New methods to compute the generalized chi-square distribution

Abhranil Das

stat.COApr 7, 2024

New methods to compute the generalized chi-square distribution

Authors: Abhranil Das

Organizations: Center for Perceptual Systems, and Center for Theoretical and Computational Neuroscience, The University of Texas at Austin

Abstract

We present four new mathematical methods, two exact and two approximate, along with open-source software, to compute the cdf, pdf and inverse cdf of the generalized chi-square distribution. Some methods are geared for speed, while others are designed to be accurate far into the tails, using which we can also measure large values of the discriminability index $d'$ between multivariate normal distributions. We compare the accuracy and speed of these and previous methods, characterize their advantages and limitations, and identify the best methods to use in different cases.

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Dec 23, 2020 · stat.MLJ/K move · Enter open · S save

Methods to integrate multinormals and compute classification measures

Abhranil Das, Wilson S Geisler

Department of Physics, The University of Texas at Austin · Center for Perceptual Systems, The University of Texas at Austin · Center for Theoretical and Computational Neuroscience, The University of Texas at Austin · Department of Psychology, The University of Texas at Austin

Univariate and multivariate normal probability distributions are widely used when modeling decisions under uncertainty. Computing the performance of such models requires integrating these distributions over specific domains, which can vary widely across models. Besides some special cases, there exist no general analytical expressions, standard numerical methods or software for these integrals. Here we present mathematical results and open-source software that provide (i) the probability in any domain of a normal in any dimensions with any parameters, (ii) the probability density, cumulative distribution, and inverse cumulative distribution of any function of a normal vector, (iii) the classification errors among any number of normal distributions, the Bayes-optimal discriminability index and relation to the operating characteristic, (iv) ways to scale the discriminability of two distributions, (v) dimension reduction and visualizations for such problems, and (vi) tests for how reliably these methods may be used on given data. We demonstrate these tools with vision research applications of detecting occluding objects in natural scenes, and detecting camouflage.

Dimensionality Reduction Camouflage

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Abstract

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