2 Introduction to Response Surface Methodology

Transcription

1 2 Introduction to Response Surface Methodology 2.1 Goals of Response Surface Methods The experimenter is often interested in 1. Finding a suitable approximating function for the purpose of predicting a future response. 2. Determining what values of the x 1, x 2,..., x k s are optimum with respect to the response y. Optimum is used in the sense of finding the x 1, x 2,..., x k s which would yield one of the three experimental goals: a maximum, minimum, or a specific (target or aim) value of the response. Response surface procedures are primarily used to either (i) determine what are the optimum operating conditions which minimize or maximize a response or (ii) determine an operating region in the design variable space for which certain operating specifications are met. Although the eventual goal is usually to answer certain questions regarding operating conditions, it is extremely important that the decision be made, at the outset, regarding what experimental design be used for data collection, that is, what factor levels should be considered in the experimental process. Because model coefficients are estimated from experimental data, it is obvious that this estimation can be accomplished effectively if proper thought is given to the question of what experimental design to use. It is assumed that the experimenter is concerned with a system involving a response y which depends on the input variables ξ 1, ξ 2,..., ξ k. It is assumed that there exists an exact functional relationship E(y) = η(ξ) where E(y) is the expected response at ξ = (ξ 1, ξ 2,..., ξ k ). This relationship, however, is usually unknown. The ξ i s, also called the natural or uncoded variables. It is assumed that the ξ i s can be controlled by the experimenter with negligible error. The ξ i s are usually transformed using a linear transformation which center and scale each of the ξ i s. The transformed variables x 1, x 2,..., x k are called the design or coded variables. ξ i x i where x i = Because the true form of η is unknown and is perhaps extremely complicated, the success of RSM depends on the approximation of η by a simpler function restricted to some region of the independent ξ i variables. Low order polynomials are most often used because of their local smoothness properties. Mathematically, we are assuming that over a limited region of interest R(ξ), the main characteristics of the underlying true function η(ξ) are adequately represented by the low order terms in a Taylor series approximation of η(ξ). 1. If the approximating model is a linear model with only first order design variable terms (first-order model), then f(x) = b 0 + k b i x i = β 0 + i=1 15 k β i x i. i=1

2 This model may be useful when the experimenter is interested in studying η in narrow regions of x = (x 1, x 2,..., x k ) where little or no curvature or interactions are present. This model is also used in situations where small pilot experiments are used as preliminary experiments leading to a more extensive experimental exploration. These experiments are often sequential in nature, where the procedure leads the experimenter toward the general region containing optimal values of (x 1, x 2,..., x k ). One such procedure is the path of steepest ascent (descent). 2. If the approximating model is a linear model with first order design variable terms and their pairwise interactions (interaction model), then f(x) = b 0 + = β 0 + k b i x i + i=1 k β i x i + i=1 This model may be useful when the experimenter is interested in studying η in narrow regions of x 1, x 2,..., x k, that is, where little or no curvature is present. 3. If the approximating model is a linear model with all first and second order design variable terms (second-order or quadratic model), then f(x) = b 0 + = β 0 + k b i x i + k b ij x i x j + i=1 i<j k β i x i + k β ij x i x j + i=1 This model may be useful when the experimenter is interested in studying η in wider regions of x 1, x 2,..., x k where curvature is expected to be present. 4. Occasionally, models of order greater than 2 (such as, cubic models) are used. It should be understood that the region of interest R(ξ) lies within a larger region called the region of operability O(ξ) which is defined as the region over which experiments could be conducted. Example: You are interested in determining the water temperature ξ which extracts the maximum amount of caffeine η from a particular brand of coffee. The region of operability O(ξ) is between 0 C and 100 C. The region of interest R(ξ) is limited to temperatures near boiling, say between 95 C and 100 C. The fact that the polynomial is an approximation does not necessarily detract from its usefulness. If a model provides an adequate representation in the region of interest, then analysis of a model fitted from data will approximate an analysis of the physical system. i<j 16

3 2.2 Approximating Function Example A two-factor (3 3 or 3 2 ) factorial experiment with n = 2 replicates was run. Factor x 1 represents temperature and x 2 represents process time. The factor levels for x 1 and x 2 were coded as 1, 0, and 1. The process yield is labelled y. The true functional relationship between the response y and the variables x 1 and x 2 is y = η(x 1, x 2 ) = An experiment was run yielding the following experimental data: η(x 1, x 2 ) η(x 1, x 2 ) + ɛ x 1 x 2 Rep (True y) (Observed y) A second-order regression model f(x 1, x 2 ) was fit to the data. The fitted model was f(x 1, x 2 ) = b 0 + b 1 x 1 + b 2 x 2 + b 12 x 1 x 2 + b 11 x b 22 x 2 2 = x x x 1 x x x 2 2 The plots show the relationship between the fitted regression model f(x 1, x 2 ) and the true functional form η(x 1, x 2 ). The first set of plots contain the three-dimensional surface plots of the true function and the approximating function η(x 1, x 2 ) = f(x 1, x 2 ) = x x x 1 x x x 2 2 The second set of plots are contour plots of the same two surfaces. 17

