Design and Analysis of Experiments

Transcription

1 Design and Analysis of Experiments Part IX: Response Surface Methodology Prof. Dr. Anselmo E de Oliveira anselmo.quimica.ufg.br anselmo.disciplinas@gmail.com

2 Methods Math Statistics Models/Analyses Response (multifactor) Objective Response Optimization y = f x 1, x 2 + ε η = f x 1, x 2 η = E y response surface expected response

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6 Models Response well modeled by a linear function of the independent variables First-order model y = β 0 + β 1 x 1 + β 2 x β k x k + ε If there is a curvature in the system Second-order model y = β 0 + k i=1 β i x i k + β ii x i 2 i=1 + β ij x i x j Method of least squares is used to estimate the parameters in the approximating polynomials Response surface analysis is performed using the fitted surface Response surface designs i<j + ε

7 RSM is a sequential procedure First-order model The objective is to lead the experimenter rapidly and efficiently along a path of improvement toward the general vicinity of the optimum Second-order model Climbing a hill

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9 For example... An experimenter is interested in determining the operating conditions that maximize the yield of a procedure. Two variables influence process yield Reaction time (t) Temperature (T) Current operating conditions: 75 min and 130 o C s = 1.5 for the yield Experimental Design 2 2 full factorial design t = 70 (-) and 80 (+) min T = (-) and (+) o C

10 2 2 factorial augmented by three center points run t /min T / o C y /g run order: 5,4,2,6,1,7,3 Natural variables: Time and temperature Coded variables: x 1 and x 2 x 1 = time 75 5 x 2 = temperature

11 First-order model y = β 0 + β 1 x 1 + β 2 x 2 + β 12 x 1 x 2 Least square fit > x1<-c(-1,1,-1,1,0,0,0) > x2<-c(-1,-1,1,1,0,0,0) > y<-c(54.3,60.3,64.6,68,60.3,64.3,62.3) > reg1<-lm(y~x1*x2) Call: lm.default(formula = y ~ x1 * x2) Coefficients: (Intercept) x1 x2 x1:x y = x x x 1 x 2

12 Factorial design effects t = 4.7 T = 9.0 tt = -1.3

13 ANOVA for the first-order model > anova(reg1) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x x * x1:x Residuals Signif. codes: 0 *** ** 0.01 * The first-order model assumes that the variables x 1 and x 2 have an additive effect on the response. Interaction between the variables would be represented by the coefficient β 12 of the crossproduct term x 1 x 2 added to the model. y = x x 2

14 Check for a pure quadratic curvature effect y average response at F y C the four points in the factorial portion of the design average response at the design center

15 run t /min T / o C y /g y F = y C = y F y c = using σ = 1.5 V y F y C = V y F + V y C = σ N F 2 + σ NC 2 V y F y C = V y F y C = 1.31 s yf y C = These two results (y F y c = 0.50 and s yf y C = 1.15) indicate that there is no evidence of second-order curvature in the response over the region of exploration

16 > reg1<-lm(y~x1+x2) # model without b12 > coefficients(reg1) # model coefficients (Intercept) x1 x > confint(reg1, level=0.95) # CIs for model parameters 2.5 % 97.5 % (Intercept) x x > fitted(reg1) # predicted values > residuals(reg1) # residuals

17 > layout(matrix(c(1,2,3,4),2,2)) # optional 4 graphs/page > plot(reg1)

18 library(plot3d) x<-seq(-1,1,by=.1) y<-seq(-1,1,by=.1) M<-mesh(x,y) z= *m$x+4.5*m$y par(mar = c(2, 2, 2, 2)) surf3d(x=m$x,y=m$y,z, bty="b2", xlab="x1",ylab="x2",zlab="y")

19 contour2d(x=x,y=y,z, xlab="x1",ylab="x2",lwd=3)

20 > library(rsm) > reg1.rsm<-rsm(y~fo(x1,x2)) > contour(reg1.rsm,~x1+x2, image = TRUE)

21 The Method of Steepest Ascent the path of steepest ascent is the line through the center of the region of interest and normal to the fitted surface

