14.0 RESPONSE SURFACE METHODOLOGY (RSM)

Transcription

1 4. RESPONSE SURFACE METHODOLOGY (RSM) (Updated Spring ) So far, we ve focused on experiments that: Identify a few important variables from a large set of candidate variables, i.e., a screening experiment. Ascertain how a few variables impact the response Now we want to answer the question: What specific levels of the important variables produce an optimum response? x ( x, x ) ( x, x ) x How do we get from starting position (, ) to optimum position ( x, x ), top of the hill? x x Our approach will be based on characterizing the true response surface, y η + ε where η is the mean response and ε is the error, with a model, ŷ. RSM: An experimental optimization procedure:. Plan and run a factorial (or fractional factorial) design near/at our starting point. ( x, x, ). Fit a linear model (no interaction or quadratic terms) to the data. 3. Determine path of steepest ascent (PSA) - quick way to move to the optimum - gradient based 4. Run tests on the PSA until response no longer improves. 9

2 5. If curvature of surface large go to step 6, else go to step 6. Neighborhood of optimum - design, run, and fit (using least squares) a nd order model. 7. Based on nd order model - pick optimal settings of independent variables. Example: Froth Flotation Process - Desire to whiten a wood pulp mixture. independent variables: () Concentration of environmentally safe bleach () Mixture temperature. Current operating point: %Bleach 4%, Temp 8 o F. we believe we are well away from the optimum. Allowable variable ranges: -% Bleach 6-3 Temp. An initial factorial design Variable Low Level Midpoint High Level ()% Bleach 4 6 () Temp x ŷ 9 ŷ Averge3.5 E 4 E E (It is small relative to E and E, so the response surface is fairly linear) x

3 Linear model: y b + b x + b x + ε Fitted model is: ŷ x + 5.5x A one unit change in x is a change of % in Bleach. A one unit change in x is a change of 5 deg. in Temperature. %Bleach-4 x , Bleach% x +4 Temp - 8 x , Temp 5x +8 5 We want to move from our starting point (x, x or % Bleach 4, Temp 8) in the direction that will increase the response the fastest. Thus, we want to move in a direction normal to the contours. Path of steepest ascent (PSA): line passing through point (x, x ) that is perpendicular to the contours. Define contour passing through (,) Relation between x & x for this contour 7x + 5.5x or x 7x ŷ x + 5.5x 3.5 Line normal to this one (slope of PSA ) contour slope x 5.5x Interpretation: for a one unit change in x, x must change by 5.5/7. units to remain on the PSA. A 7 unit change in x -> a 5.5 unit change in x. Gradient of ŷ may also be used to determine PSA. Let s define some points (candidate tests) on the path of steepest ascent (PSA) 94

4 Candidate Test x x 5.5x %Bleach x +4 Temp 5x +8 y Since it appears that the response surface is fairly linear (E is small) where we conducted tests, no reason to examine/conduct test or. Run st test on or beyond boundary defined by x, x +. Continue running tests on the PSA until response no longer increases. Highest response occurs at (Bleach 3%, Temp 98 o ). Run new factorial at this point. Ran initial factorial design - defined path of steepest ascent - moved on PSA to Bleach3%, Temperature 98 o, Whiteness 86%. Conduct another factorial design near this point 95

5 Temp x Averge79 E 6 E -5 E -7(It is fairly large) x 4 Bleach ŷ x.5x describes response surface with a plane. At x, x, ŷ 79 The combinations of (x, x ) that give ŷ 79 are specified by: 3x -.5 x 3 x x < (x, x ) relationship along 79 contour.5 ŷ.5 PSA: x x or x -.x 3.833x Let s follow the PSA but take steps in the x direction since is bigger than E. E 96

6 Candidate Test x x -.833x %Bleach 3+x Temp 97+5x y Highest response at Bleach 4.5%, Temp 9.8 o -----> 9% Whiteness. Note that we weren t able to move as far along the path before we fell off. We now believe we are fairly close to the optimum, so we must characterize the response surface with more than a plane, perhaps a model of the form: y b + b x + b x + b x x + b x + b x + ε To estimate the b s in this model, we need more than a - level factorial design. A Central Composite Design (CCD) is more efficient than a 3- level factorial design. Given the results of a nd order design - must use least squares (linear regression) to estimate the b s. We will examine the topic of regression after we finish up the RSM procedure for our example. Fit the following model to the response surface near the point: 97

7 %Bleach4.5, Temp 9.8, Whiteness9% Note a small change in nomenclature: use B s as the model parameter to be estimated, use b s as the estimate of the parameter. So, we are assuming the response surface can be characterized by the model: y B + B x + B x + B x x + B x + B x +ε The fitted model will be of the form: ŷ b + b x + b x + b x x + b x +b x b b b b b b b ( x T x ) T ỹ x Three-level Factorial Design x x + + x 3 - x x 3 k For three-level factorial designs, or designs, the number of tests in the 98

8 design is, where k is the number of variables being examined in the experiment. 3 k Variables Tests Central Composite Design x x +α + +α + x 3 - -α x - -α - -α + +α -α - + +α -α - + +α x In general, a central composite design (CCD) consists of points that form the base design, k star points, and several center points. Central Composite Designs (adapted from Montgomery) k Factorial Portion F Axial Points α Center Points (unif. prec.) n Center Points (orthog.) n Total Tests (unif. prec.) N Total Tests (orthog.) N k Number of Factorial Points, F k, α ( F) 4 : rotatability property - variance contours of ŷ are concentric circles. Unif. Prec.: variance at origin same as variance a unit away from origin. 99

