Given the above backdrop, I will now show an application of this method when generating three two-dimensional uniform numbers. Once this is done, the process is then repeated over all the remaining strata in each of these dimensions, so as to end up with n d-dimensional numbers – which was what McKay, Conover, and Beckman proposed with their latin hypercube sampling method. If only n (and not nd) d-dimensional numbers are required, one can randomly pick a stratum (from the n strata) for each of the d dimensions and then select a random number from each of the selected stratum to end up with a d-dimensional uniform generated number. Clearly as the dimension grows, so does the amount of computational time required to generate such numbers – resulting in an inefficient way of extending the stratified method.
#Multilevel latin hypercube sampling pdf#
Thus, if one needs to generate a variate from a multidimensional uniform pdf (with dimension d) one can simply divide the (0,1) interval across each dimension into n equal-sized strata and then generate a number in each stratum – resulting in nd tuplets (where each tuplet is d-dimensional). To generate uniform numbers for higher dimensions, one could simply (and blindly) extend the stratified sampling method to higher dimensions. In fact, Latin hypercube sampling tends to be a more powerful and efficient method than the stratified sampling method when generating multidimensional variables. While this method is identical to the stratified sampling method when generating variables from univariate distributions, the process undertaken in this method to generate a multidimensional variate is slightly different from that using a stratified sampling method. McKay, Conover, and Beckman first advocated this sampling method in 1979 to deal with the generation of variables from multivariate distributions.