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A method to integrate and classify normal distributions

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 where these...

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Detalles Bibliográficos
Autores principales: Das, Abhranil, Geisler, Wilson S.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: The Association for Research in Vision and Ophthalmology 2021
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8419883/
https://www.ncbi.nlm.nih.gov/pubmed/34468706
http://dx.doi.org/10.1167/jov.21.10.1
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author Das, Abhranil
Geisler, Wilson S.
author_facet Das, Abhranil
Geisler, Wilson S.
author_sort Das, Abhranil
collection PubMed
description 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 where these integrals are easy to calculate, 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 (a) the probability in any domain of a normal in any dimensions with any parameters; (b) the probability density, cumulative distribution, and inverse cumulative distribution of any function of a normal vector; (c) the classification errors among any number of normal distributions, the Bayes-optimal discriminability index, and relation to the receiver operating characteristic (ROC); (d) dimension reduction and visualizations for such problems; and (e) 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.
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spelling pubmed-84198832021-09-22 A method to integrate and classify normal distributions Das, Abhranil Geisler, Wilson S. J Vis Article 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 where these integrals are easy to calculate, 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 (a) the probability in any domain of a normal in any dimensions with any parameters; (b) the probability density, cumulative distribution, and inverse cumulative distribution of any function of a normal vector; (c) the classification errors among any number of normal distributions, the Bayes-optimal discriminability index, and relation to the receiver operating characteristic (ROC); (d) dimension reduction and visualizations for such problems; and (e) 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. The Association for Research in Vision and Ophthalmology 2021-09-01 /pmc/articles/PMC8419883/ /pubmed/34468706 http://dx.doi.org/10.1167/jov.21.10.1 Text en Copyright 2021 The Authors https://creativecommons.org/licenses/by/4.0/This work is licensed under a Creative Commons Attribution 4.0 International License.
spellingShingle Article
Das, Abhranil
Geisler, Wilson S.
A method to integrate and classify normal distributions
title A method to integrate and classify normal distributions
title_full A method to integrate and classify normal distributions
title_fullStr A method to integrate and classify normal distributions
title_full_unstemmed A method to integrate and classify normal distributions
title_short A method to integrate and classify normal distributions
title_sort method to integrate and classify normal distributions
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8419883/
https://www.ncbi.nlm.nih.gov/pubmed/34468706
http://dx.doi.org/10.1167/jov.21.10.1
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