freealg.kde

freealg.kde(eig, xs, lam_m, lam_p, h, kernel='beta', plot=False)

Kernel density estimation of eigenvalues.

Parameters:

eignumpy.array

1D array of samples of size n.

xsnumpy.array

1D array of evaluation grid (must lie within [lam_m, lam_p])

lam_mfloat

Lower end of the support endpoints with lam_m < lam_p.

lam_pfloat

Upper end of the support endpoints with lam_m < lam_p.

hfloat

Kernel bandwidth in rescaled units where 0 < h < 1.

kernel{'gaussian', 'beta'}, default= 'beta'

Kernel function using either Gaussian or Beta distribution.

plotbool, default=False

If True, the KDE is plotted.

Returns:

pdfnumpy.ndarray

Probability distribution function with the same length as xs.

See also

freealg.supp

freealg.sample

Notes

In Beta kernel density estimation, the shape parameters “a” and “b” of the Beta(a, b)) distribution are computed for each data point “u” as:

a = (u / h) + 1.0 b = ((1.0 - u) / h) + 1.0

This is a standard way of using Beta kernel (see R-package documentation: https://search.r-project.org/CRAN/refmans/DELTD/html/Beta.html

These equations are derived from “moment matching” method, where

Mean(Beta(a,b)) = u Var(Beta(a,b)) = (1-u) u h

Solving these two equations for “a” and “b” yields the relations above. See paper (page 134) https://www.songxichen.com/Uploads/Files/Publication/Chen-CSD-99.pdf