Contributions to some statistical problems in advanced manufacturing

Part I deals with optimal design of experiments for modeling processes with feedback variables. Feedback control schemes have been widely used in many engineering applications for a long time. To design a feedback control system, one has to first develop a model for the relationships among the input variables, potential feedback variables, and the output variables and then select the appropriate variables for feedback control. This is typically done through statistically designed experiments where the input variables are systematically varied, and the experiment is run in an initial, open-loop mode without feedback control. In this part, we study the optimal design of experiments for modeling such a process. The paper considers a general statistical formulation of the problem and studies the properties of optimal designs in the linear case. Locally optimal designs under the D-optimality criteria are studied in detail, while results for A-optimality and Bayesian optimal designs are also developed. These results are used to characterize the potential loss in efficiency, under various situations, in using the classical optimal designs for this problem.; Part II deals with modeling and detecting spatial clusters with applications to semiconductor manufacturing. Yield modeling and prediction have been major topics of interest in the semiconductor manufacturing industry. Early yield models were based on the Poisson distributions which essentially assumed that the defects occurred in a spatially random manner. These models turned out to be inadequate as the spatial distribution of defects on the wafer map exhibit substantial spatial clustering. These spatial defect patterns contain considerable information for potential process problems and process improvement. So there is substantial value in developing methods to identify the key spatial patterns and using them for process improvement. We consider methods for modeling and detecting spatial patterns for different types of spatial data on lattices: binary data, “composite” binary data, count data and multicolored data. Our methodology is based on spatial mixture models. A Bayesian approach with priors given by suitable Markov random field models is described. The iterative conditional mode (ICM) algorithm is used to recover spatial patterns of defect clustering under these models. The properties of these algorithms are studied and the results are compared to other results in the literature. Estimation of some of the underlying parameters in the model using Gibbs sampling is also discussed.