Low-discrepancy sequence
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy.
Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the sequence falling into an arbitrary set B is close to proportional to the measure of B, as would happen on average (but not for particular samples) in the case of an equidistributed sequence. Specific definitions of discrepancy differ regarding the choice of B (hyperspheres, hypercubes, etc.) and how the discrepancy for every B is computed (usually normalized) and combined (usually by taking the worst value).
Low-discrepancy sequences are also called quasi-random or sub-random sequences, due to their common use as a replacement of uniformly distributed random numbers. The "quasi" modifier is used to denote more clearly that the values of a low-discrepancy sequence are neither random nor pseudorandom, but such sequences share some properties of random variables and in certain applications such as the quasi-Monte Carlo method their lower discrepancy is an important advantage.
Subrandom numbers have an advantage over pure random numbers in that they cover the domain of interest quickly and evenly. They have an advantage over purely deterministic methods in that deterministic methods only give high accuracy when the number of datapoints is pre-set whereas in using subrandom sequences the accuracy typically improves continually as more datapoints are added, with full reuse of the existing points. On the other hand, subrandom sets can have a significant lower discrepancy for a given number of points than subrandom sequences.
Two useful applications are in finding the characteristic function of a probability density function, and in finding the derivative function of a deterministic function with a small amount of noise. Subrandom numbers allow higher-order moments to be calculated to high accuracy very quickly.
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