spkit.transfer_entropy

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

Transfer Entropy \(TE_{X->Y}\)

Transfer Entropy

\(TE_{X->Y} = I(Y_{i+1}, X_i | Y_i)\)

\(TE_{X->Y} = H(Y_{i+1} | Y_i) - H(Y_{i+1} | X_i, Y_i)\) [Eq1]

\(TE_{X->Y} = H(Y_{i+1}, Y_i) - H(Y_i) - H(Y_{i+1},X_i,Y_i) + H(X_i,Y_i)\)

\(TE_{X->Y} = H(X_i,Y_i) + H(Y_{i+1}, Y_i) - H(Y_{i+1},X_i,Y_i) - H(Y_i)\) [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

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}')