International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Improved formulation of Audze-Eglājs criterion for space-filling designs Vořechovský, Miroslav; Eliáš, Jan Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Improved Formulation Of Audze–Egla¯js Criterion For Space-FillingDesignsMiroslav VorˇechovskýProfessor, Inst. of Structural Mechanics, Brno Univ. of Technology, Brno, Czech RepublicJan EliášAssist. Prof., Inst. of Struct. Mechanics, Brno Univ. of Technology, Brno, Czech RepublicABSTRACT: The Audze–Egla¯js (AE) criterion was developed to achieve uniform distribution of ex-perimental points in a hypercube. However, the paper shows that the AE criterion provides stronglynonuniform designs due to the effect of boundaries of the hypercube. We propose a simple remedy thatlies in the assumption of periodic boundary conditions. The biased behavior of the original AE criterionand excellent performance of the modified criterion is demonstrated using simple numerical examplesfocused on (i) uniformity of the samples density over the design space and, (ii) statistical sampling effi-ciency measured through the ability to correctly estimate statistics of functions of random variables.1. INTRODUCTIONThis article, which is a promotion of a recent ar-ticle by Eliáš and Vorˇechovský (2015), considersthe choice of an experimental design for computerexperiments. The choice of experimental points isan important issue in planning an efficient computerexperiment. The methods used for formulating theplan/experimental points are collectively known asDesign of Experiments (DoE). DoE is a crucial pro-cess in many engineering tasks. Its purpose is toprovide a set of points lying inside a chosen designdomain that are optimally distributed; the optimal-ity of the experimental points depends on the natureof the problem. Various authors have suggested in-tuitive goals for good designs, including “good cov-erage”, the ability to fit complex models, many lev-els for each factor, and good projection properties.At the same time, a number of different mathemat-ical criteria have been put forth for comparing de-signs.The selection of the sampling points is impor-tant when evaluating approximations to integrals asis performed in Monte Carlo simulations (numer-ical integration), where equal sampling probabili-ties inside the design domain are required. Such adesign of experiments for Monte Carlo Samplingis typically performed in a hyper-cubical domainof Nvar dimensions, where each dimension/variable,Uv, ranges between zero and one (v = 1, . . . ,Nvar).In this paper, the design domain is a classical Nvar-dimensional unit hypercube. This design domain isto be covered by Nsim points as evenly as possible.The process of finding the experimental pointscan be understood as an optimization problem: weare searching for a design that minimizes an objec-tive function, E. After an initial set of experimentalpoints have been generated (typically via a pseudo-random generator), some modifications of them areperformed in sequential steps to find the minimumof the objective function. The quality of the designis controlled by a chosen objective function (or de-sign criterion).Several criteria (objective functions) have beendeveloped and used, e.g. the Audze-Egla¯js(AE) criterion (Audze and Egla¯js, 1977), theEuclidean MaxiMin and MinMax distance be-tween points, Modified L2 discrepancy, Wrap-Around L2-Discrepancy, Centered L2-discrepancy,the D−optimality criterion, criteria based on cor-relation (orthogonality), Voronoi tessellation, the φcriterion, dynamic modeling of an expanding lat-tice, designs maximizing entropy, integrated mean-squared error, and many others. Some authors be-112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015lieve that in order to obtain a versatile (robust) de-sign, several criteria should be used simultaneously.Several authors have proposed a combination ofuniformity criteria with Latin Hypercube Sampling(LHS) as a representative of variance reductiontechniques (these designs are sometimes named op-timal LHS). When optimizing an existing LH sam-ple, the coordinates for each variable have alreadybeen selected, so the remaining task is to performpairing (changing mutual orderings = shuffling) inorder to minimize the DoE criterion. LHS is a typeof stratified sampling technique; the coordinatesof Nsim experimental points (simulations) are sam-pled from Nsim equidistant subintervals of length1/Nsim so that every subinterval contains one andonly one