HALTON—Generate low-discrepancy sequence
Posted: Wed Jun 29, 2011 11:51 am
This procedure generates an n-vector of the Halton sequence for the given base (which should be a prime number). Halton sequences are deterministic "low-discrepancy" sequences which fill the unit cube relatively uniformly (with a different base for each dimension) allowing numerical integrals to be estimated more accurately for a given number of function evaluations than would be possible with truly random numbers.
Halton, J. (1964), "Algorithm 247: Radical-inverse quasi-random point sequence", ACM, p. 701
This is an example (from Greene's Econometric Analysis) which uses the procedure to do a "random effects" panel data estimation for a geometric model, which optimization over a function which requires a numerical integral. You can't use standard MC integration to do the integral since each function evaluation would give slightly different results. This particular example probably would be better done with Gauss-Hermite integration since it's a one-dimensional integral over a standard Normal, which can be done with fewer function evaluations using the weighted function values of Gauss-Hermite rather than the flat weights of the Halton sequence.
Halton, J. (1964), "Algorithm 247: Radical-inverse quasi-random point sequence", ACM, p. 701
This is an example (from Greene's Econometric Analysis) which uses the procedure to do a "random effects" panel data estimation for a geometric model, which optimization over a function which requires a numerical integral. You can't use standard MC integration to do the integral since each function evaluation would give slightly different results. This particular example probably would be better done with Gauss-Hermite integration since it's a one-dimensional integral over a standard Normal, which can be done with fewer function evaluations using the weighted function values of Gauss-Hermite rather than the flat weights of the Halton sequence.