entropy.spectral_entropy

entropy.spectral_entropy(x, sf, method='fft', nperseg=None, normalize=False, axis=- 1)

Spectral Entropy.

Parameters

xlist or np.array

1D or N-D data.

sffloat

Sampling frequency, in Hz.

methodstr

Spectral estimation method:

• 'fft' : Fourier Transform (scipy.signal.periodogram())

• 'welch' : Welch periodogram (scipy.signal.welch())

npersegint or None

Length of each FFT segment for Welch method. If None (default), uses scipy default of 256 samples.

normalizebool

If True, divide by log2(psd.size) to normalize the spectral entropy between 0 and 1. Otherwise, return the spectral entropy in bit.

axisint

The axis along which the entropy is calculated. Default is -1 (last).

Returns

sefloat

Spectral Entropy

Notes

Spectral Entropy is defined to be the Shannon entropy of the power spectral density (PSD) of the data:

\[H(x, sf) = -\sum_{f=0}^{f_s/2} P(f) \log_2[P(f)]\]

Where \(P\) is the normalised PSD, and \(f_s\) is the sampling frequency.

References

• Inouye, T. et al. (1991). Quantification of EEG irregularity by use of the entropy of the power spectrum. Electroencephalography and clinical neurophysiology, 79(3), 204-210.

• https://en.wikipedia.org/wiki/Spectral_density

• https://en.wikipedia.org/wiki/Welch%27s_method

Examples

Spectral entropy of a pure sine using FFT

>>> import numpy as np
>>> import entropy as ent
>>> sf, f, dur = 100, 1, 4
>>> N = sf * dur # Total number of discrete samples
>>> t = np.arange(N) / sf # Time vector
>>> x = np.sin(2 * np.pi * f * t)
>>> np.round(ent.spectral_entropy(x, sf, method='fft'), 2)
0.0

Spectral entropy of a random signal using Welch’s method

>>> np.random.seed(42)
>>> x = np.random.rand(3000)
>>> ent.spectral_entropy(x, sf=100, method='welch')
6.980045662371389

Normalized spectral entropy

>>> ent.spectral_entropy(x, sf=100, method='welch', normalize=True)
0.9955526198316071

Normalized spectral entropy of 2D data

>>> np.random.seed(42)
>>> x = np.random.normal(size=(4, 3000))
>>> np.round(ent.spectral_entropy(x, sf=100, normalize=True), 4)
array([0.9464, 0.9428, 0.9431, 0.9417])

Fractional Gaussian noise with H = 0.5

>>> 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.spectral_entropy(x, sf=100, normalize=True):.4f}")
0.9505

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.spectral_entropy(x, sf=100, normalize=True):.4f}")
0.8477

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.spectral_entropy(x, sf=100, normalize=True):.4f}")
0.9248