[MATH-977] Added implementation of a HaltonSequence.
git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1485346 13f79535-47bb-0310-9956-ffa450edef68
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Thomas Neidhart
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/*
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* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with
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* this work for additional information regarding copyright ownership.
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* (the "License"); you may not use this file except in compliance with
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* the License. You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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package org.apache.commons.math3.random;
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import org.apache.commons.math3.exception.DimensionMismatchException;
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import org.apache.commons.math3.exception.NotPositiveException;
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import org.apache.commons.math3.exception.NullArgumentException;
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import org.apache.commons.math3.exception.OutOfRangeException;
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import org.apache.commons.math3.util.MathUtils;
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/**
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* Implementation of a Halton sequence.
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* <p>
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* A Halton sequence is a low-discrepancy sequence generating points in the interval [0, 1] according to
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* <pre>
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* H(n) = d_0 / b + d_1 / b^2 .... d_j / b^j+1
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*
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* with
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*
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* n = d_j * b^j-1 + ... d_1 * b + d_0 * b^0
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* </pre>
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* For higher dimensions, subsequent prime numbers are used as base, e.g. { 2, 3, 5 } for a Halton sequence in R^3.
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* <p>
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* Halton sequences are known to suffer from linear correlation for larger prime numbers, thus the individual digits
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* are usually scrambled. This implementation already comes with support for up to 40 dimensions with optimal weight
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* numbers from <a href="http://etd.lib.fsu.edu/theses/available/etd-07062004-140409/unrestricted/dissertation1.pdf">
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* H. Chi: Scrambled quasirandom sequences and their applications</a>.
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* <p>
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* The generator supports two modes:
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* <ul>
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* <li>sequential generation of points: {@link #nextVector()}</li>
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* <li>random access to the i-th point in the sequence: {@link #skipTo(int)}</li>
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* </ul>
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*
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* @see <a href="http://en.wikipedia.org/wiki/Halton_sequence">Halton sequence (Wikipedia)</a>
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* @see <a href="https://lirias.kuleuven.be/bitstream/123456789/131168/1/mcm2005_bartv.pdf">
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* On the Halton sequence and its scramblings</a>
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* @version $Id$
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* @since 4.0
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*/
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public class HaltonSequenceGenerator implements RandomVectorGenerator {
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/** The first 40 primes. */
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private static final int[] PRIMES = new int[] {
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2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
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71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139,
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149, 151, 157, 163, 167, 173
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};
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/** The optimal weights used for scrambling of the first 40 dimension. */
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private static final int[] WEIGHTS = new int[] {
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1, 2, 3, 3, 8, 11, 12, 14, 7, 18, 12, 13, 17, 18, 29, 14, 18, 43, 41,
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44, 40, 30, 47, 65, 71, 28, 40, 60, 79, 89, 56, 50, 52, 61, 108, 56,
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66, 63, 60, 66
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};
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/** Space dimension. */
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private final int dimension;
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/** The current index in the sequence. */
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private int count = 0;
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/** The base numbers for each component. */
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private final int[] base;
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/** The scrambling weights for each component. */
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private final int[] weight;
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/**
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* Construct a new Halton sequence generator for the given space dimension.
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*
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* @param dimension the space dimension
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* @throws OutOfRangeException if the space dimension is outside the allowed range of [1, 40]
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*/
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public HaltonSequenceGenerator(final int dimension) throws OutOfRangeException {
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this(dimension, PRIMES, WEIGHTS);
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}
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/**
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* Construct a new Halton sequence generator with the given base numbers and weights for each dimension.
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* The length of the bases array defines the space dimension and is required to be > 0.
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*
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* @param dimension the space dimension
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* @param bases the base number for each dimension, entries should be (pairwise) prime, may not be null
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* @param weights the weights used during scrambling, may be null in which case no scrambling will be performed
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* @throws NullArgumentException if base is null
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* @throws OutOfRangeException if the space dimension is outside the range [1, len], where
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* len refers to the length of the bases array
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* @throws DimensionMismatchException if weights is non-null and the length of the input arrays differ
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*/
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public HaltonSequenceGenerator(final int dimension, final int[] bases, final int[] weights)
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throws NullArgumentException, OutOfRangeException, DimensionMismatchException {
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MathUtils.checkNotNull(bases);
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if (dimension < 1 || dimension > bases.length) {
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throw new OutOfRangeException(dimension, 1, PRIMES.length);
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}
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if (weights != null && weights.length != bases.length) {
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throw new DimensionMismatchException(weights.length, bases.length);
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}
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this.dimension = dimension;
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this.base = bases.clone();
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this.weight = weights == null ? null : weights.clone();
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count = 0;
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}
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/** {@inheritDoc} */
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public double[] nextVector() {
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final double[] v = new double[dimension];
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for (int i = 0; i < dimension; i++) {
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int index = count;
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double f = 1.0 / base[i];
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int j = 0;
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while (index > 0) {
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final int digit = scramble(i, j, base[i], index % base[i]);
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v[i] += f * digit;
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index /= base[i]; // floor( index / base )
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f /= base[i];
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}
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}
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count++;
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return v;
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}
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/**
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* Performs scrambling of digit {@code d_j} according to the formula:
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* <pre>
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* ( weight_i * d_j ) mod base
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* </pre>
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* Implementations can override this method to do a different scrambling.
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*
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* @param i the dimension index
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* @param j the digit index
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* @param b the base for this dimension
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* @param digit the j-th digit
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* @return the scrambled digit
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*/
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protected int scramble(final int i, final int j, final int b, final int digit) {
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return weight != null ? (weight[i] * digit) % b : digit;
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}
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/**
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* Skip to the i-th point in the Halton sequence.
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* <p>
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* This operation can be performed in O(1).
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*
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* @param index the index in the sequence to skip to
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* @return the i-th point in the Halton sequence
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* @throws NotPositiveException if index < 0
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*/
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public double[] skipTo(final int index) throws NotPositiveException {
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count = index;
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return nextVector();
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}
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/**
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* Returns the index i of the next point in the Halton sequence that will be returned
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* by calling {@link #nextVector()}.
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*
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* @return the index of the next point
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*/
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public int getNextIndex() {
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return count;
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}
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}
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