Then, the reservation problem can be solved by graph theory. The relationship of original sampling points in new LHS is expressed by a graph. As the generation of new sampling points is almost the same as integral-multiple extension, the reservation of original sampling points is the main problem to discuss. The extension algorithm includes two parts: the reservation of original sampling points and the generation of new ones.
In this paper, we would like to obtain a strict extension of LHS (ELHS) rather than an approximate one and the new LHS contains original sampling points as many as possible. The approximate LHS does not satisfy the definition and is harmful to the extension with some criteria, such as correlated variables, maximizing minimum distance, orthogonal array, and so on. However, a sample is a LHS if (and only if) there is only one point in each variable interval. Wei also proposed a general extension algorithm to get an approximate LHS, which might have no point falling into a variable interval. Wang and Blatman and Sudret obtained an approximate LHS of a larger size, which might have two or more original points falling into the same variable interval. In this study, we consider the general extension algorithm of LHS where the new sample size is more controllable and the algorithm can be applied more widely. The integral-multiple extension algorithms have a good feature that can obtain a strict LHS of larger size and simultaneously preserve all the original sampling points. Some related papers were produced by Vorechovsky, where the new sampling size is multiple times more than the original sampling size. A special integral-multiple extension method called -extended LHS method was illustrated, where the new LHS contains smaller LHSs. A nested Latin hypercube design with two layers is defined to be a Latin hypercube design that contains a smaller Latin hypercube design as a subset. Later, two related techniques appeared in the papers named “nested Latin hypercube design” and “nested orthogonal array-based Latin hypercube design”. gave a two-multiple extension algorithm of LHS with correlated variables. Tong proposed integral-multiple extension algorithms for stratified sampling methods including LHS. But the LHS structure makes it difficult to increase the size based on an original LHS while simultaneously keeping the stratification properties of LHS.Ī special extension case is the integral-multiple extension where the new LHS is integral times the size of the original sampling. The other is to consider the extension of LHS when the original LHS was subsequently determined to be too small and a new LHS of a larger size without original sampling points might be time consuming. One is sequential sampling for sequential analysis, adaptive metamodeling, and so on. There are at least two situations that need the extension of sampling, especially for time consuming simulation systems. The extension of LHS is to obtain a LHS of a larger size that reserves the preexisting LHS (or the original LHS). Compared with other random or stratified sampling algorithms, LHS has a better space filling effect, better robustness, and better convergence character. Latin Hypercube Sampling (LHS) is one of the most popular sampling approaches, which is widely used in the fields of simulation experiment design, uncertainty analysis, adaptive metamodeling, reliability analysis, and probabilistic load flow analysis. These algorithms are illustrated by an example and applied to evaluating the sample means to demonstrate the effectiveness. Therefore, a general extension algorithm based on greedy algorithm is proposed to reduce the extension time, which cannot guarantee to contain the most original points. The basic general extension algorithm is proposed to reserve the most original points, but it costs too much time. The relationship of original sampling points in the new LHS structure is shown by a simple undirected acyclic graph. In order to get a strict LHS of larger size, some original points might be deleted. The extension algorithms start with an original LHS of size and construct a new LHS of size that contains the original points as many as possible. For reserving original sampling points to reduce the simulation runs, two general extension algorithms of Latin Hypercube Sampling (LHS) are proposed.