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Approximation of bounds on mixed-level orthogonal arrays

Part of: Hamilton-Jacobi theories, including dynamic programming Design of experiments Stochastic systems and control Designs and configurations Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 July 2016

Ali Devin Sezer  and
Ferruh Özbudak
Ali Devin Sezer*
Affiliation:
Middle East Technical University
Ferruh Özbudak*
Affiliation:
Middle East Technical University
*
∗ Postal address: Institute of Applied Mathematics, Middle East Technical University, Eskisehir Yolu, Ankara 06531, Turkey.
∗ Postal address: Institute of Applied Mathematics, Middle East Technical University, Eskisehir Yolu, Ankara 06531, Turkey.
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Abstract

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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 the original combinatorial representations of 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 provides 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 use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

Keywords

Mixed-level orthogonal arrayRao boundGilbert-Varshamov bounderror block codecountingimportance samplinglarge deviationoptimal controlasymptotic analysissubsolution approachHamilton-Jacobi-Bellman equation

MSC classification

Primary: 05B15: Orthogonal arrays, Latin squares, Room squares 62K99: None of the above, but in this section 65C05: Monte Carlo methods
Secondary: 93E20: Optimal stochastic control 49L99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 399 - 421
Copyright
Copyright © Applied Probability Trust 2011 

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