The Research And Applications Of Several Kinds Of Problems In Fractional Factorial Designs

This paper primarily studies several kinds of problems in fractional factorial designs. Based on the theory of matrix image, we have studied the alias measure in nonregular fractional factorial designs and the corresponding applications in global sensitivity analysis. Based on the theory of matrix image, the generalized indices of resolution have been established. We have given a frame work on the theory of generalized variable resolution designs and provide several constructions. We have completely discussed the existences and constructions of the four kinds of designs containing partially clear main effects. We have provided several constructions of main-effect plans orthogonal through the block factor. Furthermore, we have studied the optimal designs under the criterion of uniformity. The main contents are arranged as follows:Chapter 1 is devoted to introducing elementary concepts and vital theory on design of experiment. It includes the development of research on fractional factorial designs and designs of factor grouping. At the end of the first chapter, an outline of this paper about our major work is given, also encompassing innovative keys.Chapter 2 studies the matrix image theory and presents a new method for distinguishing and assessing nonregular designs with complex alias structure, which works for all symmetrical and asymmetrical, regular and nonregular orthogonal arrays. Empirical studies show that the proposed method has a more precise differentiation capacity when comparing with some other criteria. On the other hand, Chapter 2 considers the use of orthogonal arrays with strength lower than the required strength in global sensitivity analysis. Based on matrix image, we first generalize the alias matrix for ANOVA high-dimensional model representation and then by sequentially minimizing the squared alias degrees, we present a approach for the estimation of sensitivity indices. A two-level orthogonal array with 16 runs and a four-level orthogonal array with 64 runs are studied for estimating both low-order and high-order significant sensitivity indices. Moreover, models containing larger than 10 input factors are also investigated. All cases show that designs with smaller squared alias degree have less bias and variance for the estimations of global sensitivity indices.Chapter 3 deals with the concept of generalized variable resolution designs D(n, (m1, m2), (τ1,τ2)τ3,τ) with nonnegligible interactions between groups. The conditions for the existence of generalized variable resolution designs are discussed. Connections between different generalized variable resolution designs and compromise plans, clear compromise plans and designs containing partially clear two-factor interactions are explored. Finally, a general construction method for the proposed designs is also discussed.Chapter 4 shows that generalized variable resolution designs are A-optimal for estimating the parameters in ANOVA high dimension model representation. Both cases including interac-tions or not are considered in the model. The properties are also illustrated through a simulation study.Chapter 5 is concerned with the theory of designs containing partially clear main effects. Clear effects criterion is an important rule for selecting designs. All main effects in a strength 3 orthogonal array are clear. If certain knowledge is available, such as certain two-factor in-teractions are assumed to be negligible, then additional factors can be studied through designs containing partially clear main effects. All possible types of them are discussed in this paper. Three types of them are found less useful comparing with strength 3 orthogonal arrays, and only the fourth type can overcome this drawback. We study the existence and construction of such designs.Chapter 6 gives several constructions of main-effect plans orthogonal through the block factor (POTB), in which the treatment factors are pairwise orthogonal through the block factor. However, less construction methods are available in the literature, we present several new construction approaches for saturated POTBs with small runs and mixed levels. Moreover, all of them are connected and variance-balanced.Chapter 7 demonstrates that the orthogonal property of main effects plans orthogonal through the block factor remains unchanged under level permutation. However, level permu-tation of factors could alter their geometrical structures and statistical properties. Hence uni-formity is used to further distinguish main effects plans orthogonal through the block factor (POTB). A modified optimization algorithm is proposed to search uniform or nearly uniform POTBs and many new optimal POTBs with lower-discrepancy are obtained.