Extension of Latin Hypercube Samples with Correlated Variables

C. J. SALLABERRY,1* J. C. HELTON2 – S. C. HORA3
aSandia National Laboratories, New Mexico, USA
bDepartment of Mathematics and Statistics, Arizona State University, Temp, USA
cUniversity of Hawaii at Hilo, USA
cnsalla@sandia.gov

Latin Hypercube Sampling (LHS) [1] is widely used as sampling based method for probabilistic calculations in support of both uncertainty analyses and sensitivity analyses ([2], [3]). This method has some clear advantages over classical random sampling (RS) that derive from its efficient stratification properties. However, LHS is not perfect. One of its limitations is that it is not possible to extend the size of an initial sample by simply adding new simulations, as this will lead to a loss of the efficient stratification associated with LHS. We describe a new method to extend the size of an LHS to n (>=2) times its original size while preserving both the LHS structure and any induced correlations between the input parameters. This method involves introducing a refined grid for the original sample (Figure 1) and then filling in empty rows and columns with new data in a way that conserves both the LHS structure and any induced correlations (Figure 2). An estimate of the bounds of the resulting correlation between two variables is derived for n=2. This result shows that the final correlation gets closer to the average of the correlations from the original sample and the new sample used in the infilling of the empty rows and columns indicated above as the sample size increase.

Figure 1: refinement of original grid and selection of an area for extended LHS, respecting correlation

Figure 2: Random selection of points in authorized area for extended LHS

Acknowledgement:
Work performed for Sandia National Laboratories (SNL), which is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Security Administration under contract DE-AC04-94AL-85000. Review at SNL provided by L. Swiler and R. Jarek. Editorial support provided by F. Puffer and J. Ripple of Tech Reps, a division of Ktech Corporation.

References

[1] McKay MD, Beckman RJ, Conover WJ: A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code. Technometrics 21(2), 239-245 (1979)

[2] Saltelli A, Tarantola S, Chan KP-S: A Quantitative Model-Independent Method for Global Sensitivity Analysis of Model Output. Technometrics 41(1), 39-56 (1999)
[3] Helton JC and FJ Davis: Sampling-Based Methods, In: Sensitivity Analysis. Eds. A. Saltelli, K. Chan, and E.M. Scott. New York, NY: Wiley. 2000, pp. 101-153.