freealg.distributions.MarchenkoPastur#

class freealg.distributions.MarchenkoPastur(lam)#

Marchenko-Pastur distribution.

Parameters:
lamfloat

Parameter \(\lambda\) of the distribution. See Notes.

Notes

The Marchenko-Pastur distribution has the absolutely-continuous density

\[\mathrm{d} \rho(x) = \frac{1}{2 \pi} \frac{\sqrt{(\lambda_{+} - x) (x - \lambda_{-})}}{\lambda x} \mathbf{1}_{x \in [\lambda_{-}, \lambda_{+}]} \mathrm{d}{x}\]

where

  • \(\lambda_{\pm} = (1 \pm \sqrt{\lambda})^2\) are the edges of the support.

  • \(\lambda > 0\) is the shape parameter of the density.

References

[1]

Marcenko, V. A., Pastur, L. A. (1967). Distribution of eigenvalues for some sets of random matrices. Mathematics of the USSR-Sbornik, 1(4), 457

Examples

>>> from freealg.distributions import MarchenkoPastur
>>> mp = MarchenkoPastur()

Methods

density([x, plot, latex, save, eig])

Density of distribution.

hilbert([x, plot, latex, save])

Hilbert transform of the distribution.

stieltjes([x, y, plot, on_disk, latex, save])

Stieltjes transform of distribution.

sample(size[, x_min, x_max, method, seed, ...])

Sample from distribution.

matrix(size[, seed])

Generate matrix with the spectral density of the distribution.