22 path of steepest ascent x 2 First-order fitted surface x 1

23 y = x x 2 To move away from the design center the point (x 1 = 0, x 2 = 0) along the path of steepest ascent, we would to move 2.35 units in the x 1 direction for every 4.50 units in the x 2 direction. Thus, the path of steepest ascent passes through the point (x 1 = 0, x 2 = 0) and has a slope 4.50/2.35. Using the relationship between time and temperature we see that 5 min of reaction time is equivalent to a step in the coded variable x 1 of x 1 = 1 Steps along the path of steepest ascent: x 1 = 1 x 2 = = 1.91

24 > steps<-steepest(reg1.rsm); steps Path of steepest ascent from ridge analysis: dist x1 x2 yhat > contour(reg1.rsm,~x1+x2, image = TRUE, + xlim=c(-1,3),ylim=c(-1,4)) > lines(steps$x1,steps$x2,col="red", type ="b", pch=19)

25 center path of steepest ascent x 1 x 2 t T run y ,6,

26 Second Design Highest yield: run 10 t = 90 min T = 145 o C New design 2 2 factorial augmented by two center points Coded variables x 1 = time x 2 = temperature x 1 x 2 t T run y center points

27 First-order model y = β 0 + β 1 x 1 + β 2 x 2 + β 12 x 1 x 2 Least square fit > x1<-c(-1,1,-1,1,0,0) > x2<-c(-1,-1,1,1,0,0) > y<-c(78.8,84.5,91.2,77.4,89.7,86.8) > lm(y~x1*x2) Call: lm.default(formula = y ~ x1 * x2) Coefficients: (Intercept) x1 x2 x1:x y = x x x 1 x 2

28 Factorial design effects t = T = 2.65 tt = first design t = 4.7 T = 9.0 tt = -1.3

29 ANOVA for the first-order model > anova(reg2) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x x x1:x Residuals There is no significant term for the first-order model!

30 Check for a pure quadratic curvature effect x 1 x 2 t T run y y F = y F y c = y C = using σ = 1.5 V y F y C = V y F + V y C = σ N F 2 + σ NC 2 V y F y C = V y F y C = 1.69 s yf y C = The first-order model is not adequate to represent the response over this region of exploration! These two results (y F y c = 5.28 and s yf y C = 1.30) indicate that there is evidence of second-order curvature in the response over the region of exploration

31 Analysis of a Second-Order Response Surface Close to the optimum a model that incorporates curvature is required k 2 y = β 0 + β i x i + β ii x i + β ij x i x j + ε i=1 k i=1 i<j y = β 0 + β 1 x 1 + β 2 x 2 + β 11 x β 22 x β 12 x 1 x 2 06 parameters: β 0, β 1, β 2, β 11, β 22 and β leves: --, -+, +-, ++, 00 Since the number of parameters is higher than the levels, the design must be augmented Central Composite Design

32 Central Composite Design (CCD) a = 1.414

33 The initial design is augmented with a group of star points 04 axial points 02 central points , ,

34 x 1 x 2 t T run y obs The design matrix consists of runs 11 to , ,

35 > library(rsm) > design.ccd<-ccd(basis=~x1+x2,n0=2,alpha="orthogonal",randomize=false) > design.ccd run.order std.order x1.as.is x2.as.is Block y Data are stored in coded form using these coding formulas... x1 ~ x1.as.is x2 ~ x2.as.is

36 > model<-lm(y ~ x1 + x2 + I(x1^2) + I(x2^2) + x1:x2,data=design.ccd) > summary(model) Call: lm.default(formula = y ~ x1 + x2 + I(x1^2) + I(x2^2) + x1:x2, data = design.ccd) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-10 *** x x I(x1^2) * I(x2^2) ** x1:x ** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 6 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 5 and 6 DF, p-value: b 2 (P 0.1) is not significant. The resulting surface model equation is y = x x x x 1 x 2

37 > model.rsm <- rsm (y ~ SO(x1,x2),data=design.ccd) > model.rsm$coefficients (Intercept) FO(x1, x2)x1 FO(x1, x2)x2 TWI(x1, x2) PQ(x1, x2)x1^2 PQ(x1, x2)x2^ > model.rsm$coefficients[3] FO(x1, x2)x > model.rsm$coefficients[3]<-c(0) > model.rsm$coefficients[3] FO(x1, x2)x2 0 > model.rsm$coefficients (Intercept) FO(x1, x2)x1 FO(x1, x2)x2 TWI(x1, x2) PQ(x1, x2)x1^2 PQ(x1, x2)x2^ y = x x x x 1 x 2

