spkit.transfer_entropy

spkit.transfer_entropy(X, Y, present_first=True)

Transfer Entropy $T{E}_{X->Y}$

Transfer Entropy

$T{E}_{X->Y}=I(Y_{i+1},X_i|Y_i)$

$T{E}_{X->Y}=H(Y_{i+1}|Y_i)-H(Y_{i+1}|X_i,Y_i)$ [Eq1]

$T{E}_{X->Y}=H(Y_{i+1},Y_i)-H(Y_i)-H(Y_{i+1},X_i,Y_i)+H(X_i,Y_i)$

$T{E}_{X->Y}=H(X_i,Y_i)+H(Y_{i+1},Y_i)-H(Y_{i+1},X_i,Y_i)-H(Y_i)$ [Eq2]

Using:

• $H(X_{i+1}|X_i)=H(X_{i+1},X_i)-H(X_i)$ - entropy_diff_cond_self(X)

• $H(X_{i+1}|X_i,Y_i)=H(X_{i+1},X_i,Y_i)-H(X_i,Y_i)$ - entropy_diff_cond(X,Y)

Parameters:

X: 2d-array,

• multi-dimentional signal space, where each column (axis=1) are the delayed signals

Y: 2d-array,

• multi-dimentional signal space, where each column (axis=1) are the delayed signals

present_first: bool, default=True

• if True, X[:,0] is present, and X[:,1:] is past, in incresing order

• if True, X[:,-1] is present, and X[:,:-1] is past

Returns:

TE_x2y: scaler

• Transfer Entropy from x to y

See also

transfer_entropy_cond

Conditional Transfer Entropy

partial_transfer_entropy

Partial Transfer Entropy

entropy_granger_causality

Granger Causality based on Differential Entropy

References

• wikipedia - https://en.wikipedia.org/wiki/Transfer_entropy

Examples

#sp.transfer_entropy
import numpy as np
import matplotlib.pyplot as plt
import spkit as sp
X, fs, ch_names = sp.data.eeg_sample_14ch()
X = X - X.mean(1)[:, None]
# Example 1
X1 = sp.signal_delayed_space(X[:,0].copy(),emb_dim=5,delay=2)
Y1 = sp.signal_delayed_space(X[:,2].copy(),emb_dim=5,delay=2)
Y2 = sp.add_noise(Y1,snr_db=0)
TE_x_y1 = sp.transfer_entropy(X1,Y1)
TE_x_y2 = sp.transfer_entropy(X1,Y2)
TE_y1_x = sp.transfer_entropy(Y1,X1)
TE_y2_x = sp.transfer_entropy(Y2,X1)
TE_y1_y2 = sp.transfer_entropy(Y1,Y2)
print('Transfer Entropy')
print(f'- TE(X1->Y1) = {TE_x_y1}')
print(f'- TE(X1->Y2) = {TE_x_y2}')
print(f'- TE(Y1->X1) = {TE_y1_x}')
print(f'- TE(Y2->X1) = {TE_y2_x}')
print(f'- TE(Y1->Y2) = {TE_y1_y2}')