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Experimental Design and Robust Regression Rochester Institute of Technology RIT Scholar Works Theses 12-15-2017 Experimental Design and Robust Regression Pranay Kumar [email protected] Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Kumar, Pranay, "Experimental Design and Robust Regression" (2017). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Rochester Institute of Technology Experimental Design and Robust Regression A Thesis Submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering in the Department of Industrial & Systems Engineering Kate Gleason College of Engineering by Pranay Kumar December 15, 2017 DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING KATE GLEASON COLLEGE OF ENGINEERING ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NEW YORK CERTIFICATE OF APPROVAL M.S. DEGREE THESIS The M.S. Degree Thesis of Pranay Kumar has been examined and approved by the thesis committee as satisfactory for the thesis requirement for the Master of Science degree Approved by: ____________________________________ Dr. Rachel Silvestrini, Thesis Advisor ____________________________________ Dr. Brian Thorn Experimental Design and Robust Regression Abstract: Design of Experiments (DOE) is a very powerful statistical methodology, especially when used with linear regression analysis. The use of ordinary least squares (OLS) estimation of linear regression parameters requires the errors to have a normal distribution. However, there are numerous situations when the error distribution is non- normal and using OLS can result in inaccurate parameter estimates. Robust regression is a useful and effective way to estimate the parameters of a regression model in the presence of non-normally distributed residuals. An extensive literature review suggests that there are limited studies comparing the performance of different robust estimators in conjunction with different experimental design sizes, models, and error distributions. The research in this thesis is an attempt to bridge this gap. The performance of the popular robust estimators is compared over different experimental design sizes, models, and error distributions and the results are presented and discussed. The results evaluating the performance of the robust estimator with OLS after performing Box-Cox transformation are also presented in this research. Table of Contents 1. Introduction 1 2. Background and Current Gap 4 3. Robust Regression 10 3.1 LAV Estimator 12 3.2 Least Median of Squares 12 3.3 M-estimator 13 3.4 MM-estimator 14 3.5 Summary of Robust Estimator 16 4. Methodology 17 5. Results 23 5.1 Comparing the performance of Robust Regression on Random versus Factorial Experimental Design 23 5.2 Comparing the performance of OLS and Robust Regression 26 5.3 Preliminary analysis of Robust Estimators 30 5.4 Detailed Comparison of the best three Robust Estimators 38 5.5 Comparing MM-estimator (Tukey’s Biweight) and OLS after transformation 48 6. Conclusion and Future Scope 50 References 51 Appendix I 54 List of Figures Figure 1: Fitted regression using OLS on data with and without outlier 6 Figure 2: Residual plots to check if the OLS residual assumptions are being met 7 Figure 3: Fitted regression using OLS and Robust Regression methods 10 Figure 4: Flowchart of the thesis research Methodology 18 Figure 5: Histogram of hundred randomly generated values having a log-normal distribution with a standard deviation of 1 20 Figure 6: Histogram of hundred randomly generated values having an exponential distribution with a standard deviation of 1 21 ̂ Figure 7: Histogram comparing the 훽1 values obtained using OLS (left) and Robust Regression (right) 25 ̂ Figure 8: Histogram comparing the 훽2 values obtained using OLS (left) and Robust Regression (right) 26 Figure 9: Comparison of performance between OLS and Robust Regression (M.H.) for parameter estimate β1 for main effects model, log-normal error distribution, = 1 28 Figure 10: Comparison of performance between OLS and Robust Regression (M.H) for parameter estimate β2 for main effects model, log-normal error distribution, = 1 28 Figure 11: Comparison of standard error values obtained by OLS and Robust Regression for parameter estimate β2 for main effects model, log-normal error distribution, = 1 29 Figure 12: Comparison of performance between MH, MT, and MMT for parameter estimate β1 for main effects model, log-normal error distribution, = 1 32 Figure 13: Comparison of performance between MH, MT, and MMT for parameter estimate β2 for main effects model, log-normal error distribution, = 1 33 Figure 14: Comparison of standard error values obtained by MH, MT, and MMT for