point. LHS guarantees the uniform dis-tribution of experimental points along each dimen-sion where it is used, typically along all Nvar dimen-sions. The strongest LHS requirement samples onlyin an a priory chosen set of coordinates, most oftenthe centers of the intervals (called LHS-median byVorˇechovský and Novák (2009)) with coordinates(i− 0.5)/Nsim for i ∈ 〈1, 2, . . . , Nsim〉. Such a typeof LHS will be used in this paper.This paper is focused on the performance of thewidely used Audze-Egla¯js (AE) criterion and itsimprovement. It is shown that the original AE cri-terion provides designs that are not uniform. Asimple explanation for this bias that arises fromthe presence of hypercube boundaries. Therefore,a remedy leading to uniform designs that involvethe assumption of periodicity is introduced. Theremedy does not increase computational complex-ity and is extremely easy to implement in sourcecodes that already contain an evaluation of the orig-inal AE criterion. Three simple numerical exam-ples are performed to show that (i) the samplingbias in the original AE criterion leads to errors inthe estimation of moments of statistical models and(ii) the improved periodic criterion provides correctvalues with low variance.2. REVIEW OF THE ORIGINAL AE CRITERIONThe AE criterion was developed by Audze andEgla¯js (1977). The authors claimed that the cri-terion may be understood to express the poten-tial energy of a system of particles with repulsiveforces between each pair of them; minimizationof this potential energy optimizes the spatial ar-rangement of the points. The repulsive forces be-tween pairs of points are functions of their distance.The Euclidean distance, Li j, between points (real-izations) ui = (ui,1,ui,2, . . . ,ui,Nvar) and u j in Nvar-dimensional space can be expressed as a functionof their coordinatesL2i j = L2 (ui,u j)=Nvar∑v=1(∆i j,v)2(1)where ∆i j,v = |ui,v− u j,v| is the distance betweentwo points measured along (or projected onto)axis/dimension v (difference in variable Uv); |X |stands for the absolute value of X . Each variableUv ranges between zero and one, therefore ∆i j,v hasthe same limits: ∆i j,v ∈ 〈0, 1〉. The Audze-Egla¯jscriterion is defined using the squared Euclidean dis-tances between all pairs of experimental points asEAE =Nsim∑i=1Nsim∑j=i+11L2i j(2)3. OPTIMIZATION OF A SAMPLE USING THE AECRITERIONOne of the possible applications of the AE crite-rion is in the optimization of samples used in MonteCarlo numerical integration, e.g. statistical analy-ses of functions involving random variables. It iswell recognized that simple random sampling of theMonte Carlo type does not perform well when itcomes to uniformly spreading out the sample pointswith respect to the target density function. This is apronounced issue especially for small sample sizes.An improvement in reducing the variance of esti-mated statistics can be achieved by LHS. The AEcriterion can then be employed in combination withthe LHS strategy, the concept for which appeared in(Bates et al., 2003).In sampling analyses, the preparation of a sam-ple is, in fact, the preparation of a sampling plan,i.e. a matrix of size Nsim×Nvar. Fig. 1 left showsa sampling plan for two variables and six simula-tions in the form of a table. When combining sam-pling strategies with given coordinates for each sep-arate variable (as in the case of LHS), the only way212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 1: Swap (exchange) of coordinates i and j ofvariable U2 in the sampling plan (left); in a bivari-ate scatterplot (middle); affected elements in a squaresymmetrical matrix of inverted squared pair distancesbetween the points (right).to optimize the sample with respect to a particu-lar criterion (correlation, AE criterion) is to changethe mutual ordering of these samples. In this pa-per, we focus on LHS, in which each variable (di-mension) has Nsim fixed coordinates (e.g. centers ofNsim equidistant intervals). To minimize EAE, onecan search among all (Nsim!)Nvar−1 possible mutualorderings of these fixed coordinates. As shown byBates et al. (2003), genetic algorithms can be effec-tively utilized. However, we use simulated anneal-ing optimization (Vorˇechovský and Novák, 2009)to search for a good solution instead. It involves thesubsequent swapping of the coordinates of a pair ofpoints (see the exchange of coordinates in Fig. 1middle). Details regarding this heuristic optimiza-tion algorithm can be found in (Vorˇechovský andNovák, 2009).At this point, we should mention that