbor approach based on Singh et al. (2003), with logarithms to the base specified in definition.

    w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    Gao et al. (09–12 May 2015) give two variants of this estimator. If corrected == false, then the uncorrected version is used. If corrected == true, then the corrected version is used, which ensures that the estimator is asymptotically unbiased.

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. KozachenkoLeonenko estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

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    ComplexityMeasures.GoriaType
    Goria <: DifferentialInfoEstimator
    -Goria(measure = Shannon(); k = 1, w = 0)

    The Goria estimator (Goria et al., 2005) computes the Shannon differential information of a multi-dimensional StateSpaceSet, with logarithms to the base specified in definition.

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. Goria estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

    Specifically, let $\bf{n}_1, \bf{n}_2, \ldots, \bf{n}_N$ be the distance of the samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ to their k-th nearest neighbors. Next, let the geometric mean of the distances be

    \[\hat{\rho}_k = \left( \prod_{i=1}^N \right)^{\dfrac{1}{N}}\]

    Goria et al. (2005)'s estimate of Shannon differential entropy is then

    \[\hat{H} = m\hat{\rho}_k + \log(N - 1) - \psi(k) + \log c_1(m),\]

    where $c_1(m) = \dfrac{2\pi^\frac{m}{2}}{m \Gamma(m/2)}$ and $\psi$ is the digamma function.

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    ComplexityMeasures.LordType
    Lord <: DifferentialInfoEstimator
    +p_i \Gamma \left( \dfrac{\eta + 1}{\eta} \right),\]

    where $\eta \geq 0$, $\Gamma(\cdot, \cdot)$ is the upper incomplete Gamma function, and $\Gamma(\cdot) = \Gamma(\cdot, 0)$ is the Gamma function. Reduces to Shannon entropy for η = 1.0.

    The maximum entropy for StrechedExponential is a rather complicated expression involving incomplete Gamma functions (see source code).

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    Other information measures

    ComplexityMeasures.ShannonExtropyType
    ShannonExtropy <: InformationMeasure
    +ShannonExtropy(; base = 2)

    The Shannon extropy (Lad et al., 2015), used with information to compute

    \[J(x) = -\sum_{i=1}^N (1 - p[i]) \log{(1 - p[i])},\]

    for a probability distribution $P = \{p_1, p_2, \ldots, p_N\}$, with the $\log$ at the given base.

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    ComplexityMeasures.RenyiExtropyType
    RenyiExtropy <: InformationMeasure
    +RenyiExtropy(; q = 1.0, base = 2)

    The Rényi extropy (Liu and Xiao, 2023).

    Description

    RenyiExtropy is used with information to compute

    \[J_R(P) = \dfrac{-(n - 1) \log{(n - 1)} + (n - 1) \log{ \left( \sum_{i=1}^N {(1 - p[i])}^q \right)} }{q - 1}\]

    for a probability distribution $P = \{p_1, p_2, \ldots, p_N\}$, with the $\log$ at the given base. Alternatively, RenyiExtropy can be used with information_normalized, which ensures that the computed extropy is on the interval $[0, 1]$ by normalizing to to the maximal Rényi extropy, given by

    \[J_R(P) = (N - 1)\log \left( \dfrac{n}{n-1} \right) .\]

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    ComplexityMeasures.TsallisExtropyType
    TsallisExtropy <: InformationMeasure
    +TsallisExtropy(; base = 2)

    The Tsallis extropy (Xue and Deng, 2023).

    Description

    TsallisExtropy is used with information to compute

    \[J_T(P) = k \dfrac{N - 1 - \sum_{i=1}^N ( 1 - p[i])^q}{q - 1}\]

    for a probability distribution $P = \{p_1, p_2, \ldots, p_N\}$, with the $\log$ at the given base. Alternatively, TsallisExtropy can be used with information_normalized, which ensures that the computed extropy is on the interval $[0, 1]$ by normalizing to to the maximal Tsallis extropy, given by

    \[J_T(P) = \dfrac{(N - 1)N^{q - 1} - (N - 1)^q}{(q - 1)N^{q - 1}}\]

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    ComplexityMeasures.FluctuationComplexityType
    FluctuationComplexity <: InformationMeasure
    +FluctuationComplexity(; definition = Shannon(; base = 2), base = 2)

    The "fluctuation complexity" quantifies the standard deviation of the information content of the states $\omega_i$ around some summary statistic (InformationMeasure) of a PMF. Specifically, given some outcome space $\Omega$ with outcomes $\omega_i \in \Omega$ and a probability mass function $p(\Omega) = \{ p(\omega_i) \}_{i=1}^N$, it is defined as

    \[\sigma_I(p) := \sqrt{\sum_{i=1}^N p_i(I_i - H_*)^2}\]

    where $I_i = -\log_{base}(p_i)$ is the information content of the i-th outcome. The type of information measure $*$ is controlled by definition.

    The base controls the base of the logarithm that goes into the information content terms. Make sure that you pick a base that is consistent with the base chosen for the definition (relevant for e.g. Shannon).

    Properties

    If definition is the Shannon entropy, then we recover the Shannon-type information fluctuation complexity from (Bates and Shepard, 1993). Then the fluctuation complexity is zero for PMFs with only a single non-zero element, or for the uniform distribution.

    If definition is not Shannon entropy, then the properties of the measure varies, and does not necessarily share the properties (Bates and Shepard, 1993).

    Potential for new research

    As far as we know, using other information measures besides Shannon entropy for the fluctuation complexity hasn't been explored in the literature yet. Our implementation, however, allows for it. Please inform us if you try some new combinations!

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    Discrete information estimators

    ComplexityMeasures.DiscreteInfoEstimatorType
    DiscreteInfoEstimator

    The supertype of all discrete information measure estimators, which are used in combination with a ProbabilitiesEstimator as input to information or related functions.

    The first argument to a discrete estimator is always an InformationMeasure (defaults to Shannon).

    Description

    A discrete InformationMeasure is a functional of a probability mass function. To estimate such a measure from data, we must first estimate a probability mass function using a ProbabilitiesEstimator from the (encoded/discretized) input data, and then apply the estimator to the estimated probabilities. For example, the Shannon entropy is typically computed using the RelativeAmount estimator to compute probabilities, which are then given to the PlugIn estimator. Many other estimators exist, not only for Shannon entropy, but other information measures as well.

    We provide a library of both generic estimators such as PlugIn or Jackknife (which can be applied to any measure), as well as dedicated estimators such as MillerMadow, which computes Shannon entropy using the Miller-Madow bias correction. The list below gives a complete overview.

    Implementations

    The following estimators are generic and can compute any InformationMeasure.

    • PlugIn. The default, generic plug-in estimator of any information measure. It computes the measure exactly as stated in the definition, using the computed probability mass function.
    • Jackknife. Uses the a combination of the plug-in estimator and the jackknife principle to estimate the information measure.

    Shannon entropy estimators

    The following estimators are dedicated Shannon entropy estimators, which provide improvements over the naive PlugIn estimator.

    Info

    Any of the implemented DiscreteInfoEstimators can be used in combination with any ProbabilitiesEstimator as input to information. What this means is that every estimator actually comes in many different variants - one for each ProbabilitiesEstimator. For example, the MillerMadow estimator of Shannon entropy is typically calculated with RelativeAmount probabilities. But here, you can use for example the BayesianRegularization or the Shrinkage probabilities estimators instead, i.e. information(MillerMadow(), RelativeAmount(outcome_space), x) and information(MillerMadow(), BayesianRegularization(outcomes_space), x) are distinct estimators. This holds for all DiscreteInfoEstimators. Many of these estimators haven't been explored in the literature before, so feel free to explore, and please cite this software if you use it to explore some new estimator combination!

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    ComplexityMeasures.PlugInType
    PlugIn(e::InformationMeasure) <: DiscreteInfoEstimatorGeneric

    The PlugIn estimator is also called the empirical/naive/"maximum likelihood" estimator, and is used with information to any discrete InformationMeasure.

    It computes any quantity exactly as given by its formula. When computing an information measure, which here is defined as a probabilities functional, it computes the quantity directly from a probability mass function, which is derived from maximum-likelihood (RelativeAmount estimates of the probabilities.

    Bias of plug-in estimates

    The plugin-estimator of Shannon entropy underestimates the true entropy, with a bias that grows with the number of distinct outcomes (Arora et al., 2022)(Arora et al., 2022),

    \[bias(H_S^{plugin}) = -\dfrac{K-1}{2N} + o(N^-1).\]

    where K is the number of distinct outcomes, and N is the sample size. Many authors have tried to remedy this by proposing alternative Shannon entropy estimators. For example, the MillerMadow estimator is a simple correction to the plug-in estimator that adds back the bias term above. Many other estimators exist; see DiscreteInfoEstimators for an overview.

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    ComplexityMeasures.MillerMadowType
    MillerMadow <: DiscreteInfoEstimatorShannon
    +MillerMadow(measure::Shannon = Shannon())

    The MillerMadow estimator is used with information to compute the discrete Shannon entropy according to Miller (1955).

    Description

    The Miller-Madow estimator of Shannon entropy is given by

    \[H_S^{MM} = H_S^{plugin} + \dfrac{m - 1}{2N},\]

    where $H_S^{plugin}$ is the Shannon entropy estimated using the PlugIn estimator, m is the number of bins with nonzero probability (as defined in Paninski (2003)), and N is the number of observations.

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    ComplexityMeasures.GeneralizedSchuermannType
    GeneralizedSchuermann <: DiscreteInfoEstimatorShannon
    +GeneralizedSchuermann(definition = Shannon(); a = 1.0)

    The GeneralizedSchuermann estimator is used with information to compute the discrete Shannon entropy with the bias-corrected estimator given in Grassberger (2022).

    The "generalized" part of the name, as opposed to the Schürmann (2004) estimator (Schuermann), is due to the possibility of picking difference parameters $a_i$ for different outcomes. If different parameters are assigned to the different outcomes, a must be a vector of parameters of length length(outcomes), where the outcomes are obtained using outcomes. See Grassberger (2022) for more information. If a is a real number, then $a_i = a \forall i$, and the estimator reduces to the Schuermann estimator.

    Description

    For a set of $N$ observations over $M$ outcomes, the estimator is given by

    \[H_S^{opt} = \varphi(N) - \dfrac{1}{N} \sum_{i=1}^M n_i G_{n_i}(a_i),\]

    where $n_i$ is the observed frequency of the i-th outcome,

    \[G_n(a) = \varphi(n) + (-1)^n \int_0^a \dfrac{x^{n - 1}}{x + 1} dx,\]

    $G_n(1) = G_n$ and $G_n(0) = \varphi(n)$, and

    \[G_n = \varphi(n) + (-1)^n \int_0^1 \dfrac{x^{n - 1}}{x + 1} dx.\]

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    ComplexityMeasures.JackknifeType
    Jackknife <: DiscreteInfoEstimatorGeneric
    +Jackknife(definition::InformationMeasure = Shannon())

    The Jackknife estimator is used with information to compute any discrete InformationMeasure.

    The Jackknife estimator uses the generic jackknife principle to reduce bias. Zahl (1977) was the first to apply the jaccknife technique in the context of Shannon entropy estimation. Here, we've generalized his estimator to work with any InformationMeasure.

    Description

    As an example of the jackknife technique, here is the formula for a jackknife estimate of Shannon entropy

    \[H_S^{J} = N H_S^{plugin} - \dfrac{N-1}{N} \sum_{i=1}^N {H_S^{plugin}}^{-\{i\}},\]

    where $N$ is the sample size, $H_S^{plugin}$ is the plugin estimate of Shannon entropy, and ${H_S^{plugin}}^{-\{i\}}$ is the plugin estimate, but computed with the $i$-th sample left out.

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    ComplexityMeasures.HorvitzThompsonType
    HorvitzThompson <: DiscreteInfoEstimatorShannon
    +HorvitzThompson(measure::Shannon = Shannon())

    The HorvitzThompson estimator is used with information to compute the discrete Shannon entropy according to Horvitz and Thompson (1952).

    Description

    The Horvitz-Thompson estimator of Shannon entropy is given by

    \[H_S^{HT} = -\sum_{i=1}^M \dfrac{p_i \log(p_i) }{1 - (1 - p_i)^N},\]

    where $N$ is the sample size and $M$ is the number of outcomes. Given the true probability $p_i$ of the $i$-th outcome, $1 - (1 - p_i)^N$ is the probability that the outcome appears at least once in a sample of size $N$ (Arora et al., 2022). Dividing by this inclusion probability is a form of weighting, and compensates for situations where certain outcomes have so low probabilities that they are not often observed in a sample, for example in power-law distributions.

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    ComplexityMeasures.ChaoShenType
    ChaoShen <: DiscreteInfoEstimatorShannon
    +ChaoShen(definition::Shannon = Shannon())

    The ChaoShen estimator is used with information to compute the discrete Shannon entropy according to Chao and Shen (2003).

    Description

    This estimator is a modification of the HorvitzThompson estimator that multiplies each plugin probability estimate by an estimate of sample coverage. If $f_1$ is the number of singletons (outcomes that occur only once) in a sample of length $N$, then the sample coverage is $C = 1 - \dfrac{f_1}{N}$. The Chao-Shen estimator of Shannon entropy is then

    \[H_S^{CS} = -\sum_{i=1}^M \left( \dfrac{C p_i \log(C p_i)}{1 - (1 - C p_i)^N} \right),\]

    where $N$ is the sample size and $M$ is the number of outcomes. If $f_1 = N$, then $f_1$ is set to $f_1 = N - 1$ to ensure positive entropy (Arora et al., 2022).

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    Differential information estimators

    ComplexityMeasures.DifferentialInfoEstimatorType
    DifferentialInfoEstimator

    The supertype of all differential information measure estimators. These estimators compute an information measure in various ways that do not involve explicitly estimating a probability distribution.

    Each DifferentialInfoEstimators uses a specialized technique to approximate relevant densities/integrals, and is often tailored to one or a few types of information measures. For example, Kraskov estimates the Shannon entropy.

    See information for usage.

    Implementations

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    ComplexityMeasures.KraskovType
    Kraskov <: DifferentialInfoEstimator
    +Kraskov(definition = Shannon(); k::Int = 1, w::Int = 0)

    The Kraskov estimator computes the Shannon differential information of a multi-dimensional StateSpaceSet using the k-th nearest neighbor searches method from Kraskov et al. (2004), with logarithms to the base specified in definition.

    w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. Kraskov estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

    See also: information, KozachenkoLeonenko, DifferentialInfoEstimator.

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    ComplexityMeasures.KozachenkoLeonenkoType
    KozachenkoLeonenko <: DifferentialInfoEstimator
    +KozachenkoLeonenko(definition = Shannon(); w::Int = 0)

    The KozachenkoLeonenko estimator (Kozachenko and Leonenko, 1987) computes the Shannon differential information of a multi-dimensional StateSpaceSet, with logarithms to the base specified in definition.

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. KozachenkoLeonenko estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))]\]

    using the nearest neighbor method from Kozachenko and Leonenko (1987), as described in Charzyńska and Gambin (2016).

    w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    In contrast to Kraskov, this estimator uses only the closest neighbor.

    See also: information, Kraskov, DifferentialInfoEstimator.

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    ComplexityMeasures.ZhuType
    Zhu <: DifferentialInfoEstimator
    +Zhu(; definition = Shannon(), k = 1, w = 0)

    The Zhu estimator (Zhu et al., 2015) is an extension to KozachenkoLeonenko, and computes the Shannon differential information of a multi-dimensional StateSpaceSet, with logarithms to the base specified in definition.

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. Zhu estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))]\]

    by approximating densities within hyperrectangles surrounding each point xᵢ ∈ x using using k nearest neighbor searches. w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    See also: information, KozachenkoLeonenko, DifferentialInfoEstimator.

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    ComplexityMeasures.ZhuSinghType
    ZhuSingh <: DifferentialInfoEstimator
    +ZhuSingh(definition = Shannon(); k = 1, w = 0)

    The ZhuSingh estimator (Zhu et al., 2015) computes the Shannon differential information of a multi-dimensional StateSpaceSet, with logarithms to the base specified in definition.

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. ZhuSingh estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

    Like Zhu, this estimator approximates probabilities within hyperrectangles surrounding each point xᵢ ∈ x using using k nearest neighbor searches. However, it also considers the number of neighbors falling on the borders of these hyperrectangles. This estimator is an extension to the entropy estimator in Singh et al. (2003).

    w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    See also: information, DifferentialInfoEstimator.

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    ComplexityMeasures.GaoType
    Gao <: DifferentialInfoEstimator
    +Gao(definition = Shannon(); k = 1, w = 0, corrected = true)

    The Gao estimator (Gao et al., 09–12 May 2015) computes the Shannon differential information, using a k-th nearest-neighbor approach based on Singh et al. (2003), with logarithms to the base specified in definition.

    w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    Gao et al. (09–12 May 2015) give two variants of this estimator. If corrected == false, then the uncorrected version is used. If corrected == true, then the corrected version is used, which ensures that the estimator is asymptotically unbiased.

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. KozachenkoLeonenko estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

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    ComplexityMeasures.GoriaType
    Goria <: DifferentialInfoEstimator
    +Goria(measure = Shannon(); k = 1, w = 0)

    The Goria estimator (Goria et al., 2005) computes the Shannon differential information of a multi-dimensional StateSpaceSet, with logarithms to the base specified in definition.

    Description

    Assume we have samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function$f : \mathbb{R}^d \to \mathbb{R}$. Goria estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

    Specifically, let $\bf{n}_1, \bf{n}_2, \ldots, \bf{n}_N$ be the distance of the samples $\{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ to their k-th nearest neighbors. Next, let the geometric mean of the distances be

    \[\hat{\rho}_k = \left( \prod_{i=1}^N \right)^{\dfrac{1}{N}}\]

    Goria et al. (2005)'s estimate of Shannon differential entropy is then

    \[\hat{H} = m\hat{\rho}_k + \log(N - 1) - \psi(k) + \log c_1(m),\]

    where $c_1(m) = \dfrac{2\pi^\frac{m}{2}}{m \Gamma(m/2)}$ and $\psi$ is the digamma function.

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    ComplexityMeasures.LordType
    Lord <: DifferentialInfoEstimator
     Lord(measure = Shannon(); k = 10, w = 0)

    The Lord estimator (Lord et al., 2018) estimates the Shannon differential information using a nearest neighbor approach with a local nonuniformity correction (LNC), with logarithms to the base specified in definition.

    w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    Description

    Assume we have samples $\bar{X} = \{\bf{x}_1, \bf{x}_2, \ldots, \bf{x}_N \}$ from a continuous random variable $X \in \mathbb{R}^d$ with support $\mathcal{X}$ and density function $f : \mathbb{R}^d \to \mathbb{R}$. Lord estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))],\]

    by using the resubstitution formula

    \[\hat{\bar{X}, k} = -\mathbb{E}[\log(f(X))] -\approx \sum_{i = 1}^N \log(\hat{f}(\bf{x}_i)),\]

    where $\hat{f}(\bf{x}_i)$ is an estimate of the density at $\bf{x}_i$ constructed in a manner such that $\hat{f}(\bf{x}_i) \propto \dfrac{k(x_i) / N}{V_i}$, where $k(x_i)$ is the number of points in the neighborhood of $\bf{x}_i$, and $V_i$ is the volume of that neighborhood.

    While most nearest-neighbor based differential entropy estimators uses regular volume elements (e.g. hypercubes, hyperrectangles, hyperspheres) for approximating the local densities $\hat{f}(\bf{x}_i)$, the Lord estimator uses hyperellopsoid volume elements. These hyperellipsoids are, for each query point xᵢ, estimated using singular value decomposition (SVD) on the k-th nearest neighbors of xᵢ. Thus, the hyperellipsoids stretch/compress in response to the local geometry around each sample point. This makes Lord a well-suited entropy estimator for a wide range of systems.

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    ComplexityMeasures.LeonenkoProzantoSavaniType
    LeonenkoProzantoSavani <: DifferentialInfoEstimator
    -LeonenkoProzantoSavani(definition = Shannon(); k = 1, w = 0)

    The LeonenkoProzantoSavani estimator (Leonenko et al., 2008) computes the Shannon, Renyi, or Tsallis differential information of a multi-dimensional StateSpaceSet, with logarithms to the base specified in definition.

    Description

    The estimator uses k-th nearest-neighbor searches. w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    For details, see Leonenko et al. (2008).

