NLSA

This project provides a MATLAB implementation of nonlinear Laplacian spectral analysis (NLSA) and related kernel algorithms for feature extraction and prediction of observables of dynamical systems.

Usage

1. Clone down the project repository:

git clone https://github.com/dg227/NLSA

2. Launch MATLAB, cd into the project's directory, and add /nlsa to the MATLAB search path. This can be done by executing the MATLAB command:

addpath(genpath('nlsa'))

Examples

• Rectification of variable-speed periodic oscillator using Koopman eigenfunctions:

/examples/circle/demoKoopman.m

• Extraction of an approximately cyclical observable of the Lorenz 63 (L63) chaotic system using kernel integral operators with delays:

/examples/circle/demoNLSA.m

• Kernel analog forecasting of the L63 state vector components:

/examples/l63/demoKAF.m

Implementation

NLSA implements a MATLAB class nlsaModel which encodes the attributes of the machine learning procedure to be carried out. This includes:

• Specification of training and test data.
• Delay-coordinate embedding.
• Pairwise distance functions.
• Density estimation for variable-bandwidth kernels.
• Symmetric and non-symmetric Markov kernels.
• Koopman operators.
• Projection and reconstruction of target data.
• Nystrom out-of-sample extension.

Each of the elements above are implemented as MATLAB classes. See /nlsa/classes for further information and basic documentation.

Results from each stage of the computation are written on disk in a directory tree with (near-) unique names based on the nlsaModel parameters.

Known issues

• In Windows environments, errors can occur due to long file/directory names.

Acknowledgment

Research funded by the National Science Foundation (grants DMS-1521775, 1842538, DMS-1854383) and Office of Naval Research (grants N00014-14-1-0150, N00014-16-1-2649, N00014-19-1-2421).