the AE cri-terion can also be used when optimizing samplesobtained by crude Monte Carlo Sampling (indepen-dent sampling on the 〈0, 1〉 interval).4. BIASED DESIGNThe supposed uniformity of the original AE designis critically evaluated in this section. The unifor-mity of point distribution can be measured as fol-lows. The probability that the i-th experimentalpoint will be located inside some chosen subset ofthe domain must be equal to VS/VD, with VS beingthe subset volume and VD the volume of the wholedomain (for unconstrained design VD = 1).Since we are using LHS, the coordinates of thepoints are known and the whole unit hypercube ofvolume VD = 1Nvar = 1 can be divided into NsimNvarFigure 2: The considered design domain – a unit hy-percube (Nvar = 3) divided into bins of equal volumes.Boxes with Nsim = 4 experimental points are high-lighted.bins of the same volume using the grid of equidis-tant coordinates along each dimension, see Fig. 2.A uniform design is achieved if the probability offilling each of these bins is identical. In order toperform a numerical test of AE-optimized LHS de-signs, Nrun designs (sampling plans of dimensionsNvar × Nsim) have been simulated and optimized.After generating the Nrun designs, the total numberof sampled points is NsimNrun. The average numberof points inside one bin should be for uniform de-sign fa = NsimNrun/NsimNvar = Nrun/NsimNvar−1. Foreach bin, we now count the actual frequency of oc-currence of the points inside that bin, f . Finally, wedefine a variable f¯ (a normalized frequency) thatcan be calculated for each bin using the ratiof¯ = f1fa= fNsimNvar−1Nrun(3)An ideal design criterion should produce f¯ = 1 forevery possible bin.The results of the numerical study are shown inFig. 3 for various numbers of samples Nsim and di-mensions Nvar in 2D images. The number of repet-itive optimized designs used, Nrun = 107, is highenough to reveal unwanted patterns. The gray colorrepresents the f¯ value at individual LHS points(bins). The first dimension (variable) is associatedwith the horizontal axis, the second variable withthe vertical axis, and the third variable (if present) iscaptured by repetitive 2D images (slices) producedfor different values of the third coordinate. Simi-larly, the fourth dimension (if present) is shown byrepetitive views of 3D plots made for different val-ues of the fourth coordinate.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150.01 0.34 1.89 1.40 1.73 1.40 1.89 0.34 0.010.34 1.86 1.29 0.94 0.15 0.94 1.29 1.86 0.341.89 1.29 0.18 0.49 1.32 0.49 0.18 1.29 1.891.40 0.94 0.49 1.31 0.72 1.31 0.49 0.94 1.401.73 0.15 1.32 0.72 1.16 0.72 1.32 0.15 1.731.40 0.94 0.49 1.31 0.72 1.31 0.49 0.94 1.401.89 1.29 0.18 0.49 1.32 0.49 0.18 1.29 1.890.34 1.86 1.29 0.94 0.15 0.94 1.29 1.86 0.340.01 0.34 1.89 1.40 1.73 1.40 1.89 0.34 0.01Figure 3: LHS designs using the original Audze–Egla¯js (AE) criterion. Relative frequencies f¯ calculated fromNrun = 107 designs with various numbers of simulations Nsim. Top row: Nvar = 2 variables. Bottom row: Nvar = 3and 4. The proposed PAE criterion leads to uniform gray color in all cases.The figures clearly show the non-uniformity ofpoint density in the design domain when the orig-inal AE criterion is used for optimization. In 2Dspace, the corners are not sampled at all, but there isan area of highly probable points close to them fol-lowed again by an improbable region. A similar be-havior is observed in 3D and 4D spaces, where thecorners of the domain are always sampled poorly.The plot for Nvar = 3 shows a formation resem-bling a sphere with an empty interior (low Nsim)or with a less-accentuated second interior sphere(higher Nsim). For Nvar = 4, a similar hypersphereforms. Generally, the tendency to avoid the cornersof the hypercube is apparent. The more coordinatesthat reach its extremes (0 or 1) at the corner/edge,the more pronounced the effect, i.e. in 3D, cor-ners are more “repellent” than edges. Appendix Aof the paper by Eliáš and Vorˇechovský (2015) pro-vides proof that the original AE criterion deliversa non-uniform distribution of points over the designspace, and thus explains the source of the bias.After the above demonstration of the undesirednon-uniformity of the AE criterion, we continuewith a description of a simple and computationallycheap remedy that provides uniform designs whilekeeping the concept of the AE criterion (the anal-ogy between the point distribution and minimizingthe energy of a system of particles) unchanged.