/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.random;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.NotPositiveException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.util.MathUtils;
/**
* Implementation of a Halton sequence.
* <p>
* A Halton sequence is a low-discrepancy sequence generating points in the interval [0, 1] according to
* <pre>
* 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
* </pre>
* For higher dimensions, subsequent prime numbers are used as base, e.g. { 2, 3, 5 } for a Halton sequence in R^3.
* <p>
* 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 <a href="http://etd.lib.fsu.edu/theses/available/etd-07062004-140409/unrestricted/dissertation1.pdf">
* H. Chi: Scrambled quasirandom sequences and their applications</a>.
* <p>
* The generator supports two modes:
* <ul>
* <li>sequential generation of points: {@link #nextVector()}</li>
* <li>random access to the i-th point in the sequence: {@link #skipTo(int)}</li>
* </ul>
*
* @see <a href="http://en.wikipedia.org/wiki/Halton_sequence">Halton sequence (Wikipedia)</a>
* @see <a href="https://lirias.kuleuven.be/bitstream/123456789/131168/1/mcm2005_bartv.pdf">
* On the Halton sequence and its scramblings</a>
* @since 3.3
*/
public class HaltonSequenceGenerator implements RandomVectorGenerator {
/** The first 40 primes. */
private static final int[] PRIMES = new int[] {
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139,
149, 151, 157, 163, 167, 173
};
/** The optimal weights used for scrambling of the first 40 dimension. */
private static final int[] WEIGHTS = new int[] {
1, 2, 3, 3, 8, 11, 12, 14, 7, 18, 12, 13, 17, 18, 29, 14, 18, 43, 41,
44, 40, 30, 47, 65, 71, 28, 40, 60, 79, 89, 56, 50, 52, 61, 108, 56,
66, 63, 60, 66
};
/** Space dimension. */
private final int dimension;
/** The current index in the sequence. */
private int count = 0;
/** The base numbers for each component. */
private final int[] base;
/** The scrambling weights for each component. */
private final int[] weight;
/**
* Construct a new Halton sequence generator for the given space dimension.
*
* @param dimension the space dimension
* @throws OutOfRangeException if the space dimension is outside the allowed range of [1, 40]
*/
public HaltonSequenceGenerator(final int dimension) throws OutOfRangeException {
this(dimension, PRIMES, WEIGHTS);
}
/**
* 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.
*
* @param dimension the space dimension
* @param bases the base number for each dimension, entries should be (pairwise) prime, may not be null
* @param weights the weights used during scrambling, may be null in which case no scrambling will be performed
* @throws NullArgumentException if base is null
* @throws OutOfRangeException if the space dimension is outside the range [1, len], where
* len refers to the length of the bases array
* @throws DimensionMismatchException if weights is non-null and the length of the input arrays differ
*/
public HaltonSequenceGenerator(final int dimension, final int[] bases, final int[] weights)
throws NullArgumentException, OutOfRangeException, DimensionMismatchException {
MathUtils.checkNotNull(bases);
if (dimension < 1 || dimension > bases.length) {
throw new OutOfRangeException(dimension, 1, PRIMES.length);
}
if (weights != null && weights.length != bases.length) {
throw new DimensionMismatchException(weights.length, bases.length);
}
this.dimension = dimension;
this.base = bases.clone();
this.weight = weights == null ? null : weights.clone();
count = 0;
}
/** {@inheritDoc} */
public double[] nextVector() {
final double[] v = new double[dimension];
for (int i = 0; i < dimension; i++) {
int index = count;
double f = 1.0 / base[i];
int j = 0;
while (index > 0) {
final int digit = scramble(i, j, base[i], index % base[i]);
v[i] += f * digit;
index /= base[i]; // floor( index / base )
f /= base[i];
}
}
count++;
return v;
}
/**
* Performs scrambling of digit {@code d_j} according to the formula:
* <pre>
* ( weight_i * d_j ) mod base
* </pre>
* Implementations can override this method to do a different scrambling.
*
* @param i the dimension index
* @param j the digit index
* @param b the base for this dimension
* @param digit the j-th digit
* @return the scrambled digit
*/
protected int scramble(final int i, final int j, final int b, final int digit) {
return weight != null ? (weight[i] * digit) % b : digit;
}
/**
* Skip to the i-th point in the Halton sequence.
* <p>
* This operation can be performed in O(1).
*
* @param index the index in the sequence to skip to
* @return the i-th point in the Halton sequence
* @throws NotPositiveException if index < 0
*/
public double[] skipTo(final int index) throws NotPositiveException {
count = index;
return nextVector();
}
/**
* Returns the index i of the next point in the Halton sequence that will be returned
* by calling {@link #nextVector()}.
*
* @return the index of the next point
*/
public int getNextIndex() {
return count;
}
}