This is done by random assigning the order numbers of layers to the individual input quantities. If the output variable ydepends on several input quantities, x 1, x 2., x m, it is necessary that each quantity is assigned values of all layers and that the quantities and layers of individual variables are randomly combined. This approach is called stratified sampling. With reasonably high number of layers (tens or hundreds), the created quantity xwill have the proper probability distribution. The interval (0 1) is divided into several layers of the same width, and the xvalues are calculated via the inverse transformation ( F –1) for the Fvalues corresponding to the center of each layer. The difference is that LHS creates the values of Fnot by generating random numbers dispersed in chaotic way in the interval (0 1), but by assigning them certain fix values. The basic idea of LHS is similar to the generation of random numbers via the inverse probabilistic transformation (3) and Figure 2 shown in Chapter 15. This problem can appear especially if the output function depends on many input variables.Ī method called Latin Hypercube Sampling (LHS) removes this drawback. Sometimes, more numbers are generated in one region than in others, and the generated quantity has thus somewhat different distribution than demanded. Second, it can happen that the generated random numbers of distribution function F(which serves for the creation of random numbers with nonstandard distributions) are not distributed sufficiently and regularly in the definition interval (0 1). If the output quantity must be obtained by time-consuming numerical computations, the simulations can last a very long time, and the response surface method is not always usable. First, it usually needs a very high number of simulations. Also available as ArXiv article 0710.The Monte Carlo method has two disadvantages. Bayesian treed Gaussian process models with an application to computer modeling Journal of the American Statistical Association, to appear. tgp: An R Package for Bayesian Nonstationary, Semiparametric Nonlinear Regression and Design by Treed Gaussian Process Models. See Also: lhs, exp2d, exp2d.Z, btgp, and other b* functions Zero-mean normal noise with sd=0.001 is added to the responses Z and ZZ Thus creating the outputs Ztrue and ZZtrue. In all three cases, the response is evaluated as A total of lh*dopt candidates will be used for sequential D-optimal subsampling for predictive locations XX in all four quadrants assuming the already-sampled X locations will be in the design. Similarly n2*dopt in the rest of the un-interesting region. Of the n1 + n2 = 2 then n1*dopt LH candidates are used for to get a D-optimal subsample of size n1 from the first (interesting) quadrant. Numeric vector describing the responses (without noise) at the XX predictive locationsĭetails: When is.null(lh), data is subsampled without replacement from data(exp2d). Plot(edlhd$X, main="D-optimally Sampled Inputs") Persp(edlhd.g, main="loess surface", theta=-30, phi=20, # D-optimal subsample with a factor of 10 (more) candidates Plot(edlh$X, main="Latin Hypercube Sampled Inputs") Persp(edlh.g, main="loess surface", theta=-30, phi=20, Plot(eds$X, main="Randomly Subsampled Inputs")Įdlh.g <- interp.loess(edlh$X, edlh$X, edlh$Z, span=0.5) Persp(eds.g, main="loess surface", theta=-30, phi=20, # perspective plot, and plot of the input (X & XX) locations # higher span = 0.5 required because the data is sparseĮds.g <- interp.loess(eds$X, eds$X, eds$Z, span=0.5) This argument only makes sense when !is.null(lh) If dopt >= 2 then d-optimal subsampling from LH candidates of the multiple indicated by the value of dopt will be used.
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