Walsh functions, scrambled $(0,m,s)$-nets, and negative covariance: applying symbolic computation to quasi-Monte Carlo integration

TL;DR

This paper uses symbolic computation to analyze Walsh functions and scrambled nets, demonstrating conditions under which the quasi-Monte Carlo estimator has lower variance than Monte Carlo, with negative covariance results.

Contribution

It introduces a symbolic computation approach to analyze Walsh functions in scrambled nets, deriving covariance expressions and establishing negative covariance conditions.

Findings

Covariance of the estimator can be expressed via Walsh coefficients.

Negative covariance occurs under specific decay conditions of Walsh coefficients.

Symbolic algorithms help find closed-form expressions for covariance signs.

Abstract

We investigate base $b$ Walsh functions for which the variance of the integral estimator based on a scrambled $(0,m,s)$-net in base $b$ is less than or equal to that of the Monte-Carlo estimator based on the same number of points. First we compute the Walsh decomposition for the joint probability density function of two distinct points randomly chosen from a scrambled $(t,m,s)$-net in base $b$ in terms of certain counting numbers and simplify it in the special case $t$ is zero. Using this, we obtain an expression for the covariance of the integral estimator in terms of the Walsh coefficients of the function. Finally, we prove that the covariance of the integral estimator is negative when the Walsh coefficients of the function satisfy a certain decay condition. To do this, we use creative telescoping and recurrence solving algorithms from symbolic computation to find a sign equivalent closed form expression for the covariance term.