Optimal experimental design for correlated errors and its expert system

在析因实验中，众所周知，在给定大小和约束的所有析因设计中，强度为2的正交表对主效应的估计是普遍最优的。在这个项目中，我们考虑了实验观测相互关联的情况。我们假设两个观测值之间的协方差只取决于相应组件的汉明距离。在这种协方差结构下，我们证明了如果一个强度为2的正交表的每一行向量的最小距离超过给定的距离d，那么在协方差结构下的每个正交表中的最小行向量距离是全局最优的，使得只有当两个组件的汉明距离小于d时才会发生相关。此外，还证明了如果正交表是线性的，则最大似然序列和最小似然序列的协方差矩阵是对角的，并且它们是重合的。利用这一结果，我们表明，对于某些协方差结构，最小距离大的线性正交表是所有线性正交表中最优的，专家系统的构建目前仍在进行中，因为在本项目期间，我们主要致力于析因设计的理论方面。

In factorial experiments, it is well-known that an orthogonal array of strength 2 is universally optimum for the estimation of main effects among the all factorial designs with given size and constraints. In this project, we consider the case when the observations of experiments are correlated. We assume that the covariance between two observation depends only on the Hamming distance of the corresponding assemblies. Under this covariance structure, we showed that if the minimum distance of each row vectors of an orthogonal array of strength 2 exceed a given distance d, then it is universally optimum for OLSE among every orthogonal arrays under the covariance structure such that correlation may occur only when the Hamming distance of two assembly is less than d. Furthermore, it was shown that if the orthogonal array is linear, then the covariance matrices of OLSE and GLSE are diagonal and they coincide. Using this result we showed that a linear orthogonal array with large minimum distance is optimum among all linear orthogonal array for some covariance structures,The construction of an expert system is now still preceeding, since in the period of this project we mainly devote ourselves to the theoretical aspects of factorial designs.