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    ComplexityMeasures.VasicekType
    Vasicek <: DifferentialInfoEstimator
    +\approx \sum_{i = 1}^N \log(\hat{f}(\bf{x}_i)),\]

    where $\hat{f}(\bf{x}_i)$ is an estimate of the density at $\bf{x}_i$ constructed in a manner such that $\hat{f}(\bf{x}_i) \propto \dfrac{k(x_i) / N}{V_i}$, where $k(x_i)$ is the number of points in the neighborhood of $\bf{x}_i$, and $V_i$ is the volume of that neighborhood.

    While most nearest-neighbor based differential entropy estimators uses regular volume elements (e.g. hypercubes, hyperrectangles, hyperspheres) for approximating the local densities $\hat{f}(\bf{x}_i)$, the Lord estimator uses hyperellopsoid volume elements. These hyperellipsoids are, for each query point xᵢ, estimated using singular value decomposition (SVD) on the k-th nearest neighbors of xᵢ. Thus, the hyperellipsoids stretch/compress in response to the local geometry around each sample point. This makes Lord a well-suited entropy estimator for a wide range of systems.

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    ComplexityMeasures.LeonenkoProzantoSavaniType
    LeonenkoProzantoSavani <: DifferentialInfoEstimator
    +LeonenkoProzantoSavani(definition = Shannon(); k = 1, w = 0)

    The LeonenkoProzantoSavani estimator (Leonenko et al., 2008) computes the Shannon, Renyi, or Tsallis differential information of a multi-dimensional StateSpaceSet, with logarithms to the base specified in definition.

    Description

    The estimator uses k-th nearest-neighbor searches. w is the Theiler window, which determines if temporal neighbors are excluded during neighbor searches (defaults to 0, meaning that only the point itself is excluded when searching for neighbours).

    For details, see Leonenko et al. (2008).

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    ComplexityMeasures.VasicekType
    Vasicek <: DifferentialInfoEstimator
     Vasicek(definition = Shannon(); m::Int = 1)

    The Vasicek estimator computes the Shannon differential information of a timeseries using the method from Vasicek (1976), with logarithms to the base specified in definition.

    The Vasicek estimator belongs to a class of differential entropy estimators based on order statistics, of which Vasicek (1976) was the first. It only works for timeseries input.

    Description

    Assume we have samples $\bar{X} = \{x_1, x_2, \ldots, x_N \}$ from a continuous random variable $X \in \mathbb{R}$ with support $\mathcal{X}$ and density function$f : \mathbb{R} \to \mathbb{R}$. Vasicek estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

    However, instead of estimating the above integral directly, it makes use of the equivalent integral, where $F$ is the distribution function for $X$,

    \[H(X) = \int_0^1 \log \left(\dfrac{d}{dp}F^{-1}(p) \right) dp\]

    This integral is approximated by first computing the order statistics of $\bar{X}$ (the input timeseries), i.e. $x_{(1)} \leq x_{(2)} \leq \cdots \leq x_{(n)}$. The Vasicek Shannon differential entropy estimate is then

    \[\hat{H}_V(\bar{X}, m) = \dfrac{1}{n} -\sum_{i = 1}^n \log \left[ \dfrac{n}{2m} (\bar{X}_{(i+m)} - \bar{X}_{(i-m)}) \right]\]

    Usage

    In practice, choice of m influences how fast the entropy converges to the true value. For small value of m, convergence is slow, so we recommend to scale m according to the time series length n and use m >= n/100 (this is just a heuristic based on the tests written for this package).

    See also: information, Correa, AlizadehArghami, Ebrahimi, DifferentialInfoEstimator.

    source
    ComplexityMeasures.AlizadehArghamiType
    AlizadehArghami <: DifferentialInfoEstimator
    +\sum_{i = 1}^n \log \left[ \dfrac{n}{2m} (\bar{X}_{(i+m)} - \bar{X}_{(i-m)}) \right]\]

    Usage

    In practice, choice of m influences how fast the entropy converges to the true value. For small value of m, convergence is slow, so we recommend to scale m according to the time series length n and use m >= n/100 (this is just a heuristic based on the tests written for this package).

    See also: information, Correa, AlizadehArghami, Ebrahimi, DifferentialInfoEstimator.

    source
    ComplexityMeasures.AlizadehArghamiType
    AlizadehArghami <: DifferentialInfoEstimator
     AlizadehArghami(definition = Shannon(); m::Int = 1)

    The AlizadehArghami estimator computes the Shannon differential information of a timeseries using the method from Alizadeh and Arghami (2010), with logarithms to the base specified in definition.

    The AlizadehArghami estimator belongs to a class of differential entropy estimators based on order statistics. It only works for timeseries input.

    Description

    Assume we have samples $\bar{X} = \{x_1, x_2, \ldots, x_N \}$ from a continuous random variable $X \in \mathbb{R}$ with support $\mathcal{X}$ and density function$f : \mathbb{R} \to \mathbb{R}$. AlizadehArghami estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

    However, instead of estimating the above integral directly, it makes use of the equivalent integral, where $F$ is the distribution function for $X$:

    \[H(X) = \int_0^1 \log \left(\dfrac{d}{dp}F^{-1}(p) \right) dp.\]

    This integral is approximated by first computing the order statistics of $\bar{X}$ (the input timeseries), i.e. $x_{(1)} \leq x_{(2)} \leq \cdots \leq x_{(n)}$. The AlizadehArghami Shannon differential entropy estimate is then the the Vasicek estimate $\hat{H}_{V}(\bar{X}, m, n)$, plus a correction factor

    \[\hat{H}_{A}(\bar{X}, m, n) = \hat{H}_{V}(\bar{X}, m, n) + -\dfrac{2}{n}\left(m \log(2) \right).\]

    See also: information, Correa, Ebrahimi, Vasicek, DifferentialInfoEstimator.

    source
    ComplexityMeasures.EbrahimiType
    Ebrahimi <: DifferentialInfoEstimator
     Ebrahimi(definition = Shannon(); m::Int = 1)

    The Ebrahimi estimator computes the Shannon information of a timeseries using the method from Ebrahimi et al. (1994), with logarithms to the base specified in definition.

    The Ebrahimi estimator belongs to a class of differential entropy estimators based on order statistics. It only works for timeseries input.

    Description

    Assume we have samples $\bar{X} = \{x_1, x_2, \ldots, x_N \}$ from a continuous random variable $X \in \mathbb{R}$ with support $\mathcal{X}$ and density function$f : \mathbb{R} \to \mathbb{R}$. Ebrahimi estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

    However, instead of estimating the above integral directly, it makes use of the equivalent integral, where $F$ is the distribution function for $X$,

    \[H(X) = \int_0^1 \log \left(\dfrac{d}{dp}F^{-1}(p) \right) dp\]

    This integral is approximated by first computing the order statistics of $\bar{X}$ (the input timeseries), i.e. $x_{(1)} \leq x_{(2)} \leq \cdots \leq x_{(n)}$. The Ebrahimi Shannon differential entropy estimate is then

    \[\hat{H}_{E}(\bar{X}, m) = \dfrac{1}{n} \sum_{i = 1}^n \log \left[ \dfrac{n}{c_i m} (\bar{X}_{(i+m)} - \bar{X}_{(i-m)}) \right],\]

    where

    \[c_i = @@ -67,9 +67,9 @@ 1 + \frac{i - 1}{m}, & 1 \geq i \geq m \\ 2, & m + 1 \geq i \geq n - m \\ 1 + \frac{n - i}{m} & n - m + 1 \geq i \geq n -\end{cases}.\]

    See also: information, Correa, AlizadehArghami, Vasicek, DifferentialInfoEstimator.

    source
    ComplexityMeasures.CorreaType
    Correa <: DifferentialInfoEstimator
     Correa(definition = Shannon(); m::Int = 1)

    The Correa estimator computes the Shannon differential information of a timeseries using the method from Correa (1995), with logarithms to the base specified in definition.

    The Correa estimator belongs to a class of differential entropy estimators based on order statistics. It only works for timeseries input.

    Description

    Assume we have samples $\bar{X} = \{x_1, x_2, \ldots, x_N \}$ from a continuous random variable $X \in \mathbb{R}$ with support $\mathcal{X}$ and density function$f : \mathbb{R} \to \mathbb{R}$. Correa estimates the Shannon differential entropy

    \[H(X) = \int_{\mathcal{X}} f(x) \log f(x) dx = \mathbb{E}[-\log(f(X))].\]

    However, instead of estimating the above integral directly, Correa makes use of the equivalent integral, where $F$ is the distribution function for $X$,

    \[H(X) = \int_0^1 \log \left(\dfrac{d}{dp}F^{-1}(p) \right) dp\]

    This integral is approximated by first computing the order statistics of $\bar{X}$ (the input timeseries), i.e. $x_{(1)} \leq x_{(2)} \leq \cdots \leq x_{(n)}$, ensuring that end points are included. The Correa estimate of Shannon differential entropy is then

    \[H_C(\bar{X}, m, n) = \dfrac{1}{n} \sum_{i = 1}^n \log \left[ \dfrac{ \sum_{j=i-m}^{i+m}(\bar{X}_{(j)} - \tilde{X}_{(i)})(j - i)}{n \sum_{j=i-m}^{i+m} (\bar{X}_{(j)} - \tilde{X}_{(i)})^2} -\right],\]

    where

    \[\tilde{X}_{(i)} = \dfrac{1}{2m + 1} \sum_{j = i - m}^{i + m} X_{(j)}.\]

    See also: information, AlizadehArghami, Ebrahimi, Vasicek, DifferentialInfoEstimator.

    source

    Table of differential information measure estimators

    The following estimators are differential information measure estimators, and can also be used with information.

    Each DifferentialInfoEstimators uses a specialized technique to approximate relevant densities/integrals, and is often tailored to one or a few types of information measures. For example, Kraskov estimates the Shannon entropy.

    EstimatorPrincipleInput dataShannonRenyiTsallisKaniadakisCuradoStretchedExponential
    KozachenkoLeonenkoNearest neighborsStateSpaceSetxxxxx
    KraskovNearest neighborsStateSpaceSetxxxxx
    ZhuNearest neighborsStateSpaceSetxxxxx
    ZhuSinghNearest neighborsStateSpaceSetxxxxx
    GaoNearest neighborsStateSpaceSetxxxxx
    GoriaNearest neighborsStateSpaceSetxxxxx
    LordNearest neighborsStateSpaceSetxxxxx
    VasicekOrder statisticsVectorxxxxx
    EbrahimiOrder statisticsVectorxxxxx
    CorreaOrder statisticsVectorxxxxx
    AlizadehArghamiOrder statisticsVectorxxxxx
    +\right],\]

    where

    \[\tilde{X}_{(i)} = \dfrac{1}{2m + 1} \sum_{j = i - m}^{i + m} X_{(j)}.\]

    See also: information, AlizadehArghami, Ebrahimi, Vasicek, DifferentialInfoEstimator.

    source

    Table of differential information measure estimators

    The following estimators are differential information measure estimators, and can also be used with information.

    Each DifferentialInfoEstimators uses a specialized technique to approximate relevant densities/integrals, and is often tailored to one or a few types of information measures. For example, Kraskov estimates the Shannon entropy.

    EstimatorPrincipleInput dataShannonRenyiTsallisKaniadakisCuradoStretchedExponential
    KozachenkoLeonenkoNearest neighborsStateSpaceSetxxxxx
    KraskovNearest neighborsStateSpaceSetxxxxx
    ZhuNearest neighborsStateSpaceSetxxxxx
    ZhuSinghNearest neighborsStateSpaceSetxxxxx
    GaoNearest neighborsStateSpaceSetxxxxx
    GoriaNearest neighborsStateSpaceSetxxxxx
    LordNearest neighborsStateSpaceSetxxxxx
    VasicekOrder statisticsVectorxxxxx
    EbrahimiOrder statisticsVectorxxxxx
    CorreaOrder statisticsVectorxxxxx
    AlizadehArghamiOrder statisticsVectorxxxxx
    diff --git a/previews/PR430/measure_count/index.html b/previews/PR430/measure_count/index.html index 0ff2ad0f..e966ad1c 100644 --- a/previews/PR430/measure_count/index.html +++ b/previews/PR430/measure_count/index.html @@ -100,4 +100,4 @@ n_discrete_info + n_diff_info + n_complexity_measures_total + - n_extra_prob_measures
    1638
    + n_extra_prob_measures
    1638
    diff --git a/previews/PR430/multiscale/index.html b/previews/PR430/multiscale/index.html index 1da58043..9d32b524 100644 --- a/previews/PR430/multiscale/index.html +++ b/previews/PR430/multiscale/index.html @@ -13,12 +13,12 @@ # Differential kNN-based estimator: hest = LeonenkoProzantoSavani(Tsallis(q = 1.5), k = 10) # 10 neighbors multiscale(downsampling, hest, x)

    Multiscale variants of any ComplexityEstimator are also trivial to compute. Let's compute the "generalized multiscale sample entropy (Costa and Goldberger, 2015)" using the second-order moment.

    using ComplexityMeasures, Statistics
    -multiscale(CompositeDownsampling(; f = Statistics.var), SampleEntropy(x), x)
    source
    ComplexityMeasures.RegularDownsamplingType
    RegularDownsampling <: MultiScaleAlgorithm
     RegularDownsampling(; f::Function = Statistics.mean, scales = 1:8)

    The original multi-scale algorithm for multiscale entropy analysis (Costa et al., 2002), which yields a single downsampled time series per scale s.

    Description

    Given a scalar-valued input time series x, the Regular multiscale algorithm downsamples and coarse-grains x by splitting it into non-overlapping windows of length s, and then constructing a new downsampled time series $D_t(s, f)$ by applying the function f to each of the resulting length-s windows.

    The downsampled time series D_t(s) with t ∈ [1, 2, …, L], where L = floor(N / s), is given by:

    \[\{ D_t(s, f) \}_{t = 1}^{L} = \left\{ f \left( \bf x_t \right) \right\}_{t = 1}^{L} = \left\{ {f\left( (x_i)_{i = (t - 1)s + 1}^{ts} \right)} -\right\}_{t = 1}^{L}\]

    where f is some summary statistic applied to the length-ts-((t - 1)s + 1) tuples xₖ. Different choices of f have yield different multiscale methods appearing in the literature. For example:

    • f == Statistics.mean yields the original first-moment multiscale sample entropy (Costa et al., 2002).
    • f == Statistics.var yields the generalized multiscale sample entropy (Costa and Goldberger, 2015), which uses the second-moment (variance) instead of the mean.

    Keyword Arguments

    • scales. The downsampling levels. If scales is set to an integer, then this integer is taken as maximum number of scales (i.e. levels of downsampling), and downsampling is done over levels 1:scales. Otherwise, downsampling is done over the provided scales (which may be a range, or some specific scales (e.g. scales = [1, 5, 6]). The maximum scale level is length(x) ÷ 2, but to avoid applying the method to time series that are extremely short, consider limiting the maximum scale (e.g. scales = length(x) ÷ 5).

    See also: CompositeDownsampling.

    source
    ComplexityMeasures.CompositeDownsamplingType
    CompositeDownsampling <: MultiScaleAlgorithm
    +\right\}_{t = 1}^{L}\]

    where f is some summary statistic applied to the length-ts-((t - 1)s + 1) tuples xₖ. Different choices of f have yield different multiscale methods appearing in the literature. For example:

    • f == Statistics.mean yields the original first-moment multiscale sample entropy (Costa et al., 2002).
    • f == Statistics.var yields the generalized multiscale sample entropy (Costa and Goldberger, 2015), which uses the second-moment (variance) instead of the mean.

    Keyword Arguments

    • scales. The downsampling levels. If scales is set to an integer, then this integer is taken as maximum number of scales (i.e. levels of downsampling), and downsampling is done over levels 1:scales. Otherwise, downsampling is done over the provided scales (which may be a range, or some specific scales (e.g. scales = [1, 5, 6]). The maximum scale level is length(x) ÷ 2, but to avoid applying the method to time series that are extremely short, consider limiting the maximum scale (e.g. scales = length(x) ÷ 5).

    See also: CompositeDownsampling.

    source
    ComplexityMeasures.CompositeDownsamplingType
    CompositeDownsampling <: MultiScaleAlgorithm
     CompositeDownsampling(; f::Function = Statistics.mean, scales = 1:8)

    Composite multi-scale algorithm for multiscale entropy analysis (Wu et al., 2013), used with multiscale to compute, for example, composite multiscale entropy (CMSE).

    Description

    Given a scalar-valued input time series x, the composite multiscale algorithm, like RegularDownsampling, downsamples and coarse-grains x by splitting it into non-overlapping windows of length s, and then constructing downsampled time series by applying the function f to each of the resulting length-s windows.

    However, Wu et al. (2013) realized that for each scale s, there are actually s different ways of selecting windows, depending on where indexing starts/ends. These s different downsampled time series D_t(s, f) at each scale s are constructed as follows:

    \[\{ D_{k}(s) \} = \{ D_{t, k}(s) \}_{t = 1}^{L}, = \{ f \left( \bf x_{t, k} \right) \} = \left\{ {f\left( (x_i)_{i = (t - 1)s + k}^{ts + k - 1} \right)} -\right\}_{t = 1}^{L},\]

    where L = floor((N - s + 1) / s) and 1 ≤ k ≤ s, such that $D_{i, k}(s)$ is the i-th element of the k-th downsampled time series at scale s.

    Finally, compute $\dfrac{1}{s} \sum_{k = 1}^s g(D_{k}(s))$, where g is some summary function, for example information or complexity.

    Keyword Arguments

    • scales. The downsampling levels. If scales is set to an integer, then this integer is taken as maximum number of scales (i.e. levels of downsampling), and downsampling is done over levels 1:scales. Otherwise, downsampling is done over the provided scales (which may be a range, or some specific scales (e.g. scales = [1, 5, 6]). The maximum scale level is length(x) ÷ 2, but to avoid applying the method to time series that are extremely short, consider limiting the maximum scale (e.g. scales = length(x) ÷ 5).
    Relation to RegularDownsampling

    The downsampled time series $D_{t, 1}(s)$ constructed using the composite multiscale method is equivalent to the downsampled time series $D_{t}(s)$ constructed using the RegularDownsampling method, for which k == 1 is fixed, such that only a single time series is returned.

    See also: RegularDownsampling.

    source

    Example literature methods

    A non-exhaustive list of literature methods, and the syntax to compute them, are listed below. Please open an issue or make a pull-request to ComplexityMeasures.jl if you find a literature method missing from this list, or if you publish a paper based on some new multiscale combination.

    MethodSyntax exampleReference
    Refined composite multiscale dispersion entropymultiscale(CompositeDownsampling(), Dispersion(), est, x, normalized = true)(Azami et al., 2017)
    Multiscale sample entropy (first moment)multiscale(RegularDownsampling(f = mean), SampleEntropy(x), x)(Costa et al., 2002)
    Generalized multiscale sample entropy (second moment)multiscale(RegularDownsampling(f = std), SampleEntropy(x), x)(Costa and Goldberger, 2015)
    +\right\}_{t = 1}^{L},\]

    where L = floor((N - s + 1) / s) and 1 ≤ k ≤ s, such that $D_{i, k}(s)$ is the i-th element of the k-th downsampled time series at scale s.

    Finally, compute $\dfrac{1}{s} \sum_{k = 1}^s g(D_{k}(s))$, where g is some summary function, for example information or complexity.

    Keyword Arguments

    • scales. The downsampling levels. If scales is set to an integer, then this integer is taken as maximum number of scales (i.e. levels of downsampling), and downsampling is done over levels 1:scales. Otherwise, downsampling is done over the provided scales (which may be a range, or some specific scales (e.g. scales = [1, 5, 6]). The maximum scale level is length(x) ÷ 2, but to avoid applying the method to time series that are extremely short, consider limiting the maximum scale (e.g. scales = length(x) ÷ 5).
    Relation to RegularDownsampling

    The downsampled time series $D_{t, 1}(s)$ constructed using the composite multiscale method is equivalent to the downsampled time series $D_{t}(s)$ constructed using the RegularDownsampling method, for which k == 1 is fixed, such that only a single time series is returned.

    See also: RegularDownsampling.

    source

    Example literature methods

    A non-exhaustive list of literature methods, and the syntax to compute them, are listed below. Please open an issue or make a pull-request to ComplexityMeasures.jl if you find a literature method missing from this list, or if you publish a paper based on some new multiscale combination.

