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INFORMATICS AND MATHEMATICAL MODELLING IMM Technical University of Denmark DK-2800 Kgs. Lyngby – Denmark DACE A MATLAB Kriging Toolbox Version 2.0, August 1, 2002 Søren N. Lophaven Hans Bruun Nielsen Jacob Søndergaard 46 44 42 40 38 36 34 100 80 60 40 20 20 0 0 100 80 60 40 Technical Report IMM-TR-2002-12 Please direct communication to Hans Bruun Nielsen (hbn@imm.dtu.dk)

Contents 1. Introduction 2. Modelling and Prediction 2.1. The Kriging Predictor . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Correlation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Generalized Least Squares Fit 1 1 2 5 6 9 3.1. Computational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4. Experimental Design 12 4.1. Rectangular Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2. Latin Hypercube Sampling . . . . . . . . . . . . . . . . . . . . . . . . 12 5. Reference Manual 13 5.1. Model Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2. Evaluate the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.3. Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.4. Correlation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.5. Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.6. Auxiliary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.7. Data Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6. Examples of Usage 21 6.1. Work-through Example . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.2. Adding a Regression Function . . . . . . . . . . . . . . . . . . . . . . 24 7. Notation 24

1. Introduction This report describes the background for and use of the software package DACE (Design and Analysis of Computer Experiments), which is a Matlab toolbox for working with kriging approximations to computer models. The typical use of this software is to construct a kriging approximation model based on data from a computer experiment, and to use this approximation model as a surrogate for the computer model. Here, a computer experiment is a collection of pairs of input and responses from runs of a computer model. Both the input and the response from the computer model are likely to be high dimensional. The computer models we address are deterministic, and thus a response from a model lacks random error, i.e., repeated runs for the same input parameters gives the same response from the model. Often the approximation models are needed as a part of a design problem, in which the best set of parameters for running the computer model is determined. This is for example problems where a computer model is ﬁtted to physical data. This design problem is related to the more general problem of predicting output from a computer model at untried inputs. In Section 2 we consider models for computer experiments and eﬃcient predictors, Section 3 discusses generalized least squares and implementation aspects, and in Section 4 we consider experimental design for the predictors. Section 5 is a reference manual for the toolbox, and ﬁnally examples of usage and list of notation are given in Sections 6 and 7. 2. Modelling and Prediction V V S:,j, S:,j Y:,j, Y:,j ··· sm] = 1, = 1, Given a set of m design sites S = [s1 Y = [y1 ··· ym] conditions1 with si ∈ IRn and responses with yi ∈ IRq. The data is assumed to satisfy the normalization ·,· where X:,j is the vector given by the jth column in matrix X, and µ[· ] and V denote respectively the mean and the covariance. Following [9] we adopt a model ˆy that expresses the deterministic response y(x)∈ IRq, for an n dimensional input x∈D ⊆ IRn, as a realization of a regression model F and j = 1, . . . , n , j = 1, . . . , q , µ[S:,j] = 0, µ[Y:,j] = 0, (2.1) 1The user does not have to think of this: The ﬁrst step in the model construction is to normalize the given S, Y so that (2.1) is satisﬁed, see (5.1) below. 1

