7.3 Ridge Analysis of the Response Surface

Transcription

1 7.3 Ridge Analysis of the Response Surface When analyzing a fitted response surface, the researcher may find that the stationary point is outside of the experimental design region, but the researcher wants to locate the optimal response value (and its associated variable levels) within the boundaries of the region. Example: For a rising or falling ridge system, the researcher may want to calculate the absolute maximum or minimum value of the estimated response ŷ over the experimental region. The method of ridge analysis is a response surface method that enables the researcher to search for an optimal ŷ value in the experimental design region given that the canonical analysis does not provide a stationary point that is contained in the region or when the stationary point is a saddle point. Specifically, the method of ridge analysis is used to find the absolute maximum or minimum of ŷ on concentric spheres of varying radii, say R l (l = 1, 2,...) which are centered at the coded variable origin (x 1, x 2,..., x k ) = (0, 0,..., 0). For example, suppose there are two design variables, x 1 and x 2. In the following figure, three circles (l = 1, 2, 3) are drawn with different radii. The objective is to find the maximum value of ŷ on the circumference of each circle. Recall the fitted second-order response surface model in k variables is ŷ = b 0 + x b + x Bx Thus, our objective is to search for the coordinates of the variables x 1, x 2,..., x k on a sphere of radius R which maximize (or minimize) ŷ subject to the constraint k x 2 i = R 2 or equivalently x x R 2 = 0 i=1 We use the method of Lagrange multipliers to solve this problem. Let L = b 0 + x b + x Bx where µ is the Lagrange multiplier. Then we differentiate L with respect to the vector x and equate it to 0: This is equivalent to L x = Thus, for a fixed µ, a solution x to this equation is a constrained stationary point on the sphere of radius R = (x x) 1/2. 135

2 The problem is to choose µ which guarantees finding an absolute maximum or minimum ŷ in the experimental design region (and not a local optimum). Here are the following rules for selecting the value of µ: 1. To find an absolute maximum ŷ on a sphere of radius R, choose any value µ such that µ is 2. To find an absolute minimum ŷ on a sphere of radius R, choose any value µ such that µ is Warning: intermediate values of µ maximum or minimum. will direct us to a local and not an absolute Computer ridge analysis algorithms (such as RSREG in SAS) require the user to specify values of R (usually within the design region). The algorithm then converts these R values to µ values and then solves for x. The output of the ridge analysis is the set of coordinates of the maxima or minima along with the predicted response ŷ for each R. The results are displayed graphically by plotting the maximum and minimum ŷ against R and by plotting the optimum conditions against R. 136

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4 If you wanted to minimize response y, then choose some µ < (which is the smallest eigenvalue), and repeat the process to find (x 1, x 2 ) ŷ, and R associated with that µ. Then ŷ is the absolute minimum predicted response value for all points that are a distance R from (0, 0) Two Ridge Analysis Examples Ridge Analysis for Example 1 Data: We will perform a ridge analysis of the data from Example 1. Based on the canonical analysis, the stationary point is a maximum and occurs at (x 1, x 2, x 3 ) = (.601,.155,.405) This point is R = (.601) 2 + (.155) 2 + (.405) 2 =.741. The minimum predicted response occurs on the boundary R = 3 at (x 1, x 2, x 3 ) = (.307,.338, 1.619) Both of these points can be observed on the ridge analysis plots. SAS Code For Ridge Analysis for Example 1 DM LOG; CLEAR; OUT; CLEAR; ; % ODS PRINTER PDF file= C:\COURSES\ST578\SAS\ridge1.PDF ; OPTIONS NODATE NONUMBER LS=78 PS=54; ODS LISTING; DATA DSGN; INPUT X1 X2 X3 LINES; ; PROC RSREG DATA=DSGN PLOTS=RIDGE; *PLOTS=(DIAGNOSTICS RIDGE SURFACE); MODEL STRENGTH = X1 X2 X3 / LACKFIT; RIDGE MAX MIN RADIUS=0 TO 1 BY.05; TITLE Ridge Analysis for Example 1 ; RUN; 138

