Dimensional Analysis for Response Surface Methodology
Open Access
- Author:
- Yang, Ching Chi
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 07, 2019
- Committee Members:
- Dennis Kon-Jin Lin, Dissertation Advisor/Co-Advisor
Dennis Kon-Jin Lin, Committee Chair/Co-Chair
Ephraim Mont Hanks, Committee Member
Le Bao, Committee Member
Fuqing Zhang, Outside Member - Keywords:
- Dimension reduction
Function approximation
Optimization
Variable selection - Abstract:
- Dimensional analysis is a widely-employed methodology in physics and engineering. Its advantages include, but not limited to: (i) the essential information extraction, (ii) the interpretability of the variables, and (iii) invariant to units of the system. Dimensional analysis transforms a dimensionally homogeneous model into its simplest form; the number of input variables can be reduced. Such a reduction is helpful in statistical analysis. The Buckingham pi theorem provides a method of transforming the model based on the basis quantities and the dimensionless variables. However, the choice of the basis quantities is not unique. In practice, different choices of the basis quantities may fluctuate performances in statistical analysis. The newly proposed criterion can be used to select the optimal basis quantities regarding minimal bias. Under normality, deriving the distribution of the proposed criterion helps in distinguishing the difference among the choices of the basis quantities. Response surface methodology is a well-developed methodology in chemical engineering and industrial processes. The objective of response surface methodology is to obtain the inputs' values such that the response is optimal. The choice of every experiment requires a thoughtful and carefully plan. The experiments are usually conducted sequentially; the later experiments can be planned delicately based on the knowledge gained from previous results. To the best of our knowledge, this dissertation proposes the first general methodology of utilizing dimensional analysis for response surface methodology. Response surface methodology can gain many benefits via dimensional analysis including: (1) the optimal design in response surface methodology based on dimensional analysis is efficient, (2) the surrogate model build under the concept of dimensional analysis can adequately represent the underlying model, and (3) by the proposed methodology, the optimal response can be obtained via a smaller number of experiments. Besides, the model based on the dimensionless variables is free from constraints in physics. The main difficulty of utilizing dimensional analysis in response surface methodology is that dimensional analysis might transform the original response as well as the input variables. The inputs' values for the optimal response might not be directly obtained. Besides, the projection from the original inputs to the transformed inputs is not one-to-one. Once the optimal set of the transformed inputs' values is obtained, the optimal set of the original inputs' values is not unique. It is proved that the optimal response obtained by the proposed method is still the optimal response in the original space. Because response surface methodology mainly consists of the three steps experimental design, model building, and optimization -- the proposed strategies can be adapted to other statistical methods correspondingly. Although the objective of other statistical methods might be different from the response surface methodology (the performance of the proposed strategy should be carefully studied), the proposed methods could still be an essential foundation. Relevant case studies are provided within each Chapter. Along with the study of dimensional analysis in response surface methodology, the lead time with order crossover study is also in our attention. Lead time plays an essential role in many areas, including supply chain, economics, and marketing. A conventional assumption in most stochastic lead-time inventory models is that the lead times are independent and identically distributed (i.i.d.). However, it can be shown that applying such an assumption on practical lead time may not be valid in case of order crossover. An order crossover occurred when a later order received earlier, which becomes a common phenomenon in many business applications. Based on different system setup, the joint distribution of the practical lead times is different. We proposed a general procedure which reveals the joint distribution. An exponential distributed lead-time case study is used to demonstrate the use of the proposed method and the risk of mis-use i.i.d. practical lead times. Other phenomena can be studied by similar derivation. Note that dimensional analysis can also be used to quantify the similarity among different system setup. Many statistical areas might gain benefits from dimensional analysis. The concepts of dimensional analysis also can help in analyzing the similarity of objects. The similarities in dimensional analysis are not limited to the geometric similarity. Dimensional analysis can also make a connection between two different domains. After transforming the response by dimensional analysis, the connected domains will yield similar properties of the transformed response. Instead of fitting statistical models directly to the data, dimensional analysis provides a critical way to analyze the variables and to extract the dimensionless variables. By extracting the dimensionless variables from the original variables in statistical methodology, the strategies can work more efficiently and better approximate the underlying model. How to utilize dimensional analysis for the models which involve the variables with the non-physical dimension is still an essential topic. Besides the models in physics and engineering, many other models exist to explain the real-world behaviors, such as economic models and social sciences models. If dimensional analysis can be utilized, the concepts in dimensional analysis can help in studying the quantities based on the unit of measurements. It leads researchers and practitioners to reconsider the quantities used in the model. The models will be invariant of the measurement system.