Approximation of Bounds on Mixed Level Orthogonal Arrays

TL;DR

This paper introduces three algorithms to compute bounds on mixed level orthogonal arrays, addressing computational complexity issues through recursive, asymptotic, and simulation methods based on stochastic processes.

Contribution

It presents novel algorithms for efficiently approximating bounds on mixed level orthogonal arrays using stochastic control and large deviation techniques.

Findings

Recursive algorithm computes bounds exactly.

Asymptotic analysis provides approximate bounds for large parameters.

Simulation algorithm using importance sampling estimates bounds efficiently.

Abstract

Mixed level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao and Gilbert-Varshamov type bounds for mixed level orthogonal arrays. The computational complexity of the terms involved in these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provide the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm uses a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton Jacobi Bellman equation associated with the limit problem.