On the optimal designs for the prediction of Ornstein-Uhlenbeck sheets

Statistics and Probability Letters 83 (2013) 1580–1587 Contents lists available at SciVerse ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro On the optimal designs for the prediction of Ornstein–Uhlenbeck sheets Sándor Barana, Kinga Sikolyaa, Milan Stehlíkb,∗ aFaculty of Informatics, University of Debrecen, Kassai út 26, H-4028 Debrecen, Hungary bInstitut für Angewandte Statistik, Johannes Kepler University in Linz, Altenberger Straße 69, A-4040 Linz a. D., Austria article info Article history: Received 8 November 2012 Received in revised form 5 March 2013 Accepted 6 March 2013 Available online 15 March 2013 MSC: primary 62K05 secondary 62M30 Keywords: Ornstein–Uhlenbeck sheet Integrated mean square prediction error Entropy Fisher information abstract Computer simulations are often used to replace physical experiments for exploring the complex relationships between input and output variables. We study the optimal design problem for the prediction of a stationary Ornstein–Uhlenbeck sheet on a monotonic set with respect to the integrated mean square prediction error criterion and the entropy criterion. We show that there is a substantial difference between the shapes of optimal designs for Ornstein–Uhlenbeck processes and sheets. In particular, we show that the optimal prediction based on the integrated mean square prediction error does not necessarily lead to space-filling designs. ©2013 Elsevier B.V. All rights reserved. 1. Introduction 1 A common problem in spatial statistics is deciding how to choose a set of sample locations in order to predict a 2 random process in an optimal way. In the present paper we study the problem of optimal design for the prediction of an 3 Ornstein–Uhlenbeck (OU) sheet on a monotonic set with respect to the integrated mean square prediction error (IMSPE) 4 criterion and the entropy criterion. 5 We consider the stationary process 6 Y(s,t)=θ+ε(s,t)(1.1) 7 with the design points taken from a compact design space X= [a1,b1]×[a2,b2], where a1<b1and a2<b2and ε(s,t), 8 s,t∈R, is a stationary OU sheet, that is a zero-mean Gaussian process with covariance structure 9 Eε(s1,t1)ε(s2,t2)=σ2 4αβ exp (−α|s1−s2|−β|t1−t2|),(1.2) 10 where α > 0, β > 0,σ > 0. We remark that ε(s,t)can also be represented as 11 ε(s,t)=σ 2√αβ e−αs−βtWe2αs,e2βt,12 where W(s,t), s,t∈R, is a standard Brownian sheet (Baran et al.,2003;Baran and Sikolya, 2012). 13 ∗Corresponding author. Tel.: +43 732 2468 6806; fax: +43 732 2468 9846. E-mail address: [email protected] (M. Stehlík). 