[Paper Review] Methods to integrate multinormals and compute classification measures

Abhranil Das, Wilson S. Geisler|arXiv (Cornell University)|Dec 23, 2020

Bayesian Methods and Mixture Models21 references50 citations

TL;DR

The paper develops mathematical methods and open-source software to integrate multivariate normal distributions over arbitrary domains and to compute classification measures for Bayes-optimal and other classifiers.

ABSTRACT

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.

Motivation & Objective

• Motivate the need to compute performance of models under uncertainty when using normal distributions.
• Provide general mathematical frameworks to integrate normals over arbitrary domains and to handle functions of normal vectors.
• Develop algorithms and software to compute probability, pdf, cdf, and inverse cdf for normals in complex domains and for their quadratic forms.
• Derive Bayes-optimal decision rules for classification among multiple normal distributions and relate to discriminability measures like d'.

Proposed method

• Generalized chi-square method to integrate multinormals over quadratic domains and derive distributions of quadratic forms.
• Ray-trace method to integrate normals over any domain by standardizing to spherical coordinates and tracing boundary crossings along rays.
• Computation of pdf, cdf, and inverse cdf for any function of a normal vector via function-probability mapping.
• Derivation of Bayes-optimal classification rules for two or more normal classes and expression of error rates via generalized chi-square distributions.
• Dimension reduction and visualization techniques for high-dimensional normal integrations.
• Software implementation as a Matlab toolbox ‘Integrate and classify normal distributions’ with source at GitHub.

Experimental results

Research questions

• RQ1 How can normals with arbitrary means and covariances be integrated over arbitrary domains in any dimension?
• RQ2 How can we compute the distribution (pdf, cdf, inverse cdf) of any function of a normal vector?
• RQ3 How can we quantify classification performance when discriminating among any number of normal distributions under a given prior and cost structure?
• RQ4 What are efficient methods to reduce dimensionality and visualize high-dimensional normal integration problems?
• RQ5 How accurate and fast are the generalized chi-square and ray-trace methods across representative problems?

Key findings

• The paper provides methods to compute probabilities of normal vectors in arbitrary quadratic domains and more general domains.
• The generalized chi-square method yields a 1D integral for the probability, enabling practical computation of complex quadratics.
• The ray-trace method enables integration over any domain by tracing radial crosses and summing probabilities across rays.
• Bayes-optimal discrimination boundaries for two normals are derived as a quadratic form, enabling computation of error rates.
• The authors demonstrate accuracy and speed of both methods across multiple applied problems in vision science, including object occlusion and camouflage detection.
• The work provides open-source Matlab toolbox and GitHub source code for practical use.