9 Variable Levels -α α + Bleach (%) Temp (deg) Test Plan Test x x %Bleach Temp Calculating the Parameter Estimates x ỹ

10 T ( ) x x T ( ) x x T ( ỹ) x b b b b b b b ( x T x ) T ỹ x The fitted model is shown in the graph below

11 ŷ x x Linear Regression y x To describe the data above, propose the model: y B + B x + ε Fitted model will then be ŷ b + b x

12 n 6 Want to select values for b &b that minimize ( y i ŷ i ). i Define n 6 Sb (, b ) y i ŷ i i ( ) : the model residual Sum of Squares. Minimize n 6 ( y i b b x i ) Sb (, b ) ( y i ŷ i ) i n 6 i To find minimum S, take partial derivatives of S with respect to b &b, set these equal to zero, and solve for b &b. Sb (, b b ) Σ( y i b b x i )( ) Sb (, b b ) Σ( y i b b x i )( x i ) Σy i + Σb o + Σb x i Simplifying, we obtain: Σx i y i + Σb o x i + Σb x i nb + b Σx i Σy i These two equations are known as Normal Equations. The values of b &b that satisfy the Normal Equations are the least squares estimates -- they are the values that give a minimum S. In matrix form b Σx i + b Σx i Σx i y i 3

13 n Σx i Σx i Σx i b Σ y i b Σ x i y i b b Least Squares Estimates n Σx i Σx i Σx i Σ y i Σ x i y i b b Σx nσx i ( Σx i ) i Σx i Σx i n Σ y i Σ x i y i Σx i Σyi Σx i Σx i y i nσx i ( Σx i ) Σx i Σy i + nσx i y i b b b, are the values of b & b that minimize S, the Residual Sum of b Squares. b Bˆ an estimate of B b Bˆ an estimate of B 4

14 Y X Matrix Approach 3 ỹ : Vector of Observations 5, : Matrix of Independent Variables (Design Matrix) x ŷ ŷ ŷ Vector of Observations : : x b ŷ n b coefficients b b 5

15 ẽ Vector of Prediction Errors e e : : e n ẽ ỹ ŷ Want to Min ẽ T ẽ or Min ( ỹ ) x b T ( ỹ ) x b Take derivative with respect to b s and set T T T ( ỹ ) ỹ + ( x x b x x x )b T ( x x )b x T ỹ Therefore, b T ( ) x x T ỹ x It is analogous to b n Σx b Σx Σx Σy Σxy Re-run experiments several times b.4667 b.539 b.55,, b b b If true model is y B +B x + ε Then E(b ) B, E(b )B, E[ ] b B 6

16 b B σ y T Var( ) ( ) b x x σ y where describes the experimental error variation in the y s( ). σ ε Var( b For our example, Var( ) ) Cov( b, b ). b Cov( b, b ) Var( b ) σ y σ ε If (or ) is unknown, we can estimate it with s ( y ŷ)t ( y ŷ) e T e ( n - # of parameters) n p and for the example, S res ν s y n 6 i ( ˆ ) y i y i n.6769 s b T Var( ) ( ) b x x s y ).586, s b.767 standard error of b s b.39, s b.97 standard error of b Marginal Confidence Interval for Parameter Values Given our parameter estimates, we can develop a (-α)% confidence intervals for the unknown parameter values. The confidence intervals take 7

17 the form: b i ± t α ν, -- s bi For our example (t 4, ), For B :.4667 ± (.776) (.767),.4667 ±.5 For B :.943 ± (.776) (.97),.943 ±.546 Note that these confidence intervals do not consider the fact that the parameter estimates may be correlated. Joint Confidence Interval for Parameter Values Joint Confidence Interval (approximately) is bounded by a Sum of Squares Contour, S. Consider all coefficients simultaneously, p S S R + n F p pn, p, α where S R ( ) ( y Sb i ŷ i ) n i Sum of Squares contours are only circles if b &b, are uncorrelated - (x T x) - is diagonal. Sum of Squares often contours appear as ellipses. Confidence Interval for Mean Response The predicted response at an arbitrary point, o, is ŷ. Due to the x o x b uncertainty in the b s, there is uncertainty in (remember is the best estimate available for the mean response at o ). x ŷ o Var( ŷ ) x ( x T x) x T σy ŷ o 8

18 of course, if the variance must be estimated from the data, we obtain: Var( ŷ ) x ( x T x) ) x T sy We can develop a ( α )% C.I. for the true mean response at x µ ( ): ŷ ± t α[ Var( ŷ )] α, -- ) Y X Prediction Interval Even if we know with certainty the ŷ o for a given x, there would still be variation in the responses due to system noise(ε). The variance of the mean of g future trials at is: x ) Var( y ) s Var( ŷ g ) ) We can define limits within which the mean of g future observations will fall. These limits for a future outcome are known as a (-α)% prediction interval. 9

19 ˆ ± t α ν y, -- s Var( ŷ g ) ) Y X Model Adequacy Checking. Residual Analysis The model adequacy can be checked by looking at the residuals. The normal probability plot should not show violations of the normality assumption. If there are items not included in the model that are of potential interest, then the residuals should be plotted against these omitted factors. Any structure in such a plot would indicate that the model could be improved by the addition of that factor.. F test If the sum of square of the residuals of the model is S. The sum of squares of the residuals of a new model with s factors added is S, then S ν F( ν S ν, ν ) and ν n p ν n p s where n is the number of total observations, and p is the number of parameters in the original model.