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Latin Hypercube Sampling

McKay, Conover, and Beckman first advocated this sampling method in 1979 to deal with the generation of variables from multivariate distributions. 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. In fact, Latin hypercube sampling tends to be a more powerful and efficient method than the stratified sampling method when generating multidimensional variables.

To generate uniform numbers for higher dimensions, one could simply (and blindly) extend the stratified sampling method to higher dimensions. 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). 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.

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. 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.

Given the above backdrop, I will now show an application of this method when generating three two-dimensional uniform numbers. As in the stratified sampling method, there would be three equal strata defined by the intervals (0, 1 /3), (1 /3,2/3), and (2/3, 1) in both the dimensions. One can randomly pick a stratum in each dimension – ending with a randomly selected stratum pair. Repeating this selection process over the remaining strata two more times, one can arrive at another two pairs of the two-dimensional strata. To get the actual two-dimensional variates, one randomly generates a number from each randomly sampled stratum – eventually ending with three pairs of two-dimensional, uniform numbers. The implementation of this is shown in Table 4.3.

As can be seen from Table 4.3, the 3 simulated pairs are (0.74685, 0.34243), (0.04615, 0.04105), and (0.41683, 0.74714) – where the strata are randomly selected in rows 2 to 4 and 9 to 11. Furthermore, what is presented in Table 4.3 is just one way to implement this in Microsoft Excel although, in practice, there are other, better ways to do this especially when the size of strata selected (n) gets large. The three sampled pairs can be intuitively represented as shown in Table 4.4.

As can be seen in Table 4.4, any selected pair provides the only contribution to the stratum inherent in the pair. For example, since one of the selected pair comes from stratum 1 in both the dimensions, there would not be any further sampled pair that would contain stratum 1 even as an ordinate of the pair.

In implementing the Latin hypercube method, I have conveniently assumed that the variates generated across each dimension are done independently. In practice, depending on the multivariates that need to be generated, it may be necessary for these variates to have a degree of dependence (something that I have not discussed in my implementation).

The sampling methods described above are just some examples of how practitioners generate uniform numbers. Other methods that are also used in practice include low discrepancy sequences, importance sampling, and randomized Monte Carlo sampling. The interested reader is again referred to Glasserman (2003) for more details.

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