7 The Analysis of Response Surfaces

Transcription

1 7 The Analysis of Response Surfaces Goal: the researcher is seeking the experimental conditions which are most desirable, that is, determine optimum design variable levels. Once the researcher has determined the experimental design region by selecting ranges of the experimental variables, an experimental design is chosen and the experimental data is collected. The coefficients in the model are then estimated. We are considering the fitted second-order model or the fitted response surface: ŷ = b + k b i x i i= + k b ij x i x j + i<j k b ii x 2 i (9) i= where the b s are the estimated coefficients. Eq (9) is used to predict the response y for given x, x 2,..., x k and enables the researcher to conduct an analysis of the fitted response surface. In matrix notation Eq (9) is given by where x = x x 2 x k b = Note: B is a symmetric matrix. ŷ = () b b b 2 /2 b k /2 b 2 b 22 b 2k /2 B = b b k,k /2 k b kk Suppose the goal of the experimenter is to estimate the conditions on x, x 2,..., x k which maximize the response y. Then, the maximum, if it exists, will be a set of conditions on (x, x 2,..., x k ) such that the partial derivatives are simultaneously zero. ŷ/ x, ŷ/ x 2,..., ŷ/ x k This point, say x s = [x,, x 2,,..., x k, ], is called the stationary point of the fitted response surface. We apply the rules for differentiating with respect to a column vector and then equate the result to zero: ŷ = ] [b + x b + x Bx = x x Solving for x yields the stationary point x s = This point may or may not be the point which maximizes the response. The point falls into one of three categories:. x s is the point at which the response surface attains a maximum. 2. x s is the point at which the response surface attains a minimum. 3. x s is a saddle point. That is, the response could either increase or decrease as you move away from x s (depending on what direction you move). 4

2 7. Canonical Analysis The goal is to determine the nature of the stationary point and the entire response surface. The nature of the stationary point is determined by the signs of the eigenvalues of B. The type of response surface is determined by examining the magnitudes of the eigenvalues. Let P be the k k matrix whose columns are the normalized eigenvectors associated with the eigenvalues λ, λ 2,..., λ k of B. Then where Λ = Diag(λ, λ 2,..., λ k ) is a diagonal matrix of the eigenvalues of B. To understand a response surface, it is necessary to translate the matrix form in () into another form which is easier to interpret. The analysis begins with a translation of response function in (9) from the origin (x, x 2,..., x k ) = (,,..., ) to a new origin at the stationary point x s. Let z = (z, z 2,..., z k ) be the vector of re-centered x coordinates: z = x x s (Shift) Now we want to rotate the axes to align with the contours of the response surface. We do this by use of the eigenvalues λ, λ 2,..., λ k and the matrix of eigenvectors P. The response function is expressed in terms of new variables w, w 2,..., w k which correspond to the principal axes. That is, rotate the original axes about x, x 2,..., x k by transforming z to w = ( w, w 2,..., w k ) via the matrix of normalized eigenvectors: w = P z (Rotation) 5

3 In terms of predicted values, this translation yields ŷ = b + x b + x Bx where ŷ s is the predicted response at the stationary point x s. Because PP = I, we see P = P, and because w = P w, we have z = Pw. Therefore, ŷ = ŷ s + z Bz The w-axes are the principal axes of the contour system. The variables w, w 2,..., w k are called the canonical variables. The canonical form of the response surface is obtained by rewriting Eq () as: () ŷ = ŷ s + k λ i wi 2 (2) i= If λ i >, then ŷ will increase as you move along w i in either direction from x s. If λ i <, then ŷ will decrease as you move along w i in either direction from x s. The analysis after reduction of the response surface to canonical form is called a canonical analysis. 6