0167-7152/$ – see front matter ©2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.03.003 S. Baran et al. / Statistics and Probability Letters 83 (2013) 1580–1587 1581 In order to apply the usual notation of spatial modelling (Kiseľák and Stehlík, 2008) we introduce σ:= σ/(2√αβ) and1 instead of (1.2) we investigate2 Eε(s1,t1)ε(s2,t2)=σ2exp (−α|s1−s2|−β|t1−t2|),(1.3)3 where σis considered as a nuisance parameter.4 Further, we require that the shifted OU sheet (1.1) is measured at the points constituting a monotonic set. A monotonic5 set can be defined in an arbitrary Hilbert space H, with real or complex scalars. For x,y∈H, we denote by ⟨x,y⟩the real part6 of the inner product. A set E⊂H×His called monotonic (see Minty (1962,1963)) provided that for all (x1,y1), (x2,y2)∈E7 we have ⟨x1−x2,y1−y2⟩ ≥ 0. A practical example of such a set is provided by the measurements on isotherms of a8 stationary temperature field, with several applications in thermal slab modelling (see e.g. Babiak et al. (2005)orKoizumi9 and Jin (2012)). In this paper we consider the following version of a monotonic set:10 Condition D. The design points {(s1,t1), (s2,t2), . . . , (sn,tn)}⊂X,n∈N,are not overlapping (observations without11 repetitions); moreover, 0<s1<s2<··· <snand 0<t1<t2<··· <tnhold.12 For the shifted OU sheet (1.1) observed at points satisfying Condition D,Baran and Stehlík (submitted for publication)13 derived the exact form of the Fisher information (matrices) corresponding to the trend parameter θand to covariance14 parametersr=(α, β), and the authors investigated the problem ofD-optimal designs forthe estimation ofthese quantities.15 We derive the exact form of the IMSPE recently used in several papers (see e.g. Crary (2002) or Sacks et al. (1989)), and16 show that in contrast to the case for the OU process on a compact interval investigated by Baldi Antognini and Zagoraiou17 (2010), the equidistant design is usually not optimal. This is an important observation, and also consistent with the results18 of Pronzato and Müller (2012), since statistical folklore is that space-filling designs have superior properties when it comes19 to the prediction of emulator functions. We also investigate the properties of the optimal design with respect to the entropy20 criterion (Baldi Antognini and Zagoraiou, 2010;Shewry and Wynn, 1987).21 Observe that by condition Condition D, the results presented here are not applicable for space-filling designs. In the22 generalcasethe mainproblemisthedifficultyoffindingtheinverse ofthecovariancematrixoftheobservations.Theauthors23 have some preliminary results for regular grid designs, where the covariance matrix is the Kronecker product of two simple24 covariance matrices corresponding to the two spatial dimensions, and this special form allows finding a reasonable form25 of the inverse and calculating the objective function for the IMSPE criterion. However, the presentation of these results is26 beyond the space limitations of this work.27 2. Optimal design with respect to the IMSPE criterion28 Suppose we observe our process Y(s,t)at the design points {(s1,t1), (s2,t2), . . . , (sn,tn)}satisfying Condition D. The29 main aim of the kriging technique consists in predicting the output of the simulator on the experimental region, and for30 any untried location x=(x1,x2)∈Xthe estimation procedure is focused on the best linear unbiased estimator of Y(x)31 given by Y(x)= θ+Q⊤(x)C−1(n,r)(Y−1n θ), where