    MethodSyntax exampleReference
    Refined composite multiscale dispersion entropymultiscale(CompositeDownsampling(), Dispersion(), est, x, normalized = true)(Azami et al., 2017)
    Multiscale sample entropy (first moment)multiscale(RegularDownsampling(f = mean), SampleEntropy(x), x)(Costa et al., 2002)
    Generalized multiscale sample entropy (second moment)multiscale(RegularDownsampling(f = std), SampleEntropy(x), x)(Costa and Goldberger, 2015)
    diff --git a/previews/PR430/probabilities/index.html b/previews/PR430/probabilities/index.html index e9b32cf1..ba708bba 100644 --- a/previews/PR430/probabilities/index.html +++ b/previews/PR430/probabilities/index.html @@ -1,20 +1,20 @@ -Probabilities · ComplexityMeasures.jl

    Probabilities

    Note

    Be sure you have gone through the Tutorial before going through the API here to have a good idea of the terminology used in ComplexityMeasures.jl.

    ComplexityMeasures.jl implements an interface for probabilities that exactly follows the mathematically rigorous formulation of probability spaces. Probability spaces are formalized by an OutcomeSpace $\Omega$. Probabilities are extracted from data then by referencing an outcome space in the functions counts and probabilities. The mathematical formulation of probabilities spaces is further enhanced by ProbabilitiesEstimator and its subtypes, which may correct theoretically known biases when estimating probabilities from finite data.

    In reality, probabilities can be either discrete (mass functions) or continuous (density functions). Currently in ComplexityMeasures.jl, only probability mass functions (i.e., countable $\Omega$) are implemented explicitly. Quantities that are estimated from probability density functions (i.e., uncountable $\Omega$) also exist and are implemented in ComplexityMeasures.jl. However, these are estimated by a one-step processes without the intermediate estimation of probabilities.

    If $\Omega$ is countable, the process of estimating the outcomes from input data is also called discretization of the input data.

    Outcome spaces

    ComplexityMeasures.OutcomeSpaceType
    OutcomeSpace

    The supertype for all outcome space implementation.

    Description

    In ComplexityMeasures.jl, an outcome space defines a set of possible outcomes $\Omega = \{\omega_1, \omega_2, \ldots, \omega_L \}$ (some form of discretization). In the literature, the outcome space is often also called an "alphabet", while each outcome is called a "symbol" or an "event".

    An outcome space also defines a set of rules for mapping input data to to each outcome $\omega_i$, a processes called encoding or symbolizing or discretizing in the literature (see encodings). Some OutcomeSpaces first apply a transformation, e.g. a delay embedding, to the data before discretizing/encoding, while other OutcomeSpaces discretize/encode the data directly.

    Implementations

    Outcome spacePrincipleInput dataCounting-compatible
    UniqueElementsCount of unique elementsAny
    ValueBinningBinning (histogram)Vector, StateSpaceSet
    OrdinalPatternsOrdinal patternsVector, StateSpaceSet
    SpatialOrdinalPatternsOrdinal patterns in spaceArray
    DispersionDispersion patternsVector
    SpatialDispersionDispersion patterns in spaceArray
    CosineSimilarityBinningCosine similarityVector
    BubbleSortSwapsSwap counts when sortingVector
    SequentialPairDistancesSequential state vector distancesVector, StateSpaceSet
    TransferOperatorBinning (transfer operator)Vector, StateSpaceSet
    NaiveKernelKernel density estimationStateSpaceSet
    WeightedOrdinalPatternsOrdinal patternsVector, StateSpaceSet
    AmplitudeAwareOrdinalPatternsOrdinal patternsVector, StateSpaceSet
    WaveletOverlapWavelet transformVector
    PowerSpectrumFourier transformVector

    In the column "input data" it is assumed that the eltype of the input is <: Real.

    Usage

    Outcome spaces are used as input to

    Counting-compatible vs. non-counting compatible outcome spaces

    There are two main types of outcome spaces.

    • Counting-compatible outcome spaces have a well-defined way of counting how often each point in the (encoded) input data is mapped to a particular outcome $\omega_i$. These outcome spaces use encode to discretize the input data. Examples are OrdinalPatterns (which encodes input data into ordinal patterns) or ValueBinning (which discretizes points onto a regular grid). The table below lists which outcome spaces are counting compatible.
    • Non-counting compatible outcome spaces have no well-defined way of counting explicitly how often each point in the input data is mapped to a particular outcome $\omega_i$. Instead, these outcome spaces returns a vector of pre-normalized "relative counts", one for each outcome $\omega_i$. Examples are WaveletOverlap or PowerSpectrum.

    Counting-compatible outcome spaces can be used with any ProbabilitiesEstimator to convert counts into probability mass functions. Non-counting-compatible outcome spaces can only be used with the maximum likelihood (RelativeAmount) probabilities estimator, which estimates probabilities precisely by the relative frequency of each outcome (formally speaking, the RelativeAmount estimator also requires counts, but for the sake of code consistency, we allow it to be used with relative frequencies as well).

    The function is_counting_based can be used to check whether an outcome space is based on counting.

    Deducing the outcome space (from data)

    Some outcome space models can deduce $\Omega$ without knowledge of the input, such as OrdinalPatterns. Other outcome spaces require knowledge of the input data for concretely specifying $\Omega$, such as ValueBinning with RectangularBinning. If o is some outcome space model and x some input data, then outcome_space(o, x) returns the possible outcomes $\Omega$. To get the cardinality of $\Omega$, use total_outcomes.

    Implementation details

    The element type of $\Omega$ varies between outcome space models, but it is guaranteed to be hashable and sortable. This allows for conveniently tracking the counts of a specific event across experimental realizations, by using the outcome as a dictionary key and the counts as the value for that key (or, alternatively, the key remains the outcome and one has a vector of probabilities, one for each experimental realization).

    source
    ComplexityMeasures.outcomesFunction
    outcomes(o::OutcomeSpace, x)

    Return all (unique) outcomes that appear in the (encoded) input data x, according to the given OutcomeSpace. Equivalent to probabilities_and_outcomes(o, x)[2], but for some estimators it may be explicitly extended for better performance.

    source
    ComplexityMeasures.outcome_spaceFunction
    outcome_space(o::OutcomeSpace, x) → Ω

    Return a sorted container containing all possible outcomes of o for input x.

    For some estimators the concrete outcome space is known without knowledge of input x, in which case the function dispatches to outcome_space(o). In general it is recommended to use the 2-argument version irrespectively of estimator.

    source
    ComplexityMeasures.total_outcomesFunction
    total_outcomes(o::OutcomeSpace, x)

    Return the length (cardinality) of the outcome space $\Omega$ of est.

    For some OutcomeSpace, the cardinality is known without knowledge of input x, in which case the function dispatches to total_outcomes(est). In general it is recommended to use the 2-argument version irrespectively of estimator.

    source

    Count occurrences

    ComplexityMeasures.UniqueElementsType
    UniqueElements()

    An OutcomeSpace based on straight-forward counting of distinct elements in a univariate time series or multivariate dataset. This is the same as giving no estimator to probabilities.

    Outcome space

    The outcome space is the unique sorted values of the input. Hence, input x is needed for a well-defined outcome_space.

    Implements

    • codify. Used for encoding inputs where ordering matters (e.g. time series).
    source

    Histograms

    ComplexityMeasures.ValueBinningType
    ValueBinning(b::AbstractBinning) <: OutcomeSpace

    An OutcomeSpace based on binning the values of the data as dictated by the binning scheme b and formally computing their histogram, i.e., the frequencies of points in the bins. An alias to this is VisitationFrequency. Available binnings are subtypes of AbstractBinning.

    The ValueBinning estimator has a linearithmic time complexity (n log(n) for n = length(x)) and a linear space complexity (l for l = dimension(x)). This allows computation of probabilities (histograms) of high-dimensional datasets and with small box sizes ε without memory overflow and with maximum performance. For performance reasons, the probabilities returned never contain 0s and are arbitrarily ordered.

    ValueBinning(ϵ::Union{Real,Vector})

    A convenience method that accepts same input as RectangularBinning and initializes this binning directly.

    Outcomes

    The outcome space for ValueBinning is the unique bins constructed from b. Each bin is identified by its left (lowest-value) corner, because bins are always left-closed-right-open intervals [a, b). The bins are in data units, not integer (cartesian indices units), and are returned as SVectors, i.e., same type as input data.

    For convenience, outcome_space returns the outcomes in the same array format as the underlying binning (e.g., Matrix for 2D input).

    For FixedRectangularBinning the outcome_space is well-defined from the binning, but for RectangularBinning input x is needed as well.

    Implements

    • codify. Used for encoding inputs where ordering matters (e.g. time series).
    source
    ComplexityMeasures.RectangularBinningType
    RectangularBinning(ϵ, precise = false) <: AbstractBinning

    Rectangular box partition of state space using the scheme ϵ, deducing the histogram extent and bin width from the input data.

    RectangularBinning is a convenience struct. It is re-cast into FixedRectangularBinning once the data are provided, so see that docstring for info on the bin calculation and the meaning of precise.

    Binning instructions are deduced from the type of ϵ as follows:

    1. ϵ::Int divides each coordinate axis into ϵ equal-length intervals that cover all data.
    2. ϵ::Float64 divides each coordinate axis into intervals of fixed size ϵ, starting from the axis minima until the data is completely covered by boxes.
    3. ϵ::Vector{Int} divides the i-th coordinate axis into ϵ[i] equal-length intervals that cover all data.
    4. ϵ::Vector{Float64} divides the i-th coordinate axis into intervals of fixed size ϵ[i], starting from the axis minima until the data is completely covered by boxes.

    RectangularBinning ensures all input data are covered by extending the created ranges if need be.

    source
    ComplexityMeasures.FixedRectangularBinningType
    FixedRectangularBinning <: AbstractBinning
    -FixedRectangularBinning(ranges::Tuple{<:AbstractRange...}, precise = false)

    Rectangular box partition of state space where the partition along each dimension is explicitly given by each range ranges, which is a tuple of AbstractRange subtypes. Typically, each range is the output of the range Base function, e.g., ranges = (0:0.1:1, range(0, 1; length = 101), range(2.1, 3.2; step = 0.33)). All ranges must be sorted.

    The optional second argument precise dictates whether Julia Base's TwicePrecision is used for when searching where a point falls into the range. Useful for edge cases of points being almost exactly on the bin edges, but it is exactly four times as slow, so by default it is false.

    Points falling outside the partition do not contribute to probabilities. Bins are always left-closed-right-open: [a, b). This means that the last value of each of the ranges dictates the last right-closing value. This value does not belong to the histogram! E.g., if given a range r = range(0, 1; length = 11), with r[end] = 1, the value 1 is outside the partition and would not attribute any increase of the probability corresponding to the last bin (here [0.9, 1))!

    Equivalently, the size of the histogram is histsize = map(r -> length(r)-1, ranges)!

    FixedRectangularBinning leads to a well-defined outcome space without knowledge of input data, see ValueBinning.

    source

    Symbolic permutations

    ComplexityMeasures.OrdinalPatternsType
    OrdinalPatterns <: OutcomeSpace
    +Probabilities · ComplexityMeasures.jl

    Probabilities

    Note

    Be sure you have gone through the Tutorial before going through the API here to have a good idea of the terminology used in ComplexityMeasures.jl.

    ComplexityMeasures.jl implements an interface for probabilities that exactly follows the mathematically rigorous formulation of probability spaces. Probability spaces are formalized by an OutcomeSpace $\Omega$. Probabilities are extracted from data then by referencing an outcome space in the functions counts and probabilities. The mathematical formulation of probabilities spaces is further enhanced by ProbabilitiesEstimator and its subtypes, which may correct theoretically known biases when estimating probabilities from finite data.

    In reality, probabilities can be either discrete (mass functions) or continuous (density functions). Currently in ComplexityMeasures.jl, only probability mass functions (i.e., countable $\Omega$) are implemented explicitly. Quantities that are estimated from probability density functions (i.e., uncountable $\Omega$) also exist and are implemented in ComplexityMeasures.jl. However, these are estimated by a one-step processes without the intermediate estimation of probabilities.

    If $\Omega$ is countable, the process of estimating the outcomes from input data is also called discretization of the input data.

    Outcome spaces

    ComplexityMeasures.OutcomeSpaceType
    OutcomeSpace

    The supertype for all outcome space implementation.

    Description

    In ComplexityMeasures.jl, an outcome space defines a set of possible outcomes $\Omega = \{\omega_1, \omega_2, \ldots, \omega_L \}$ (some form of discretization). In the literature, the outcome space is often also called an "alphabet", while each outcome is called a "symbol" or an "event".

    An outcome space also defines a set of rules for mapping input data to to each outcome $\omega_i$, a processes called encoding or symbolizing or discretizing in the literature (see encodings). Some OutcomeSpaces first apply a transformation, e.g. a delay embedding, to the data before discretizing/encoding, while other OutcomeSpaces discretize/encode the data directly.

    Implementations

    Outcome spacePrincipleInput dataCounting-compatible
    UniqueElementsCount of unique elementsAny
    ValueBinningBinning (histogram)Vector, StateSpaceSet
    OrdinalPatternsOrdinal patternsVector, StateSpaceSet
    SpatialOrdinalPatternsOrdinal patterns in spaceArray
    DispersionDispersion patternsVector
    SpatialDispersionDispersion patterns in spaceArray
    CosineSimilarityBinningCosine similarityVector
    BubbleSortSwapsSwap counts when sortingVector
    SequentialPairDistancesSequential state vector distancesVector, StateSpaceSet
    TransferOperatorBinning (transfer operator)Vector, StateSpaceSet
    NaiveKernelKernel density estimationStateSpaceSet
    WeightedOrdinalPatternsOrdinal patternsVector, StateSpaceSet
    AmplitudeAwareOrdinalPatternsOrdinal patternsVector, StateSpaceSet
    WaveletOverlapWavelet transformVector
    PowerSpectrumFourier transformVector

    In the column "input data" it is assumed that the eltype of the input is <: Real.

    Usage

    Outcome spaces are used as input to

    Counting-compatible vs. non-counting compatible outcome spaces

    There are two main types of outcome spaces.

    • Counting-compatible outcome spaces have a well-defined way of counting how often each point in the (encoded) input data is mapped to a particular outcome $\omega_i$. These outcome spaces use encode to discretize the input data. Examples are OrdinalPatterns (which encodes input data into ordinal patterns) or ValueBinning (which discretizes points onto a regular grid). The table below lists which outcome spaces are counting compatible.
    • Non-counting compatible outcome spaces have no well-defined way of counting explicitly how often each point in the input data is mapped to a particular outcome $\omega_i$. Instead, these outcome spaces returns a vector of pre-normalized "relative counts", one for each outcome $\omega_i$. Examples are WaveletOverlap or PowerSpectrum.

    Counting-compatible outcome spaces can be used with any ProbabilitiesEstimator to convert counts into probability mass functions. Non-counting-compatible outcome spaces can only be used with the maximum likelihood (RelativeAmount) probabilities estimator, which estimates probabilities precisely by the relative frequency of each outcome (formally speaking, the RelativeAmount estimator also requires counts, but for the sake of code consistency, we allow it to be used with relative frequencies as well).

    The function is_counting_based can be used to check whether an outcome space is based on counting.

    Deducing the outcome space (from data)

    Some outcome space models can deduce $\Omega$ without knowledge of the input, such as OrdinalPatterns. Other outcome spaces require knowledge of the input data for concretely specifying $\Omega$, such as ValueBinning with RectangularBinning. If o is some outcome space model and x some input data, then outcome_space(o, x) returns the possible outcomes $\Omega$. To get the cardinality of $\Omega$, use total_outcomes.

    Implementation details

    The element type of $\Omega$ varies between outcome space models, but it is guaranteed to be hashable and sortable. This allows for conveniently tracking the counts of a specific event across experimental realizations, by using the outcome as a dictionary key and the counts as the value for that key (or, alternatively, the key remains the outcome and one has a vector of probabilities, one for each experimental realization).

    source
    ComplexityMeasures.outcomesFunction
    outcomes(o::OutcomeSpace, x)

    Return all (unique) outcomes that appear in the (encoded) input data x, according to the given OutcomeSpace. Equivalent to probabilities_and_outcomes(o, x)[2], but for some estimators it may be explicitly extended for better performance.

    source
    ComplexityMeasures.outcome_spaceFunction
    outcome_space(o::OutcomeSpace, x) → Ω

    Return a sorted container containing all possible outcomes of o for input x.

    For some estimators the concrete outcome space is known without knowledge of input x, in which case the function dispatches to outcome_space(o). In general it is recommended to use the 2-argument version irrespectively of estimator.

    source
    ComplexityMeasures.total_outcomesFunction
    total_outcomes(o::OutcomeSpace, x)

    Return the length (cardinality) of the outcome space $\Omega$ of est.

    For some OutcomeSpace, the cardinality is known without knowledge of input x, in which case the function dispatches to total_outcomes(est). In general it is recommended to use the 2-argument version irrespectively of estimator.

    source

    Count occurrences

    ComplexityMeasures.UniqueElementsType
    UniqueElements()

    An OutcomeSpace based on straight-forward counting of distinct elements in a univariate time series or multivariate dataset. This is the same as giving no estimator to probabilities.

    Outcome space

    The outcome space is the unique sorted values of the input. Hence, input x is needed for a well-defined outcome_space.

    Implements

    • codify. Used for encoding inputs where ordering matters (e.g. time series).
    source

    Histograms

    ComplexityMeasures.ValueBinningType
    ValueBinning(b::AbstractBinning) <: OutcomeSpace

    An OutcomeSpace based on binning the values of the data as dictated by the binning scheme b and formally computing their histogram, i.e., the frequencies of points in the bins. An alias to this is VisitationFrequency. Available binnings are subtypes of AbstractBinning.

    The ValueBinning estimator has a linearithmic time complexity (n log(n) for n = length(x)) and a linear space complexity (l for l = dimension(x)). This allows computation of probabilities (histograms) of high-dimensional datasets and with small box sizes ε without memory overflow and with maximum performance. For performance reasons, the probabilities returned never contain 0s and are arbitrarily ordered.

    ValueBinning(ϵ::Union{Real,Vector})

    A convenience method that accepts same input as RectangularBinning and initializes this binning directly.

    Outcomes

    The outcome space for ValueBinning is the unique bins constructed from b. Each bin is identified by its left (lowest-value) corner, because bins are always left-closed-right-open intervals [a, b). The bins are in data units, not integer (cartesian indices units), and are returned as SVectors, i.e., same type as input data.

    For convenience, outcome_space returns the outcomes in the same array format as the underlying binning (e.g., Matrix for 2D input).

    For FixedRectangularBinning the outcome_space is well-defined from the binning, but for RectangularBinning input x is needed as well.

    Implements

    • codify. Used for encoding inputs where ordering matters (e.g. time series).
    source
    ComplexityMeasures.RectangularBinningType
    RectangularBinning(ϵ, precise = false) <: AbstractBinning

    Rectangular box partition of state space using the scheme ϵ, deducing the histogram extent and bin width from the input data.

    RectangularBinning is a convenience struct. It is re-cast into FixedRectangularBinning once the data are provided, so see that docstring for info on the bin calculation and the meaning of precise.

    Binning instructions are deduced from the type of ϵ as follows:

    1. ϵ::Int divides each coordinate axis into ϵ equal-length intervals that cover all data.
    2. ϵ::Float64 divides each coordinate axis into intervals of fixed size ϵ, starting from the axis minima until the data is completely covered by boxes.
    3. ϵ::Vector{Int} divides the i-th coordinate axis into ϵ[i] equal-length intervals that cover all data.
    4. ϵ::Vector{Float64} divides the i-th coordinate axis into intervals of fixed size ϵ[i], starting from the axis minima until the data is completely covered by boxes.

    RectangularBinning ensures all input data are covered by extending the created ranges if need be.

    source
    ComplexityMeasures.FixedRectangularBinningType
    FixedRectangularBinning <: AbstractBinning
    +FixedRectangularBinning(ranges::Tuple{<:AbstractRange...}, precise = false)

    Rectangular box partition of state space where the partition along each dimension is explicitly given by each range ranges, which is a tuple of AbstractRange subtypes. Typically, each range is the output of the range Base function, e.g., ranges = (0:0.1:1, range(0, 1; length = 101), range(2.1, 3.2; step = 0.33)). All ranges must be sorted.

    The optional second argument precise dictates whether Julia Base's TwicePrecision is used for when searching where a point falls into the range. Useful for edge cases of points being almost exactly on the bin edges, but it is exactly four times as slow, so by default it is false.

