Halton Sequence

# Halton Sequence

In statisticsHalton sequences are sequences used
to generate points in space for numerical methods such asMonte Carlo simulations.
Although these sequences are deterministic they are
of low discrepancy, that is, appear to be random for
many purposes. They were first introduced in 1960 and are an example of a quasi-random
number
sequence. They generalise the one-dimensional van der Corput sequences.

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## Example of Halton sequence used to generate points in (0, 1) × (0, 1) in R2

The Halton sequence is constructed according to a deterministic method that uses a prime number as its base. As a
simple example, let’s take one dimension of the Halton sequence to be based on 2 and the other on 3. To generate the sequence for 2, we start by dividing the interval (0,1) in half, then in fourths, eighths, etc., which generates

12143418583878116916,…

and to generate the sequence for 3, we divide the interval (0,1) in thirds, then ninths, twenty-sevenths, etc., which generates

1323194979295989127,…

When we pair them up, we get a sequence of points in a unit square:

(1213),
(1423),
(3419),
(1849),
(5879),
(3829),
(7859),
(11689),
(916127).

Even though standard Halton sequences perform very well in low dimensions, correlation problems have been noted between sequences generated from higher primes. For example if we started with the primes 17 and 19, the first 16 pairs of points would have perfect linear
correlation
. To avoid this, it is common to drop the first 20 entries, or some other predetermined number depending on the primes chosen. In order to deal with this problem, various other methods have been proposed; one of the most prominent solutions
is the scrambled Halton sequence, which uses permutations of the coefficients used in the construction of the standard sequence.

## Implementation in Pseudo Code

```   FUNCTION (index, base)
BEGIN
result = 0;
f = 1 / base;
i = index;
WHILE (i > 0)
BEGIN
result = result + f * (i % base);
i = FLOOR(i / base);
f = f / base;
END
RETURN result;
END```