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20155. AE CRITERION IN PERIODIC SPACEThe remedy proposed by Eliáš and Vorˇechovský(2015), that leads to uniform designs, is based onperiodic repetition of the unit hypercube (see Fig. 2)containing the experimental points, along all direc-tions/variables of the design domain. In order tosimplify the situation with a large number of (oreven infinitely many) periodic images of each point,the authors shown in (Eliáš and Vorˇechovský, 2015)that it suffices to take only the nearest image ofpoint. The proposed simplification lies in consider-ing only the shortest distance, Li j, among all thesepairs, see the very thick line in Fig. 4 for the case ofNvar = 2. The squared shortest distance Li j betweenFigure 4: Periodic space, the shortest distance Li j andall pairs in the first layer around point ui.two points i and all its periodic images is given byexpressionL2i j =Nvar∑v=1[min(∆i j,v,1−∆i j,v)]2 (4)6. UNIFORMITY OF THE PROPOSED PERIODICAUDZE-EGLA¯JS CRITERIONMotivated by the observation described in the pre-vious section, an improvement of the AE criterionis proposed. To distinguish between the original AEformulation (Audze and Egla¯js, 1977) and the pro-posed one based on periodic space, we will call thenew formulation the Periodic Audze-Egla¯js (PAE)criterion. The proposed PAE criterion has the fol-lowing formEPAE =Nsim∑i=1Nsim∑j=i+11L2i j, (5)By comparing Eq. (5) with the original formula-tion in Eq. (2) one can see that the only differ-ence between them lies in selecting the minima be-tween (∆i j,v) and (1−∆i j,v) along each coordinatev ∈ 〈1,Nvar〉 instead of using the coordinate differ-ence (∆i j,v) directly. Technically, this improvementis very easy to implement in computer programsand the additional computer time necessary to per-form the comparison and selection of the minimais inconsiderable. Fig. 3 calculated with the pro-posed PAE criterion is just a uniformly gray rectan-gle clearly demonstrating that the new formulationprovides truly uniform designs.The source of uniformity actually lies in the in-variance of PAE with respect to translation. If allthe points in periodic space are shifted by an ar-bitrary vector, the PAE value remains unchanged.The invariance with respect to translation is simpleto show (Eliáš and Vorˇechovský, 2015).7. APPLICATION TO STATISTICAL ANALYSESOF FUNCTIONS OF RANDOM VARIABLESAs mentioned above, one of the frequent uses ofDoE is in statistical sampling for Monte Carlo in-tegration. We present the application of statis-tical sampling to the problem of estimating sta-tistical moments of a function of random vari-ables. In particular, a deterministic function, Z =g(X ), is considered, which can be a computa-tional model or a physical experiment. Z is theuncertain response variable (or generally a vec-tor of the outputs). The vector X ∈ RNvar is con-sidered to be a random vector of Nvar continuousmarginals (input random variables describing un-certainties/randomness) with a given joint probabil-ity density function (PDF).Estimation of the statistical moments of variableZ = g(X ) is, in fact, an estimation of integrals over512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015domains of random variables weighted by a givenjoint PDF of the input random vector fX (x). Weseek the statistical parameters of Z = g(X ) in theform of the following integral:E[S [g(X )]] =∞∫−∞. . .∞∫−∞S [g(x)] dFX (x) (6)where dFX (x) = fX (x) · dx1 dx2 · · · dxNvar is theinfinitesimal probability (FX denotes the joint cu-mulative density function) and where the particularform of the function S [g(·)] depends on the statis-tical parameter of interest. For example, to gainthe mean value, S[g(·)] = g(·). Computation ofthe variance involves additionally the second powerS[g(·)] = g2 (·).In Monte Carlo sampling, which is the mostprevalent statistical sampling technique, the aboveintegrals are numerically estimated via calculationof averages of functions of random variables ob-tained using Nsim sampled points (realizations, asample) that are selected with the same probabil-ity of occurrence 1/Nsim. Practically, this can beachieved by reproducing a uniform distribution inthe design space (unit hypercube) that representsthe space of sampling probabilities. We now limitourselves to independent random variables in vec-tor X .We now assume an estimate of integral in Eq. (6)by the following statistic (the average computed us-ing Nsim realizations of U , namely the samplingpoints u j ( j = 1, . . . ,Nsim))E[S [g(X )]]≈1NsimNsim∑i=1S [g(xi)] (7)where the sampling points xi ={xi,1, . . . ,xi,v, . . . ,xi,Nvar} are selected using thetransformation xi,v = F−1v (ui,v), in which weassume that each of the Nsim sampling pointsui (i = 1, . . . ,Nsim) were selected with the sameprobability of 1/Nsim. Violation of the uniformityof the distribution of points ui in the unit hypercubemay lead to erroneous estimations of the integrals.We will show that the original definition of the AEcriterion suffers from this problem.7.1. Numerical examplesThis section continues with two numerical ex-amples presenting transformations of standard in-dependent Gaussian random variables Xv, v =1, . . . ,Nvar. The non-uniformity of the original AEcriterion and also the improved performance of theproposed PAE criterion will be demonstrated byshowing the ability of the optimized samples to es-timate the mean value and standard deviation (de-noted as µZ , σZ) of the transformed variable Z =g(X ).We consider two functions (transformations ofinput random variables):Zsum = gsum(X ) =Nvar∑v=1X2v (8)Zexp = gexp (X) =Nvar∑v=1exp(−X2v)(9)where the input variables Xv, v = 1, . . . ,Nvar are in-dependent standard Gaussian variables.The first random variable Zsum has a chi-squareddistribution (also chi-square or χ2-distribution)with Nvar degrees of freedom. The standard de-viation of this distribution is well known: σsum =√2Nvar. The chi-squared distribution slowly con-verges to a Gaussian distribution as Nvar→ ∞.The second random variable Zexp has the ex-act statistical moments derived in (Eliáš and Vorˇe-chovský, 2015). The approximate standard devia-tion is σexp ≈ 0.337461√Nvar.The quality of sampling is measured through thedifference between the theoretical and estimatedstatistical parameters of Z. Since the placement ofNsim design points into an Nvar-dimensional hyper-cube is random (it depends on sequences generatedby a pseudo-random number generator), the esti-mated statistical parameter can also be viewed asa realization of a random variable. The simulatedannealing algorithm by Vorˇechovský and Novák(2009) has been used for the optimization of themutual ordering of LHS samples for three DoE cri-teria: AE, PAE and COR (Pearson’s correlation co-efficient). A relatively high number of Nrun = 103designs were optimized for the same settings (cri-terion, Nsim and Nvar) and the mean value and stan-612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015dard deviation of the estimated statistical parame-ters have been plotted in the form of graphs – de-pendencies on the sample size Nsim; see Fig. 5. Thegraphs show the exact solution for each selectedstatistical parameter via a dashed line. The aver-age result of its estimation is plotted by a full linesurrounded by a scatter-band representing the av-erage ± one standard deviation. Such graphs givean idea about the convergence of the average es-timation and also the variance of the estimation.Crude Monte Carlo results are not presented as theestimates exhibit a large variance. In LHS, thevariability of the estimate is never higher than incrude Monte Carlo sampling because the selectionof sampling probabilities is deterministic and theonly variability arises from random mutual pairing.The ability to estimate the mean value is pur-posely not presented. The reason is that samplesoptimized with all three criteria (COR, AE andPAE) provide exactly the same estimates of µsumand µexp. These estimates have no variability as thefunctions are additive and, in the LHS method, theaverages are independent of the mutual ordering ofsamples.The general trends are as follows:• The AE criterion yields, on average, erroneousestimates (with almost no variability) for thestudied statistical characteristics of the inves-tigated three g functions. The error becomespronounced for higher Nvar. Increasing thesample size Nsim does not help. Both the incor-rect means and the low variance of estimatesare consequences of non-uniform sampling:some regions are under-represented (such ascorners) and others are over-represented (asdocumented in Fig. 3 where the AE samplesoccur only in hyper-spheres inside the hyper-cube).• The PAE criterion yields a uniform distribu-tion of sample points and therefore the estima-tors converge to the exact values.• The COR criterion yields a uniform distribu-tion of sample points for Nsim→ ∞; however,for small Nsim the algorithm selects from a lim-ited number of optimal arrangements (Vorˇe-chovský, 2012) that are not uniform.