Y=(Y(s1,t1), . . . , Y(sn,tn))⊤is the vector of observations, 1nis the32 column vector with entries 1 of length n,C(n,r)is the covariance matrix of Y,Q(x)is the vector of correlations between33 Y(x)and Ydefined by Q(x)=(ϱ(x,s1,t1), . . . , ϱ(x,sn,tn))⊤with ϱ(x,si,ti):= exp (−α|x1−si|−β|x2−ti|)and  θis34 the generalized least squares estimator of θ, that is θ=1⊤ nC−1(n,r)1n−11⊤ nC−1(n,r)Y. Usually, correlation parameters35 α, β are unknown and will be estimated by the maximum likelihood method. Thus, the kriging predictor is obtained by36 substituting the maximum likelihood estimators (MLE) (α, β) for (α, β) and in such a case Y(x)is called the MLE–empirical37 best linear unbiased predictor (Santner et al., 2003).38 Thus natural criteria will minimize suitable functionals of the mean squared prediction error (MSPE) given by39 MSPE Y(x):= σ21−1,Q⊤(x)01⊤ n 1nC(n,r)−11,Q⊤(x)⊤.(2.1)40 Since the prediction accuracy is often related to the entire prediction region X, the design criterion IMSPE is given by41 IMSPE  Y:= σ−2X MSPE  Y(x)dx.42 Without loss of generality, we can assume that the design space X= [0,1]2. Therefore, as extrapolative prediction is not43 advisable in kriging, we can set t1=s1=0 and tn=sn=1.44 Theorem 2.1. In our set-up,45 MSPE  Y(x)=σ21−ϱ(x,sn,tn)2− n−1  i=1 (ϱ(x,si,ti)−ϱ(x,si+1,ti+1)qi)2 1−q2 i 46 +1+ n−1  i=1 1−qi 1+qi−11−ϱ(x,sn,tn)− n−1  i=1 ϱ(x,si,ti)−ϱ(x,si+1,ti+1)qi 1+qi2 ,47 1582 S. Baran et al. / Statistics and Probability Letters 83 (2013) 1580–1587 Fig. 1. IMSPE corresponding to the optimal design for the prediction and Fisher information on θas functions of correlation parameters (α, β) for n=3. where qi:= exp(−αdi−βδi)with di:= si+1−siand δi:= ti+1−ti,i=1,2,...,n−1. Further, 1 IMSPE  Y=1−An+1+ n−1  i=1 1−qi 1+qi−1 Bn,(2.2) 2 where 3 An=R1,1+ n−1  i=1 Ri,iq2 i−2Ri+1,iqi+Ri+1,i+1 1−q2 i ,(2.3) 4 Bn=1−2ωn+Rn,n+ n−1  i=1Ri,i−2Ri+1,iqi+Ri+1,i+1q2 i (1+qi)2+2(Rn,i−ωi)−(Rn,i+1−ωi+1)qi 1+qi +2 n−1  i=2 i−1  j=1 Ri,j−Ri+1,jqi−Ri,j+1qj+Ri+1,j+1qiqj (1+qi)(1+qj),(2.4) 5 with 6 ωi=1 αβ 2−e−α(d1+···+di−1)−e−α(di+···+dn−1)2−e−β(δ1+···+δi−1)−e−β(δi+···+δn−1),(2.5) 7 Ri,j=1 2α2e−α(dj+···+di−1)−e−α(2d1+···+2dj−1+dj+···+di−1)−e−α(dj+···+di−1+2di+···+2dn−1) +(dj+···+di−1)e−α(dj+···+di−1) ×1 2β2e−β(δj+···+δi−1)−e−β(2δ1+···+2δj−1+δj+···+δi−1)−e−β(δj+···+δi−1+2δi+···+2δn−1) +(δj+···+δi−1)e−β(δj+···+δi−1),i,j∈N,1≤j≤i≤n.(2.6) 8 Example 2.2. Consider a three-point design, that is n=3, s1=t1=0,s2:= d,t2:= δ, s3=t3=1. For this particular 9 design we obviously have d1=d,d2=1−d, δ1=δ, δ2=1−δ. By analysing the partial derivatives of IMSPE  Yone 10 can easily show that the minimum of IMSPE is reached at d=1/2, δ =1/2, so for n=3 the equidistant design is optimal. 