entropy.sample_entropy

entropy.sample_entropy(x, order=2, metric='chebyshev')

Sample Entropy.

Parameters

xlist or np.array

One-dimensional time series of shape (n_times).

orderint

Embedding dimension. Default is 2.

metricstr

Name of the distance metric function used with sklearn.neighbors.KDTree. Default is to use the Chebyshev distance.

Returns

sefloat

Sample Entropy.

Notes

Sample entropy is a modification of approximate entropy, used for assessing the complexity of physiological time-series signals. It has two advantages over approximate entropy: data length independence and a relatively trouble-free implementation. Large values indicate high complexity whereas smaller values characterize more self-similar and regular signals.

The sample entropy of a signal \(x\) is defined as:

\[H(x, m, r) = -\log\frac{C(m + 1, r)}{C(m, r)}\]

where \(m\) is the embedding dimension (= order), \(r\) is the radius of the neighbourhood (default = \(0.2 * \text{std}(x)\)), \(C(m + 1, r)\) is the number of embedded vectors of length \(m + 1\) having a Chebyshev distance inferior to \(r\) and \(C(m, r)\) is the number of embedded vectors of length \(m\) having a Chebyshev distance inferior to \(r\).

Note that if metric == 'chebyshev' and len(x) < 5000 points, then the sample entropy is computed using a fast custom Numba script. For other distance metric or longer time-series, the sample entropy is computed using a code from the mne-features package by Jean-Baptiste Schiratti and Alexandre Gramfort (requires sklearn).

References

Richman, J. S. et al. (2000). Physiological time-series analysis using approximate entropy and sample entropy. American Journal of Physiology-Heart and Circulatory Physiology, 278(6), H2039-H2049.

https://scikit-learn.org/stable/modules/generated/sklearn.neighbors.DistanceMetric.html

Examples

Fractional Gaussian noise with H = 0.5

>>> import numpy as np
>>> import entropy as ent
>>> import stochastic.processes.noise as sn
>>> rng = np.random.default_rng(seed=42)
>>> x = sn.FractionalGaussianNoise(hurst=0.5, rng=rng).sample(10000)
>>> print(f"{ent.sample_entropy(x, order=2):.4f}")
2.1819

Same with order = 3 and using the Euclidean distance

>>> print(f"{ent.sample_entropy(x, order=3, metric='euclidean'):.4f}")
2.6806

Fractional Gaussian noise with H = 0.9

>>> rng = np.random.default_rng(seed=42)
>>> x = sn.FractionalGaussianNoise(hurst=0.9, rng=rng).sample(10000)
>>> print(f"{ent.sample_entropy(x):.4f}")
1.9078

Fractional Gaussian noise with H = 0.1

>>> rng = np.random.default_rng(seed=42)
>>> x = sn.FractionalGaussianNoise(hurst=0.1, rng=rng).sample(10000)
>>> print(f"{ent.sample_entropy(x):.4f}")
2.0555

Random

>>> rng = np.random.default_rng(seed=42)
>>> print(f"{ent.sample_entropy(rng.random(1000)):.4f}")
2.2017

Pure sine wave

>>> x = np.sin(2 * np.pi * 1 * np.arange(3000) / 100)
>>> print(f"{ent.sample_entropy(x):.4f}")
0.1633

Linearly-increasing time-series

>>> x = np.arange(1000)
>>> print(f"{ent.sample_entropy(x):.4f}")
-0.0000