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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...
Autores principales: | , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
The Association for Research in Vision and Ophthalmology
2021
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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. |
format | Online Article Text |
id | pubmed-8419883 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | The Association for Research in Vision and Ophthalmology |
record_format | MEDLINE/PubMed |
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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