38 > persp (model.rsm, ~x1+x2, at = xs(model.rsm), + contours = "col", col = rainbow(40), zlab = "Yield", + xlabs = c("x1", "x2"))

39 st design: T and t y 2nd design: T and t y

40 Location of the Stationary Points Stationary point Let X be a function differentiable and continuous, and X D a subset of R n x s D is a stationary point if x s = 0 x x s = x k x 0,1, x 0,2,, x 0,n = 0, k = 1,2,, n k x 1, x 2,, x k y = y = = y = 0 x 1 x 2 x k stationary point coordinates Maximum response Minimum response Saddle point Response Surface

41 Fitted second-order model in a matrix notation y = β 0 + x t + x t Bx x = x 1 x 2 x k b = y x β 1 β 2 β k B = = b + 2Bx = 0 x s = 1 2 B 1 b β 11 β 1k /2 β kk y s = β x s t b

42 > model.rsm$coefficients (Intercept) FO(x1, x2)x1 FO(x1, x2)x2 TWI(x1, x2) PQ(x1, x2)x1^2 PQ(x1, x2)x2^ > b<-c(model.rsm$coefficients[2],model.rsm$coefficients[3]) β 1 > b<-matrix(b,nrow=2,ncol=1); b [,1] [1,] [2,] > b<-c(model.rsm$coefficients[2],model.rsm$coefficients[3]) > b<-matrix(b,nrow=2,ncol=1) > B<-c(model.rsm$coefficients[5],model.rsm$coefficients[4]/2, + model.rsm$coefficients[4]/2,model.rsm$coefficients[6]) > B<-matrix(B,nrow=2,ncol=2); B [,1] [,2] [1,] [2,] > xs=-0.5*solve(b)%*%b; xs [,1] [1,] [2,] > ys<-model.rsm$coefficients[1]+0.5*t(xs)%*%b; ys [,1] [1,] x s = 1 2 B 1 b B = b = β 2 β k β 11 β 1k /2 β kk y s = β x s t b x 1 = x 2 = The maximum is out of the design region x 1 = time 90 min 10 min x 2 = Temperature 145 o C 5 o C

43 Canonical analysis New coordinate system with the origin at the stationary point x s Rotate the axis of this system until they are parallel to the principal axes of the fitted response surface Canonical form of the model y = y s + λ 1 w λ 2 w λ k w k 2 where {w i } are the transformed independent variables and the {λ i } are constants. {λ i } are the eigenvalues or characteristic roots of B {λ i } are all positive: x s is a point of minimum response {λ i } are all negative: x s is a point of maximum response {λ i } have different signs: x s is a saddle point > yca<-eigen(b) > yca$val [1]

44 > steps.can<-canonical.path(model.rsm);steps.can dist x1 x2 x1.as.is x2.as.is yhat > contour(model.rsm,~x1+x2,image=true, + xlabs=c("x1", "x2"), + xlim=c(-8,1.5),ylim=c(-1.5,6.5)) > lines(steps.can$x1,steps.can$x2, + col="red",type ="b",pch=19)

45 canonical path stationary point

46 y x 2 x 1

47 New Model y = x x 1 2 2x 2 2 2x 1 x 2 > b<-c(-1.381,0) > b<-matrix(b,nrow=2,ncol=1); b [,1] [1,] [2,] > B<-c(-5.14,-2/2,-2/2,-2) > B<-matrix(B,nrow=2,ncol=2); B [,1] [,2] [1,] [2,] > xs=-0.5*solve(b)%*%b; xs [,1] [1,] [2,] > ys<-model.rsm$coefficients[1]+0.5*t(xs)%*%b; ys [,1] [1,] > yca<-eigen(b) > yca$val [1] stationary point coordinates response at the stationary point x s is a point of maximum response

