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 \(\mathrm{Beta}(a, b)\) 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 [1]). These equations are derived from moment matching method, where

\[\mathrm{Mean}(\mathrm{Beta}(a,b)) = u \mathrm{Var}(\mathrm{Beta}(a,b)) = (1-u) u h\]

Solving these two equations for \(a\) and \(b\) yields the relations above. See [2] (page 134).

References

[1]

R-package documentation for Beta kernel

[2]

Chen, S. X. (1999). Beta Kernel estimators for density functions. Computational Statistics and Data Analysis 31 p. 131–145.