    Points falling outside the partition do not contribute to probabilities. Bins are always left-closed-right-open: [a, b). This means that the last value of each of the ranges dictates the last right-closing value. This value does not belong to the histogram! E.g., if given a range r = range(0, 1; length = 11), with r[end] = 1, the value 1 is outside the partition and would not attribute any increase of the probability corresponding to the last bin (here [0.9, 1))!

    Equivalently, the size of the histogram is histsize = map(r -> length(r)-1, ranges)!

    FixedRectangularBinning leads to a well-defined outcome space without knowledge of input data, see ValueBinning.

    source

    Symbolic permutations

    ComplexityMeasures.OrdinalPatternsType
    OrdinalPatterns <: OutcomeSpace
     OrdinalPatterns{m}(τ = 1, lt::Function = ComplexityMeasures.isless_rand)

    An OutcomeSpace based on lengh-m ordinal permutation patterns, originally introduced in Bandt and Pompe (2002)'s paper on permutation entropy. Note that m is given as a type parameter, so that when it is a literal integer there are performance accelerations.

    When passed to probabilities the output depends on the input data type:

    • Univariate data. If applied to a univariate timeseries (AbstractVector), then the timeseries is first embedded using embedding delay τ and dimension m, resulting in embedding vectors $\{ \bf{x}_i \}_{i=1}^{N-(m-1)\tau}$. Then, for each $\bf{x}_i$, we find its permutation pattern $\pi_{i}$. Probabilities are then estimated as the frequencies of the encoded permutation symbols by using UniqueElements. When giving the resulting probabilities to information, the original permutation entropy is computed (Bandt and Pompe, 2002).
    • Multivariate data. If applied to a an D-dimensional StateSpaceSet, then no embedding is constructed, m must be equal to D and τ is ignored. Each vector $\bf{x}_i$ of the dataset is mapped directly to its permutation pattern $\pi_{i}$ by comparing the relative magnitudes of the elements of $\bf{x}_i$. Like above, probabilities are estimated as the frequencies of the permutation symbols. The resulting probabilities can be used to compute multivariate permutation entropy (He et al., 2016), although here we don't perform any further subdivision of the permutation patterns (as in Figure 3 of He et al. (2016)).

    Internally, OrdinalPatterns uses the OrdinalPatternEncoding to represent ordinal patterns as integers for efficient computations.

    See WeightedOrdinalPatterns and AmplitudeAwareOrdinalPatterns for estimators that not only consider ordinal (sorting) patterns, but also incorporate information about within-state-vector amplitudes. For a version of this estimator that can be used on spatial data, see SpatialOrdinalPatterns.

    Handling equal values in ordinal patterns

    In Bandt and Pompe (2002), equal values are ordered after their order of appearance, but this can lead to erroneous temporal correlations, especially for data with low amplitude resolution (Zunino et al., 2017). Here, by default, if two values are equal, then one of the is randomly assigned as "the largest", using lt = ComplexityMeasures.isless_rand. To get the behaviour from Bandt and Pompe (2002), use lt = Base.isless.

    Outcome space

    The outcome space Ω for OrdinalPatterns is the set of length-m ordinal patterns (i.e. permutations) that can be formed by the integers 1, 2, …, m. There are factorial(m) such patterns.

    For example, the outcome [2, 3, 1] corresponds to the ordinal pattern of having the smallest value in the second position, the next smallest value in the third position, and the next smallest, i.e. the largest value in the first position. See also OrdinalPatternEncoding.

    In-place symbolization

    OrdinalPatterns also implements the in-place probabilities! for StateSpaceSet input (or embedded vector input) for reducing allocations in looping scenarios. The length of the pre-allocated symbol vector must be the length of the dataset. For example

    using ComplexityMeasures
     m, N = 2, 100
     est = OrdinalPatterns{m}(τ)
     x = StateSpaceSet(rand(N, m)) # some input dataset
     πs_ts = zeros(Int, N) # length must match length of `x`
    -p = probabilities!(πs_ts, est, x)
    source
    ComplexityMeasures.WeightedOrdinalPatternsType
    WeightedOrdinalPatterns <: OutcomeSpace
    -WeightedOrdinalPatterns{m}(τ = 1, lt::Function = ComplexityMeasures.isless_rand)

    A variant of OrdinalPatterns that also incorporates amplitude information, based on the weighted permutation entropy (Fadlallah et al., 2013). The outcome space and arguments are the same as in OrdinalPatterns.

    Description

    For each ordinal pattern extracted from each state (or delay) vector, a weight is attached to it which is the variance of the vector. Probabilities are then estimated by summing the weights corresponding to the same pattern, instead of just counting the occurrence of the same pattern.

    An implementation note

    Note: in equation 7, section III, of the original paper, the authors write

    \[w_j = \dfrac{1}{m}\sum_{k=1}^m (x_{j-(k-1)\tau} - \mathbf{\hat{x}}_j^{m, \tau})^2.\]

    *But given the formula they give for the arithmetic mean, this is not the variance of the delay vector $\mathbf{x}_i$, because the indices are mixed: $x_{j+(k-1)\tau}$ in the weights formula, vs. $x_{j+(k+1)\tau}$ in the arithmetic mean formula. Here, delay embedding and computation of the patterns and their weights are completely separated processes, ensuring that we compute the arithmetic mean correctly for each vector of the input dataset (which may be a delay-embedded timeseries).

    source
    ComplexityMeasures.WeightedOrdinalPatternsType
    WeightedOrdinalPatterns <: OutcomeSpace
    +WeightedOrdinalPatterns{m}(τ = 1, lt::Function = ComplexityMeasures.isless_rand)

    A variant of OrdinalPatterns that also incorporates amplitude information, based on the weighted permutation entropy (Fadlallah et al., 2013). The outcome space and arguments are the same as in OrdinalPatterns.

    Description

    For each ordinal pattern extracted from each state (or delay) vector, a weight is attached to it which is the variance of the vector. Probabilities are then estimated by summing the weights corresponding to the same pattern, instead of just counting the occurrence of the same pattern.

    An implementation note

    Note: in equation 7, section III, of the original paper, the authors write

    \[w_j = \dfrac{1}{m}\sum_{k=1}^m (x_{j-(k-1)\tau} - \mathbf{\hat{x}}_j^{m, \tau})^2.\]

    *But given the formula they give for the arithmetic mean, this is not the variance of the delay vector $\mathbf{x}_i$, because the indices are mixed: $x_{j+(k-1)\tau}$ in the weights formula, vs. $x_{j+(k+1)\tau}$ in the arithmetic mean formula. Here, delay embedding and computation of the patterns and their weights are completely separated processes, ensuring that we compute the arithmetic mean correctly for each vector of the input dataset (which may be a delay-embedded timeseries).

    source
    ComplexityMeasures.AmplitudeAwareOrdinalPatternsType
    AmplitudeAwareOrdinalPatterns <: OutcomeSpace
     AmplitudeAwareOrdinalPatterns{m}(τ = 1, A = 0.5, lt = ComplexityMeasures.isless_rand)

    A variant of OrdinalPatterns that also incorporates amplitude information, based on the amplitude-aware permutation entropy (Azami and Escudero, 2016). The outcome space and arguments are the same as in OrdinalPatterns.

    Description

    Similarly to WeightedOrdinalPatterns, a weight $w_i$ is attached to each ordinal pattern extracted from each state (or delay) vector $\mathbf{x}_i = (x_1^i, x_2^i, \ldots, x_m^i)$ as

    \[w_i = \dfrac{A}{m} \sum_{k=1}^m |x_k^i | + \dfrac{1-A}{d-1} -\sum_{k=2}^d |x_{k}^i - x_{k-1}^i|,\]

    with $0 \leq A \leq 1$. When $A=0$ , only internal differences between the elements of $\mathbf{x}_i$ are weighted. Only mean amplitude of the state vector elements are weighted when $A=1$. With, $0<A<1$, a combined weighting is used.

    source

    Dispersion patterns

    ComplexityMeasures.DispersionType
    Dispersion(; c = 5, m = 2, τ = 1, check_unique = true)

    An OutcomeSpace based on dispersion patterns, originally used by Rostaghi and Azami (2016) to compute the "dispersion entropy", which characterizes the complexity and irregularity of a time series.

    Recommended parameter values (Li et al., 2019) are m ∈ [2, 3], τ = 1 for the embedding, and c ∈ [3, 4, …, 8] categories for the Gaussian symbol mapping.

    Description

    Assume we have a univariate time series $X = \{x_i\}_{i=1}^N$. First, this time series is encoded into a symbol timeseries $S$ using the Gaussian encoding GaussianCDFEncoding with empirical mean μ and empirical standard deviation σ (both determined from $X$), and c as given to Dispersion.

    Then, $S$ is embedded into an $m$-dimensional time series, using an embedding lag of $\tau$, which yields a total of $N - (m - 1)\tau$ delay vectors $z_i$, or "dispersion patterns". Since each element of $z_i$ can take on c different values, and each delay vector has m entries, there are c^m possible dispersion patterns. This number is used for normalization when computing dispersion entropy.

    The returned probabilities are simply the frequencies of the unique dispersion patterns present in $S$ (i.e., the UniqueElements of $S$).

    Outcome space

    The outcome space for Dispersion is the unique delay vectors whose elements are the the symbols (integers) encoded by the Gaussian CDF, i.e., the unique elements of $S$.

    Data requirements and parameters

    The input must have more than one unique element for the Gaussian mapping to be well-defined. Li et al. (2019) recommends that x has at least 1000 data points.

    If check_unique == true (default), then it is checked that the input has more than one unique value. If check_unique == false and the input only has one unique element, then a InexactError is thrown when trying to compute probabilities.

    Why 'dispersion patterns'?

    Each embedding vector is called a "dispersion pattern". Why? Let's consider the case when $m = 5$ and $c = 3$, and use some very imprecise terminology for illustration:

    When $c = 3$, values clustering far below mean are in one group, values clustered around the mean are in one group, and values clustering far above the mean are in a third group. Then the embedding vector $[2, 2, 2, 2, 2]$ consists of values that are close together (close to the mean), so it represents a set of numbers that are not very spread out (less dispersed). The embedding vector $[1, 1, 2, 3, 3]$, however, represents numbers that are much more spread out (more dispersed), because the categories representing "outliers" both above and below the mean are represented, not only values close to the mean.

    For a version of this estimator that can be used on high-dimensional arrays, see SpatialDispersion.

    Implements

    • codify. Used for encoding inputs where ordering matters (e.g. time series).
    source

    Transfer operator

    ComplexityMeasures.TransferOperatorType
    TransferOperator <: OutcomeSpace
    +\sum_{k=2}^d |x_{k}^i - x_{k-1}^i|,\]

    with $0 \leq A \leq 1$. When $A=0$ , only internal differences between the elements of $\mathbf{x}_i$ are weighted. Only mean amplitude of the state vector elements are weighted when $A=1$. With, $0<A<1$, a combined weighting is used.

    source

    Dispersion patterns

    ComplexityMeasures.DispersionType
    Dispersion(; c = 5, m = 2, τ = 1, check_unique = true)

    An OutcomeSpace based on dispersion patterns, originally used by Rostaghi and Azami (2016) to compute the "dispersion entropy", which characterizes the complexity and irregularity of a time series.

    Recommended parameter values (Li et al., 2019) are m ∈ [2, 3], τ = 1 for the embedding, and c ∈ [3, 4, …, 8] categories for the Gaussian symbol mapping.

    Description

    Assume we have a univariate time series $X = \{x_i\}_{i=1}^N$. First, this time series is encoded into a symbol timeseries $S$ using the Gaussian encoding GaussianCDFEncoding with empirical mean μ and empirical standard deviation σ (both determined from $X$), and c as given to Dispersion.

    Then, $S$ is embedded into an $m$-dimensional time series, using an embedding lag of $\tau$, which yields a total of $N - (m - 1)\tau$ delay vectors $z_i$, or "dispersion patterns". Since each element of $z_i$ can take on c different values, and each delay vector has m entries, there are c^m possible dispersion patterns. This number is used for normalization when computing dispersion entropy.

    The returned probabilities are simply the frequencies of the unique dispersion patterns present in $S$ (i.e., the UniqueElements of $S$).

    Outcome space

    The outcome space for Dispersion is the unique delay vectors whose elements are the the symbols (integers) encoded by the Gaussian CDF, i.e., the unique elements of $S$.

    Data requirements and parameters

    The input must have more than one unique element for the Gaussian mapping to be well-defined. Li et al. (2019) recommends that x has at least 1000 data points.

    If check_unique == true (default), then it is checked that the input has more than one unique value. If check_unique == false and the input only has one unique element, then a InexactError is thrown when trying to compute probabilities.

    Why 'dispersion patterns'?

    Each embedding vector is called a "dispersion pattern". Why? Let's consider the case when $m = 5$ and $c = 3$, and use some very imprecise terminology for illustration:

    When $c = 3$, values clustering far below mean are in one group, values clustered around the mean are in one group, and values clustering far above the mean are in a third group. Then the embedding vector $[2, 2, 2, 2, 2]$ consists of values that are close together (close to the mean), so it represents a set of numbers that are not very spread out (less dispersed). The embedding vector $[1, 1, 2, 3, 3]$, however, represents numbers that are much more spread out (more dispersed), because the categories representing "outliers" both above and below the mean are represented, not only values close to the mean.

    For a version of this estimator that can be used on high-dimensional arrays, see SpatialDispersion.

    Implements

    • codify. Used for encoding inputs where ordering matters (e.g. time series).
    source

    Transfer operator

    ComplexityMeasures.TransferOperatorType
    TransferOperator <: OutcomeSpace
     TransferOperator(b::AbstractBinning; warn_precise = true, rng = Random.default_rng())

    An OutcomeSpace based on binning data into rectangular boxes dictated by the given binning scheme b.

    When used with probabilities, then the transfer (Perron-Frobenius) operator is approximated over the bins, then bin probabilities are estimated as the invariant measure associated with that transfer operator. Assumes that the input data are sequential (time-ordered).

    This implementation follows the grid estimator approach in Diego et al. (2019).

    Precision

    The default behaviour when using RectangularBinning or FixedRectangularBinning is to accept some loss of precision on the bin boundaries for speed-ups, but this may lead to issues for TransferOperator where some points may be encoded as the symbol -1 ("outside the binning"). The warn_precise keyword controls whether the user is warned when a less precise binning is used.

    Outcome space

    The outcome space for TransferOperator is the set of unique bins constructed from b. Bins are identified by their left (lowest-value) corners, are given in data units, and are returned as SVectors.

    Bin ordering

    Bins returned by probabilities_and_outcomes are ordered according to first appearance (i.e. the first time the input (multivariate) timeseries visits the bin). Thus, if

    b = RectangularBinning(4)
     est = TransferOperator(b)
     probs, outcomes = probabilities_and_outcomes(x, est) # x is some timeseries

    then probs[i] is the invariant measure (probability) of the bin outcomes[i], which is the i-th bin visited by the timeseries with nonzero measure.

    Description

    The transfer operator $P^{N}$is computed as an N-by-N matrix of transition probabilities between the states defined by the partition elements, where N is the number of boxes in the partition that is visited by the orbit/points.

    If $\{x_t^{(D)} \}_{n=1}^L$ are the $L$ different $D$-dimensional points over which the transfer operator is approximated, $\{ C_{k=1}^N \}$ are the $N$ different partition elements (as dictated by ϵ) that gets visited by the points, and $\phi(x_t) = x_{t+1}$, then

    \[P_{ij} = \dfrac {\#\{ x_n | \phi(x_n) \in C_j \cap x_n \in C_i \}} -{\#\{ x_m | x_m \in C_i \}},\]

    where $\#$ denotes the cardinal. The element $P_{ij}$ thus indicates how many points that are initially in box $C_i$ end up in box $C_j$ when the points in $C_i$ are projected one step forward in time. Thus, the row $P_{ik}^N$ where $k \in \{1, 2, \ldots, N \}$ gives the probability of jumping from the state defined by box $C_i$ to any of the other $N$ states. It follows that $\sum_{k=1}^{N} P_{ik} = 1$ for all $i$. Thus, $P^N$ is a row/right stochastic matrix.

    Invariant measure estimation from transfer operator

    The left invariant distribution $\mathbf{\rho}^N$ is a row vector, where $\mathbf{\rho}^N P^{N} = \mathbf{\rho}^N$. Hence, $\mathbf{\rho}^N$ is a row eigenvector of the transfer matrix $P^{N}$ associated with eigenvalue 1. The distribution $\mathbf{\rho}^N$ approximates the invariant density of the system subject to binning, and can be taken as a probability distribution over the partition elements.

    In practice, the invariant measure $\mathbf{\rho}^N$ is computed using invariantmeasure, which also approximates the transfer matrix. The invariant distribution is initialized as a length-N random distribution which is then applied to $P^{N}$. For reproducibility in this step, set the rng. The resulting length-N distribution is then applied to $P^{N}$ again. This process repeats until the difference between the distributions over consecutive iterations is below some threshold.

    See also: RectangularBinning, FixedRectangularBinning, invariantmeasure.

    source

    Utility methods/types

    ComplexityMeasures.invariantmeasureFunction
    invariantmeasure(x::AbstractStateSpaceSet, binning::RectangularBinning;
    +{\#\{ x_m | x_m \in C_i \}},\]

    where $\#$ denotes the cardinal. The element $P_{ij}$ thus indicates how many points that are initially in box $C_i$ end up in box $C_j$ when the points in $C_i$ are projected one step forward in time. Thus, the row $P_{ik}^N$ where $k \in \{1, 2, \ldots, N \}$ gives the probability of jumping from the state defined by box $C_i$ to any of the other $N$ states. It follows that $\sum_{k=1}^{N} P_{ik} = 1$ for all $i$. Thus, $P^N$ is a row/right stochastic matrix.

    Invariant measure estimation from transfer operator

    The left invariant distribution $\mathbf{\rho}^N$ is a row vector, where $\mathbf{\rho}^N P^{N} = \mathbf{\rho}^N$. Hence, $\mathbf{\rho}^N$ is a row eigenvector of the transfer matrix $P^{N}$ associated with eigenvalue 1. The distribution $\mathbf{\rho}^N$ approximates the invariant density of the system subject to binning, and can be taken as a probability distribution over the partition elements.

    In practice, the invariant measure $\mathbf{\rho}^N$ is computed using invariantmeasure, which also approximates the transfer matrix. The invariant distribution is initialized as a length-N random distribution which is then applied to $P^{N}$. For reproducibility in this step, set the rng. The resulting length-N distribution is then applied to $P^{N}$ again. This process repeats until the difference between the distributions over consecutive iterations is below some threshold.

    See also: RectangularBinning, FixedRectangularBinning, invariantmeasure.

    source

    Utility methods/types

    ComplexityMeasures.invariantmeasureFunction
    invariantmeasure(x::AbstractStateSpaceSet, binning::RectangularBinning;
         rng = Random.default_rng()) → iv::InvariantMeasure

    Estimate an invariant measure over the points in x based on binning the data into rectangular boxes dictated by the binning, then approximate the transfer (Perron-Frobenius) operator over the bins. From the approximation to the transfer operator, compute an invariant distribution over the bins. Assumes that the input data are sequential.

    Details on the estimation procedure is found the TransferOperator docstring.

    Example

    using DynamicalSystems
     henon_rule(x, p, n) = SVector{2}(1.0 - p[1]*x[1]^2 + x[2], p[2]*x[1])
     henon = DeterministicIteratedMap(henon_rule, zeros(2), [1.4, 0.3])
    @@ -24,7 +24,7 @@
     iv = invariantmeasure(orbit, RectangularBinning(15))
     
     # Get the probabilities and bins
    -invariantmeasure(iv)

    Probabilities and bin information

    invariantmeasure(iv::InvariantMeasure) → (ρ::Probabilities, bins::Vector{<:SVector})

    From a pre-computed invariant measure, return the probabilities and associated bins. The element ρ[i] is the probability of visitation to the box bins[i].

    Transfer operator approach vs. naive histogram approach

    Why bother with the transfer operator instead of using regular histograms to obtain probabilities?

    In fact, the naive histogram approach and the transfer operator approach are equivalent in the limit of long enough time series (as $n \to \intfy$), which is guaranteed by the ergodic theorem. There is a crucial difference, however:

    The naive histogram approach only gives the long-term probabilities that orbits visit a certain region of the state space. The transfer operator encodes that information too, but comes with the added benefit of knowing the transition probabilities between states (see transfermatrix).