a random function (stochastic process), ˆy(x) = F(β:,, x) + z(x), = 1, . . . , q . (2.2) We use a regression model which is a linear combination of p chosen functions fj : IRn → IR, F(β:,, x) = β1,f1(x) + ··· βp,fp(x) = [f1(x) ··· fp(x)] β:, ≡ f (x) β:, . The coeﬃcients {βk,} are regression parameters. The random process z is assumed to have mean zero and covariance E z(w)z(x) = σ2 R(θ, w, x), = 1, . . . , q (2.3) (2.4) between z(w) and z(x), where σ2 is the process variance for the th component of the response and R(θ, w, x) is the correlation model with parameters θ. An interpretation of the model (2.2) is that deviations from the regression model, though the response is deterministic, may resemble a sample path of a (suitably chosen) stochastic process z. In the following we will focus on the kriging predictor for y. First, however, we must bear in mind that the true value can be written as y(x) = F(β:,, x) + α(β:,, x) , (2.5) where α is the approximation error. The assumption is that by proper choice of β this error behaves like “white noise” in the region of interest, i.e., for x∈D. F = [f (s1) ··· f (sm)] , 2.1. The Kriging Predictor For the set S of design sites we have the expanded m×p design matrix F with Fij = fj(si), (2.6) with f (x) deﬁned in (2.3). Further, deﬁne R as the matrix R of stochastic-process correlations between z’s at design sites, Rij = R(θ, si, sj), i, j = 1, . . . , m . At an untried point x let r(x) = [R(θ, s1, x) ··· R(θ, sm, x)] (2.7) (2.8) be the vector of correlations between z’s at design sites and x. Now, for the sake of convenience, assume that q = 1, implying that β = β:,1 and Y = Y:,1, and consider the linear predictor ˆy(x) = cY , 2 (2.9)

ϕ(x) = E F c(x) = f (x) . 2 2 (ˆy(x) − y(x)) cZ − z z2 + cZZc − 2cZz 1 + cRc − 2cr . − λ 1 + cRc − 2cr = E = E = σ2 (2.10) (2.11) with c = c(x)∈ IRm. The error is ˆy(x) − y(x) = cY − y(x) = c = cZ − z + (F β + Z) − (f (x) F c − f (x) β + z) β , where Z = [z1 . . . zm] unbiased we demand that F c − f (x) = 0, or are the errors at the design sites. To keep the predictor Under this condition the mean squared error (MSE) of the predictor (2.9) is The Lagrangian function for the problem of minimizing ϕ with respect to c and subject to the constraint (2.10) is L(c, λ) = σ2 (F c − f ) . (2.12) The gradient of (2.12) with respect to c is c(c, λ) = 2σ2(Rc − r) − F λ , L and from the ﬁrst order necessary conditions for optimality (see e.g. [7, Section 12.2]) we get the following system of equations R F F 0 c ˜λ = r f , (2.13) where we have deﬁned The solution to (2.13) is ˜λ = − λ 2σ2 . ˜λ = (F R−1F ) c = R−1(r − F ˜λ) . −1(F R−1r − f ) , (2.14) The matrix R and therefore R−1 is symmetric, and by means of (2.9) we ﬁnd 3

ˆy(x) = (r − F ˜λ) R−1Y = rR−1Y − (F R−1r − f ) (F R−1F ) −1F R−1Y . (2.15) In Section 3 we show that for the regression problem F β Y the generalized least squares solution (with respect to R) is β∗ = (F R−1F ) −1F R−1Y , and inserting this in (2.15) we ﬁnd the predictor ˆy(x) = rR−1Y − (F R−1r − f ) + rR−1(Y − F β∗ ) β∗ = fβ∗ = f (x) + r(x) γ∗ . β∗ (2.16) = Y − F β∗ For multiple responses (q > 1) the relation (2.16) hold for each column in Y , so that (2.16) holds with β∗ ∈ IRp×q given by (2.15) and γ∗ ∈ IRm×q computed via the residuals, Rγ∗ Note that for a ﬁxed set of design data the matrices β∗ are ﬁxed. For every new x we just have to compute the vectors f (x) ∈ IRp and r(x) ∈ IRm and add two simple products. and γ∗ . Getting an estimate of the error involves a larger computational work. Again we ﬁrst let q = 1, and from (2.11) and (2.14) we get the following expression for the MSE of the predictor, (Rc − 2r) 1 + c 1 + (F ˜λ − r) R−1(F ˜λ + r) 1 + ˜λF R−1F ˜λ − rR−1r −1u − rR−1r 1 + u (F R−1F ) . ϕ(x) = σ2 = σ2 = σ2 = σ2 (2.17) where u = F R−1r − f and σ2 is found by means of (3.7) below. This expression generalizes immediately to the multiple response case: for the th response function we replace σ by σ, the process variance for th response function. Computational aspects are given in Section 3.1. Remark 2.1.1. Let x = si, the ith design site. Then r(x) = R:,i, the ith column of R, and R−1r(x) = ei, the ith column of the unit matrix, ei = I:,i. Using these relations and (2.6) in (2.16) we ﬁnd ˆy(si) = f (si) = f (si) = f (si) β∗ β∗ β∗ + r(si) i (Y − F β∗ + e + yi − Fi,:β∗ ) = yi . R−1(Y − F β∗ ) 4