48 y x 1 x 2

49 y x 2 x 1

50 Response Surfaces maximum

51 Saddle point

52 minimum

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54 Multiple Response Optimization Desirability Objective Function It is one of the most widely used methods in industry for the optimization of multiple response processes For each response y i x, a desirability function d i y i assigns numbers between 0 and 1 to the possible values of y i d i y i = 0 represents a completely undesirable value of y i d i y i = 1 represents a completely desirable or ideal response value

55 The individual desirabilities are then combined using the geometric mean, which gives the overall desirability n D = d i y i i=1 1 n with n denoting the number of responses It determines the best combination of responses The desirability approach consists of the following steps Conduct experiments and fit response models for all n responses Define individual desirability functions for each response Maximize the overall desirability D with respect to the controllable factors

56 Octave online

57 Octave: 4D Plots x1=-1:.1:1; x2=x1; x3=x1; [X1,X2,X3]=meshgrid(x1,x2,x3); Y= *X1+7*X2+8.25*X3-9.25*X1.*X2+9.5*X1.*X3+2.82*X1.*X1-6.18*X2.*X2; slice(x1,x2,x3,y,[-1. 1.],[-1. 1.],[-1. 1.]) colorbar () xlabel("x1"); ylabel("x2"); zlabel("x3"); view(120,35)

58 Experimental Designs for Fitting Response Surfaces Orthogonal First-Order Designs Simplex design y = b 0 + k i=1 b i x i + e k = 2 k = 3

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61 Variable-size

62 Spyridon Konstantinidis, Sunil Chhatre, Ajoy Velayudhan, Eva Heldin, Nigel Titchener-Hooker, Analytica Chimica Acta 743 (2012) 19 32

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64 4 vertexes x 2 x 1 x 3

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66 Second-Order Designs Central Composite Design 2 k Factorial (or fractional factorial of resolution V) n f factorial runs 2 k axial or star runs n c center runs K = 2 K = 3

67 Rotability V y x is the same at all points x that are at the same distance from the design center V y x = σ 2 x t X t X 1 x α = n f 1 4, where nf is the number of points used in the factorial portion of the design

68 n f = 4 α = 2 1.4

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70 circumscribed face centered inscribed

71 Box-Behnken Design Spherical design 2 radius Very efficient in terms of the number of required runs It does not contain any points at the vertices This could be advantageous when the points on the corners are prohibitively expensive or impossible to test

72 X 1 X 2 X 3 X 1 X 2 X

73 Other Designs Equiradial designs hexagon pentagon

74 Other Designs Optimal designs Irregular experimental region (not a cube or a sphere) Constraint on the design variables x 1 + x 2 or condition

75 Nonstandard model y = β 0 + β 1 x 1 + β 2 x 2 + β 11 x β 22 x β 12 x 1 x 2 + β 112 x 1 2 x 2 + β 1112 x 3 2 x 2 Unusual sample size requirements The number of design runs can be extremely expensive or time-consuming Small composite design Fractional factorial design in the cube of resolution III hybrid design Irregular levels

76 Box-Behnken 3 K design

77 Full 3 k 27 run Worker Bottle Shelf Run Order x 1 x 2 x 3 y >library(doe.base) >design<-fac.design(nlevels=3, nfactors=3, factor.names = list(x1=c(-1,0,1), x2=c(-1,0,1), x3=c(-1,0,1)), replications=1, randomize=false) >x1<-as.numeric(levels(design$x1)[design$x1]) >x2<-as.numeric(levels(design$x2)[design$x2]) >x3<-as.numeric(levels(design$x3)[design$x3]) >model<-lm(y~x1*x2*x3+i(x1^2)+i(x2^2)+i(x3^2)) >summary(model)

78 Box-Behnken 14 runs (12 +2 pc) Worker Bottle Shelf Run Order x 1 x 2 x 3 y x 1 >library(rsm) >design.bbd<-bbd(3,n0=2,randomize=false) >model<-lm(y~x1*x2*x3+i(x1^2)+i(x2^2)+i(x3^2),data=design.bbd) >summary(model) x 2 x 3

79 Full 3 k x 3 = 0.66 x 1 x 3 = 0.23 x 1 2 = 0.80 x 3 2 = 0.31 Box-Behnken x 3 = 0.57 x 1 2 = 0.70 x 3 2 = 0.5