Discussion: Sobol sequence

Description and background

A Sobol sequence is a sequence of points in a design space which has the general property of being space filling. The procedure page ProcSobolSequence provides a detailed algorithm for constructing Sobol sequences based on Sobol’s original description, and a worked example. Sobol’s construction is based on bit-wise XOR operations between special generators called direction numbers. The first impression is that the construction of a Sobol sequence is a complex task. However, by constructiong a few Sobol numbers by hand, using the procedure in ProcSobolSequence, it is easy to understand the underlying structure.

Discussion

For users of the R programming language, we suggest the function runif.sobol(n,d) from the package fOptions in the R repository.

A well known concern about Sobol sequences is the correlation between points in high dimensional sequences. A solution to this is by scrambling the sequence. The R package above described allows for the sequence to be scrambled, if desired. In this case, either of the two functions described above can be called with the arguments

(n,d,scrambling,seed,init)

The argument scrambling takes an integer which is 0 if no scrambling is to be used, 1 for Owen type of scrambling, 2 for Faure-Tezuka scrambling and 3 for a combination of both. The argument seed is the integer seed to trigger the scrambling process and init is a logical flag which allows the Sobol sequence to restart if set to TRUE.