Latin Hypercube Sampling On The Simplex And Its Improvement

How to improve the efficiency of design and sampling is an important research topic on approximate calculation of numerical integration. Generally speaking, a good design has low discrepancy, but it is not random, so it is hard to carry out statistical analysis. On the other hand, a sampling is random, however, there often exist bad samples. So we can combine the two together and apply it to the numerical integration. For example, Latin hypercube sampling is a combination of design and sampling. Many scholars have improved Latin hypercube sampling, resulting in uniform design sampling, OA-Based uniform Latin hypercube sampling and other methods. The efficiency of these methods to calculate the integral is higher than that of Latin hypercube sampling. These sampling methods are generally defined on the unit cube. For other special domains, although the predecessors have put forward the idea of transformation, but did not give a specific proof and analysis. Based on the above knowledge, this paper summarizes a general method from design to sampling. Then we try to apply Latin hypercube sampling and its improvement to the simplex by transformations. What’s more, we emphatically analyze the properties of these sampling methods on the simplex. According to the general Koksma-Hlawka inequality and the properties of total variation in sense of Hardy and Krause, we prove that these sampling methods still have good properties on the simplex, and the good transformation does not change the efficiency of sampling to calculate the integral. Finally, we use R software to verify this conclusion by simulation.