• Estimates obtained with PAE are, on average,never worse than with COR – in most of thecases they are better. Also, the variance of theestimates obtained with PAE is never higherthan that gained from COR. In other words,the estimates converge faster with smaller vari-ance.The best of the three methods seems to be thePAE criterion.More numerical examples showing also the abil-ity to estimate higher statistical moments are avail-able in the paper by Eliáš and Vorˇechovský (2015).8. CONCLUSIONSIt has been shown that the original Audze-Egla¯jscriterion used for optimization of the design ofexperiments provides a non-uniform experimentalpoint distribution. Though this feature is not im-portant in several applications of Design of Ex-periments, it is the crucial property in numericalMonte-Carlo type integration. Similarly, the cor-relation criterion of optimization (COR) also yieldsnonuniform coverage of the design domain; how-ever, the problem disappears when sample size Nsimincreases. Since the widely used AE criterion sam-ples more frequently in some subregions of thedesign domain while leaving other areas under-represented, numerical integration using the AE cri-terion provides incorrect results.A simple remedy based on considering the pe-riodicity of the design space was proposed, and itwas demonstrated that the modified version – thePeriodic Audze-Egla¯js (PAE) criterion – providesa truly uniform distribution of points in the designdomain. The PAE and AE criteria have the samecomputational complexity, so no additional effort isassociated with considering the proposed scheme.The proposed PAE criterion is invariant with re-spect to shifts of the whole sample in any direction.The numerical studies presented in this paperwere performed with Latin Hypercube Samples(optimized using three criteria). However, the crit-icism of the AE criterion and the remedy usingthe proposed PAE criterion also holds for crude712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015 0 0.2 0.4 0.6 0.8 1 1.2 1.41 101102103Nvar = 9 0 0.2 0.4 0.6 0.8 1 Nvar = 5 0 0.2 0.4 0.6 0.8 Nvar = 3 0 0.2 0.4 0.6Nvar = 2COR  AE  PAE  Sample size  NsimStandard deviation  σexp 0.4 0  3  6  9 0.4 0  1  2  5 3  4 0.5 0  1  2  3 1 0  1  2μσ σ 0 1 2 3 41 101102103Sample size  N 0 1 2 3 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2COR  AE  PAE  Nvar = 9Nvar = 5Nvar = 3Nvar = 2simStandard deviation  σsumFigure 5: Convergence of estimated standard deviations σsum and σexp with increasing sample size Nsim. Theinsets visualize the PDF of Zexp.Monte Carlo Sampling (independent sampling onthe 〈0, 1〉 interval).The proposed criterion is implemented in FReETsoftware (Novák et al., 2014). It has also been im-plemented for the optimization of a sample in thesample size extension of an existing LH sample (amethod proposed by Vorˇechovský (2015)).ACKNOWLEDGEMENTSThis work has been supported by the Czech ScienceFoundation under projects Nos. 13-19416J and 15-19865Y and a Specific University Research projectMŠMT No. FAST-J-15-2868. The support is grate-fully acknowledged.9. REFERENCESAudze, P. and Egla¯js, V. (1977). “New approach forplanning out of experiments.” Problems of Dynamicsand Strengths, 35, 104–107 (in Russian).Bates, S., Sienz, J., and Langley, D. (2003). “For-mulation of the Audze–Eglais uniform Latin Hyper-cube design of experiments.” Advances in Engineer-ing Software, 34(8), 493–506.Eliáš, J. and Vorˇechovský, M. (2015). “Modification ofthe Audze–Egla¯js criterion to achieve a uniform dis-tribution of sampling points.” Computer-Aided Civiland Infrastructure Engineering, in review.Novák, D., Vorˇechovský, M., and Teplý, B. (2014).“FReET: Software for the statistical and reliabil-ity analysis of engineering problems and FReET-D:Degradation module.” Advances in Engineering Soft-ware (Elsevier), 72, 179–192 Special Issue dedicatedto Professor Zdeneˇk Bittnar on the occasion of hisSeventieth Birthday: Part 2.Vorˇechovský, M. (2012). “Correlation control in smallsample Monte Carlo type simulations II: Analysisof estimation formulas, random correlation and per-fect uncorrelatedness.” Probabilistic Engineering Me-chanics, 29, 105–120.Vorˇechovský, M. (2015). “Hierarchical refinement oflatin hypercube samples.” Computer-Aided Civil andInfrastructure Engineering, 30(5), 1–18.Vorˇechovský, M. and Novák, D. (2009). “Correlationcontrol in small sample Monte Carlo type simulationsI: A Simulated Annealing approach.” ProbabilisticEngineering Mechanics, 24(3), 452–462.8

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