function eig_corr

deeptime.numeric.eig_corr(C0, Ct, epsilon=1e-10, method='QR', canonical_signs=False, return_rank=False)

Solve generalized eigenvalue problem with correlation matrices C0 and Ct

Numerically robust solution of a generalized Hermitian (symmetric) eigenvalue problem of the form

Ctri=C0rili\mathbf{C}_t \mathbf{r}_i = \mathbf{C}_0 \mathbf{r}_i l_i

Computes mm dominant eigenvalues lil_i and eigenvectors ri\mathbf{r}_i, where mm is the numerical rank of the problem. This is done by first conducting a Schur decomposition of the symmetric positive matrix C0\mathbf{C}_0, then truncating its spectrum to retain only eigenvalues that are numerically greater than zero, then using this decomposition to define an ordinary eigenvalue Problem for Ct\mathbf{C}_t of size mm, and then solving this eigenvalue problem.

Parameters:

  • C0 (ndarray (n,n)) – time-instantaneous correlation matrix. Must be symmetric positive definite

  • Ct (ndarray (n,n)) – time-lagged correlation matrix. Must be symmetric

  • epsilon (float) – eigenvalue norm cutoff. Eigenvalues of C0 with norms <= epsilon will be cut off. The remaining number of Eigenvalues define the size of the output.

  • method (str) –

    Method to perform the decomposition of WW before inverting. Options are:

    • ’QR’: QR-based robust eigenvalue decomposition of W

    • ’schur’: Schur decomposition of W

  • canonical_signs (bool) – If True, re-scale each eigenvector such that its entry with maximal absolute value is positive.

  • return_rank (bool, default=False) – If True, return the rank of generalized eigenvalue problem.

Returns:

  • l (ndarray (m)) – The first m generalized eigenvalues, sorted by descending norm

  • R (ndarray (n,m)) – The first m generalized eigenvectors, as a column matrix.

  • rank (int) – Rank of C0−0.5C0^{-0.5}, if return_rank is True.