freealg.distributions.DeformedMarchenkoPastur.matrix#

DeformedMarchenkoPastur.matrix(size, seed=None)#

Generate matrix with the spectral density of the distribution.

Parameters:
sizeint

Size \(n\) of the matrix.

seedint, default=None

Seed for random number generator.

Returns:
Anumpy.ndarray

A matrix of the size \(n \times n\).

Notes

Generate an \(n x n\) sample covariance matrix \(\mathbf{S}\) whose ESD converges to \(H \boxtimes MP_c\), where \(H = \sum_i w_i \delta_{t_i}\).

Finite \(n\) construction:

  • \(m\) is chosen so that \(n/m\) approx \(c\) (when \(c>0\)),

  • \(Z\) has i.i.d. \(N(0,1)\),

  • \(\boldsymbol{\Sigma}\) has eigenvalues \(t_i\) with proportions \(w_i\),

  • \(\mathbf{S} = (1/m) \boldsymbol{\Sigma}^{1/2} \mathbf{Z} \mathbf{Z}^T \boldsymbol{\Sigma}^{1/2}\).