    See also: InvariantMeasure.

    source
    ComplexityMeasures.transfermatrixFunction
    transfermatrix(iv::InvariantMeasure) → (M::AbstractArray{<:Real, 2}, bins::Vector{<:SVector})

    Return the transfer matrix/operator and corresponding bins. Here, bins[i] corresponds to the i-th row/column of the transfer matrix. Thus, the entry M[i, j] is the probability of jumping from the state defined by bins[i] to the state defined by bins[j].

    See also: TransferOperator.

    source

    Kernel density

    ComplexityMeasures.NaiveKernelType
    NaiveKernel(ϵ::Real; method = KDTree, w = 0, metric = Euclidean()) <: OutcomeSpace

    An OutcomeSpace based on a "naive" kernel density estimation approach (KDE), as discussed in Prichard and Theiler (1995).

    Probabilities $P(\mathbf{x}, \epsilon)$ are assigned to every point $\mathbf{x}$ by counting how many other points occupy the space spanned by a hypersphere of radius ϵ around $\mathbf{x}$, according to:

    \[P_i( X, \epsilon) \approx \dfrac{1}{N} \sum_{s} B(||X_i - X_j|| < \epsilon),\]

    where $B$ gives 1 if the argument is true. Probabilities are then normalized.

    Keyword arguments

    • method = KDTree: the search structure supported by Neighborhood.jl. Specifically, use KDTree to use a tree-based neighbor search, or BruteForce for the direct distances between all points. KDTrees heavily outperform direct distances when the dimensionality of the data is much smaller than the data length.
    • w = 0: the Theiler window, which excludes indices $s$ that are within $|i - s| ≤ w$ from the given point $x_i$.
    • metric = Euclidean(): the distance metric.

    Outcome space

    The outcome space Ω for NaiveKernel are the indices of the input data, eachindex(x). Hence, input x is needed for a well-defined outcome_space. The reason to not return the data points themselves is because duplicate data points may not get assigned same probabilities (due to having different neighbors).

    source

    Timescales

    ComplexityMeasures.WaveletOverlapType
    WaveletOverlap([wavelet]) <: OutcomeSpace

    An OutcomeSpace based on the maximal overlap discrete wavelet transform (MODWT).

    When used with probabilities, the MODWT is applied to a signal, then probabilities are computed as the (normalized) energies at different wavelet scales. These probabilities are used to compute the wavelet entropy according to Rosso et al. (2001). Input timeseries x is needed for a well-defined outcome space.

    By default the wavelet Wavelets.WT.Daubechies{12}() is used. Otherwise, you may choose a wavelet from the Wavelets package (it must subtype OrthoWaveletClass).

    Outcome space

    The outcome space for WaveletOverlap are the integers 1, 2, …, N enumerating the wavelet scales. To obtain a better understanding of what these mean, we prepared a notebook you can view online. As such, this estimator only works for timeseries input and input x is needed for a well-defined outcome_space.

    source
    ComplexityMeasures.PowerSpectrumType
    PowerSpectrum() <: OutcomeSpace

    An OutcomeSpace based on the power spectrum of a timeseries (amplitude square of its Fourier transform).

    If used with probabilities, then the spectrum normalized to sum = 1 is returned as probabilities. The Shannon entropy of these probabilities is typically referred in the literature as spectral entropy, e.g. Llanos et al. (2017) and Tian et al. (2017).

    The closer the spectrum is to flat, i.e., white noise, the higher the entropy. However, you can't compare entropies of timeseries with different length, because the binning in spectral space depends on the length of the input.

    Outcome space

    The outcome space Ω for PowerSpectrum is the set of frequencies in Fourier space. They should be multiplied with the sampling rate of the signal, which is assumed to be 1. Input x is needed for a well-defined outcome_space.

    source

    Cosine similarity binning

    ComplexityMeasures.CosineSimilarityBinningType
    CosineSimilarityBinning(; m::Int, τ::Int, nbins::Int)

    A OutcomeSpace based on the cosine similarity (Wang et al., 2020).

    It can be used with information to compute the "diversity entropy" of an input timeseries (Wang et al., 2020).

    The implementation here allows for τ != 1, which was not considered in the original paper.

    Description

    CosineSimilarityBinning probabilities are computed as follows.

    1. From the input time series x, using embedding lag τ and embedding dimension m, construct the embedding $Y = \{\bf x_i \} = \{(x_{i}, x_{i+\tau}, x_{i+2\tau}, \ldots, x_{i+m\tau - 1}\}_{i = 1}^{N-mτ}$.
    2. Compute $D = \{d(\bf x_t, \bf x_{t+1}) \}_{t=1}^{N-mτ-1}$, where $d(\cdot, \cdot)$ is the cosine similarity between two m-dimensional vectors in the embedding.
    3. Divide the interval [-1, 1] into nbins equally sized subintervals (including the value +1).
    4. Construct a histogram of cosine similarities $d \in D$ over those subintervals.
    5. Sum-normalize the histogram to obtain probabilities.

    Outcome space

    The outcome space for CosineSimilarityBinning is the bins of the [-1, 1] interval, and the return configuration is the same as in ValueBinning (left bin edge).

    Implements

    • codify. Used for encoding inputs where ordering matters (e.g. time series).
    source

    Sequential pair distances

    ComplexityMeasures.SequentialPairDistancesType
    SequentialPairDistances <: CountBasedOutcomeSpace
    +invariantmeasure(iv)

    Probabilities and bin information

    invariantmeasure(iv::InvariantMeasure) → (ρ::Probabilities, bins::Vector{<:SVector})

    From a pre-computed invariant measure, return the probabilities and associated bins. The element ρ[i] is the probability of visitation to the box bins[i].

    Transfer operator approach vs. naive histogram approach

    Why bother with the transfer operator instead of using regular histograms to obtain probabilities?

    In fact, the naive histogram approach and the transfer operator approach are equivalent in the limit of long enough time series (as $n \to \intfy$), which is guaranteed by the ergodic theorem. There is a crucial difference, however:

    The naive histogram approach only gives the long-term probabilities that orbits visit a certain region of the state space. The transfer operator encodes that information too, but comes with the added benefit of knowing the transition probabilities between states (see transfermatrix).

    See also: InvariantMeasure.

    source
    ComplexityMeasures.transfermatrixFunction
    transfermatrix(iv::InvariantMeasure) → (M::AbstractArray{<:Real, 2}, bins::Vector{<:SVector})

    Return the transfer matrix/operator and corresponding bins. Here, bins[i] corresponds to the i-th row/column of the transfer matrix. Thus, the entry M[i, j] is the probability of jumping from the state defined by bins[i] to the state defined by bins[j].

    See also: TransferOperator.

    source

    Kernel density

    ComplexityMeasures.NaiveKernelType
    NaiveKernel(ϵ::Real; method = KDTree, w = 0, metric = Euclidean()) <: OutcomeSpace

    An OutcomeSpace based on a "naive" kernel density estimation approach (KDE), as discussed in Prichard and Theiler (1995).

    Probabilities $P(\mathbf{x}, \epsilon)$ are assigned to every point $\mathbf{x}$ by counting how many other points occupy the space spanned by a hypersphere of radius ϵ around $\mathbf{x}$, according to:

    \[P_i( X, \epsilon) \approx \dfrac{1}{N} \sum_{s} B(||X_i - X_j|| < \epsilon),\]

    where $B$ gives 1 if the argument is true. Probabilities are then normalized.

    Keyword arguments

    • method = KDTree: the search structure supported by Neighborhood.jl. Specifically, use KDTree to use a tree-based neighbor search, or BruteForce for the direct distances between all points. KDTrees heavily outperform direct distances when the dimensionality of the data is much smaller than the data length.
    • w = 0: the Theiler window, which excludes indices $s$ that are within $|i - s| ≤ w$ from the given point $x_i$.
    • metric = Euclidean(): the distance metric.

    Outcome space

    The outcome space Ω for NaiveKernel are the indices of the input data, eachindex(x). Hence, input x is needed for a well-defined outcome_space. The reason to not return the data points themselves is because duplicate data points may not get assigned same probabilities (due to having different neighbors).

    source

    Timescales

    ComplexityMeasures.WaveletOverlapType
    WaveletOverlap([wavelet]) <: OutcomeSpace

    An OutcomeSpace based on the maximal overlap discrete wavelet transform (MODWT).

    When used with probabilities, the MODWT is applied to a signal, then probabilities are computed as the (normalized) energies at different wavelet scales. These probabilities are used to compute the wavelet entropy according to Rosso et al. (2001). Input timeseries x is needed for a well-defined outcome space.

    By default the wavelet Wavelets.WT.Daubechies{12}() is used. Otherwise, you may choose a wavelet from the Wavelets package (it must subtype OrthoWaveletClass).

    Outcome space

    The outcome space for WaveletOverlap are the integers 1, 2, …, N enumerating the wavelet scales. To obtain a better understanding of what these mean, we prepared a notebook you can view online. As such, this estimator only works for timeseries input and input x is needed for a well-defined outcome_space.

    source
    ComplexityMeasures.PowerSpectrumType
    PowerSpectrum() <: OutcomeSpace

    An OutcomeSpace based on the power spectrum of a timeseries (amplitude square of its Fourier transform).

    If used with probabilities, then the spectrum normalized to sum = 1 is returned as probabilities. The Shannon entropy of these probabilities is typically referred in the literature as spectral entropy, e.g. Llanos et al. (2017) and Tian et al. (2017).

    The closer the spectrum is to flat, i.e., white noise, the higher the entropy. However, you can't compare entropies of timeseries with different length, because the binning in spectral space depends on the length of the input.

    Outcome space

    The outcome space Ω for PowerSpectrum is the set of frequencies in Fourier space. They should be multiplied with the sampling rate of the signal, which is assumed to be 1. Input x is needed for a well-defined outcome_space.

    source

    Cosine similarity binning

    ComplexityMeasures.CosineSimilarityBinningType
    CosineSimilarityBinning(; m::Int, τ::Int, nbins::Int)

    A OutcomeSpace based on the cosine similarity (Wang et al., 2020).

    It can be used with information to compute the "diversity entropy" of an input timeseries (Wang et al., 2020).

    The implementation here allows for τ != 1, which was not considered in the original paper.

    Description

    CosineSimilarityBinning probabilities are computed as follows.

    1. From the input time series x, using embedding lag τ and embedding dimension m, construct the embedding $Y = \{\bf x_i \} = \{(x_{i}, x_{i+\tau}, x_{i+2\tau}, \ldots, x_{i+m\tau - 1}\}_{i = 1}^{N-mτ}$.
    2. Compute $D = \{d(\bf x_t, \bf x_{t+1}) \}_{t=1}^{N-mτ-1}$, where $d(\cdot, \cdot)$ is the cosine similarity between two m-dimensional vectors in the embedding.
    3. Divide the interval [-1, 1] into nbins equally sized subintervals (including the value +1).
    4. Construct a histogram of cosine similarities $d \in D$ over those subintervals.
    5. Sum-normalize the histogram to obtain probabilities.

    Outcome space

    The outcome space for CosineSimilarityBinning is the bins of the [-1, 1] interval, and the return configuration is the same as in ValueBinning (left bin edge).

    Implements

    • codify. Used for encoding inputs where ordering matters (e.g. time series).
    source

    Sequential pair distances

    ComplexityMeasures.SequentialPairDistancesType
    SequentialPairDistances <: CountBasedOutcomeSpace
     SequentialPairDistances(x::AbstractVector; n = 3, metric = Chebyshev(), m = 3, τ = 1)
     SequentialPairDistances(x::AbstractStateSpaceSet; n = 3, metric = Chebyshev())

    An outcome space based on the distribution of distances of sequential pairs of points.

    This outcome space appears implicitly as part of the "distribution entropy" introduced by Li et al. (2015), which of course can be reproduced here (see example below). We've generalized the method to be used with any InformationMeasure and DiscreteInfoEstimator, and with valid distance metric (from Distances.jl).

    Input data x are needed for initialization, because distances must be pre-computed to know the minimum/maximum distances needed for binning the distribution of pairwise distances. If the input is an AbstractVector, then the vector is embedded before computing distances. If the input is an AbstractStateSpaceSet, then the embedding step is skipped and distances are computed directly on each state vector xᵢ ∈ x.

    Description

    SequentialPairDistances does the following:

    • Transforms the input timeseries x by first embedding it using embedding dimension m and embedding lag τ (or skip this step if the input is already embedded).
    • Computes the distances ds between sequential pairs of points according to the given metric.
    • Divides the interval [minimum(ds), maximum(ds)] into n equal-size bins by using RectangularBinEncoding, then maps the distances onto these bins.

    Outcome space

    The outcome space Ω for SequentialPairDistances are the bins onto which the pairwise distances are mapped, encoded as the integers 1:n. If you need the actual bin coordinates, these can be recovered with decode (see example below).

    Implements

    • codify. Note that the input x is ignored when calling codify, because the input data is already handled when constructing a SequentialPairDistances.

    Examples

    The outcome bins can be retrieved as follows.

    using ComplexityMeasures
     x = rand(100)
    @@ -32,7 +32,7 @@
     cts, outs = counts_and_outcomes(o, x)

    Computing the "distribution entropy" with n = 3 bins for the distance histogram:

    using ComplexityMeasures
     x = rand(1000000)
     o = SequentialPairDistances(x, n = 3, metric = Chebyshev()) # metric from original paper
    -h = information(Shannon(base = 2), o, x)
    source

    Bubble sort swaps

    Bubble sort swaps

    ComplexityMeasures.BubbleSortSwapsType
    BubbleSortSwaps <: CountBasedOutcomeSpace
     BubbleSortSwaps(; m = 3, τ = 1)

    The BubbleSortSwaps outcome space is based on Manis et al. (2017)'s paper on "bubble entropy".

    Description

    BubbleSortSwaps does the following:

    • Embeds the input data using embedding dimension m and embedding lag τ
    • For each state vector in the embedding, counting how many swaps are necessary for the bubble sort algorithm to sort state vectors.

    For counts_and_outcomes, we then define a distribution over the number of necessary swaps. This distribution can then be used to estimate probabilities using probabilities_and_outcomes, which again can be used to estimate any InformationMeasure. An example of how to compute the "Shannon bubble entropy" is given below.

    Outcome space

    The outcome_space for BubbleSortSwaps are the integers 0:N, where N = (m * (m - 1)) / 2 + 1 (the worst-case number of swaps). Hence, the number of total_outcomes is N + 1.

    Implements

    • codify. Returns the number of swaps required for each embedded state vector.

    Examples

    With the BubbleSortSwaps outcome space, we can easily compute a "bubble entropy" inspired by (Manis et al., 2017). Note: this is not actually a new entropy - it is just a new way of discretizing the input data. To reproduce the bubble entropy complexity measure from (Manis et al., 2017), see BubbleEntropy.

    Examples

    using ComplexityMeasures
     x = rand(100000)
     o = BubbleSortSwaps(; m = 5) # 5-dimensional embedding vectors
    @@ -40,8 +40,8 @@
     
     # We can also compute any other "bubble quantity", for example the 
     # "Tsallis bubble extropy", with arbitrary probabilities estimators:
    -information(TsallisExtropy(), BayesianRegularization(), o, x)
    source

    Spatial outcome spaces

    ComplexityMeasures.SpatialOrdinalPatternsType
    SpatialOrdinalPatterns <: OutcomeSpaceModel
    -SpatialOrdinalPatterns(stencil, x; periodic = true)

    A symbolic, permutation-based OutcomeSpace for spatiotemporal systems that generalises OrdinalPatterns to high-dimensional arrays. The order m of the permutation pattern is extracted from the stencil, see below.

    SpatialOrdinalPatterns is based on the 2D and 3D spatiotemporal permutation entropy estimators by Ribeiro et al. (2012) and Schlemmer et al. (2018), respectively, but is here implemented as a pure probabilities probabilities estimator that is generalized for D-dimensional input array x, with arbitrary regions (stencils) to get patterns form and (possibly) periodic boundary conditions.

    See below for ways to specify the stencil. If periodic = true, then the stencil wraps around at the ends of the array. If false, then collected regions with indices which exceed the array bounds are skipped.

    In combination with information and information_normalized, this probabilities estimator can be used to compute generalized spatiotemporal permutation InformationMeasure of any type.

    Outcome space

    The outcome space Ω for SpatialOrdinalPatterns is the set of length-m ordinal patterns (i.e. permutations) that can be formed by the integers 1, 2, …, m, ordered lexicographically. There are factorial(m) such patterns. Here m refers to the number of points included in stencil.

    Stencils

    The stencil defines what local area to use to group hypervoxels. Each grouping of hypervoxels is mapped to an order-m permutation pattern, which is then mapped to an integer as in OrdinalPatterns. The stencil is moved around the input array, in a sense "scanning" the input array, to collect all possible groupings allowed by the boundary condition (periodic or not).

    Stencils are passed in one of the following three ways:

    1. As vectors of CartesianIndex which encode the offset of indices to include in the stencil, with respect to the current array index when scanning over the array. For example stencil = CartesianIndex.([(0,0), (0,1), (1,1), (1,0)]). Don't forget to include the zero offset index if you want to include the hypervoxel itself, which is almost always the case. Here the stencil creates a 2x2 square extending to the bottom and right of the pixel (directions here correspond to the way Julia prints matrices by default). When passing a stencil as a vector of CartesianIndex, m = length(stencil).
    2. As a D-dimensional array (where D matches the dimensionality of the input data) containing 0s and 1s, where if stencil[index] == 1, the corresponding pixel is included, and if stencil[index] == 0, it is not included. To generate the same estimator as in 1., use stencil = [1 1; 1 1]. When passing a stencil as a D-dimensional array, m = sum(stencil)
    3. As a Tuple containing two Tuples, both of length D, for D-dimensional data. The first tuple specifies the extent of the stencil, where extent[i] dictates the number of hypervoxels to be included along the ith axis and lag[i] the separation of hypervoxels along the same axis. This method can only generate (hyper)rectangular stencils. To create the same estimator as in the previous examples, use here stencil = ((2, 2), (1, 1)). When passing a stencil using extent and lag, m = prod(extent).
    source

    Spatial outcome spaces

    ComplexityMeasures.SpatialOrdinalPatternsType
    SpatialOrdinalPatterns <: OutcomeSpaceModel
    +SpatialOrdinalPatterns(stencil, x; periodic = true)

    A symbolic, permutation-based OutcomeSpace for spatiotemporal systems that generalises OrdinalPatterns to high-dimensional arrays. The order m of the permutation pattern is extracted from the stencil, see below.

    SpatialOrdinalPatterns is based on the 2D and 3D spatiotemporal permutation entropy estimators by Ribeiro et al. (2012) and Schlemmer et al. (2018), respectively, but is here implemented as a pure probabilities probabilities estimator that is generalized for D-dimensional input array x, with arbitrary regions (stencils) to get patterns form and (possibly) periodic boundary conditions.

    See below for ways to specify the stencil. If periodic = true, then the stencil wraps around at the ends of the array. If false, then collected regions with indices which exceed the array bounds are skipped.

    In combination with information and information_normalized, this probabilities estimator can be used to compute generalized spatiotemporal permutation InformationMeasure of any type.

    Outcome space

    The outcome space Ω for SpatialOrdinalPatterns is the set of length-m ordinal patterns (i.e. permutations) that can be formed by the integers 1, 2, …, m, ordered lexicographically. There are factorial(m) such patterns. Here m refers to the number of points included in stencil.

    Stencils

    The stencil defines what local area to use to group hypervoxels. Each grouping of hypervoxels is mapped to an order-m permutation pattern, which is then mapped to an integer as in OrdinalPatterns. The stencil is moved around the input array, in a sense "scanning" the input array, to collect all possible groupings allowed by the boundary condition (periodic or not).

