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Entropy - EEG Signal - Multi-Channel¶

An example of Entropy computation (Mutual Information, Joint Information, Cross-entropy and KL-Divergence Entropy) among multi-channels of a sample EEG signals.

Mutual Information
Joint Information
Cross-entropy and
KL-Divergence Entropy

spkit-version : 0.0.9.7
(2048, 14)
['AF3', 'F7', 'F3', 'FC5', 'T7', 'P7', 'O1', 'O2', 'P8', 'T8', 'FC6', 'F4', 'F8', 'AF4']
Selecting two channels AF3 (Frontal Lobe)and O1 (Occipital Lobe)
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Shannan entropy
Entropy of AF3: H(AF3)=  4.9573307496464665
Entropy of O1 : H(O1) =  4.697278111823318
-
Rényi entropy
Entropy of AF3: H(AF3)=  4.375932807816424
Entropy of O1 : H(O1) =  4.145472461333239
-
Joint Entropy: H(AF3,O1)    =    3.750169316339896
Mutual Information I(AF3,O1)=    1.2071614333065703
Conditional Entropy of : H(AF3|O1)=      3.750169316339896
Conditional Entropy of : H(O1|AF3)=      3.4901166785167455
-
Cross Entropy of : H(AF3,O1)=    5.732514591834208
Kullback–Leibler divergence : Dkl(AF3,O1)=       0.7751838421877422
Entropy of Each channel

import numpy as np
import matplotlib.pyplot as plt
import spkit as sp
print('spkit-version :', sp.__version__)

X,fs,ch_names = sp.data.eeg_sample_14ch()
t = np.arange(X.shape[0])/fs
nC = len(ch_names)
print(X.shape)
print(ch_names)

print('Selecting two channels AF3 (Frontal Lobe)and O1 (Occipital Lobe)')
AF3 = X[:,0] #'AF3' - Frontal Lobe
O1  = X[:,6] #'O1'  - Occipital Lobe
bins = int(max(sp.bin_width(AF3)[1], sp.bin_width(O1)[1]))

#Shannan entropy
H_x1= sp.entropy(AF3,alpha=1,bins=bins)
H_x2= sp.entropy(O1,alpha=1,bins=bins)

#Rényi entropy
Hr_x1= sp.entropy(AF3,alpha=2,bins=bins)
Hr_x2= sp.entropy(O1,alpha=2,bins=bins)

#Joint entropy
H_x12= sp.entropy_joint(AF3,O1,bins=bins)

#Conditional Entropy
H_x12= sp.entropy_cond(AF3,O1,bins=bins)
H_x21= sp.entropy_cond(O1,AF3,bins=bins)

#Mutual Information
I_x12 = sp.mutual_info(AF3,O1,bins=bins)

#Cross Entropy
H_x12_cross= sp.entropy_cross(AF3,O1,bins=bins)

#Diff Entropy
D_x12= sp.entropy_kld(AF3,O1,bins=bins)

print('-')
print('Shannan entropy')
print('Entropy of AF3: H(AF3)=\t',H_x1)
print('Entropy of O1 : H(O1) =\t',H_x2)
print('-')
print('Rényi entropy')
print('Entropy of AF3: H(AF3)=\t',Hr_x1)
print('Entropy of O1 : H(O1) =\t',Hr_x2)
print('-')
print('Joint Entropy: H(AF3,O1)    =\t',H_x12)
print('Mutual Information I(AF3,O1)=\t',I_x12)
print('Conditional Entropy of : H(AF3|O1)=\t',H_x12)
print('Conditional Entropy of : H(O1|AF3)=\t',H_x21)
print('-')
print('Cross Entropy of : H(AF3,O1)=\t',H_x12_cross)
print('Kullback–Leibler divergence : Dkl(AF3,O1)=\t',D_x12)


print('Entropy of Each channel')

bins = int(max(sp.bin_width(ch)[1] for ch in X.T))

H_X= [sp.entropy(x,alpha=1,bins=bins) for x in X.T]

MI = np.zeros([nC,nC])
JE = np.zeros([nC,nC])
CE = np.zeros([nC,nC])
KL = np.zeros([nC,nC])
for i in range(nC):
    x1 = X[:,i]
    for j in range(nC):
        x2 = X[:,j]

        #Mutual Information
        MI[i,j] = sp.mutual_info(x1,x2,bins=bins)

        #Joint entropy
        JE[i,j]= sp.entropy_joint(x1,x2,bins=bins)

        #Cross Entropy
        CE[i,j]= sp.entropy_cross(x1,x2,bins=bins)

        #Diff Entropy
        KL[i,j]= sp.entropy_kld(x1,x2,bins=bins)


MI_d = MI - np.diag(np.diag(MI))
plt.figure(figsize=(8,7))
plt.subplot(221)
plt.imshow(MI_d,origin='lower')
plt.yticks(np.arange(nC),ch_names)
plt.xticks(np.arange(nC),ch_names,rotation=90)
plt.title('Mutual Information')
plt.colorbar()
plt.subplot(222)
plt.imshow(JE,origin='lower')
plt.yticks(np.arange(nC),ch_names)
plt.xticks(np.arange(nC),ch_names,rotation=90)
plt.title('Joint Entropy')
plt.colorbar()
plt.subplot(223)
plt.imshow(CE,origin='lower')
plt.yticks(np.arange(nC),ch_names)
plt.xticks(np.arange(nC),ch_names,rotation=90)
plt.title('Cross Entropy')
plt.colorbar()
plt.subplot(224)
plt.imshow(KL,origin='lower')
plt.yticks(np.arange(nC),ch_names)
plt.xticks(np.arange(nC),ch_names,rotation=90)
plt.title('KL-Divergence')
plt.colorbar()
plt.tight_layout()
plt.show()

plt.figure(figsize=(12,3))
plt.bar(np.arange(nC)*2,H_X)
plt.xticks(np.arange(nC)*2,ch_names)
plt.ylabel('Entropy (bits)')
plt.xlabel('Channel')
plt.tight_layout()
plt.show()

plt.figure(figsize=(12,3))
plt.plot(t,X+np.arange(nC)*200)
plt.grid()
plt.xlim(t[0],t[-1])
plt.yticks(np.arange(nC)*200, ch_names)
plt.xlabel('time (s)')
plt.tight_layout()
plt.show()