MULTIVARIATE PROBABILITY DENSITY ESTIMATION USING PIECEWISE AFFINE FUNCTIONS
Samudio, Gabriel M.
Royset, Johannes O.
Koyak, Robert A.
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Continuous multivariate data is ubiquitous in U.S. Military Operations Research. Since continuous distributions are fully characterized by their probability density functions, we concentrate on estimating such functions. Current estimation methods perform well for low dimensions; however, they can be too restrictive to capture the actual data characteristics, and can become intractable beyond three dimensions. This work develops a new estimation technique that seeks to increase flexibility and mitigate the curse of dimensionality. We achieve both by modeling the actual density using piecewise affine functions; however, we impose a nonconvex maximum likelihood optimization problem. The problem includes nine parameters, which can each affect the resulting estimate likelihood value and computation time. We conduct case studies on estimating the density for data up to five dimensions on sample sizes as low as 100 points. The results indicate progress in moderating the nonconvexity challenge to optimize the likelihood, and demonstrate potential advantages over currently used methods.
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