A method to integrate and classify normal distributions
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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 exists no general analytical expression, standard numerical method or software for these integrals. Here we present mathematical results and software that provide (i) the probability in any domain of a normal in any dimensions with any parameters, (ii) the probability density, distribution, and percentage points of any function of a normal vector, (iii) quantities, such as the error matrix and discriminability, which summarize classification performance amongst any number of normal distributions, (iv) dimension reduction and visualizations for all such problems, and (v) tests for how reliably these methods can be used on given data. We illustrate these tools with models for detecting occluding targets in natural scenes and for detecting camouflage.
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