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5 7.2 Interpretation of the Response Surface The canonical form in (2) describes the nature of the stationary point and the nature of the response surface about the stationary point:. If λ, λ 2,..., λ k are all negative, then the stationary point is a point of. k Note that if all λ i < then λ i wi 2 < for any nonzero w. Thus, if we move away from x s in any direction, ŷ decreases. i= 2. If λ, λ 2,..., λ k are all positive, then the stationary point is a point of. k Note that if all λ i > then λ i wi 2 > for any nonzero w. Thus, if we move away from x s in any direction, ŷ increases. i= 3. If λ, λ 2,..., λ k are mixed in sign, then the stationary point is a. Note k that if the λ i are mixed in sign then λ i wi 2 can be positive or negative. Thus, an increase or decrease in ŷ depends upon the direction traveled away from x s. We now consider three other types of response surfaces: i=. The stationary ridge system indicates that there is a region, rather than a single stationary point, where there is approximately a maximum or minimum response. Figure a shows a maximum stationary ridge along w and Figure b shows a minimum stationary ridge along w. 8

6 2. The rising ridge system has its stationary point remote from the experimental region and the estimated response increases as you move along the axis toward the stationary point. See the dotted line in the Figure 2a. 3. The falling ridge system has its stationary point remote from the experimental region and the estimated response decreases as you move along the axis toward the stationary point. See the dotted line in the Figure 2b. To determine whether a stationary, rising, or falling ridge describes the response surface in the experimental region, we need to simultaneously study the location of the stationary point and the magnitude of the eigenvalues. The magnitude of the eigenvalues provides information about the shape of the response surface. This is best exemplified in response surfaces involving k = 2 variables (and, hence, two eigenvalues): Case I: Suppose λ and λ 2 are both negative and λ 2 is considerably greater than λ. See Figure 3a. The stationary point is a maximum but note the sensitivity of ŷ with respect to the two canonical variables w and w 2. As you move along the w axis away from the stationary point in either direction, there is less of a change (decrease) in the response relative to the change (decrease) that occurs when moving a similar distance along the w 2 axis away from the stationary point in either direction. Case II: Suppose λ and λ 2 are both positive and λ 2 is considerably greater than λ. See Figure 3b. The stationary point is a minimum but note the sensitivity of ŷ with respect to the two canonical variables w and w 2. As you move along the w axis away from the stationary point in either direction, there is less of a change (increase) in the response relative to the change (increase) that occurs when moving a similar distance along the w 2 axis away from the stationary point in either direction. In other words, for Cases I & II, the response surface is elongated in the w direction and is insensitive to small changes in movement along the w axis. 9

7 Case III: Suppose λ and λ 2 have different signs such that λ < and λ 2 >, and λ 2 is considerably greater than λ. See Figure 4a. The stationary point is a saddle point but note the sensitivity of ŷ with respect to the two canonical variables w and w 2. As you move along the w axis away from the stationary point in either direction, there is less of a change (decrease) in the response relative to the change (increase) that occurs when moving a similar distance along the w 2 axis away from the stationary point in either direction. Case IV: Suppose λ and λ 2 have different signs such that λ <, and λ 2 > and λ 2 is considerably smaller than λ. See Figure 4b. The stationary point is a saddle point but note the sensitivity of ŷ with respect to the two canonical variables w and w 2. As you move along the w 2 axis away from the stationary point in either direction, there is less of a change (increase) in the response relative to the change (decrease) that occurs when moving a similar distance along the w axis away from the stationary point in either direction. In general, for any response surface with k eigenvalues, if λ j is considerably greater than λ i for two eigenvalues λ i and λ j, the response surface contours are elongated along w i and are narrower along w j. In general, for any response surface with k eigenvalues, if λ i λ j for two eigenvalues λ i and λ j, the response surface contours are nearly evenly spaced along w i and w j. That is, change in ŷ along w change in ŷ along w 2 when moving similar distances along the axes away from the stationary point. Near-Stationary Ridges The most extreme case with two variables is the stationary ridge. An exact stationary ridge occurs when one of the eigenvalues is zero. For the stationary ridges shown in Figures a and b, we have λ =. Consider the stationary ridge systems in Figures A and B. If λ, then ŷ = ŷ s + λ w 2 + λ 2 w 2 2 ŷ s + λ w 2 2. Therefore, changing w will have little effect on the predicted value ŷ. In practice, if one of the eigenvalues is very small, then the response surface approximates a stationary ridge. 2