This shows that the Kriging predictor interpolates the design data. Further, in (2.17) we get u = F ei − f (si) = 0 and the associated MSE ϕ(si) = σ2(1 − R:,i ei) = σ2(1 − Rii) = 0 , since Rii = 1. Remark 2.1.2. As indicated by the name MSE (mean squared error) we expect that ϕ(x) ≥ 0, but in (2.17) it may happen that rR−1r > 1+u −1u, in which case ϕ(x) < 0. This point needs further investigation, but as a ﬁrst explanation we oﬀer the following: Equation (2.11) is based on the assumption that the diﬀerence between the regression model and the true value is “white noise”, and if there is signiﬁcant approximation error, (2.5), then this assumption and its implications do not hold. (F R−1F ) Remark 2.1.3. From (2.16) it follows that the gradient ˆy = ∂ ˆy ∂x1 ··· ∂ ˆy ∂xn can be expressed as (x) = Jf (x) γ∗ , where Jf and Jr is the Jacobian of f and r, respectively, ∂R ∂xj (Jf (x))ij = (Jr(x))ij = β∗ + Jr(x) ∂fi ∂xj (x), (2.18) (θ, si, x) . (2.19) From (2.17) it follows that the gradient of the MSE can be expressed as ϕ (x) = 2σ2 = 2σ2 (F R−1F ) (F R−1F ) −1u − R−1r −1(F R−1r − Jf β∗ ) − R−1r ˆy . (2.20) These expressions are implemented in the toolbox, see Section 5.2. 2.2. Regression Models The toolbox provides regression models with polynomials of orders 0, 1 and 2. More speciﬁc, with xj denoting the jth component of x, Constant, p = 1 : f1(x) = 1 , (2.21) Linear, p = n+1 : f1(x) = 1, f2(x) = x1, . . . , fn+1(x) = xn , (2.22) 5

Quadratic, p = 1 2(n+1)(n+2) : f1(x) = 1 f2(x) = x1, . . . , fn+1(x) = xn fn+2(x) = x2 f2n+2(x) = x2 ··· 1, . . . , f2n+1(x) = x1xn 2, . . . , f3n(x) = x2xn ··· fp(x) = x2 n . (2.23) The corresponding Jacobians are (index n×q denotes the size of the matrix and O is the matrix of all zeros) constant : Jf = [On×1] , linear : Jf = [On×1 In×n] , quadratic : Jf = [On×1 In×n H] , where we illustrate H ∈ IRn×(p−n−1) by 2x1 x2 0   2x1 x2 x3 0 x1 2x2 0 0 x1 0 0 x1 , 0 0 2x2 x3 0 0 0 x2 2x3   . n = 2 : H = n = 3 : H = 2.3. Correlation Models As [9] we restrict our attention to correlations of the form Rj(θ, wj − xj) , R(θ, w, x) = n j=1 expg Name exp i.e., to products of stationary, one-dimensional correlations. More speciﬁc, the tool- box contains the following 7 choices Rj(θ, dj) exp(−θj|dj|) exp(−θj|dj|θn+1), exp(−θjd2 j ) max{0, 1−θj|dj|} 1 − 1.5ξj + 0.5ξ3 j , 1 − 3ξ2 j + 2ξ3 j , ς(ξj), (2.24) ξj = min{1, θj|dj|} ξj = min{1, θj|dj|} ξj = θj|dj| 0 < θn+1 ≤ 2 spherical spline gauss cubic lin Table 2.1. Correlation functions. dj = wj − xj 6

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