Construction And Application Of Three-level Triple Designs

Experimental design is one of the most important branches of statistics, it enables an investigator to conduct better experiments and efficiently analyze data. With the development of science and technology, a experiment includes more and more factors, and the level of each factor are large. In this case, the number of runs in a full factorial designs greatly exceed the researcher's tolerance. For reasons of economic run size, fractional factorial design is widely used, which is a part of the full factorial design. The two-level and the three-level fractional factorial designs are the most widely used in practice. One of the most important tasks in experimental design is to find "Good" designs from many factorial designs, such that obtaining the most useful information by design with the least run size. As for the "Good" designs, various optimality criteria have been proposed from different angels for design comparison. Among them, resolution criterion and minimum generalized aberration criterion are the most popular ones.A method called doubling has been recently used in constructing two-level frac-tional factorial designs, in particular, in constructing those of large designs, doubling is a simple and very powerful method. In constructing design of resolution IV, dou-bling is also very useful. Recently, the properties and applications of double design have been discussed in many literatures. On the other hand, But double design is only suitable for two-level design, so we consider the possibility of extending the concept of double design, such that the construction of three-level or higher-level large designs becomes simple and effective. Moreover, foldover is a special kind of level permutation which have been widely applied in constructing design, hence the concept of double design is extended based on level permutation with more choices. However, after exploring, the columns in the initial design and the resulted columns are not orthogonal if only the number of row is double in any case of the five level permutation methods, that is, the constructed design is not good. Finally, after careful study we found that if the number of row is triple may avoided this weak-ness. Therefore, the concept of double design is extended based on level permutation three-level triple design.Uniform designs and uniform theory firstly presented by the scholars of our country are a brand-new fractional factorial design theory and methods. They have been widely used in these domains, such as the computer simulation experiment, na-tional defense, agriculture, industry, medicine and high-new technological innovation and so on. In addition, they have generated notable economical and social benefits. Discrepancy is a measure of uniform designs, the common used discrepancies are the centered L2-discrepancy, the wrap-around L2-discrepancy, the symmetric L2-discrcpancy, the discrete discrepancy and the Lee discrepancy and so on.Based on the above discussions, the dissertation is devoted to the following researches:(1) The form and construction of triple designs T(A) and structure of three-level triple designs are discussed. Based on level permutation, some properties of T(A) are discussed by indicator function.(2) The uniformity of T(A) under the wrap-around L2-discrepancy are dis-cussed:an analytic relationship between the wrap-around L2-discrepancy of T(A) and the aberration of design A, and a tight lower bound of the wrap-around L2-discrepancy of T(A) is obtained by using Taylor formula.(3) The application of T(A) and its projection design in constructing minimum aberration design are discussed.In the following, let us introduce the contents of each chapter in brief.Some related background of experimental design, the innovation and structure of this dissertation are summarized in Chapter 1.Some basic concepts, formulas that will be used in the other chapters are briefly introduced in Chapter 2.The structure of three-level triple designs T(A) based on level per-mutation and analyses some excellent properties of T(A) by indicator function are discussed in chapter 3. The resolution of T(A) is ? if the reso-lution of A is ?. Furthermore, the resolution of several kinds of projection design of T(A) is ? or ? if the resolution of A is ? or ?.The aberration and uniformity measured by the wrap-around L2-discrepancy of T(A) are discussed in chapter 4. An analytic relationship be-tween the aberration of A and uniformity of T(A) is established, the wrap-around L2-discrepancy and aberration of T(A) is more less if design A has smaller aber-ration. Finally, a tight lower bound of the wrap-around L2-discrepancy of T(A) is obtained, which is convenient to search uniform design.Some applications of T(A) and it's projection design are discussed in chapter 5. For several selected initial designs A, T(A) and it's projection design have good uniformity measured by the wrap-around L2-discrepancy, and some constructed designs are even better than some uniform designs in the UD home page, and all of them have minimum aberration.The dissertation and prospects the future work are summarized in chapter 6.