spkit.entropy_spectral

spkit.entropy_spectral(x, fs, method='fft', alpha=1, base=2, normalize=True, axis=-1, nperseg=None, esp=1e-10)

Spectral Entropy \(H_f(X)\)

Measure of the uncertainity of frequency components in a signal. Spectral entropy a signal drawn from uniform distribution will be usually quite similar to one that is drawn from gaussian distrobutation. Since both have almost flat-like spectrum.

\[H_f(x) = H(F(x))\]

\(F(x)\) - FFT of x

Parameters:
x: 1d array
fs = sampling frequency
method: ‘fft’ use periodogram and ‘welch’ uses wekch method
base: base of log, 2 for bit, 10 for nats
Returns:
H_fx: scalar

  • Spectral entropy

See also

entropy

Entropy

entropy_sample

Sample Entropy

entropy_approx

Approximate Entropy

dispersion_entropy

Dispersion Entropy

entropy_svd

SVD Entropy

entropy_permutation

Permutation Entropy

entropy_differential

Differential Entropy

Examples

>>> sp.entropy_spectral
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> import spkit as sp
>>> np.random.seed(1)
>>> fs = 1000
>>> t = np.arange(1000)/fs
>>> x1 = np.random.randn(len(t))
>>> x2 = np.cos(2*np.pi*10*t)+np.cos(2*np.pi*30*t)+np.cos(2*np.pi*20*t)
>>> Hx1 = sp.entropy(x1)
>>> Hx2 = sp.entropy(x2)
>>> Hx1_se = sp.entropy_spectral(x1,fs=fs,method='welch',normalize=False)
>>> Hx2_se = sp.entropy_spectral(x2,fs=fs,method='welch',normalize=False)
>>> print('Spectral Entropy:')
>>> print(r' - H_f(x1) = ',Hx1_se,)
>>> print(r' - H_f(x1) = ',Hx2_se)
>>> print('Shannon Entropy:')
>>> print(r' - H(x1) = ',Hx1)
>>> print(r' - H(x1) = ',Hx2)
>>> print('-')
>>> Hx1_n = sp.entropy(x1,normalize=True)
>>> Hx2_n = sp.entropy(x2,normalize=True)
>>> Hx1_se_n = sp.entropy_spectral(x1,fs=fs,method='welch',normalize=True)
>>> Hx2_se_n = sp.entropy_spectral(x2,fs=fs,method='welch',normalize=True)
>>> print('Spectral Entropy (Normalised)')
>>> print(r' - H_f(x1) = ',Hx1_se_n,)
>>> print(r' - H_f(x1) = ',Hx2_se_n,)
>>> print('Shannon Entropy (Normalised)')
>>> print(r' - H_f(x1) = ',Hx1_n)
>>> print(r' - H_f(x1) = ',Hx2_n)
>>> np.random.seed(None)
>>> plt.figure(figsize=(11,4))
>>> plt.subplot(121)
>>> plt.plot(t,x1,label='x1: Gaussian Noise',alpha=0.8)
>>> plt.plot(t,x2,label='x2: Sinusoidal',alpha=0.8)
>>> plt.xlim([t[0],t[-1]])
>>> plt.xlabel('time (s)')
>>> plt.ylabel('x1')
>>> plt.legend(bbox_to_anchor=(1, 1.2),ncol=2,loc='upper right')
>>> plt.subplot(122)
>>> label1 = f'x1: Gaussian Noise \n H(x): {Hx1.round(2)}, H_f(x): {Hx1_se.round(2)}'
>>> label2 = f'x2: Sinusoidal \n H(x): {Hx2.round(2)}, H_f(x): {Hx2_se.round(2)}'
>>> P1x,f1q = sp.periodogram(x1,fs=fs,show_plot=True,label=label1)
>>> P2x,f2q = sp.periodogram(x2,fs=fs,show_plot=True,label=label2)
>>> plt.legend(bbox_to_anchor=(0.4, 0.4))
>>> plt.grid()
>>> plt.tight_layout()
>>> plt.show()
Spectral Entropy:
- H_f(x1) =  6.889476342103717
- H_f(x1) =  2.7850662938305786
Shannon Entropy:
- H_f(x1) =  3.950686888274901
- H_f(x1) =  3.8204484006660255
-
Spectral Entropy (Normalised)
- H_f(x1) =  0.9826348642134711
- H_f(x1) =  0.3972294995395931
Shannon Entropy (Normalised)
- H_f(x1) =  0.8308686349524237
- H_f(x1) =  0.8445663920995877