混合水平正交表上界的逼近
Approximation of Bounds on Mixed Level Orthogonal Arrays
摘要：
混合水平正交表是试验设计的基本结构。我们开发了三个算法，计算Rao和Gilbert-Varshamov型边界的混合水平正交表。随着阵列参数的增加，这些边界中所涉及的项的计算复杂性会快速增长，这证明了这些算法的构造是合理的。第一个是一个递归算法，精确计算的界限，第二个是基于渐近分析，第三个是一个模拟算法。它们都是基于代表性的组合表达式，出现在边界的期望，涉及一个对称的随机游动。基础随机游动的马尔可夫性给出了计算期望值的递归公式。期望值的大偏差（LD）分析提供了渐近算法。相同期望的渐近最优重要抽样（IS）提供了模拟算法。LD分析和IS算法的构造都使用这些问题的表示作为收敛到极限变分问题的随机最优控制问题的序列。建设的IS算法使用最近发现的方法，使用子解的Hamilton Jacobi Bellman方程与极限问题。
英文摘要：
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.
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