5 RIDGE ANALYSIS USING RSREG Response Surface for Variable STRENGTH Response Mean Root MSE R-Square Coefficient of Variation Sum of Residual DF Squares Mean Square F Value Pr > F Lack of Fit Pure Error Total Error Parameter Estimate Standard from Coded Parameter DF Estimate Error t Value Pr > t Data Intercept < X X X X1*X X2*X X2*X X3*X X3*X X3*X Canonical Analysis of Response Surface Based on Coded Data Critical Value Factor Coded Uncoded X X X Predicted value at stationary point: Eigenvectors Eigenvalues X1 X2 X Stationary point is a maximum. 139

6 Estimated Ridge of Minimum Response for Variable STRENGTH Estimated Standard Uncoded Factor Values Coded Radius Response Error X1 X2 X > ^^^^^^^^ Absolute minimum Estimated Ridge of Maximum Response for Variable STRENGTH Estimated Standard Uncoded Factor Values Coded Radius Response Error X1 X2 X > ^^^^^^^^ Absolute maximum 140

7 Ridge Analysis for Example 1 The RSREG Procedure Ridge of Minimum STRENGTH Estimated STRENGTH Coded Factor Levels Radius Ridge X1 X2 X3 141

8 Ridge Analysis for Example 1 The RSREG Procedure Ridge of Maximum STRENGTH Estimated STRENGTH Coded Factor Levels Radius Ridge X1 X2 X3 142

9 Ridge Analysis for Example 3 Data: We will perform a ridge analysis of the data from Example 3. Based on the canonical analysis, the stationary point is a saddle point in the design space. Therefore, the minimum and maximum predicted values will occur on the boundary. The maximum predicted response occurs on the boundary R = 3 at (x 1, x 2, x 3 ) = (.544, 1.589,.089) The minimum predicted response occurs on the boundary R = 3 at (x 1, x 2, x 3 ) = (1.356,.248,.964) Both of these points can be observed on the ridge analysis plots. SAS Code for Ridge Analysis of Example 3 Data DM LOG; CLEAR; OUT; CLEAR; ; ODS LISTING; % ODS PRINTER PDF file= C:\COURSES\ST578\sas\ridge2.pdf ; OPTIONS NODATE NONUMBER PS=54 LS=78; TITLE Ridge Analysis for Example 3 (Saddle Point Example) ; DATA DSGN; INPUT X1 X2 X3 LINES; ; PROC RSREG DATA=DSGN PLOTS = RIDGE; MODEL YIELD = X1 X2 X3 / LACKFIT ; RIDGE MAX MIN RADIUS=0 TO 1 BY.05; RUN; SAS Output Ridge Analysis for Example 3 (Saddle Point Example) Coding Coefficients for the Independent Variables Factor Subtracted off Divided by X X X

10 Degrees of Type I Sum Regression Freedom of Squares R-Square F-Ratio Prob > F Linear Quadratic Crossproduct Total Regress Degrees of Sum of Residual Freedom Squares Mean Square F-Ratio Prob > F Lack of Fit Pure Error Total Error Degrees of Parameter Standard T for H0: Parameter Freedom Estimate Error Parameter=0 Prob > T INTERCEPT X X X X1*X X2*X X2*X X3*X X3*X X3*X Canonical Analysis of Response Surface (based on coded data) Critical Value Factor Coded Uncoded X X X Predicted value at stationary point Eigenvectors Eigenvalues X1 X2 X Stationary point is a saddle point. Estimated Ridge of Minimum Response for Variable YIELD 144

11 Estimated Standard Uncoded Factor Values Coded Radius Response Error X1 X2 X Estimated Ridge of Maximum Response for Variable YIELD Estimated Standard Uncoded Factor Values Coded Radius Response Error X1 X2 X

12 Ridge Analysis for Example 3 (Saddle Point Example) The RSREG Procedure 13.0 Ridge of Maximum YIELD Estimated YIELD Coded Factor Levels Radius Ridge X1 X2 X3 146

13 Ridge Analysis for Example 3 (Saddle Point Example) The RSREG Procedure Ridge of Minimum YIELD Estimated YIELD Coded Factor Levels Radius Ridge X1 X2 X3 147