    Stencils are passed in one of the following three ways:

    1. As vectors of CartesianIndex which encode the offset of indices to include in the stencil, with respect to the current array index when scanning over the array. For example stencil = CartesianIndex.([(0,0), (0,1), (1,1), (1,0)]). Don't forget to include the zero offset index if you want to include the hypervoxel itself, which is almost always the case. Here the stencil creates a 2x2 square extending to the bottom and right of the pixel (directions here correspond to the way Julia prints matrices by default). When passing a stencil as a vector of CartesianIndex, m = length(stencil).
    2. As a D-dimensional array (where D matches the dimensionality of the input data) containing 0s and 1s, where if stencil[index] == 1, the corresponding pixel is included, and if stencil[index] == 0, it is not included. To generate the same estimator as in 1., use stencil = [1 1; 1 1]. When passing a stencil as a D-dimensional array, m = sum(stencil)
    3. As a Tuple containing two Tuples, both of length D, for D-dimensional data. The first tuple specifies the extent of the stencil, where extent[i] dictates the number of hypervoxels to be included along the ith axis and lag[i] the separation of hypervoxels along the same axis. This method can only generate (hyper)rectangular stencils. To create the same estimator as in the previous examples, use here stencil = ((2, 2), (1, 1)). When passing a stencil using extent and lag, m = prod(extent).
    source
    ComplexityMeasures.SpatialDispersionType
    SpatialDispersion <: OutcomeSpace
     SpatialDispersion(stencil, x::AbstractArray;
         periodic = true,
         c = 5,
    @@ -67,7 +67,7 @@
     est = SpatialDispersion(stencil, first(imgs))
     h_vs_t = information_normalized.(Ref(est), imgs)

    Computing generalized spatiotemporal dispersion entropy is trivial, e.g. with Renyi:

    x = reshape(repeat(1:5, 500) .+ 0.1*rand(500*5), 50, 50)
     est = SpatialDispersion(stencil, x)
    -information(Renyi(q = 2), est, x)

    See also: SpatialOrdinalPatterns, GaussianCDFEncoding, codify.

    source
    ComplexityMeasures.SpatialBubbleSortSwapsType
    SpatialBubbleSortSwaps <: SpatialOutcomeSpace
     SpatialBubbleSortSwaps(stencil, x; periodic = true)

    SpatialBubbleSortSwaps generalizes BubbleSortSwaps to high-dimensional arrays by encoding pixel/voxel/hypervoxel windows in terms of how many swap operations the bubble sort algorithm requires to sort them.

    What does this mean? For BubbleSortSwaps the input data is embedded using embedding dimension m and the number of swaps required are computed for each embedding vector. For SpatialBubbleSortSwaps, the "embedding dimension" m for is inferred from the number of elements in the stencil, and the "embedding vectors" are the hypervoxels selected by the stencil.

    Outcome space

    The outcome space Ω for SpatialBubbleSortSwaps is the range of integers 0:(n*(n-1)÷2), corresponding to the number of swaps required by the bubble sort algorithm to sort a particular pixel/voxel/hypervoxel window.

    Arguments

    • stencil. Defines what local area (hyperrectangle), or which points within this area, to include around each hypervoxel (i.e. pixel in 2D). See SpatialOrdinalPatterns and SpatialDispersion for more information about stencils and examples of how to specify them.
    • x::AbstractArray. The input data. Must be provided because we need to know its size for optimization and bound checking.

    Keyword arguments

    • periodic::Bool. If periodic == true, then the stencil should wrap around at the end of the array. If periodic = false, then pixels whose stencil exceeds the array bounds are skipped.

    Example

    using ComplexityMeasures
     using Random; rng = MersenneTwister(1234)
     
    @@ -81,7 +81,7 @@
     counts_and_outcomes(o, x)
     
     # "Spatial bubble Kaniadakis entropy", with shrinkage-adjusted probabilities
    -information(Kaniadakis(), Shrinkage(), o, x)
    source
    ComplexityMeasures.ProbabilitiesType
    Probabilities <: Array{<:AbstractFloat, N}
     Probabilities(probs::Array [, outcomes [, dimlabels]]) → p
     Probabilities(counts::Counts [, outcomes [, dimlabels]]) → p

    Probabilities stores an N-dimensional array of probabilities, while ensuring that the array sums to 1 (normalized probability mass). In most cases the array is a standard vector. p itself can be manipulated and iterated over, just like its stored array.

    The probabilities correspond to outcomes that describe the axes of the array. If p isa Probabilities, then p.outcomes[i] is an an abstract vector containing the outcomes along the i-th dimension. The outcomes have the same ordering as the probabilities, so that p[i][j] is the probability for outcome p.outcomes[i][j]. The dimensions of the array are named, and can be accessed by p.dimlabels, where p.dimlabels[i] is the label of the i-th dimension. Both outcomes and dimlabels are assigned automatically if not given. If the input is a set of Counts, and outcomes and dimlabels are not given, then the labels and outcomes are inherited from the counts.

    Examples

    julia> probs = [0.2, 0.2, 0.2, 0.2]; Probabilities(probs) # will be normalized to sum to 1
      Probabilities{Float64,1} over 4 outcomes
    @@ -92,28 +92,28 @@
      Probabilities{Float64,1} over 3 outcomes
      "out1"  0.3
      "out2"  0.4
    - "out3"  0.3
    source
    ComplexityMeasures.probabilitiesFunction
    probabilities(
         [est::ProbabilitiesEstimator], o::OutcomeSpace, x::Array_or_SSSet
    -) → p::Probabilities

    Compute the same probabilities as in the probabilities_and_outcomes function, with two differences:

    1. Do not explicitly return the outcomes.
    2. If the outcomes are not estimated for free while estimating the counts, a special integer type is used to enumerate the outcomes, to avoid the computational cost of estimating the outcomes.
    probabilities([est::ProbabilitiesEstimator], counts::Counts) → (p::Probabilities, Ω)

    The same as above, but estimate the probability directly from a set of Counts.

    source
    ComplexityMeasures.probabilities_and_outcomesFunction
    probabilities_and_outcomes(
    +) → p::Probabilities

    Compute the same probabilities as in the probabilities_and_outcomes function, with two differences:

    1. Do not explicitly return the outcomes.
    2. If the outcomes are not estimated for free while estimating the counts, a special integer type is used to enumerate the outcomes, to avoid the computational cost of estimating the outcomes.
    probabilities([est::ProbabilitiesEstimator], counts::Counts) → (p::Probabilities, Ω)

    The same as above, but estimate the probability directly from a set of Counts.

    source
    ComplexityMeasures.probabilities_and_outcomesFunction
    probabilities_and_outcomes(
         [est::ProbabilitiesEstimator], o::OutcomeSpace, x::Array_or_SSSet
     ) → (p::Probabilities, Ω)

    Estimate a probability distribution over the set of possible outcomes Ω defined by the OutcomeSpace o, given input data x. Probabilities are estimated according to the given probabilities estimator est, which defaults to RelativeAmount.

    The input data is typically an Array or a StateSpaceSet (or SSSet for short); see Input data for ComplexityMeasures.jl. Configuration options are always given as arguments to the chosen outcome space and probabilities estimator.

    Return a tuple where the first element is a Probabilities instance, which is vector-like and contains the probabilities, and where the second element Ω are the outcomes corresponding to the probabilities, such that p[i] is the probability for the outcome Ω[i].

    The outcomes are actually included in p, and you can use the outcomes function on the p to get them. probabilities_and_outcomes returns both for backwards compatibility.

    probabilities_and_outcomes(
         [est::ProbabilitiesEstimator], counts::Counts
    -) → (p::Probabilities, Ω)

    Estimate probabilities from the pre-computed counts using the given ProbabilitiesEstimator est.

    Description

    Probabilities are computed by:

    1. Discretizing/encoding x into a finite set of outcomes Ω specified by the provided OutcomeSpace o.
    2. Assigning to each outcome Ωᵢ ∈ Ω either a count (how often it appears among the discretized data points), or a pseudo-count (some pre-normalized probability such that sum(Ωᵢ for Ωᵢ in Ω) == 1).

    For outcome spaces that result in pseudo counts, such as PowerSpectrum, these pseudo counts are simply treated as probabilities and returned directly (that is, est is ignored). For counting-based outcome spaces (see OutcomeSpace docstring), probabilities are estimated from the counts using some ProbabilitiesEstimator (first signature).

    Observed vs all probabilities

    Due to performance optimizations, whether the returned probabilities contain 0s as entries or not depends on the outcome space. E.g., in ValueBinning 0s are skipped, while in PowerSpectrum 0 are not skipped, because we get them for free.

    Use allprobabilities_and_outcomes to guarantee that zero probabilities are also returned (may be slower).

    source
    ComplexityMeasures.allprobabilities_and_outcomesFunction
    allprobabilities_and_outcomes(est::ProbabilitiesEstimator, x::Array_or_SSSet) → (p::Probabilities, outs)
    -allprobabilities_and_outcomes(o::OutcomeSpace, x::Array_or_SSSet) → (p::Probabilities, outs)

    The same as probabilities_and_outcomes, but ensures that outcomes with 0 probability are explicitly added in the returned vector. This means that p[i] is the probability of ospace[i], with ospace =outcome_space(est, x).

    This function is useful in cases where one wants to compare the probability mass functions of two different input data x, y under the same estimator. E.g., to compute the KL-divergence of the two PMFs assumes that the obey the same indexing. This is not true for probabilities even with the same est, due to the skipping of 0 entries, but it is true for allprobabilities_and_outcomes.

    source

    Counts

    ComplexityMeasures.CountsType
    Counts <: Array{<:Integer, N}
    -Counts(counts [, outcomes [, dimlabels]]) → c

    Counts stores an N-dimensional array of integer counts corresponding to a set of outcomes. This is typically called a "frequency table" or "contingency table".

    If c isa Counts, then c.outcomes[i] is an abstract vector containing the outcomes along the i-th dimension, where c[i][j] is the count corresponding to the outcome c.outcomes[i][j], and c.dimlabels[i] is the label of the i-th dimension. Both labels and outcomes are assigned automatically if not given. c itself can be manipulated and iterated over like its stored array.

    source
    ComplexityMeasures.counts_and_outcomesFunction
    counts_and_outcomes(o::OutcomeSpace, x) → (cts::Counts, Ω)

    Discretize/encode x (which must be sortable) into a finite set of outcomes Ω specified by the provided OutcomeSpace o, and then count how often each outcome Ωᵢ ∈ Ω (i.e. each "discretized value", or "encoded symbol") appears.

    Return a tuple where the first element is a Counts instance, which is vector-like and contains the counts, and where the second element Ω are the outcomes corresponding to the counts, such that cts[i] is the count for the outcome Ω[i].

    The outcomes are actually included in cts, and you can use the outcomes function on the cts to get them. counts_and_outcomes returns both for backwards compatibility.

    counts_and_outcomes(x) → cts::Counts

    If no OutcomeSpace is specified, then UniqueElements is used as the outcome space.

    Description

    For OutcomeSpaces that uses encode to discretize, it is possible to count how often each outcome $\omega_i \in \Omega$, where $\Omega$ is the set of possible outcomes, is observed in the discretized/encoded input data. Thus, we can assign to each outcome $\omega_i$ a count $f(\omega_i)$, such that $\sum_{i=1}^N f(\omega_i) = N$, where $N$ is the number of observations in the (encoded) input data. counts returns the counts $f(\omega_i)_{obs}$ and outcomes only for the observed outcomes $\omega_i^{obs}$ (those outcomes that actually appear in the input data). If you need the counts for unobserved outcomes as well, use allcounts_and_outcomes.

    source
    ComplexityMeasures.countsFunction
    counts(o::OutcomeSpace, x) → cts::Counts

    Compute the same counts as in the counts_and_outcomes function, with two differences:

    1. Do not explicitly return the outcomes.
    2. If the outcomes are not estimated for free while estimating the counts, a special integer type is used to enumerate the outcomes, to avoid the computational cost of estimating the outcomes.
    source

    Probability estimators

    ComplexityMeasures.ProbabilitiesEstimatorType
    ProbabilitiesEstimator

    The supertype for all probabilities estimators.

    The role of the probabilities estimator is to convert (pseudo-)counts to probabilities. Currently, the implementation of all probabilities estimators assume finite outcome space with known cardinality. Therefore, ProbabilitiesEstimator accept an OutcomeSpace as the first argument, which specifies the set of possible outcomes.

    Probabilities estimators are used with probabilities and allprobabilities_and_outcomes.

    Implementations

    The default probabilities estimator is RelativeAmount, which is compatible with any OutcomeSpace. The following estimators only support counting-based outcomes.

    Description

    In ComplexityMeasures.jl, probability mass functions are estimated from data by defining a set of possible outcomes $\Omega = \{\omega_1, \omega_2, \ldots, \omega_L \}$ (by specifying an OutcomeSpace), and assigning to each outcome $\omega_i$ a probability $p(\omega_i)$, such that $\sum_{i=1}^N p(\omega_i) = 1$ (by specifying a ProbabilitiesEstimator).

    source
    ComplexityMeasures.RelativeAmountType
    RelativeAmount <: ProbabilitiesEstimator
    +) → (p::Probabilities, Ω)

    Estimate probabilities from the pre-computed counts using the given ProbabilitiesEstimator est.

    Description

    Probabilities are computed by:

    1. Discretizing/encoding x into a finite set of outcomes Ω specified by the provided OutcomeSpace o.
    2. Assigning to each outcome Ωᵢ ∈ Ω either a count (how often it appears among the discretized data points), or a pseudo-count (some pre-normalized probability such that sum(Ωᵢ for Ωᵢ in Ω) == 1).

    For outcome spaces that result in pseudo counts, such as PowerSpectrum, these pseudo counts are simply treated as probabilities and returned directly (that is, est is ignored). For counting-based outcome spaces (see OutcomeSpace docstring), probabilities are estimated from the counts using some ProbabilitiesEstimator (first signature).

    Observed vs all probabilities

    Due to performance optimizations, whether the returned probabilities contain 0s as entries or not depends on the outcome space. E.g., in ValueBinning 0s are skipped, while in PowerSpectrum 0 are not skipped, because we get them for free.

    Use allprobabilities_and_outcomes to guarantee that zero probabilities are also returned (may be slower).

    source
    ComplexityMeasures.allprobabilities_and_outcomesFunction
    allprobabilities_and_outcomes(est::ProbabilitiesEstimator, x::Array_or_SSSet) → (p::Probabilities, outs)
    +allprobabilities_and_outcomes(o::OutcomeSpace, x::Array_or_SSSet) → (p::Probabilities, outs)

    The same as probabilities_and_outcomes, but ensures that outcomes with 0 probability are explicitly added in the returned vector. This means that p[i] is the probability of ospace[i], with ospace =outcome_space(est, x).

    This function is useful in cases where one wants to compare the probability mass functions of two different input data x, y under the same estimator. E.g., to compute the KL-divergence of the two PMFs assumes that the obey the same indexing. This is not true for probabilities even with the same est, due to the skipping of 0 entries, but it is true for allprobabilities_and_outcomes.

    source

    Counts

    ComplexityMeasures.CountsType
    Counts <: Array{<:Integer, N}
    +Counts(counts [, outcomes [, dimlabels]]) → c

    Counts stores an N-dimensional array of integer counts corresponding to a set of outcomes. This is typically called a "frequency table" or "contingency table".

    If c isa Counts, then c.outcomes[i] is an abstract vector containing the outcomes along the i-th dimension, where c[i][j] is the count corresponding to the outcome c.outcomes[i][j], and c.dimlabels[i] is the label of the i-th dimension. Both labels and outcomes are assigned automatically if not given. c itself can be manipulated and iterated over like its stored array.

    source
    ComplexityMeasures.counts_and_outcomesFunction
    counts_and_outcomes(o::OutcomeSpace, x) → (cts::Counts, Ω)

    Discretize/encode x (which must be sortable) into a finite set of outcomes Ω specified by the provided OutcomeSpace o, and then count how often each outcome Ωᵢ ∈ Ω (i.e. each "discretized value", or "encoded symbol") appears.

    Return a tuple where the first element is a Counts instance, which is vector-like and contains the counts, and where the second element Ω are the outcomes corresponding to the counts, such that cts[i] is the count for the outcome Ω[i].

    The outcomes are actually included in cts, and you can use the outcomes function on the cts to get them. counts_and_outcomes returns both for backwards compatibility.

    counts_and_outcomes(x) → cts::Counts

    If no OutcomeSpace is specified, then UniqueElements is used as the outcome space.

    Description

    For OutcomeSpaces that uses encode to discretize, it is possible to count how often each outcome $\omega_i \in \Omega$, where $\Omega$ is the set of possible outcomes, is observed in the discretized/encoded input data. Thus, we can assign to each outcome $\omega_i$ a count $f(\omega_i)$, such that $\sum_{i=1}^N f(\omega_i) = N$, where $N$ is the number of observations in the (encoded) input data. counts returns the counts $f(\omega_i)_{obs}$ and outcomes only for the observed outcomes $\omega_i^{obs}$ (those outcomes that actually appear in the input data). If you need the counts for unobserved outcomes as well, use allcounts_and_outcomes.

    source
    ComplexityMeasures.countsFunction
    counts(o::OutcomeSpace, x) → cts::Counts

    Compute the same counts as in the counts_and_outcomes function, with two differences:

    1. Do not explicitly return the outcomes.
    2. If the outcomes are not estimated for free while estimating the counts, a special integer type is used to enumerate the outcomes, to avoid the computational cost of estimating the outcomes.
    source

    Probability estimators

    ComplexityMeasures.ProbabilitiesEstimatorType
    ProbabilitiesEstimator

    The supertype for all probabilities estimators.

    The role of the probabilities estimator is to convert (pseudo-)counts to probabilities. Currently, the implementation of all probabilities estimators assume finite outcome space with known cardinality. Therefore, ProbabilitiesEstimator accept an OutcomeSpace as the first argument, which specifies the set of possible outcomes.

    Probabilities estimators are used with probabilities and allprobabilities_and_outcomes.

    Implementations

    The default probabilities estimator is RelativeAmount, which is compatible with any OutcomeSpace. The following estimators only support counting-based outcomes.

    Description

    In ComplexityMeasures.jl, probability mass functions are estimated from data by defining a set of possible outcomes $\Omega = \{\omega_1, \omega_2, \ldots, \omega_L \}$ (by specifying an OutcomeSpace), and assigning to each outcome $\omega_i$ a probability $p(\omega_i)$, such that $\sum_{i=1}^N p(\omega_i) = 1$ (by specifying a ProbabilitiesEstimator).

    source
    ComplexityMeasures.RelativeAmountType
    RelativeAmount <: ProbabilitiesEstimator
     RelativeAmount()

    The RelativeAmount estimator is used with probabilities and related functions to estimate probabilities over the given OutcomeSpace using maximum likelihood estimation (MLE), also called plug-in estimation. See ProbabilitiesEstimator for usage.

    Description

    Consider a length-m outcome space $\Omega$ and random sample of length N. The maximum likelihood estimate of the probability of the k-th outcome $\omega_k$ is

    \[p(\omega_k) = \dfrac{n_k}{N},\]

    where $n_k$ is the number of times the k-th outcome was observed in the (encoded) sample.

    This estimation is known as maximum likelihood estimation. However, RelativeAmount also serves as the fall-back probabilities estimator for OutcomeSpaces that are not count-based and only yield "pseudo-counts", for example WaveletOverlap or PowerSpectrum. These outcome spaces do not yield counts, but pre-normalized numbers that can be treated as "relative frequencies" or "relative power". Hence, this estimator is called RelativeAmount.

    Examples

    using ComplexityMeasures
     x = cumsum(randn(100))
     ps = probabilities(OrdinalPatterns{3}(), x) # `RelativeAmount` is the default estimator
     ps_mle = probabilities(RelativeAmount(), OrdinalPatterns{3}(), x) # equivalent
    -ps == ps_mle # true

    See also: BayesianRegularization, Shrinkage.

    source
    ComplexityMeasures.BayesianRegularizationType
    BayesianRegularization <: ProbabilitiesEstimator
     BayesianRegularization(; a = 1.0)

    The BayesianRegularization estimator is used with probabilities and related functions to estimate probabilities an m-element counting-based OutcomeSpace using Bayesian regularization of cell counts (Hausser and Strimmer, 2009). See ProbabilitiesEstimator for usage.

    Outcome space requirements

    This estimator only works with counting-compatible outcome spaces.

    Description

    The BayesianRegularization estimator estimates the probability of the $k$-th outcome $\omega_{k}$ is

    \[\omega_{k}^{\text{BayesianRegularization}} = \dfrac{n_k + a_k}{n + A},\]

    where $n$ is the number of samples in the input data, $n_k$ is the observed counts for the outcome $\omega_{k}$, and $A = \sum_{i=1}^k a_k$.

