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

\[\mathbf{C}_t \mathbf{r}_i = \mathbf{C}_0 \mathbf{r}_i l_i \]

Computes \(m\) dominant eigenvalues \(l_i\) and eigenvectors \(\mathbf{r}_i\), where \(m\) is the numerical rank of the problem. This is done by first conducting a Schur decomposition of the symmetric positive matrix \(\mathbf{C}_0\), 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 \(\mathbf{C}_t\) of size \(m\), 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 \(W\) 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 \(C0^{-0.5}\), if return_rank is True.