11 In Fig. 1, IMSPE  Ycorresponding to the optimal design and the Fisher information on the shift parameter θare plotted 12 as functions of correlation parameters (α, β). Having a look at the figures for IMSPE and Mθone can observe that there is S. Baran et al. / Statistics and Probability Letters 83 (2013) 1580–1587 1583 Fig. 2. The values of d∗ 1,d∗ 2and δ∗ 1, δ∗ 2giving the optimum of IMSPE as functions of correlation parameters (α, β) for n=5. a substantial difference between the variations of the estimation and the prediction criterion with respect to correlation1 parameters (α, β). The equilateral lines for Mθdiffer considerably from equilateral curves for IMSPE, which shows the2 different level of sensitivity of these criteria to the misspecification of correlation parameters. For more information on3 the sensitivity of optimal design criteria with respect to parameter misspecification, see e.g. Stehlík (2004).4 Theorem 2.3. IMSPE given by (2.2) is symmetric in the sense that the interchange of pairs of distances (di, δi)and (dn−i, δn−i),5 i=1,2,...,n−1, does not change the value of IMSPE  Y.6 Theorem 2.3 gives us some idea of the behaviour of the optimal design with respect to the IMSPE criterion.7 Corollary 2.4. The optimal design with respect to the IMSPE criterion is symmetric, that is if the optimum of IMSPE  Yis reached8 at (d∗ 1,d∗ 2,...,d∗ n−1, δ∗ 1, δ∗ 2, . . . , δ∗ n−1)then d∗ i=d∗ n−iand δ∗ i=δ∗ n−i,i=1,2,...,n.9 Example 2.5. Consider a five-point design, that is n=5 and s1=t1=0,s5=t5=1. By Corollary 2.4, for the optimal10 design we have d∗ 1=d∗ 4,d∗ 2=d∗ 3and δ∗ 1=δ∗ 4, δ∗ 2=δ∗ 3. In Fig. 2 the values of d∗ 1,d∗ 2and δ∗ 1, δ∗ 2are plotted as functions of11 correlation parameters αand βboth varying between 0 and 3. We remark that by symmetry, d∗ 1+d∗ 2=δ∗ 1+δ∗ 2=1/2.12 3. The optimal information gain for a shifted OU sheet13 Another approach to optimal design is to find locations which maximize the amount of information obtained. Following14 the ideas of Shewry and Wynn (1987), one has to maximize the entropy Ent(Y)of the observations corresponding to the15 chosen design, which in the Gaussian case form an n-dimensional normal vector with covariance matrix σ2C(n,r), that is16 Ent(Y)=n 21+ln(2πσ2)+1 2ln det C(n,r).17 Theorem 3.1. Under the conditions of Theorem 2.1, entropy Ent(Y)has the form18 Ent(Y)=n 21+ln(2πσ2)+1 2 n−1  i=1 ln 1−q2 i.(3.1)19 For any sample size, the equidistant design when values αdi+βδi,i=1,2,...,n−1(and in this way qi), are constant is optimal20 with respect to the entropy criterion.21 Remark 3.2. Observe that the equispaced monotonic design d1=d2= ···dn, δ1=δ2= ··· = δnclearly optimizes the22 entropy criterion.23 Remark 3.3. If the design space Xis the unit square then the optimal value of the entropy equals24 n 21+ln(2πσ2)+n−1 2ln (1−exp (−(α +β)/(n−1))) → −∞ as n→ ∞.25 1584 S. Baran et al. / Statistics and Probability Letters 83 (2013) 1580–1587 Table 1 IMSPE values corresponding to the optimal and to the equispaced monotonic design and relative efficiency of the equispaced monotonic design. nα=0.5, β =0.8α=1, β =1α=1, β =10 α=2.5, β =1.5α=3, β =3 4 Optimal 0.2602 0.4008 0.9266 0.6549 0.8487 Equispaced 0.2693 0.4010 0.9326 0.6598 0.8493 Rel. eff. (%) 96.62 99.95 99.36 99.26 99.93 5 Optimal 0.2309 0.3699 0.8290 0.5981 0.7866 Equispaced 0.2473 0.3700 0.8409 0.6065 0.7873 