parameter estimate β1 for main effects model, log-normal error distribution, = 1 34 Figure 15: Comparison of performance between MH, MT, and MMT for parameter estimate β1 for main effects model, exponential error distribution, = 1 36 Figure 16: Comparison of performance between MH, MT, and MMT for parameter estimate β2 for main effects model, exponential error distribution, = 1 37 Figure 17: Comparison of standard error values obtained by MH, MT, and MTT for parameter estimate β1 for main effects model, exponential error distribution, = 1 38 Figure 18: Comparison of performance between MH, MT, and MMT for parameter estimate β1 for main effects model with two factor interactions, log-normal error distribution, = 1 41 Figure 19: Comparison of performance between MH, MT, and MMT for parameter estimate β2 for the main effects model with two factor interactions, log-normal error distribution, = 1 41 Figure 20: Comparison of standard error values obtained by MH, MT, and MTT for parameter estimate β1 for main effects model with two factor interactions, log-normal error distribution, = 1 42 Figure 21: Comparison of performance between MH, MT, and MMT for parameter estimate β1 for the main effects model with two factor interactions, exponential error distribution, = 1 44 Figure 22: Comparison of performance between MH, MT, and MMT for parameter estimate β2 for the main effects model with two factor interactions, exponential error distribution, = 1 44 Figure 23: Comparison of standard error values obtained by MH, MT, and MTT for parameter estimate β1 for main effects model with two factor interactions, exponential error distribution and = 1 45 Figure 24: Scores of MM-estimator (Tukey’s), M-estimator (Tukey’s), and MM-estimator (Huber’s Function) 47 Figure 25: Scores of MM-estimator (Tukey’s), M-estimator (Tukey’s), and MM-estimator (Huber’s Function) over different error distributions 48 Figure 26: Comparison of RSE values between OLS and Robust Estimator 49 List of Tables Table 1: Summary of Robust Estimators 16 Table 2: Factorial Design (-1, 1) 19 Table 3: Possible unique conditions over which all the robust estimators are compared 22 Table 4: Preliminary performance comparison between the ordinary least square (OLS) and Robust Regression (RR) 24 Table 5. Comparing the performance of Robust Regression (M-estimator (Huber)) and OLS on Full Factorial Design and Random Design. Condition: Main effects model, log-normal error distribution, σ = 1 27 Table 6. Comparing the performance of the five-robust estimator over five experimental design sizes. Condition: Main effects model, log-normal error distribution, σ = 1 30 Table 7. Comparing the performance of the five-robust estimator over five experimental design sizes. Condition: Main effects model, exponential error distribution, σ = 1 34 Table 8. Comparing the performance of the M-estimator (Huber), M-estimator (Tukey’s) and MM-estimator (Tukey’s) over five experimental design sizes. Condition: Main effects model with two factor interactions, log-normal error distribution, σ = 1 40 Table 9. Comparing the performance of the M-estimator (Huber), M-estimator (Tukey’s) and MM-estimator (Tukey’s) over five experimental design sizes. Condition: Main effects model with two factor interactions, exponential error distribution, σ = 1 43 Table 10. Scoring out of 60 obtained by M-estimator (Huber Function), M-estimator (Tukey’s) and MM-estimator (Tukey’s) 46 1. Introduction Design of Experiments (DOE) (or experimental design) is a very powerful statistical methodology. It has vast applications ranging from agriculture (Hoshmand, 2006) to chemical industry (Lazic, 2006) to aerospace industry (Khanna and Davim, 2015). Additionally, DOE is widely used in quality management and in Lean Six Sigma (Thomas et al., 2008). In many real-world applications, there are numerous factors that affect a response and it is not always possible or feasible to test every combination of factors to select the one which gives the desired response. Experimentation takes time, resource, and money, and this often limits the number of tests that can be performed. In situations such as these DOE is invaluable. DOE can be used to systematically determine which factors might have a significant effect on the response and which factor settings provide the desired response without carrying out exhaustive experimentation and trying all possible factors and scenarios. In cases where experimentation is extremely expensive and/or time- consuming, such as equipment testing for military systems, finding a small and suitable set of tests is critical. The factorial and fractional factorial
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