8 Mathematically, the stationary ridge condition arises as a limiting eigenvalue case from a maximum, minimum, or saddle point condition. Thus, it is unlikely that an exact stationary ridge condition (some λ i = ) would occur in practice. The practical implications of a near-stationary ridge (some λ i researcher. ) are important to the For example, if the stationary point is a maximum and λ is near zero, then for all practical purposes the stationary point does not clearly define optimum operating conditions. In such circumstances, the researcher has a range of potential operating conditions along the w axis, all of which yield a near-maximum response. Approximate Rising and Falling Ridges Suppose that for k = 2 variables that the stationary point is not near the experimental region and that λ < and λ 2 is near zero. The response surface approaches a rising ridge condition. See Figure 5a. The boxed area is the experimental design region. Figure 5a Suppose that for k = 2 variables that the stationary point is not near the experimental region and that λ > and λ 2 is near zero. The response surface approaches a falling ridge condition. See Figure 5b. The boxed area is the experimental design region. Figure 5b 2

9 7.2. Three Examples of Data Analysis EXAMPLE : Two types of fertilizers, a standard N-P-K combination and the other a nutritional supplement, were applied to experimental plots to assess the effects on the yield of peanuts measured in pounds per plot. The level of the amount (lb/plot) of each fertilizer applied to a plot was determined by the coordinate settings of a central composite rotatable design. The data below represent the harvested yields of peanuts taken from two replicates of each experimental combination. (Note: the uncoded fertilizer levels corresponding to coded 2 levels are rounded to decimal place.) Fertilizer Fertilizer Coded Coded Yields (lb/plot) 2 x x 2 Rep Rep DM LOG; CLEAR; OUT; CLEAR; ; ODS PRINTER PDF file= C:\COURSES\ST578\SAS\canon.pdf ; ODS LISTING; OPTIONS LS=72 NODATE NONUMBER; **********************; ***** EXAMPLE # *****; **********************; DATA IN; INPUT F F2 LINES; ; PROC RSREG DATA=IN PLOTS=ALL ; MODEL YIELD = F F2 / LACKFIT ; TITLE EXAMPLE -- USING UNCODED DATA IN PROC RSREG ; RUN; 22

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11 EXAMPLE -- USING UNCODED DATA IN PROC RSREG..5 Fit Diagnostics for YIELD RStudent - RStudent Predicted Value Predicted Value Leverage - YIELD 2 5 Cook's D Quantile Predicted Value 5 5 Observation 3 Fit Mean Percent Observations Parameters Error DF MSE R-Square Adj R-Square Proportion Less 24

12 . Plots for YIELD EXAMPLE -- USING UNCODED DATA IN PROC RSREG.5 Plots for YIELD F F F F2 25

13 EXAMPLE 2: Data taken from Response Surface Methodology by Raymond Myers, pages In this experiment, the researcher wants to gain an insight into the influence of sealing temperature (x ), cooling bar temperature (x 2 ), and % polyethylene additive (x 3 ) on the seal strength in grams per inch of a breadwrapper stock. The actual levels of the variables are coded as follows: x = seal temp x 2 = cooling temp 55 9 On the uncoded scale, five levels of each variable were used: x 3 = % polyethylene..6 The design variables and responses are: x x x x x 2 x 3 y x x 2 x 3 y From the canonical analysis, the stationary point is x s = (x, x 2, x 3 ) = (.,.26,.68). The uncoded levels are ξ = (3)(.) = Seal Temperature ξ 2 = 55 + (9)(.26) = 57.3 Cool Temperature ξ 3 =. + (.6)(.68) =.5 Polyethelene Additive SAS Code for Example 2 ******************************************************; *** EXAMPLE # FACTOR CCD -- MYERS PP ***; ******************************************************; DATA DSGN; INPUT X X2 X3 LINES; ; PROC RSREG DATA=DSGN PLOTS=(ALL SURFACE(AT(X= ))); MODEL STRENGTH = X X2 X3 / LACKFIT ; TITLE EXAMPLE 2 -- USING CODED DATA IN PROC RSREG ; RUN; 26