Rel. eff. (%) 93.37 99.97 98.58 98.62 99.91 6 Optimal 0.2130 0.3529 0.7593 0.5640 0.7502 Equispaced 0.2351 0.3530 0.7766 0.5763 0.7509 Rel. eff. (%) 90.60 99.97 97.77 97.87 99.91 7 Optimal 0.2007 0.3423 0.7066 0.5241 0.7269 Equispaced 0.2274 0.3424 0.7288 0.5571 0.7275 Rel. eff. (%) 88.26 99.97 96.95 94.08 99.92 8 Optimal 0.1692 0.3351 0.6655 0.5007 0.7111 Equispaced 0.2222 0.3352 0.6918 0.5441 0.7115 Rel. eff. (%) 76.15 99.97 96.20 92.02 99.94 9 Optimal 0.1620 0.3300 0.6325 0.4858 0.6997 Equispaced 0.2184 0.3301 0.6626 0.5348 0.7001 Rel. eff. (%) 74.18 99.97 95.46 90.84 99.94 10 Optimal 0.1570 0.3262 0.6057 0.4756 0.6912 Equispaced 0.2155 0.3262 0.6390 0.5278 0.6915 Rel. eff. (%) 72.87 99.98 94.79 90.11 99.95 Table 2 Optimalfour-pointdesigns withrespecttotheIMSPEcriterion. α=β=0.5α=β=1α=β=1.5 d1=δ10.3372 0.3401 0.3421 d2=δ20.3256 0.3199 0.3158 d3=δ30.3372 0.3401 0.3421 4. Numerical experiments 1 4.1. Comparison of the IMSPE criterion and the entropy criterion 2 In order to compare the performances of the two criteria we compare the optimal values of IMSPE( Y)calculated using 3 the fmincon function of Matlab to its values corresponding to the equispaced monotonic design which is optimal for the 4 entropy criterion. In Table 1 the values of IMSPE are given for both designs together with the relative efficiency of the 5 equispaced monotonic design with respect to the optimal one for various sample sizes and combinations of parameters 6 (α, β). Unfortunately, larger sample sizes cause technical problems in optimization, since for ndesign points one has to find 7 numerically the constrained minimum of functions with 2n−2 parameters. 8 Observe that for symmetric models when α=β, the efficiency of the equispaced monotonic design is nearly 100%, 9 while for different covariance parameters it reduces significantly. However, even in this special case the optimal designs 10 with respect to the IMSPE and entropy criteria do not coincide. As an example, consider Table 2 where distances diand δi11 corresponding to the optimal four-point design are given for three different values of α=β. Obviously, in all three cases we 12 have d1+δ1= d2+δ2, so according to Theorem 3.1 these designs cannot be optimal with respect to the entropy criterion. 13 4.2. IMSPE on a monotonic set and on a regular grid 14 In Theorem 2.1, the exact form of IMSPE is derived only for designs satisfying Condition D. However, one might ask what 15 is the relative efficiency of the optimal value of IMSPE on monotonic sets containing n=m2design points compared to the 16 IMSPE of a regular grid with the same number of points. Table 3 gives the optimal values of IMSPE on monotonic sets, IMSPE 17 values for regular designs and the relative efficiencies of the optimal IMSPE values on monotonic sets for different sample 18 sizes and combinations of parameters (α, β). Observe, that for four points the optimal monotonic design gives much better 19 IMSPE values than the regular grid and for α=β=3 the relative efficiency is slightly above 100% even in the case n=9. 