Theory And Application Of Orthogonal Array With Mixing Level And Mixing Intensity

"Multilateral Matrix Theory", inspired by the Eastern holistic thinking, is trying to provide and improve a whole set of powerful mathematical tools to handle multi-target local issues, non-uniformity problems and nonlinear problems of complex system ranging from the whole to the part with rigorous theoretical analysis and proof. The definition of multilateral matrix bases on the operation methods for frame. Designing the frame is the base of applying Multilateral Matrix Theory.This paper is organized into three parts. In the first part, frame and the operation methods for it is introduced. First, frame is defined in two forms, the set form and the matrix form. Then, based on matrix theories, Cartesian product, Generalized Kronecker product, Generalized Hadamard product, bi-space frame and vector operation method are defined. Some examples are introduced to show the process and the property of results.Since generalized Kronecker product is the most useful one in the five operation methods showed above, then I will focus on study generalized Kronecker product. I discuss whether it meets three operation laws, introduce the relationship among normal Kronecker product, Kronecker sum. Some examples are provided to describe how to design Hadamard matrix. Latin square and difference matrix. And I extend the definition of Latin matrix and difference matrix. Then another extension definition of Kronecker product, strong Kronecker product is introduced.In the second part, orthogonal array, main effect and interaction are introduced. The intensity of common used orthogonal array in DOE is2. If the level is mixed and there is interaction between factors with different levels, it’s difficult to handle with common orthogonal array. To solve the problem, I use the definition of Transect and Snick to introduce the definition of orthogonal array with every intensity and orthogonal array with mixing level and mixing intensity. Then based on the conception of matrix image in matrix theory, the form of main effect and interaction in Multilateral Matrix Theory are introduced. By using them, the sum of deviation square can be easily calculated.In the third part, I use those tools to analyze the data. I use the model in which the2-level and3-level are mixed, and there is interaction between2-level factor and3-level factor. First, do ANOVA by the orthogonal array with mixing level and mixing intensity. Second, do ANOVA by the orthogonal array with pseudo level. Finally, compare the2methods with stochastic simulation. I find that the conclusion of their significance tests are similar, but the number of experiment times in orthogonal array with mixing level and mixing intensity is very smaller so it’s significant better than the other one.Finally, the SAS code of six operation methods for frame are provided in the appendix to spread for application.