Manifold Learning module (tmrc.manifold_learning)

Manifold learning methods for parametrizing the transition manifold.

tmrc.manifold_learning.L2distance(system, cloud1, cloud2, rho, epsi)

Compute 1/rho-weighted L2 distance between two densities (given by finite samples) using Kernel density estimation. Currently implemented only for two-dimensional systems.

Parameters:
  • system (System object) – required for boundaries of the state space
  • cloud1 (np.array of shape [# points, system dimension]) – contains samples from first density
  • cloud2 (np.array of shape [# points, system dimension]) – contains samples from second density
  • rho (function(double,double)) – invariant density of the system
  • epsi (float) – bandwidth parameter for the kernel density estimation
Returns:

distance between the two densities
Return type:

float
tmrc.manifold_learning.diffusionMaps(distMat, n_components=10, epsi=1.0, alpha=0.5)

Solve diffusion map eigenproblem

Parameters:
  • distMat (symmetric array of shape (n_features, n_features)) – distance matrix
  • n_components (int) – number of eigenvalues and eigenvectors to compute NOTE: n_components must be set at least to the number of desired dimensions in the diffusion space to which the data is projected by calling evaluateDiffusionMaps() later on
  • epsi (float) – diffusion map kernel bandwidth parameter
  • alpha (float) – diffusion map normalization parameter
Returns:

the first n_components eigenpairs of the diffusion maps eigenproblem as returned by diffusionMaps()
Return type:

tuple of (eigenvalues, eigenvectors)
tmrc.manifold_learning.evaluateDiffusionMaps(eigs, n_components)

Transform data to diffusion map space

Parameters:
  • eigs (tuple of (eigenvalues, eigenvectors)) – eigenpairs of the diffusion maps eigenproblem as returned by diffusionMaps(), where eigenvalues is an array of shape (n_features,) and eigenvectors is an array of shape (n_features, n_eigenvectors)
  • n_components (int) – number of dimensions in diffusion map space retained by dimensionality reduction NOTE: n_components must be smaller or equal to n_eigenvectors
Returns:

the transformed data in diffusion space
Return type:

array of shape (n_features, n_components)