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15 EXAMPLE 2 -- USING CODED DATA IN PROC RSREG Fit Diagnostics for STRENGTH RStudent - RStudent Predicted Value Predicted Value Leverage 2 - STRENGTH Cook's D Quantile Predicted Value Observation 4 Fit Mean Percent Observations Parameters Error DF MSE R-Square Adj R-Square Proportion Less 28

16 EXAMPLE 2 -- USING CODED DATA IN PROC RSREG EXAMPLE 2 -- USING CODED DATA IN PROC RSREG 29

17 EXAMPLE 3: From Response Surfaces by Andre Khuri and John Cornell, pages In this experiment, the researcher wants to investigate the effects of three fertilizer ingredients on the yield of snap beans under field conditions. The fertilizer ingredients and actual amounts applied were (i) nitrogen (N) from.94 to 6.29 lb/plt, (ii) phosphoric acid (P 2 O 5 ) from.59 to 2.97 lb/plot, and (iii) potash (K 2 O) from.6 to 4.22 lb/plot. The response of interest is the average yield in pounds per plot of snap beans. When coded the levels of nitrogen, phosphoric acid, and potash are: x = N 3.62 x 2 = P 2O 5.78 x 3 = K 2O On the uncoded scale, five levels of each variable were used: The design variables and responses are: x x x x x 2 x 3 y x x 2 x 3 y x x 2 x 3 y SAS Code for Example 3: DM LOG; CLEAR; OUT; CLEAR; ; * ODS PRINTER PDF file= C:\COURSES\ST578\SAS\canon3p.pdf ; ODS LISTING; OPTIONS PS=54 LS=72 NODATE NONUMBER; TITLE 3 FACTOR CCD -- KHURI AND CORNELL (PAGE 9) ; DATA DSGN; INPUT X X2 X3 LINES; ; PROC RSREG DATA=DSGN PLOTS=(ALL SURFACE(AT(x2= ))); MODEL YIELD = X X2 X3 / LACKFIT ; RUN; 3

18 3 FACTOR CCD -- KHURI AND CORNELL (PAGE 9) Coding Coefficients for the Independent Variables Factor Subtracted off Divided by X.682 X2.682 X3.682 Response Surface for Variable YIELD Response Mean.98 3 FACTOR CCD -- KHURI AND CORNELL (PAGE 9) Root MSE R-Square.786 Coefficient of Variation Parameter Estimate from Type Standard I Sum Coded Parameter Regression DF Estimate DF of Squares ErrorR-Square t Value F Pr Value > t Pr > Data F Intercept Linear < X Quadratic X2 Crossproduct X3 Total Model X*X X2*X Sum of X2*X2 DF Squares.2623 Mean Square 2.4 F.576 Value Pr.5955 > F X3*X Lack of Fit X3*X2 Pure Error X3*X3 Total Error Factor DF Sum of Squares Mean Square F Value Pr > F X X X

19 3 FACTOR CCD -- KHURI AND CORNELL (PAGE 9) Parameter DF Estimate Standard Error t Value Pr > t Parameter Estimate from Coded Data Intercept < X X X X*X X2*X X2*X X3*X X3*X X3*X FACTOR CCD -- KHURI AND CORNELL (PAGE 9) Canonical Analysis Sum of Response of Surface Based on Coded Data Factor DF Squares Mean Square F Value Pr > F X Critical Value X2 4 Factor Coded Uncoded X X X X Predicted value at stationary point: Eigenvectors Eigenvalues X X2 X Stationary point is a saddle point. 32

20 3 FACTOR CCD -- KHURI AND CORNELL (PAGE 9) Fit Diagnostics for YIELD RStudent - RStudent Predicted Value Predicted Value Leverage YIELD 2 8 Cook's D Quantile 8 2 Predicted Value Observation 3 2 Fit Mean Percent 2-2 Observations Parameters Error DF MSE R-Square Adj R-Square Proportion Less 33

21 3 FACTOR CCD -- KHURI AND CORNELL (PAGE 9) 3 FACTOR CCD -- KHURI AND CORNELL (PAGE 9) 34