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W10INSE6220 - 1 3 Linear regression MATLAB example INSE...

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1 INSE 6220 -- Week 10 Advanced Statistical Approaches to Quality Experimental Design Design of Experiments in Process Improvement Factorial Experiments Factorial Design Dr. A. Ben Hamza Concordia University Factor A Factor B 2 Linear Regression and Analysis of Variance 2 2 2 1 1 1 ˆ ˆ ( ) ( ) ( ) n n n i i i i i i i T R E y y y y y y SS SS SS The total sum of squares of the observed y values is a measure of the total variability in the response: And the sample correlation coefficient is given by: Analysis of Variance for Testing Significance of Regression Source of Variation Sum of Squares Degrees of Freedom Mean Square Regression Error or residual Total 1 ˆ xy xx T xx yy S S r SS S S 1 / - 2 1 R R R E E E T SS MS MS MS SS n MS SS n 0 F 2 2 1 1 ˆ , ( ) , n xy R xy T yy i E T R i xx S SS S SS S y y SS SS SS S 3 Linear regression: MATLAB example We want to check if there is a linear relationship between y and x x = [0.1750 0.2200 0.2250 0.2260 0.2500 0.2765]'; y = [0.0480 0.0525 0.0540 0.0535 0.0570 0.0610]'; plot( x , y ,'o'); grid on; xlabel( ' x ' ); ylabel( ' y ' ); Now we want to use the regress function to estimate the model parameters and perform regression diagnostics. The regress function requires that the matrix of independent variables includes a column of ones so that the model contains a constant term, so we create a matrix X as: X = [ones(size(x)) x]; b = regress(y,X); betahat0 = b(1); betahat1 = b(2); yhat = betahat0 + betahat1*x; hold on; plot(x,yhat,'r'); 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.046 0.048 0.05 0.052 0.054 0.056 0.058 0.06 0.062 x y 0 1 ˆ ˆ ˆ i i y x Red curve: 4 Introduction to Design of Experiments An experiment is a test or a series of tests Experiments are used widely in the engineering world Process characterization & optimization Evaluation of material properties Product design & development Component & system tolerance determination “All experiments are designed experiments, some are poorly designed, some are well-designed” Reduce time to design/develop new products & processes Improve performance of existing processes Improve reliability and performance of products Achieve product & process robustness Evaluation of materials, design alternatives, setting component & system tolerances, etc.
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5 Experimental Design Objectives of an experiment may include: 1) Determining which variables are most influential on response y . 2) Determining where to set influential x ’s so that y is near nominal requirement. (which level?) 3) Determining where to set influential x ’s so that variability in y is small. 4) Determining where to set influential x ’s so that effects of uncontrollable variables z are minimized. 6 Factorial Experiments The effect of a factor is defined as the change in response produced by a change in the level of the factor. This is called a main effect because it refers to the primary factors in the study. For example, consider the data in Fig. 12-5. In this factorial design, both the factors A and B have two levels, denotes by “ - ” and “ + ”. These two levels are called “low”
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