org.apache.commons.math3.random

## Class HaltonSequenceGenerator

- java.lang.Object
- - org.apache.commons.math3.random.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:

  - sequential generation of points: `nextVector()`
  - random access to the i-th point in the sequence: `skipTo(int)`
  Since:
    3.3
  Version:
    $Id: HaltonSequenceGenerator.java 1512043 2013-08-08 21:27:57Z tn $
  See Also:
    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)` Skip to the i-th point in the Halton sequence.

    - ### 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

      Skip to the i-th point in the Halton sequence.

      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

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