freealg.distributions.FreeLevy#

class freealg.distributions.FreeLevy(t, w, lam, a=0.0, sigma=0.0)#

Free Levy distribution

Parameters:

tarray_like: Array \([t_1, \dots, t_r]\), where \(t_i\) is the jump location \(\delta_{t_i}\) in the discrete distribution \(H\) (see notes below).
warray_like: Array \([w_1, \dots, w_r]\), where \(w_i\) is the jump weights of \(\delta_{t_i}\) in the discrete distribution \(H\) (see notes below). The sum of all weights must be one: \(\sum_{i=1}^r w_i = 1\).
lamfloat: Total rate (intensity) \(\lambda > 0\).
afloat, default=0.0: Shift (drift) parameter \(a\) in \(\delta_a\) (see notes below.)
sigmafloat, default=0.0: Semicircle standard deviation \(\sigma\) (variance is \(\sigma^2\)).

See also

freealg.distributions.CompoundFreePoisson

Notes

General Free Levy Law:

This class extends the compound free Poisson law (see freealg.distributions.CompoundFreePoisson) by adding a drift and semicircle (free Gaussian) part.

This model is constructed by the free additive convolution of the following three laws (see Theorem 13.8 in [1]):

\[\mu = \mu_{\delta_a} \boxplus \mu_{\mathrm{SC}_{\sigma^2}} \boxplus \mu_{\mathrm{CFP}(\lambda, H)},\]

where

\(\delta_a\) is the shift \(x \mapsto x - a\) (given by the parameter a);
\(\mathrm{SC}_{\sigma^2}\) is the semicircle (free Gaussian) law with variance \(\sigma^2\) (give by the parameter sigma);
\(\mathrm{CFP}(\lambda, H)\) is the compound free Poisson law with the rate \(\lambda\) and jump \(H\), given by the R transform

\[R_{\mathrm{CFP}(\lambda, H)} = \lambda \int_{\mathbf{R}} \frac{x}{1 - wz} H(\mathrm{d} x).\]

Here, \(H > 0\) is a positive measure. The free Levy distribution represents all free infinitely divisible (FID) distributions.

Algebraic Free Levy Law:

In this class, we assume \(H\) is discrete atomic distribution given by

\[H = \sum_{i=1}^r w_i \delta_{t_i},\]

where \(t_i>0\) are jump sizes (given by the parameter t as a list), and \(w_i>0\) are weights with \(\sum_i w_i = 1\) (given by the parameter w as a list). This assumption on \(H\) restricts the free Levy class to algebraic FID distributions where its Stieltjes transform satisfies a polynomial constraint (see below).

The R transform the atomic free Levy law is the sum of all constituent laws

\[R = R_{\delta_a} + R_{\mathrm{SC}_{\sigma^2}} + R_{\mathrm{CFP}(\lambda, H)},\]

which for atomic \(H\) becomes

\[R(w) = a + \sigma^2 w + \lambda \sum_{i=1}^r w_i \frac{t_i}{1-t_i w}.\]

Stieltjes Transform:

The Stieltjes transform \(m(z)\) satisfies

\[R(-m(z)) = z + \frac{1}{m(z)}.\]

Solving for \(m\) and clearing the common denominators for the rational function representation, we get a polynomial \(P(z, m) = 0\) with

\[P(z, m) = \sum_{i=1}^{d_z} \sum_{j=1}^{d_m} c_{ij} z^i m^j = 0,\]

where \(d_z\) and \(d_m\) are the degrees of \(P\) in \(z\) and \(m\), respectively. The degree \(d_z\) is always 1.

For \(r\) atoms in \(H\) and \(\sigma = 0\), this yields a polynomial with degree \(d_m = r+1\) in \(m\). When \(\sigma > 0\), the polynomial is of degree \(d_m = r+2\).

When \(\sigma=0\), this model has an atom at \(x = a\) with mass \(\max(1-\lambda, 0)\).

Functions:

The coefficients \(c_{ij}\) can be obtained from poly() function.
For a given \(z\), all \(d_m\) roots (in \(m\)) of the polynomial \(P(z, m) = 0\) can be computed by roots().
Among all roots, only one root corresponds to the physical branch, known as the Stieltjes transform. This physical root can be computed by stieltjes() function.

See examples below.

References

[1]

Alexandru Nica and Roland Speicher (2006). Lectures on the Combinatorics of Free Probability. Cambridge University Press, LMS.

Examples

Here we create a distribution and plot is density and compute its support.

>>> from freealg.distributions import FreeLevy

>>> # Create an object of the class
>>> fl = FreeLevy(t=[2.0, 5.5], w=[0.75, 1-0.75], lam=0.1, a=0,
...               sigma=0.9)

>>> # Plot density
>>> rho = fl.density(plot=True)

>>> # Get the support intervals
>>> supp = fl.support()
[(-1.9990360090022503, 3.7017366841710433),
 (4.170457614403601, 7.886856714178547)]

Here we sample from this distribution: either as a matrix realization, or as an array of eigenavalues:

>>> # Generate a random matrix realization of this law
>>> A = fl.matrix(size=2000, seed=0)

>>> # Sample from eigenvalues of this law
>>> eig = fl.sample(size=2000)

Here, we compute the coefficents the polynomial \(P(z, m) = 0\), all its roots at a given point \(z\), and its physical root (Stieltjes transform):

>>> # Get the coefficients of the polynomial P(z, m) = 0
>>> coeffs = fl.poly().real
array([[ 1.    ,  7.2125, 10.71  ,  6.075 ,  8.91  ],
       [ 0.    ,  1.    ,  7.5   , 11.    , -0.    ]])

>>> # Compute all roots of the polynomial at a given z
>>> z = 2.0 + 3.0j
>>> roots = fl.roots(z)
array([-2.34243764-3.91061653e+00j, -0.50248052-2.41746562e-02j,
       -0.12025009+2.34338362e-01j, -0.18578573-3.25087974e-03j])

>>> # Compute the Stieltjes transform at z (the physical root)
>>> m = fl.stieltjes(z)
array(-0.12025009+0.23433836j)

Methods

`density`([x, eta, max_iter, tol, ac_only, ...])	Density of distribution.
`roots`(z)	Roots of polynomial implicitly representing Stieltjes transform
`stieltjes`(z[, max_iter, tol])	Stieltjes transform
`support`([eta, n_probe, thr, x_max, x_pad])	Support intervals 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.
`poly`()	Polynomial coefficients implicitly representing the Stieltjes