    Picking a

    There are many common choices of priors, some of which are listed in Hausser and Strimmer (2009). They include

    • a == 0, which is equivalent to the RelativeAmount estimator.
    • a == 0.5 (Jeffrey's prior)
    • a == 1 (Bayes-Laplace uniform prior)

    a can also be chosen as a vector of real numbers. Then, if used with allprobabilities_and_outcomes, it is required that length(a) == total_outcomes(o, x), where x is the input data and o is the OutcomeSpace. If used with probabilities, then length(a) must match the number of observed outcomes (you can check this using probabilities_and_outcomes). The choice of a can severely impact the estimation errors of the probabilities, and the errors depend both on the choice of a and on the sampling scenario (Hausser and Strimmer, 2009).

    Assumptions

    The BayesianRegularization estimator assumes a fixed and known m. Thus, using it with probabilities_and_outcomes and allprobabilities_and_outcomes will yield different results, depending on whether all outcomes are observed in the input data or not. For probabilities_and_outcomes, m is the number of observed outcomes. For allprobabilities_and_outcomes, m = total_outcomes(o, x), where o is the OutcomeSpace and x is the input data.

    Note

    If used with allprobabilities_and_outcomes, then outcomes which have not been observed may be assigned non-zero probabilities. This might affect your results if using e.g. missing_outcomes.

    Examples

    using ComplexityMeasures
     x = cumsum(randn(100))
    -ps_bayes = probabilities(BayesianRegularization(a = 0.5), OrdinalPatterns{3}(), x)

    See also: RelativeAmount, Shrinkage.

    source
    ComplexityMeasures.ShrinkageType
    Shrinkage{<:OutcomeSpace} <: ProbabilitiesEstimator
     Shrinkage(; t = nothing, λ = nothing)

    The Shrinkage estimator is used with probabilities and related functions to estimate probabilities over the given m-element counting-based OutcomeSpace using James-Stein-type shrinkage (James and Stein, 1992), as presented in Hausser and Strimmer (2009).

    Description

    The Shrinkage estimator estimates a cell probability $\theta_{k}^{\text{Shrink}}$ as

    \[\theta_{k}^{\text{Shrink}} = \lambda t_k + (1-\lambda) \hat{\theta}_k^{RelativeAmount},\]

    where $\lambda \in [0, 1]$ is the shrinkage intensity ($\lambda = 0$ means no shrinkage, and $\lambda = 1$ means full shrinkage), and $t_k$ is the shrinkage target. Hausser and Strimmer (2009) picks $t_k = 1/m$, i.e. the uniform distribution.

    If t == nothing, then $t_k$ is set to $1/m$ for all $k$, as in Hausser and Strimmer (2009). If λ == nothing (the default), then the shrinkage intensity is optimized according to Hausser and Strimmer (2009). Hence, you should probably not pick λ nor t manually, unless you know what you are doing.

    Assumptions

    The Shrinkage estimator assumes a fixed and known number of outcomes m. Thus, using it with probabilities_and_outcomes) and allprobabilities_and_outcomes will yield different results, depending on whether all outcomes are observed in the input data or not. For probabilities_and_outcomes, m is the number of observed outcomes. For allprobabilities_and_outcomes, m = total_outcomes(o, x), where o is the OutcomeSpace and x is the input data.

    Note

    If used with allprobabilities_and_outcomes, then outcomes which have not been observed may be assigned non-zero probabilities. This might affect your results if using e.g. missing_outcomes.

    Examples

    using ComplexityMeasures
     x = cumsum(randn(100))
    -ps_shrink = probabilities(Shrinkage(), OrdinalPatterns{3}(), x)

    See also: RelativeAmount, BayesianRegularization.

    source
    ComplexityMeasures.AddConstantType
    AddConstant <: ProbabilitiesEstimator
    -AddConstant(; c = 1.0)

    A generic add-constant probabilities estimator for counting-based OutcomeSpaces, where several literature estimators can be obtained tuning c. Currently $c$ can only be a scalar.

    • c = 1.0 is the Laplace estimator, or the "add-one" estimator.

    Description

    Probabilities for the $k$-th outcome $\omega_{k}$ are estimated as

    \[p(\omega_k) = \dfrac{(n_k + c)}{n + mc},\]

    where $m$ is the cardinality of the outcome space, and $n$ is the number of (encoded) input data points, and $n_k$ is the number of times the outcome $\omega_{k}$ is observed in the (encoded) input data points.

    If the AddConstant estimator used with probabilities_and_outcomes, then $m$ is set to the number of observed outcomes. If used with allprobabilities_and_outcomes, then $m$ is set to the number of possible outcomes.

    Unobserved outcomes are assigned nonzero probability!

    Looking at the formula above, if $n_k = 0$, then unobserved outcomes are assigned a non-zero probability of $\dfrac{c}{n + mc}$. This means that if the estimator is used with allprobabilities_and_outcomes, then all outcomes, even those that are not observed, are assigned non-zero probabilities. This might affect your results if using e.g. missing_outcomes.

    source

    Encodings/Symbolizations API

    Count-based OutcomeSpaces first "encode" input data into an intermediate representation indexed by the positive integers. This intermediate representation is called an "encoding". Alternative names for "encode" in the literature is "symbolize" or "codify", and in this package we use the latter.

    The encodings API is defined by:

    ComplexityMeasures.EncodingType
    Encoding

    The supertype for all encoding schemes. Encodings always encode elements of input data into the positive integers. The encoding API is defined by the functions encode and decode. Some probability estimators utilize encodings internally.

    Current available encodings are:

    source
    ComplexityMeasures.encodeFunction
    encode(c::Encoding, χ) -> i::Int

    Encode an element χ ∈ x of input data x (those given to e.g., counts) into the positive integers using encoding c. The special value of i = -1 is used as a return value for inappropriate elements χ that cannot be encoded according to c.

    source
    ComplexityMeasures.codifyFunction
    codify(o::OutcomeSpace, x::Vector) → s::Vector{Int}
    -codify(o::OutcomeSpace, x::AbstractStateSpaceSet{D}) → s::NTuple{D, Vector{Int}

    Codify x according to the outcome space o. If x is a Vector, then a Vector{<:Integer} is returned. If x is a StateSpaceSet{D}, then symbolization is done column-wise and an NTuple{D, Vector{<:Integer}} is returned, where D = dimension(x).

    Description

    The reason this function exists is that we don't always want to encode the entire input x at once. Sometimes, it is desirable to first apply some transformation to x first, then apply Encodings in a point-wise manner in the transformed space. (the OutcomeSpace dictates this transformation). This is useful for encoding timeseries data.

    The length of the returned s depends on the OutcomeSpace. Some outcome spaces preserve the input data length (e.g. UniqueElements), while some outcome spaces (e.g. OrdinalPatterns) do e.g. delay embeddings before encoding, so that length(s) < length(x).

    source

    Available encodings

    ComplexityMeasures.AddConstantType
    AddConstant <: ProbabilitiesEstimator
    +AddConstant(; c = 1.0)

    A generic add-constant probabilities estimator for counting-based OutcomeSpaces, where several literature estimators can be obtained tuning c. Currently $c$ can only be a scalar.

    • c = 1.0 is the Laplace estimator, or the "add-one" estimator.

    Description

    Probabilities for the $k$-th outcome $\omega_{k}$ are estimated as

    \[p(\omega_k) = \dfrac{(n_k + c)}{n + mc},\]

    where $m$ is the cardinality of the outcome space, and $n$ is the number of (encoded) input data points, and $n_k$ is the number of times the outcome $\omega_{k}$ is observed in the (encoded) input data points.

    If the AddConstant estimator used with probabilities_and_outcomes, then $m$ is set to the number of observed outcomes. If used with allprobabilities_and_outcomes, then $m$ is set to the number of possible outcomes.

    Unobserved outcomes are assigned nonzero probability!

    Looking at the formula above, if $n_k = 0$, then unobserved outcomes are assigned a non-zero probability of $\dfrac{c}{n + mc}$. This means that if the estimator is used with allprobabilities_and_outcomes, then all outcomes, even those that are not observed, are assigned non-zero probabilities. This might affect your results if using e.g. missing_outcomes.

    source

    Encodings/Symbolizations API

    Count-based OutcomeSpaces first "encode" input data into an intermediate representation indexed by the positive integers. This intermediate representation is called an "encoding". Alternative names for "encode" in the literature is "symbolize" or "codify", and in this package we use the latter.

    The encodings API is defined by:

    ComplexityMeasures.EncodingType
    Encoding

    The supertype for all encoding schemes. Encodings always encode elements of input data into the positive integers. The encoding API is defined by the functions encode and decode. Some probability estimators utilize encodings internally.

    Current available encodings are:

    source
    ComplexityMeasures.encodeFunction
    encode(c::Encoding, χ) -> i::Int

    Encode an element χ ∈ x of input data x (those given to e.g., counts) into the positive integers using encoding c. The special value of i = -1 is used as a return value for inappropriate elements χ that cannot be encoded according to c.

    source
    ComplexityMeasures.codifyFunction
    codify(o::OutcomeSpace, x::Vector) → s::Vector{Int}
    +codify(o::OutcomeSpace, x::AbstractStateSpaceSet{D}) → s::NTuple{D, Vector{Int}

    Codify x according to the outcome space o. If x is a Vector, then a Vector{<:Integer} is returned. If x is a StateSpaceSet{D}, then symbolization is done column-wise and an NTuple{D, Vector{<:Integer}} is returned, where D = dimension(x).

    Description

    The reason this function exists is that we don't always want to encode the entire input x at once. Sometimes, it is desirable to first apply some transformation to x first, then apply Encodings in a point-wise manner in the transformed space. (the OutcomeSpace dictates this transformation). This is useful for encoding timeseries data.

    The length of the returned s depends on the OutcomeSpace. Some outcome spaces preserve the input data length (e.g. UniqueElements), while some outcome spaces (e.g. OrdinalPatterns) do e.g. delay embeddings before encoding, so that length(s) < length(x).

    source

    Available encodings

    ComplexityMeasures.OrdinalPatternEncodingType
    OrdinalPatternEncoding <: Encoding
     OrdinalPatternEncoding{m}(lt = ComplexityMeasures.isless_rand)

    An encoding scheme that encodes length-m vectors into their permutation/ordinal patterns and then into the integers based on the Lehmer code. It is used by OrdinalPatterns and similar estimators, see that for a description of the outcome space.

    The ordinal/permutation pattern of a vector χ is simply sortperm(χ), which gives the indices that would sort χ in ascending order.

    Description

    The Lehmer code, as implemented here, is a bijection between the set of factorial(m) possible permutations for a length-m sequence, and the integers 1, 2, …, factorial(m). The encoding step uses algorithm 1 in Berger et al. (2019), which is highly optimized. The decoding step is much slower due to missing optimizations (pull requests welcomed!).

    Example

    julia> using ComplexityMeasures
     
     julia> χ = [4.0, 1.0, 9.0];
    @@ -127,7 +127,7 @@
     3-element SVector{3, Int64} with indices SOneTo(3):
      2
      1
    - 3

    If you want to encode something that is already a permutation pattern, then you can use the non-exported permutation_to_integer function.

    source
    ComplexityMeasures.GaussianCDFEncodingType
    GaussianCDFEncoding <: Encoding
    + 3

    If you want to encode something that is already a permutation pattern, then you can use the non-exported permutation_to_integer function.

    source
    ComplexityMeasures.GaussianCDFEncodingType
    GaussianCDFEncoding <: Encoding
     GaussianCDFEncoding{m}(; μ, σ, c::Int = 3)

    An encoding scheme that encodes a scalar or vector χ into one of the integers sᵢ ∈ [1, 2, …, c] based on the normal cumulative distribution function (NCDF), and decodes the sᵢ into subintervals of [0, 1] (with some loss of information).

    Initializing a GaussianCDFEncoding

    The size of the input to be encoded must be known beforehand. One must therefore set m = length(χ), where χ is the input (m = 1 for scalars, m ≥ 2 for vectors). To do so, one must explicitly give m as a type parameter: e.g. encoding = GaussianCDFEncoding{3}(; μ = 0.0, σ = 0.1) to encode 3-element vectors, or encoding = GaussianCDFEncoding{1}(; μ = 0.0, σ = 0.1) to encode scalars.

    Description

    Encoding/decoding scalars

    GaussianCDFEncoding first maps an input scalar $χ$ to a new real number $y_ \in [0, 1]$ by using the normal cumulative distribution function (CDF) with the given mean μ and standard deviation σ, according to the map

    \[x \to y : y = \dfrac{1}{ \sigma \sqrt{2 \pi}} \int_{-\infty}^{x} e^{(-(x - \mu)^2)/(2 \sigma^2)} dx.\]

    Next, the interval [0, 1] is equidistantly binned and enumerated $1, 2, \ldots, c$, and $y$ is linearly mapped to one of these integers using the linear map $y \to z : z = \text{floor}(y(c-1)) + 1$.

    Because of the floor operation, some information is lost, so when used with decode, each decoded sᵢ is mapped to a subinterval of [0, 1]. This subinterval is returned as a length-1 Vector{SVector}.

    Notice that the decoding step does not yield an element of any outcome space of the estimators that use GaussianCDFEncoding internally, such as Dispersion. That is because these estimators additionally delay embed the encoded data.

    Encoding/decoding vectors

    If GaussianCDFEncoding is used with a vector χ, then each element of χ is encoded separately, resulting in a length(χ) sequence of integers which may be treated as a CartesianIndex. The encoded symbol s ∈ [1, 2, …, c] is then just the linear index corresponding to this cartesian index (similar to how CombinationEncoding works).

    When decoded, the integer symbol s is converted back into its CartesianIndex representation, which is just a sequence of integers that refer to subdivisions of the [0, 1] interval. The relevant subintervals are then returned as a length-χ Vector{SVector}.

    Examples

    julia> using ComplexityMeasures, Statistics
     
    @@ -146,18 +146,18 @@
     julia> decode(encoding, 3)
     2-element SVector{2, Float64} with indices SOneTo(2):
      0.4
    - 0.6
    source
    ComplexityMeasures.RelativeMeanEncodingType
    RelativeMeanEncoding <: Encoding
    -RelativeMeanEncoding(minval::Real, maxval::Real; n = 2)

    RelativeMeanEncoding encodes a vector based on the relative position the mean of the vector has with respect to a predefined minimum and maximum value (minval and maxval, respectively).

    Description

    This encoding is inspired by Azami and Escudero (2016)'s algorithm for amplitude-aware permutation entropy. They use a linear combination of amplitude information and first differences information of state vectors to correct probabilities. Here, however, we explicitly encode the amplitude-part of the correction as an a integer symbol Λ ∈ [1, 2, …, n]. The first-difference part of the encoding is available as the RelativeFirstDifferenceEncoding encoding.

    Encoding/decoding

    When used with encode, an $m$-element state vector $\bf{x} = (x_1, x_2, \ldots, x_m)$ is encoded as $Λ = \dfrac{1}{N}\sum_{i=1}^m abs(x_i)$. The value of $Λ$ is then normalized to lie on the interval [0, 1], assuming that the minimum/maximum value any single element $x_i$ can take is minval/maxval, respectively. Finally, the interval [0, 1] is discretized into n discrete bins, enumerated by positive integers 1, 2, …, n, and the number of the bin that the normalized $Λ$ falls into is returned.

    When used with decode, the left-edge of the bin that the normalized $Λ$ fell into is returned.

    source
    ComplexityMeasures.RelativeMeanEncodingType
    RelativeMeanEncoding <: Encoding
    +RelativeMeanEncoding(minval::Real, maxval::Real; n = 2)

    RelativeMeanEncoding encodes a vector based on the relative position the mean of the vector has with respect to a predefined minimum and maximum value (minval and maxval, respectively).

    Description

    This encoding is inspired by Azami and Escudero (2016)'s algorithm for amplitude-aware permutation entropy. They use a linear combination of amplitude information and first differences information of state vectors to correct probabilities. Here, however, we explicitly encode the amplitude-part of the correction as an a integer symbol Λ ∈ [1, 2, …, n]. The first-difference part of the encoding is available as the RelativeFirstDifferenceEncoding encoding.

    Encoding/decoding

    When used with encode, an $m$-element state vector $\bf{x} = (x_1, x_2, \ldots, x_m)$ is encoded as $Λ = \dfrac{1}{N}\sum_{i=1}^m abs(x_i)$. The value of $Λ$ is then normalized to lie on the interval [0, 1], assuming that the minimum/maximum value any single element $x_i$ can take is minval/maxval, respectively. Finally, the interval [0, 1] is discretized into n discrete bins, enumerated by positive integers 1, 2, …, n, and the number of the bin that the normalized $Λ$ falls into is returned.

    When used with decode, the left-edge of the bin that the normalized $Λ$ fell into is returned.

    source
    ComplexityMeasures.RelativeFirstDifferenceEncodingType
    RelativeFirstDifferenceEncoding <: Encoding
     RelativeFirstDifferenceEncoding(minval::Real, maxval::Real; n = 2)

    RelativeFirstDifferenceEncoding encodes a vector based on the relative position the average of the first differences of the vectors has with respect to a predefined minimum and maximum value (minval and maxval, respectively).

    Description

    This encoding is inspired by Azami and Escudero (2016)'s algorithm for amplitude-aware permutation entropy. They use a linear combination of amplitude information and first differences information of state vectors to correct probabilities. Here, however, we explicitly encode the first differences part of the correction as an a integer symbol Λ ∈ [1, 2, …, n]. The amplitude part of the encoding is available as the RelativeMeanEncoding encoding.

    Encoding/decoding

    When used with encode, an $m$-element state vector $\bf{x} = (x_1, x_2, \ldots, x_m)$ is encoded as $Λ = \dfrac{1}{m - 1}\sum_{k=2}^m |x_{k} - x_{k-1}|$. The value of $Λ$ is then normalized to lie on the interval [0, 1], assuming that the minimum/maximum value any single $abs(x_k - x_{k-1})$ can take is minval/maxval, respectively. Finally, the interval [0, 1] is discretized into n discrete bins, enumerated by positive integers 1, 2, …, n, and the number of the bin that the normalized $Λ$ falls into is returned. The smaller the mean first difference of the state vector is, the smaller the bin number is. The higher the mean first difference of the state vectors is, the higher the bin number is.

    When used with decode, the left-edge of the bin that the normalized $Λ$ fell into is returned.

    Performance tips

    If you are encoding multiple input vectors, it is more efficient to construct a RelativeFirstDifferenceEncoding instance and re-use it:

    minval, maxval = 0, 1
     encoding = RelativeFirstDifferenceEncoding(minval, maxval; n = 4)
     pts = [rand(3) for i = 1:1000]
    -[encode(encoding, x) for x in pts]
    source
    ComplexityMeasures.UniqueElementsEncodingType
    UniqueElementsEncoding <: Encoding
     UniqueElementsEncoding(x)

    UniqueElementsEncoding is a generic encoding that encodes each xᵢ ∈ unique(x) to one of the positive integers. The xᵢ are encoded according to the order of their first appearance in the input data.

    The constructor requires the input data x, since the number of possible symbols is length(unique(x)).

    Example

    using ComplexityMeasures
     x = ['a', 2, 5, 2, 5, 'a']
     e = UniqueElementsEncoding(x)
    -encode.(Ref(e), x) == [1, 2, 3, 2, 3, 1] # true
    source
    ComplexityMeasures.PairDistanceEncodingType
    PairDistanceEncoding <: Encoding
     PairDistanceEncoding(min_dist, max_dist; n = 2, metric = Chebyshev(), precise = false)

    An encoding that encodes point pairs of the form Tuple{<:AbstractVector, <:AbstractVector} by first computing their distance using the given metric, then dividing the interval [min_dist, max_dist] into n equal-size bins, and mapping the computed distance onto one of those bins. Bins are enumerated as 1:n. When decode-ing the bin integer, the left edge of the bin is returned.

    precise has the same meaning as in RectangularBinEncoding.

    Example

    Let's create an example where the minimum and maximum allowed distance is known.

    using ComplexityMeasures, Distances, StaticArrays
     m = Chebyshev()
     y = [SVector(1.0), SVector(0.5), SVector(0.25), SVector(0.64)]
    @@ -172,11 +172,11 @@
     c2 = encode(encoding, pair2) # 1
     c3 = encode(encoding, pair3) # 3
     
    -decode(encoding, c3) ≈ [0.35] # true
    source
    ComplexityMeasures.BubbleSortSwapsEncodingType
    BubbleSortSwapsEncoding <: Encoding
     BubbleSortSwapsEncoding{m}()

    BubbleSortSwapsEncoding is used with encode to encode a length-m input vector x into an integer in the range ω ∈ 0:((m*(m-1)) ÷ 2), by counting the number of swaps required for the bubble sort algorithm to sort x in ascending order.

    decode is not implemented for this encoding.

