gx2 1.0.5

Statistics, pdf, cdf, inverse cdf and random numbers of the generalized chi-square distribution

gx2 — Generalized chi-square distribution

gx2 is a python package that computes the statistics, characteristic function, pdf, cdf, inverse cdf and random numbers of the generalized chi-square distribution. This is the python port of the MATLAB toolbox.

Author and citation

Abhranil Das, Center for Perceptual Systems, The University of Texas at Austin. Bugs / comments / questions / suggestions to [e-mail].

If you use this code, please cite:

• Methods to integrate multinormals and compute classification measures
• New methods to compute the generalized chi-square distribution: journal / arxiv

A generalized chi-square variable is a weighted sum of independent non-central chi-square variables plus a normal variable — equivalently, the quadratic form of a normal random vector. It is parametrized by:

parameter	meaning
w	weights of the non-central chi-square terms
k	their degrees of freedom
l	their non-centralities (named l because lambda is a Python keyword)
s	scale (standard deviation) of the added normal term
m	constant offset

Installation

pip install gx2

Requires numpy, scipy and mpmath. matplotlib is optional, for plotting in the getting-started notebook.

To install from a local clone instead:

pip install .
# or, for development (editable install with test/plot extras):
pip install -e ".[plot,test]"

Getting started

import gx2

w, k, l, s, m = [1, -5, 2], [1, 2, 3], [2, 3, 7], 0, 5

gx2.stat(w, k, l, s, m)          # mean and variance
gx2.cdf(25, w, k, l, s, m)       # cdf at x = 25
gx2.pdf(25, w, k, l, s, m)       # pdf at x = 25
gx2.inv(0.9, w, k, l, s, m)      # 90th percentile
gx2.rnd(w, k, l, s, m, size=5)   # random numbers

Open GettingStarted.ipynb for an interactive tour with worked examples and plots. For any function, see its full documentation with help(gx2.cdf) (or gx2.cdf? in Jupyter).

Public functions

function	purpose
stat(w, k, l, s, m)	mean and variance
char(t, w, k, l, s, m)	characteristic function
rnd(w, k, l, s, m, size=, method=)	random numbers
cdf(x, w, k, l, s, m, side=, method=, ...)	cdf
pdf(x, w, k, l, s, m, side=, method=, ...)	pdf
inv(p, w, k, l, s, m, side=, method=, ...)	inverse cdf
gx2_to_norm_quad_params(w, k, l, s, m)	gx2 → quadratic-form coefficients of a standard normal
norm_quad_to_gx2_params(mu, v, quad, merge=)	quadratic form of a normal → gx2 parameters

The individual computation routines (imhof, ruben, ifft, pearson, tail, ellipse, cdf_ray, pdf_ray, …) and numerical helpers (log_sum_exp, signed_log_sum_exp, phi_ray, …) are also exposed.

Computation methods for cdf / pdf

method='auto' (default) picks a good method for the given parameters. You can also force one:

method	notes
'imhof'	Imhof–Davies numerical integration (precision='basic' or 'vpa')
'ray'	ray-trace method (precision='basic', 'log' or 'vpa'; tune with n_rays, force_mc)
'ifft'	inverse-FFT method; x='full' returns the cdf/pdf over a spanning grid
'ruben'	Ruben's series — requires all w the same sign and s=0
'tail'	infinite-tail approximation
'pearson'	Pearson's 3-moment approximation
'ellipse'	ellipse approximation near a finite tail — requires all w the same sign and s=0

Usage notes

• cdf and pdf return just the probability/density by default. Pass full_output=True (auto-enabled for x='full') to also receive the error estimate and, for x='full', the grid of x-values.
• In the far tails, probabilities can fall below double precision (~1e-308). The 'tail', 'ellipse' and 'ray' methods then return the base-10 logarithm of such values (a negative number); inv likewise accepts a negative p as a log10 probability. precision='log' (the ray default) is the easiest way to reach this regime.
• The 'ray' method runs on the CPU with NumPy and batches automatically over rays to bound memory. precision='vpa' uses mpmath and returns mpmath.mpf objects for sub-realmin values.

License

MIT — see LICENSE.