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In the field of real world or simulation experiments, latin hypercube designs are very useful to approximate implicit given functions based on experimental designs of rectangular grid of points used e.g. for an efficient optimization of black-box experiments. The so called standard latin hypercube sampling is based on an idea of [8] to simulate correlated multivariate relationships based on uncorrelated variables. Hereby, the linear or rank correlations between the random variables can be reduced using a Cholesky decomposition of the covariance matrix. However, this algorithm requires that the number of samples is higher than the number of input variables to ensure a positive definite and non-singular covariance matrix. Furthermore, a rearrangement of the latin hypercubes so that they will have the same ordering to increase the multidimensional uniformity introduces an unwanted pseudo linear correlation of the simulated variables. In this paper, the new method of proper orthogonal decomposition methods in adaptive combination with re-sorting the rank-order of the original marginal distributions not only eliminates the drawbacks of the standard latin hypercube sampling, but also reduces the number of experimental designs necessary. In contrast to the standard latin hypercube sampling, the adaptive approach results in a further improvement of the independency and the multidimensional uniformity of the variables and shows an improved efficiency in comparison to optimization-based methods. Furthermore, a lower-rank approximation of the design matrix based on singular value decomposition ensures a positive definite and non-singular correlation matrix to reduce the correlation error in a stepwise manner during the adaptive procedure.

LATIN HYPERCUBE SAMPLING BASED ON ADAPTIVE ORTHOGONAL DECOMPOSITION