On Numerical Estimation of Joint Probability Distribution from Lebesgue Integral Quadratures

TL;DR

This paper develops a method to estimate joint probability distributions of random processes using Lebesgue integral quadratures and eigenvalue problems, introducing density-matrix correlation and extending to multiple processes.

Contribution

It introduces a novel approach to joint distribution estimation via Lebesgue quadratures and eigenproblems, including new correlation measures and extension to multiple processes.

Findings

Successfully formulated joint distribution estimators from eigenproblems.

Introduced density-matrix correlation and probability-correlation concepts.

Extended method to estimate joint probabilities of three or more processes.

Abstract

An important application of Lebesgue integral quadrature arXiv:1807.06007 is developed. Given two random processes, $f(x)$ and $g(x)$, two generalized eigenvalue problems can be formulated and solved. In addition to obtaining two Lebesgue quadratures (for $f$ and $g$) from two eigenproblems, the projections of $f$- and $g$- eigenvectors on each other allow to build a joint distribution estimator, the most general form of which is a density-matrix correlation. Examples of the density-matrix correlation can be a value-correlation $V_{f^{[i]};g^{[j]}}$, similar to a regular correlation concept, and a new one, a probability-correlation $P_{f^{[i]};g^{[j]}}$. If Christoffel function average is used instead of regular average the approach can be extended to an estimation of joint probability of three and more random processes. The theory is implemented numerically; the software is available under the GPLv3 license.