20 5. Conclusion 21 We derive the exact form of the IMSPE for an OU sheet on a monotonic set. We show that the optimal design for the 22 prediction based on IMSPE may differ substantially from the equidistant one. This is in contrast both to the optimal design 23 S. Baran et al. / Statistics and Probability Letters 83 (2013) 1580–1587 1585 Table 3 IMSPE values corresponding to the optimal monotonic and to the regular grid design and relative efficiency of the optimal monotonic design. nα=0.5, β =0.8α=1, β =1α=1, β =10 α=2.5, β =1.5α=3, β =3 4 Monotonic 0.2602 0.4008 0.9266 0.6549 0.8487 Regular 0.3580 0.5389 1.1449 0.8370 1.0094 Rel. eff. (%) 137.59 134.46 123.56 127.81 118.93 9 Monotonic 0.1620 0.3300 0.6325 0.4858 0.6997 Regular 0.1527 0.3018 0.5762 0.4850 0.7011 Rel. eff. (%) 94.26 91.45 91.10 99.84 100.20 for estimation on a monotonic set (Baran and Stehlík, submitted for publication) and to the optimal design for the prediction1 of the OU process on a compact interval investigated by Baldi Antognini and Zagoraiou (2010). We also investigate the2 properties of the optimal design with respect to the entropy criterion, where constraining sample points from a rectangle3 to a monotonic set, as expected, decreases the entropy of the Gaussian field. Simulations illustrate selected cases of optimal4 designs for small number of sampling locations. Since the above discussed designs depend on the values of the correlation5 parameters, the optimal designs obtained are only locally optimal. We briefly study the dependence of the designs obtained6 on these parameters, too. Such knowledge may be crucial for an experimenter for increasing the efficiency of a design in a7 practical set-up.8 Acknowledgements9 The authors are grateful to Lenka Filová for helpful comments during the preparation of the manuscript. This research10 was supported by the Hungarian Scientific Research Fund under Grants Nos OTKA T079128/2009 and OTKA NK101680/201211 and by the Hungarian–Austrian intergovernmental S&T cooperation program TÉT_10-1-2011-0712, and partially supported12 by the TÁMOP-4.2.2.C-11/1/KONV-2012-0001 project. The project was supported by the European Union, with co-financing13 from the European Social Fund. The third author acknowledges the support of the project DESIRE. The authors are indebted14 to the unknown referees for their useful remarks that helped a lot with improving the manuscript.15 Appendix16 A.1. Proof of Theorem 2.117 To shorten our formulae, in what follows instead of ϱ(x,si,ti)we are using simply ϱi,i=1,2,...,n. Consider first18 MSPE  Y(x)given by (2.1). Short matrix algebraic calculations show that19 01⊤ n 1nC(n,r)−1 =00⊤ n 0nC−1(n,r)−1⊤ nC−1(n,r)1n−1 1−C−1(n,r)1n⊤ −C−1(n,r)1nC−1(n,r)1nC−1(n,r)1n⊤ ,20 and according to the results of Baran and Stehlík (submitted for publication) we have 1⊤ nC−1(n,r)1n=1+n−1 i=11−qi 1+qi. We21 remark that 1⊤ nC−1(n,r)1nis the Fisher information on θbased on Y. In this way, we obtain22 MSPE  Y(x)=σ21−Q⊤(x)C−1(n,r)Q(x)+1⊤ nC−1(n,r)1n−11−Q⊤(x)C−1(n,r)1n2 23 =σ21+ϱ2 1−2ϱ1ϱ2q1 q2 1−1+ϱ2 n q2 n−1−1− n−1  i=22ϱiϱi+1qi q2 i−1+ϱ2 i(1−q2 iq2 i−1) (q2 i−1)(q2 i−1−1) 24 −1+ n−1  i=1 1−qi 1+qi−1 25 ×1+ϱ1−(ϱ1+ϱ2)q1 q2 1−1+ϱn q2 n−1−1− n−1  i=2(ϱi+ϱi+1)qi q2 i−1+ϱi(1−q2 iq2 i−1) (q2 i−1)(q2 i−1−1)2  26 =σ2 1−ϱ2 n− n−1  i=1 (ϱi−ϱi+1qi)2 1−q2 i+1+ n−1  i=1 1−qi 1+qi−11−ϱn− n−1  i=1 ϱi−ϱi+1qi 1+qi2 .27 Further, according to the definition of the IMSPE criterion, we can write28 IMSPE  Y=1−An+1+ n−1  i=1 1−qi 1+qi−1 Bn,29 1586 S. Baran et al. / Statistics and Probability Letters 83 (2013) 1580–1587 where 1 An:= X Q⊤(x)C−1(n,r)Q(x)dx=tr C−1(n,r)R,2 Bn:= 1−21⊤ nC−1(n,r)W+1⊤ nC−1(n,r)RC−1(n,r)1n,3 with 4 W=(ω1, . . . , ωn)⊤:= X Q(x)dxand R=Ri,j:= X Q(x)Q⊤(x)dx.5 Obviously, 6 ωi=1 αβ 2−e−αsi−e−α(1−si)2−e−βti−e−β(1−ti)7 =1 αβ 2−e−α(d1+···+di−1)−e−α(di+···+dn−1)2−e−β(δ1+···+δi−1)−e−β(δi+···+δn−1),8 Ri,j=1 2α2e−α|si−sj|−e−α(si+sj)−e−α(2−si−sj)+|si−sj|e−α|si−sj|9 ×1 2β2e−β|ti−tj|−e−β(ti+tj)−e−β(2−ti−tj)+|ti−tj|e−β|ti−tj|10 =1 2α2e−α(di∧j+···+di∨j−1)−e−α(2d1+···+2di∧j−1+di∧j+···+di∨j−1)−e−α(di∧j+···+di∨j−1+2di∨j+···+2dn−1)11 +(di∧j+···+di∨j−1)e−α(di∧j+···+di∨j−1)12 ×1 2β2e−β(δi∧j+···+δi∨j−1)−e−β(2δ1+···+2δi∧j−1+δi∧j+···+δi∨j−1)−e−β(δi∧j+···+δi∨j−1+2δi∨j+···+2δn−1)13 +(δi∧j+···+δi∨j−1)e−β(δi∧j+···+δi∨j−1),14 where i∧j:= min{i,j},i∨j:= max{i,j},i,j∈N, and the empty sum is defined to be zero. In this way, we obtain 15 An=Rn,n+ n−1  i=1 Ri,i−2Ri+1,iqi+Ri+1,i+1q2 i 1−q2 i=R1,1+ n−1  i=1 Ri,iq2 i−2Ri+1,iqi+Ri+1,i+1 1−q2 i ,16 Bn=1−2ωn−2 n−1  i=1 ωi−ωi+1qi 1+qi+Rn,n +2 n−1  i=1 Rn,i−Rn,i+1qi 1+qi+ n−1  i=1 n−1  j=1 Ri,j−Ri+1,jqi−Ri,j+1qj+Ri+1,j+1qiqj (1+qi)(1+qj) =1−2ωn+Rn,n+ n−1  i=1Ri,i−2Ri+1,iqi+Ri+1,i+1q2 i (1+qi)2+2(Rn,i−ωi)−(Rn,i+1−ωi+1)qi 1+qi +2 n−1  i=2 i−1  j=1 Ri,j−Ri+1,jqi−Ri,j+1qj+Ri+1,j+1qiqj (1+qi)(1+qj).17 A.2. Proof of Theorem 2.3 18 Let di:= dn−i, δi:= δn−i, qi:= exp −α di−β δi,i=1,2,...,n−1, and denote by ωkand Rk,ℓ the values calculated 19 from (2.5) and (2.6), respectively, using distances ( di, δi), i=1,2,...,n−1. Further, let qi, ωkand Rk,ℓ denote the 20 corresponding quantities calculated from the original distances (di, δi), i=1,2,...,n−1. It is easy to see that 21  qi=qn−i,i=1,2,...,n−1,and ωk=ωn−k+1, Rk,ℓ =Rn−k+1,n−ℓ+1,k, ℓ =1,2,...,n,(A.1) 22 and in this way we obtain 23 n−1  i=1 1− qi 1+ qi= n−1  i=1 1−qi 1+qi .24 S. Baran et al. / Statistics and Probability Letters 83 (2013) 1580–1587 1587 Now, denote by Anand Bnthe values calculated from (2.3) and (2.4), respectively, using distances ( di, δi), i=1,2,...,n−1.1 Then with the help of (A.1), after straightforward but tedious calculations one can show that An=Anand Bn=Bn, which2 completes the proof. 