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7 Example: Taylor Series Approximation For function η(x, y) = 5 + e.5x 1.5y and p 0 = (0, 0) in some open ball B containing p 0, define p = (x, y) and P 2 (p, p 0 ) = η(0, 0) + x η 1! x + y η 1! y + x2 2! 2 η x 2 + xy 1!1! 2 η x y + y2 2! 2 η y 2 R 2 (p, p ) = 3 k=0 x k y (3 k) k!(3 k)! k η k x (3 k) y p where p is a point on the line segment joining p and (0, 0). Taking the partial derivatives η(x, y) = 5 + e.5x 1.5y, and evaluating them at (0, 0) yields η x = =.5e 0 =.5 η y = = 1.5e 0 = η x 2 = =.25e 0 =.25 2 η y 2 = = 2.25e 0 = η x y = =.75e 0 =.75 Thus, the second-order Taylor series approximation of η(x, y) = 5 + e.5x 1.5y about the point (0, 0) is f(x, y) = x y xy + 1 (1)(1) 2 x y2 = 6 +.5x 1.5y.75xy +.125x y 2 But, this function, too, is unknown. So what to we do as statisticians? Although f(x, y) is unknown, we know a second-order Taylor series approximation exists. We then use our data to estimate the intercept and coefficients of the terms in a full second-order (quadratic) regression model above. 21

8 Recall: For our data example, the fitted model using least-squares was f(x, y) = x 2.38y 1.09xy +.28x y 2 To find the critical point, take the first partial derivatives, equate to zero, and solve: f x = f y = Setting f x = 0 and f y = 0 produces two equations and two unknowns. The solution is the critical point (x 0, y 0 ) = (.50,.65). Next, perform the second-derivative. We need to evaluate the derivatives at critical point (x 0, y 0 ) = (.50,.65): ( ) 2 2 f x y = 1.09 (.50,.65) ( ) 2 f =.56 x 2 (.50,.65) ( ) 2 f = 2.82 y 2 (.50,.65) Evaluating yields (.50,.65) = ( ) 2 2 f x y ( ) ( 2 f x 2 2 f y 2 ) (.50,.65) = ( 1.09) 2 (.56)(2.82) =.39 But 2 f x 2 > 0, so (.50,.65) is a minimum for the fitted model (response surface). 22

9 2.3 Bias We assume the polynomial response surface model is an inadequate approximation of the true model. As a result, the model coefficients are biased by terms that are of order higher than the order of the assumed model. For example, if a first-order model is fit when curvature exists, the coefficients are seriously affected by the exclusion of important interaction and squared terms. Let y = (y 1, y 2,..., y n ) be a vector of n responses. Let ɛ = (ɛ 1, ɛ 2,..., ɛ n ) be a vector of n random errors having zero vector mean. Let f(ξ) be the approximating function of η(ξ). represented by y = or, using the approximating model, by The true functional relationship can be y = where δ(ξ) = η(ξ) f(ξ) is the difference between the actual and approximating models. We would like this to be very small over the region of interest R(ξ). Thus, there are two types of errors which must be taken into account: (i) systematic, or bias errors δ(ξ) = η(ξ) f(ξ) and (ii) random errors ɛ. Although this implies that systematic errors δ(ξ) are unavoidable, there has been a tendency since the time of Gauss ( ) to ignore them and to concentrate only on the random errors ɛ. This has been done because nice mathematical and statistical results are possible, in particular, in hypothesis testing results that rely on normality of the random errors. When choosing an experimental design, ignoring systematic errors can be a serious problem because the design point selection is based on an inadequate approximating model which, in turn, can lead to misleading results. 2.4 Fitted Model Example Using SAS The following experimental data was used to fit the model y = b 0 + b 1 x + b 2 x 2. x y We will review the SAS output and what it represents. Note that σ 2 = MS E = which is our estimate of σ 2. Then σ 2 (X X) 1 The square root of the diagonal entries are the standard errors of the model parameter estimates: s.e.(b 0 ) = (.15417)(238.62) = s.e.(b 1 ) = (.15417)(.01723) =.0515 s.e.(b 2 ) = (.15417)( ) =

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