pyhctsa.operations.entropy.fuzzy_entropy¶

pyhctsa.operations.entropy.fuzzy_entropy(y, M=2, r=0.2, n=2)¶

Fuzzy entropy of a time series.

Chen et al.’s fuzzy entropy [1, 2], a smooth relative of sample entropy (sample_entropy()). The series is cut into overlapping runs of m consecutive values, and the mean of each run is subtracted from it, so runs are compared by shape and not by level. Two runs are not simply ‘matching’ or ‘not matching’, as in sample entropy: they are given a similarity exp(-(d/r)**n), where d is the largest absolute difference between corresponding values of the two (baseline-removed) runs. phi_m is the mean similarity over all pairs of distinct runs of length m, and the fuzzy entropy at dimension m is log(phi_m) - log(phi_(m+1)). Low values indicate regular, predictable series; high values irregular ones. The smooth similarity makes the measure continuous in r and defined for short series for which sample entropy would find no matches.

All runs of length 1, …, M+1 are taken from the same N - M starting points, so that successive dimensions are compared on the same footing. The distances are computed pair by pair, so memory use does not grow with the square of the series length (the run time does: it is O(N^2 M)). Port of hctsa’s EN_FuzzyEn.

The similarity is written here as exp(-(d/r)**n), so that r is a distance (in units of the standard deviation of y). Chen et al. (2007) write it as exp(-d**n/r), in which r is not a distance: for n = 2, their r = 0.2 on standardized data is a Gaussian width of sqrt(0.2) = 0.45 standard deviations, against 0.2 here. Values of r are therefore not directly comparable with those quoted in that paper. To get the fuzzy entropy of the increments of a series, give np.diff(y) as the input.

References

Parameters:¶

y : array-like¶

The input time series.

M : int, optional¶

The largest embedding dimension: the fuzzy entropy is returned for m = 1, ..., M. Default is 2.

r : float, optional¶

The width of the similarity function, as a fraction of the standard deviation of y (the width in the units of y is r * std(y), so the measure is unchanged by any rescaling of y). Default is 0.2.

n : float, optional¶

The exponent of the similarity function exp(-(d/r)**n) (larger values make the similarity closer to a hard threshold). Default is 2.

Returns:¶

Fields ‘fuzzyEn1’, ‘fuzzyEn2’, …, ‘fuzzyEnM’: the fuzzy entropy at each embedding dimension (nats). At m = 1 the run mean removed is the value itself, so phi_1 = 1 and ‘fuzzyEn1’ is -log(phi_2). All fields are NaN if the series is constant, has fewer than M + 3 points, or has no pair of runs with a nonzero similarity.

Return type:¶

dict