3 A.3. Proof of Theorem 3.14 According to the results of Baran and Stehlík (submitted for publication),5 C(n,r)=                         1q1q1q2q1q2q3··· ··· n−1  i=1 qi q11q2q2q3··· ··· n−1  i=2 qi q1q2q21q3··· ··· n−1  i=3 qi q1q2q3q2q3q31··· ··· . . . . . .. . .. . .. . ..... . . . . .. . .. . .. . ....qn−1 n−1  i=1 qi n−1  i=2 qi n−1  i=3 qi··· ··· qn−11                         .6 Hence, one can use the same type of decomposition as Baldi Antognini and Zagoraiou (2010, Lemma 3.1), that is7 C(n,r)=LDL⊤,8 where Lis a lower triangular matrix with entries 1 in its main diagonal and Dis a diagonal matrix where in the diagonal we9 have 1,1−q2 1,1−q2 2,...,1−q2 n−1. In this way10 det C(n,r)=det D= n−1  i=1 (1−q2 i),11 which proves (3.1).12 In order to find the optimal design one has to find the constrained maximum of13 F(q1,q2,...,qn−1):= n−1  i=1 ln 1−q2 iunder condition n−1  i=1 ln qi= −α−β.14 By analysing the first partial derivatives and the Hessian of the Lagrange function15 Λ(q1,q2,...,qn−1;λ):= n−1  i=1 ln 1−q2 i+λn−1  i=1 ln qi+α+β 16 one can easily see that the maximum is reached when q1=q2= ··· = qn−1, which completes the proof. 17 References18 Babiak, J., Minárová, M., Petráš, D., 2005. Principles and calculations of temperature distribution in an active slab depending up various operation modes19 of TABS using FEM software. In: Proceeding in CLIMA Congress, Lausanne.20 Baldi Antognini, A., Zagoraiou, M., 2010. Exact optimal designs for computer experiments via Kriging metamodelling. J. Statist. Plann. Inference 140,21 2607–2617.22 Baran, S., Pap, G., Zuijlen, M.V., 2003. Estimation of the mean of stationary and nonstationary Ornstein–Uhlenbeck processes and sheets. Comput. Math.23 Appl. 45, 563–579.24 Baran, S., Sikolya, K., 2012. Parameter estimation in linear regression driven by a Gaussian sheet. Acta Sci. Math. (Szeged) 78, 689–713.25 Baran, S., Stehlík, M., 2013. Optimal designs for parameters of shifted Ornstein–Uhlenbeck sheets measured on monotonic sets. Appl. Stoch. Model. Bus.26 (submitted for publication).27 Crary, S.B., 2002. Design of computer experiments for metamodel generation. Analog Integr. Circuits Signal Process. 32, 7–16.28 Kiseľák, J., Stehlík, M., 2008. Equidistant D-optimal designs for parameters of Ornstein–Uhlenbeck process. Statist. Probab. Lett. 78, 1388–1396.29 Koizumi, H., Jin, Y.H., 2012. Performance enhancement of a latent heat thermal energy storage system using curved-slab containers. Appl. Therm. Eng. 37,30 145–153.31 Minty, G.J., 1962. Monotone (non-linear) operators in Hilbert space. Duke Math. J. 29, 341–346.32 Minty, G.J., 1963. On a monotonicity method for the solution of nonlinear equations in Banach spaces. Proc. Natl. Acad. Sci. US 50, 1038–1041.33 Pronzato, L., Müller, W.G., 2012. Design of computer experiments: space filling and beyond. Stat. Comput. 22, 681–701.34 Sacks, J., Schiller, S.B., Welch, W.J., 1989. Design for computer experiments. Technometrics 31, 41–47.35 Santner, T.J., Williams, B.J., Notz, W.I., 2003. The Design and Analysis of Computer Experiments. Springer-Verlag, New York.36 Shewry, M.C., Wynn, H.P., 1987. Maximum entropy sampling. J. Appl. Stat. 14, 165–170.37 Stehlík, M., 2004. Some properties of D-optimal designs for random fields with different variograms. Research Report Series Report No. 4, Department of38 Statistics and Mathematics, Wirtschaftsuniversität Wien, Austria.39