    Example

    using ComplexityMeasures
     x = [1, 5, 3, 1, 2]
     e = BubbleSortSwapsEncoding{5}() # constructor type argument must match length of vector 
    -encode(e, x)
    source
    ComplexityMeasures.CombinationEncodingType
    CombinationEncoding <: Encoding
     CombinationEncoding(encodings)

    A CombinationEncoding takes multiple Encodings and creates a combined encoding that can be used to encode inputs that are compatible with the given encodings.

    Encoding/decoding

    When used with encode, each Encoding in encodings returns integers in the set 1, 2, …, n_e, where n_e is the total number of outcomes for a particular encoding. For k different encodings, we can thus construct the cartesian coordinate (c₁, c₂, …, cₖ) (cᵢ ∈ 1, 2, …, n_i), which can uniquely be identified by an integer. We can thus identify each unique combined encoding with a single integer.

    When used with decode, the integer symbol is converted to its corresponding cartesian coordinate, which is used to retrieve the decoded symbols for each of the encodings, and a tuple of the decoded symbols are returned.

    The total number of outcomes is prod(total_outcomes(e) for e in encodings).

    Examples

    using ComplexityMeasures
     
     # We want to encode the vector `x`.
    @@ -198,4 +198,4 @@
     # Decode symbol (into a vector of decodings, one for each encodings `e ∈ encodings`).
     # In this particular case, the first two element will be left-bin edges, and
     # the last element will be the decoded ordinal pattern (indices that would sort `x`).
    -d = decode(c, ω)
    source
    +d = decode(c, ω)
    source
    diff --git a/previews/PR430/references/index.html b/previews/PR430/references/index.html index 60fa07d7..81bfb2a6 100644 --- a/previews/PR430/references/index.html +++ b/previews/PR430/references/index.html @@ -1,2 +1,2 @@ -References · ComplexityMeasures.jl

    References

    +References · ComplexityMeasures.jl

    References

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z!xD^i*guYME-0TRj-w=bvDs#%cy^6XNa(`E`It@*yLj>9rAw>emY$xPWN)5F2@)uo tp-}O$3#F?FEsCElvUC2o{Vi*n>-e?lz3-C4EXfrfQq$U>zSlhXKLER{bm9O2 diff --git a/previews/PR430/tutorial/index.html b/previews/PR430/tutorial/index.html index 16e38dc5..8e998a20 100644 --- a/previews/PR430/tutorial/index.html +++ b/previews/PR430/tutorial/index.html @@ -5,27 +5,27 @@ ε = 0.1 # bin width o = ValueBinning(ε) o isa OutcomeSpace

    true

    Such outcome spaces may be given to probabilities_and_outcomes to estimate the probabilities and corresponding outcomes from input data:

    probs, outs = probabilities_and_outcomes(o, x);
    -probs
     Probabilities{Float64,1} over 72 outcomes
    - [-4.564085055799177]   0.0001
    - [-3.6640850557991773]  0.0001
    - [-3.5640850557991772]  0.0002
    - [-3.464085055799177]   0.0001
    - [-3.364085055799177]   0.0004
    - [-3.2640850557991774]  0.0003
    - [-3.164085055799177]   0.0005
    - [-3.0640850557991772]  0.0002
    - [-2.964085055799177]   0.0005
    - [-2.864085055799177]   0.0006
    +probs
     Probabilities{Float64,1} over 71 outcomes
    + [-3.568701989479745]   0.0001
    + [-3.2687019894797453]  0.0002
    + [-3.1687019894797452]  0.0002
    + [-3.068701989479745]   0.0001
    + [-2.968701989479745]   0.0008
    + [-2.868701989479745]   0.0009
    + [-2.7687019894797453]  0.0012
    + [-2.6687019894797452]  0.0016
    + [-2.568701989479745]   0.0024
    + [-2.468701989479745]   0.0018
      ⋮                      
    - [2.5359149442008233]   0.0012
    - [2.635914944200823]    0.0015
    - [2.7359149442008235]   0.0004
    - [2.835914944200823]    0.001
    - [2.9359149442008228]   0.0001
    - [3.0359149442008233]   0.0003
    - [3.2359149442008235]   0.0001
    - [3.335914944200823]    0.0002
    - [3.4359149442008228]   0.0003

    In this example the probabilities are the (normalized) heights of each bin of the histogram. The bins, which are the elements of the outcome space, are shown in the margin, left of the probabilities. They are also returned explicitly as outs above.

    This convenience printing syntax with outcomes and probabilities is useful for visual inspection of the probabilities data. However, don't let it worry you. Probabilities are returned as a special Probabilities type that behaves identically to a standard Julia numerical Vector. You can obtain the maximum probability

    maximum(probs)
    0.0416

    or iterate over the probabilities

    function total(probs)
    + [2.831298010520255]    0.0008
    + [2.931298010520255]    0.0004
    + [3.0312980105202554]   0.0006
    + [3.131298010520255]    0.0001
    + [3.2312980105202556]   0.0002
    + [3.331298010520255]    0.0003
    + [3.431298010520255]    0.0001
    + [3.5312980105202554]   0.0001
    + [3.831298010520255]    0.0001

    In this example the probabilities are the (normalized) heights of each bin of the histogram. The bins, which are the elements of the outcome space, are shown in the margin, left of the probabilities. They are also returned explicitly as outs above.

    This convenience printing syntax with outcomes and probabilities is useful for visual inspection of the probabilities data. However, don't let it worry you. Probabilities are returned as a special Probabilities type that behaves identically to a standard Julia numerical Vector. You can obtain the maximum probability

    maximum(probs)
    0.0434

    or iterate over the probabilities

    function total(probs)
         t = 0.0
         for p in probs
             t += p
    @@ -33,30 +33,30 @@
         return t
     end
     
    -total(probs)
    0.9999999999999997

    Notice that if you use probabilities instead of probabilities_and_outcomes, then outcomes are enumerated generically. This avoids computing outcomes explicitly, and can save some computation time in cases where you don't need the outcomes.

    probs2 = probabilities(o, x)
     Probabilities{Float64,1} over 72 outcomes
    +total(probs)
    0.9999999999999999

    Notice that if you use probabilities instead of probabilities_and_outcomes, then outcomes are enumerated generically. This avoids computing outcomes explicitly, and can save some computation time in cases where you don't need the outcomes.

    probs2 = probabilities(o, x)
     Probabilities{Float64,1} over 71 outcomes
       Outcome(1)  0.0001
    -  Outcome(2)  0.0001
    +  Outcome(2)  0.0002
       Outcome(3)  0.0002
       Outcome(4)  0.0001
    -  Outcome(5)  0.0004
    -  Outcome(6)  0.0003
    -  Outcome(7)  0.0005
    -  Outcome(8)  0.0002
    -  Outcome(9)  0.0005
    - Outcome(10)  0.0006
    +  Outcome(5)  0.0008
    +  Outcome(6)  0.0009
    +  Outcome(7)  0.0012
    +  Outcome(8)  0.0016
    +  Outcome(9)  0.0024
    + Outcome(10)  0.0018
                ⋮  
    - Outcome(64)  0.0012
    - Outcome(65)  0.0015
    - Outcome(66)  0.0004
    - Outcome(67)  0.001
    - Outcome(68)  0.0001
    - Outcome(69)  0.0003
    + Outcome(63)  0.0008
    + Outcome(64)  0.0004
    + Outcome(65)  0.0006
    + Outcome(66)  0.0001
    + Outcome(67)  0.0002
    + Outcome(68)  0.0003
    + Outcome(69)  0.0001
      Outcome(70)  0.0001
    - Outcome(71)  0.0002
    - Outcome(72)  0.0003

    Of course, if the outcomes are obtained for free while estimating the probabilities, they would be included in the return value.

    For the ValueBinning example that we use, the outcomes are the left edges of each bin. This allows us to straightforwardly visualize the results.

    using CairoMakie
    + Outcome(71)  0.0001

    Of course, if the outcomes are obtained for free while estimating the probabilities, they would be included in the return value.

    For the ValueBinning example that we use, the outcomes are the left edges of each bin. This allows us to straightforwardly visualize the results.

    using CairoMakie
     outs = outcomes(probs);
     left_edges = only.(outs) # convert `Vector{SVector}` into `Vector{Real}`
    -barplot(left_edges, probs; axis = (ylabel = "probability",))
    Example block output

    Naturally, there are other outcome spaces one may use, and one can find the list of implemented ones in OutcomeSpace. A prominent example used in the NLTS literature are ordinal patterns. The outcome space for it is OrdinalPatterns, and can be particularly useful with timeseries that come from nonlinear dynamical systems. For example, let's simulate a logistic map timeseries.

    using DynamicalSystemsBase
    +barplot(left_edges, probs; axis = (ylabel = "probability",))
    Example block output

    Naturally, there are other outcome spaces one may use, and one can find the list of implemented ones in OutcomeSpace. A prominent example used in the NLTS literature are ordinal patterns. The outcome space for it is OrdinalPatterns, and can be particularly useful with timeseries that come from nonlinear dynamical systems. For example, let's simulate a logistic map timeseries.

    using DynamicalSystemsBase
     
     logistic_rule(u, r, t) = SVector(r*u[1]*(1 - u[1]))
     ds = DeterministicIteratedMap(logistic_rule, [0.4], 4.0)
    @@ -69,23 +69,23 @@
      [2, 1, 3]  0.13051305130513052
      [2, 3, 1]  0.1996199619961996
      [3, 1, 2]  0.26572657265726574

    Comparing these probabilities with those for the purely random timeseries x,

    probsx = probabilities(o, x)
     Probabilities{Float64,1} over 6 outcomes
    - [1, 2, 3]  0.16503300660132025
    - [1, 3, 2]  0.16793358671734346
    - [2, 1, 3]  0.16613322664532906
    - [2, 3, 1]  0.16983396679335866
    - [3, 1, 2]  0.16813362672534507
    - [3, 2, 1]  0.16293258651730347

    you will notice that the probabilities computing from x has six outcomes, while the probabilities computed from the timeseries y has five outcomes.

    The reason that there are less outcomes in the y is because one outcome was never encountered in the y data. This is a common theme in ComplexityMeasures.jl: outcomes that are not in the data are skipped. This can save memory for outcome spaces with large numbers of outcomes. To explicitly obtain all outcomes, by assigning 0 probability to not encountered outcomes, use allprobabilities_and_outcomes. For OrdinalPatterns the outcome space does not depend on input data and is always the same. Hence, the corresponding outcomes matching to allprobabilities_and_outcomes, coincide for x and y, and also coincide with the output of the function outcome_space:

    o = OrdinalPatterns()
    + [1, 2, 3]  0.16543308661732348
    + [1, 3, 2]  0.16643328665733145
    + [2, 1, 3]  0.17153430686137228
    + [2, 3, 1]  0.16443288657731547
    + [3, 1, 2]  0.16963392678535708
    + [3, 2, 1]  0.16253250650130027

    you will notice that the probabilities computing from x has six outcomes, while the probabilities computed from the timeseries y has five outcomes.

    The reason that there are less outcomes in the y is because one outcome was never encountered in the y data. This is a common theme in ComplexityMeasures.jl: outcomes that are not in the data are skipped. This can save memory for outcome spaces with large numbers of outcomes. To explicitly obtain all outcomes, by assigning 0 probability to not encountered outcomes, use allprobabilities_and_outcomes. For OrdinalPatterns the outcome space does not depend on input data and is always the same. Hence, the corresponding outcomes matching to allprobabilities_and_outcomes, coincide for x and y, and also coincide with the output of the function outcome_space:

    o = OrdinalPatterns()
     probsx = allprobabilities(o, x)
     probsy = allprobabilities(o, y)
     outsx = outsy = outcome_space(o)
     # display all quantities as parallel columns
     hcat(outsx, probsx, probsy)
    6×3 Matrix{Any}:
    - [1, 2, 3]  0.165033  0.339734
    - [1, 3, 2]  0.167934  0.0644064
    - [2, 1, 3]  0.166133  0.130513
    - [2, 3, 1]  0.169834  0.19962
    - [3, 1, 2]  0.168134  0.265727
    - [3, 2, 1]  0.162933  0.0

    The number of possible outcomes, i.e., the cardinality of the outcome space, can always be found using total_outcomes:

    total_outcomes(o)
    6

    Beyond the basics: probabilities estimators

    So far we have been estimating probabilities by counting the amount of times each possible outcome was encountered in the data, then normalizing. This is called "maximum likelihood estimation" of probabilities. To get the counts themselves, use the counts or counts_and_outcomes function.

    countsy = counts(o, y)
     Counts{Int64,1} over 5 outcomes
    + [1, 2, 3]  0.165433  0.339734
    + [1, 3, 2]  0.166433  0.0644064
    + [2, 1, 3]  0.171534  0.130513
    + [2, 3, 1]  0.164433  0.19962
    + [3, 1, 2]  0.169634  0.265727
    + [3, 2, 1]  0.162533  0.0

    The number of possible outcomes, i.e., the cardinality of the outcome space, can always be found using total_outcomes:

    total_outcomes(o)
    6

    Beyond the basics: probabilities estimators

    So far we have been estimating probabilities by counting the amount of times each possible outcome was encountered in the data, then normalizing. This is called "maximum likelihood estimation" of probabilities. To get the counts themselves, use the counts or counts_and_outcomes function.

    countsy = counts(o, y)
     Counts{Int64,1} over 5 outcomes
      Outcome(1)  3397
      Outcome(2)   644
      Outcome(3)  1305
    @@ -106,7 +106,7 @@
       1.8994302469765856e-7
      -3.285014627013583e-5

    While the corrections of BayesianRegularization are small in this case, they are nevertheless measurable.

    When calling probabilities only with an outcome space instance and some input data (skipping the ProbabilitiesEstimator), then by default, the RelativeAmount probabilities estimator is used to extract the probabilities.

    Entropies

    Many compexity measures are a straightforward estimation of Shannon entropy, computed over probabilities estimated from data over some particular outcome space. For example, the well known permutation entropy (Bandt and Pompe, 2002) is exactly the Shannon entropy of the probabilities probsy we computed above based on ordinal patterns. To compute it, we use the entropy function.

    perm_ent_x = entropy(OrdinalPatterns(), x)
     perm_ent_y = entropy(OrdinalPatterns(), y)
    -(perm_ent_x, perm_ent_y)
    (2.5848295129526115, 2.139506127185978)

    As expected, the permutation entropy of the x signal is higher, because the signal is "more random". Moreover, since we have estimated the probabilities already, we could have passed these to the entropy function directly instead of recomputing them as above

    perm_ent_y_2 = entropy(probsy)
    2.139506127185978

    We crucially realize here that many quantities in the NLTS literature that are named as entropies, such as "permutation entropy", are not really new entropies. They are the good old Shannon entropy (Shannon), but calculated with new outcome spaces that smartly quantify some dynamic property in the data. Nevertheless, we acknowledge that names such as "permutation entropy" are commonplace, so in ComplexityMeasures.jl we provide convenience functions like entropy_permutation. More convenience functions can be found in the convenience documentation page.

    Beyond Shannon: more entropies

    Just like there are different outcome spaces, the same concept applies to entropy. There are many fundamentally different entropies. Shannon entropy is not the only one, just the one used most often. Each entropy is a subtype of InformationMeasure. Another commonly used entropy is the Renyi or generalized entropy. We can use Renyi as an additional first argument to the entropy function to estimate the generalized entropy:

    perm_ent_y_q2 = entropy(Renyi(q = 2.0), OrdinalPatterns(), y)
    +(perm_ent_x, perm_ent_y)
    (2.584719928958848, 2.139506127185978)

    As expected, the permutation entropy of the x signal is higher, because the signal is "more random". Moreover, since we have estimated the probabilities already, we could have passed these to the entropy function directly instead of recomputing them as above

    perm_ent_y_2 = entropy(probsy)
    2.139506127185978

    We crucially realize here that many quantities in the NLTS literature that are named as entropies, such as "permutation entropy", are not really new entropies. They are the good old Shannon entropy (Shannon), but calculated with new outcome spaces that smartly quantify some dynamic property in the data. Nevertheless, we acknowledge that names such as "permutation entropy" are commonplace, so in ComplexityMeasures.jl we provide convenience functions like entropy_permutation. More convenience functions can be found in the convenience documentation page.

    Beyond Shannon: more entropies

    Just like there are different outcome spaces, the same concept applies to entropy. There are many fundamentally different entropies. Shannon entropy is not the only one, just the one used most often. Each entropy is a subtype of InformationMeasure. Another commonly used entropy is the Renyi or generalized entropy. We can use Renyi as an additional first argument to the entropy function to estimate the generalized entropy:

    perm_ent_y_q2 = entropy(Renyi(q = 2.0), OrdinalPatterns(), y)
     (perm_ent_y_q2, perm_ent_y)
    (2.0170680478970375, 2.139506127185978)

    In fact, when we called entropy(OrdinalPatterns(), y), this dispatched to the default call of entropy(Shannon(), OrdinalPatterns(), y).

    Beyond entropies: discrete estimators

    The estimation of an entropy truly parallelizes the estimation of probabilities: in the latter, we could decide an outcome space and an estimator to estimate probabilities. The same happens for entropy: we can decide an entropy definition and an estimator of how to estimate the entropy. For example, instead of the default PlugIn estimator that we used above implicitly, we could use the Jackknife estimator.

    ospace = OrdinalPatterns()
     entdef = Renyi(q = 2.0)
     entest = Jackknife(entdef)
    @@ -114,5 +114,5 @@
     
     (perm_ent_y_q2, perm_ent_y_q2_jack)
    (2.0170680478970375, 1.2173088128474774)

    Entropy estimators always reference an entropy definition, even if they only apply to one type of entropy (typically the Shannon one). From here, it is up to the researcher to read the documentation of the plethora of estimators implemented and decide what is most suitable for their data at hand. They all can be found in DiscreteInfoEstimator.

    Beyond entropies: other information measures

    Recall that at the very beginning of this notebook we mentioned a code separation of information and complexity. We did this because there are other measures, besides entropy, that are explicit functionals of some probability mass function. One example is the Shannon extropy ShannonExtropy, the complementary dual of entropy, which could be computed as follows.

    extdef = ShannonExtropy()
     perm_ext_y = information(extdef, ospace, y)
    1.2450281736700084

    Just like the Shannon entropy, the extropy could also be estimated with a different estimator such as Jackknife.

    perm_ext_y_jack = information(Jackknife(extdef), ospace, y)
    1.126229249906828

    In truth, when we called entropy(e, o, y) it actually calls information(e, o, y), as all "information measures" are part of the same function interface. And entropy is an information measure.

    Beyond discrete: differential or continuous

    Discrete information measures are functions of probability mass functions. It is also possible to compute information measures of probability density functions. In ComplexityMeasures.jl, this is done by calling entropy (or the more general information) with a differential information estimator, a subtype of DifferentialInfoEstimator. These estimators are given directly to information without assigning an outcome space, because the probability density is approximated implicitly, not explicitly. For example, the Correa estimator approximates the differential Shannon entropy by utilizing order statistics of the timeseries data:

    diffest = Correa()
    -diffent = entropy(diffest, x)
    1.7859169459741842

    Beyond information: other complexity measures

    As discussed at the very beginning of this tutorial, there are some complexity measures that are not explicit functionals of probabilities, and hence cannot be straightforwardly related to an outcome space, in the sense of providing an instance of OutcomeSpace to the estimation function. These are estimated with the complexity function, by providing it a subtype of ComplexityEstimator. An example here is the well-known sample entropy (which isn't actually an entropy in the formal mathematical sense). It can be computed as follows.

    complest = SampleEntropy(r = 0.1)
    -sampent = complexity(complest, y)
    0.612355215298896
    +diffent = entropy(diffest, x)
    1.815849824414553

    Beyond information: other complexity measures

    As discussed at the very beginning of this tutorial, there are some complexity measures that are not explicit functionals of probabilities, and hence cannot be straightforwardly related to an outcome space, in the sense of providing an instance of OutcomeSpace to the estimation function. These are estimated with the complexity function, by providing it a subtype of ComplexityEstimator. An example here is the well-known sample entropy (which isn't actually an entropy in the formal mathematical sense). It can be computed as follows.

    complest = SampleEntropy(r = 0.1)
    +sampent = complexity(complest, y)
    0.612355215298896

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