FITTING A RECTANGULAR FUNCTION BY GAUSSIANS AND APPLICATION TO THE MULTIVARIATE NORMAL INTEGRALS

This article introduces a new scheme to express a rectangular function as a linear combination of Gaussian functions. The main idea of this scheme is based on fitting samples of the rectangular function by adapting the well-known clustering algorithm, Gaussian mixture models (GMM). This method has several advantages compared to other existing fitting algorithms. First, it incorporates an efficient algorithm that can fit more Gaussian functions. Second, weights of the linear combination are already constrained in the algorithm to lie in the interval [0,1], which avoids large/small values that cause numerical instability. Third, almost the entire fitted Gaussian functions lie within the interval of the rectangular function, which can be utilized efficiently to approximate difficult definite integrals such as the multivariate normal integral. Experiments show that it is efficient when low accuracy is required (error of order of 10^-4) especially for small values of the correlation coefficients.