Eigen Manifold Cross Mapping (EMCM) Toolbox

EMCM is a MATLAB toolbox for generating high-dimensional manifolds from time-series data using Eigen-Time-Delay Embedding and measuring their shared dynamics using Cross Mapping techniques.

Overview

1. Perform Eigen Time-Delay (ETD) embedding of a time series to generate high-dimensional manifolds (Brunton, 2017)
2. Use the eigenvalue statistics to assay complexity and capture meaningful dimensions of activity.
3. Infer shared dynamics between ETD Manifolds Using Sugihara's Convergent Cross Mapping (Sugihara, 2012).

Download

1. In the command line:

git clone https://github.com/ParhamP/EMCM.git

2. Or simply "Download Zip" from Github's "Code" tab

Install

In MATLAB:

addpath(genpath('AbsolutePathToToolbox'))

Example

The MATLAB file for this example can be found in "examples/eeg_example/" folder.

% Load two EEG time-series data
alpha_eeg = load('data/alpha_eeg.mat').first_eeg;
gamma_eeg = load('data/gamma_eeg.mat').second_eeg;


%create an EMCM object
emcm = EMCM();

alpha_number = 1;
gamma_number = 2;

% Assign the time-series to the object
emcm.set_timeseries(alpha_number, alpha_eeg, 'name', 'Alpha');
emcm.set_timeseries(gamma_number, gamma_eeg, 'name', 'Gamma');

% Visualize the time-series from within the object
emcm.visualize_time_series();

% Set the delay embedding parameters
delay_lag = 1;
embedding_dimension = 60;

% First, generate attractors from time-series data using traditional time-delay-embedding
emcm.generate_single_attractor(alpha_number, delay_lag, embedding_dimension);
emcm.generate_single_attractor(gamma_number, delay_lag, embedding_dimension);

% Visualize the attractors in 3D
emcm.visualize_attractors_3d();

% Apply PCA to get Eigen-Time_Delay Attractors.
% The threshold method determines the number of components to keep.
% The threhold method of 'one' would retain all the components with eigenvalue greater than 1. 

emcm.apply_pca_and_set_r(alpha_number, 'threshold_method', 'one');
emcm.apply_pca_and_set_r(gamma_number, 'threshold_method', 'one');


% Visualize the attractors in 3D
emcm.visualize_attractors_3d();

% Access the complexity measures for eigen attractors (i.e., their number of retained components)
[alpha_complexity, gamma_complexity] = emcm.complexities('barPlot', true);

% Calculate Convergent Cross Mapping (CCM) scores to infer shared dynamics between the two manifolds.
% The 'weigh_by_eigens' option will weigh each dimension's correlation by its eigenvalue.
[alpha_precits_gamma_score, gamma_predicts_alpha_score] = emcm.ccm('weigh_by_eigens', true, 'barPlot', true);

Collaborators

• Parham Pourdavood
• Michael Jacob, MD, PhD

References

1. Brunton, S. L., Brunton, B. W., Proctor, J. L., Kaiser, E., & Nathan Kutz, J. (2017). Chaos as an intermittently forced linear system. Nature Communications, 8(1).
2. Sugihara, G., May, R., Ye, H., Hsieh, C. H., Deyle, E., Fogarty, M., & Munch, S. (2012). Detecting causality in complex ecosystems. Science, 338(6106), 496–500.

Acknowledgement

This work was supported by grants from the Department of Veterans Affairs VA (IK1CX002089 and IK2CX002457 to MSJ)