org.apache.commons.math3.random

## Class HaltonSequenceGenerator

• All Implemented Interfaces:
RandomVectorGenerator

public class HaltonSequenceGenerator
extends Object
implements RandomVectorGenerator
Implementation of a Halton sequence.

A Halton sequence is a low-discrepancy sequence generating points in the interval [0, 1] according to

   H(n) = d_0 / b + d_1 / b^2 .... d_j / b^j+1

with

n = d_j * b^j-1 + ... d_1 * b + d_0 * b^0

For higher dimensions, subsequent prime numbers are used as base, e.g. { 2, 3, 5 } for a Halton sequence in R^3.

Halton sequences are known to suffer from linear correlation for larger prime numbers, thus the individual digits are usually scrambled. This implementation already comes with support for up to 40 dimensions with optimal weight numbers from H. Chi: Scrambled quasirandom sequences and their applications.

The generator supports two modes:

Since:
3.3
Halton sequence (Wikipedia), On the Halton sequence and its scramblings
• ### Constructor Summary

Constructors
Constructor and Description
HaltonSequenceGenerator(int dimension)
Construct a new Halton sequence generator for the given space dimension.
HaltonSequenceGenerator(int dimension, int[] bases, int[] weights)
Construct a new Halton sequence generator with the given base numbers and weights for each dimension.
• ### Method Summary

Methods
Modifier and Type Method and Description
int getNextIndex()
Returns the index i of the next point in the Halton sequence that will be returned by calling nextVector().
double[] nextVector()
Generate a random vector.
protected int scramble(int i, int j, int b, int digit)
Performs scrambling of digit d_j according to the formula:
double[] skipTo(int index)
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### HaltonSequenceGenerator

public HaltonSequenceGenerator(int dimension)
throws OutOfRangeException
Construct a new Halton sequence generator for the given space dimension.
Parameters:
dimension - the space dimension
Throws:
OutOfRangeException - if the space dimension is outside the allowed range of [1, 40]
• #### HaltonSequenceGenerator

public HaltonSequenceGenerator(int dimension,
int[] bases,
int[] weights)
throws NullArgumentException,
OutOfRangeException,
DimensionMismatchException
Construct a new Halton sequence generator with the given base numbers and weights for each dimension. The length of the bases array defines the space dimension and is required to be > 0.
Parameters:
dimension - the space dimension
bases - the base number for each dimension, entries should be (pairwise) prime, may not be null
weights - the weights used during scrambling, may be null in which case no scrambling will be performed
Throws:
NullArgumentException - if base is null
OutOfRangeException - if the space dimension is outside the range [1, len], where len refers to the length of the bases array
DimensionMismatchException - if weights is non-null and the length of the input arrays differ
• ### Method Detail

• #### nextVector

public double[] nextVector()
Generate a random vector.
Specified by:
nextVector in interface RandomVectorGenerator
Returns:
a random vector as an array of double.
• #### scramble

protected int scramble(int i,
int j,
int b,
int digit)
Performs scrambling of digit d_j according to the formula:
   ( weight_i * d_j ) mod base

Implementations can override this method to do a different scrambling.
Parameters:
i - the dimension index
j - the digit index
b - the base for this dimension
digit - the j-th digit
Returns:
the scrambled digit
• #### skipTo

public double[] skipTo(int index)
throws NotPositiveException

This operation can be performed in O(1).

Parameters:
index - the index in the sequence to skip to
Returns:
the i-th point in the Halton sequence
Throws:
NotPositiveException - if index < 0
• #### getNextIndex

public int getNextIndex()
Returns the index i of the next point in the Halton sequence that will be returned by calling nextVector().
Returns:
the index of the next point