2.1.1. Mutual Information (MI) for time delay (tau).
uses mutual information to find a suitable delay via the location of the first minima in the mutual information vs delay plot, which is calculated using multiple x(t) vs x(t+tau) plots. These plots have their individual mutual information calculated. Various methods for partitioning the x(t) vs x(t+tau) plots for calculating the mutual information.
- teaspoon.parameter_selection.MI_delay.MI_DV(x, y)[source]
This function calculates the mutual information between the time series x(t) and its delayed version x(t+tau) using adaptive partitioning of the plots of the time series x(t) and its delayed version x(t+tau). This method was developed by Georges Darbellay and Igor Vajda in 1999 and was published as Estimation of information by an adaptive partitioning of the observation.
- Parameters:
x (array) – time series x(t)
y (array) – delayed time series x(t + tau)
- Returns:
I, mutual information between x(t) and x(t+tau).
- Return type:
(float)
- teaspoon.parameter_selection.MI_delay.MI_basic(x, y, h_method='sturge', ranking=True)[source]
This function calculates the mutual information between the time series x(t) and its delayed version x(t+tau) using equi-spaced partitions. The size of the partition is based on the desired bin size method commonly selected for histograms.
- Parameters:
x (array) – time series x(t)
y (array) – delayed time series x(t + tau)
- Kwargs:
h_method (string): bin size selection method. Methods are struge, sqrt, or rice. Default is sturge.
ranking (bool): whether the ranked or unranked x and y inputs will be used. Default is ranking = True.
- Returns:
I, mutual information between x(t) and x(t+tau).
- Return type:
(float)
- teaspoon.parameter_selection.MI_delay.MI_for_delay(ts, plotting=False, method='basic', h_method='sturge', k=2, ranking=True)[source]
This function calculates the mutual information until a first minima is reached, which is estimated as a sufficient embedding dimension for permutation entropy.
- Parameters:
ts (array) – Time series (1d).
- Kwargs:
plotting (bool): Plotting for user interpretation. defaut is False.
method (string): Method for calculating MI. Options include basic, kraskov 1, kraskov 2, or adaptive partitions. Default is basic.
h_method (string): Bin size selection method for basic method. Methods are struge, sqrt, or rice. Default is sturge.
ranking (bool): Whether the ranked or unranked x and y inputs will be used for kraskov and basic methods. Default is ranking = True.
k (int): Number of nearest neighbors used in MI estimation for Kraskov methods. Default is k = 2.
- Returns:
tau, The embedding delay for permutation formation based on first mutual information minima.
- Return type:
(int)
- teaspoon.parameter_selection.MI_delay.MI_kraskov(x, y, k=2, ranking=True)[source]
This function estimates the mutual information between the time series x(t) and its delayed version x(t+tau) in two different ways. This method was developed by Alexander Kraskov, Harald Stoegbauer, and Peter Grassberger in 2003 and published as Estimating Mutual Information.
- Parameters:
x (array) – time series x(t)
y (array) – delayed time series x(t + tau)
- Kwargs:
k (int): number of nearest neighbors used in MI estimation. Default is k = 2.
ranking (bool): whether the ranked or unranked x and y inputs will be used. Default is ranking = True.
- Returns:
I1, first mutual information estimation method between x(t) and x(t+tau). (float): I2, second mutual information estimation method between x(t) and x(t+tau).
- Return type:
(float)
The following is an example implementing the MI method for selecting tau:
from teaspoon.parameter_selection.MI_delay import MI_for_delay
import numpy as np
fs = 10
t = np.linspace(0, 100, fs*100)
ts = np.sin(t) + np.sin((1/np.pi)*t)
tau = MI_for_delay(ts, plotting = True, method = 'basic', h_method = 'sturge', k = 2, ranking = True)
print('Delay from MI: ',tau)
Where the output for this example is:
Delay from MI: 23