Conjugate Unscented Transform rules for uniform probability density functions

Abstract

This paper presents a few novel quadrature rules to evaluate expectation integrals with respect to a uniform probability density function. In 1-dimensional expectation integrals the most widely used numerical method is the Gauss-Legendre quadratures as they are exact for polynomials. As for a generic N-dimensional integral, the tensor product of 1-dimensional Gauss-Legendre quadratures results in an undesirable exponential growth of the number of points. The cubature rules proposed in this paper can be used as a direct alternative to the Gauss-Legendre quadrature rules as they are also designed to exactly evaluate the integrals of polynomials but use only a small fraction of the number of points. In addition, they also have all positive weights.