    001    /*
    002     * Licensed to the Apache Software Foundation (ASF) under one or more
    003     * contributor license agreements.  See the NOTICE file distributed with
    004     * this work for additional information regarding copyright ownership.
    005     * The ASF licenses this file to You under the Apache License, Version 2.0
    006     * (the "License"); you may not use this file except in compliance with
    007     * the License.  You may obtain a copy of the License at
    008     *
    009     *      http://www.apache.org/licenses/LICENSE-2.0
    010     *
    011     * Unless required by applicable law or agreed to in writing, software
    012     * distributed under the License is distributed on an "AS IS" BASIS,
    013     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    014     * See the License for the specific language governing permissions and
    015     * limitations under the License.
    016     */
    017
    018    package org.apache.commons.math3.analysis.function;
    019
    020    import java.util.Arrays;
    021
    022    import org.apache.commons.math3.analysis.FunctionUtils;
    023    import org.apache.commons.math3.analysis.UnivariateFunction;
    024    import org.apache.commons.math3.analysis.DifferentiableUnivariateFunction;
    025    import org.apache.commons.math3.analysis.ParametricUnivariateFunction;
    026    import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
    027    import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableFunction;
    028    import org.apache.commons.math3.exception.NotStrictlyPositiveException;
    029    import org.apache.commons.math3.exception.NullArgumentException;
    030    import org.apache.commons.math3.exception.DimensionMismatchException;
    031    import org.apache.commons.math3.util.FastMath;
    032    import org.apache.commons.math3.util.Precision;
    033
    034    /**
    035     * <a href="http://en.wikipedia.org/wiki/Gaussian_function">
    036     *  Gaussian</a> function.
    037     *
    038     * @since 3.0
    039     * @version $Id: Gaussian.java 1383441 2012-09-11 14:56:39Z luc $
    040     */
    041    public class Gaussian implements UnivariateDifferentiableFunction, DifferentiableUnivariateFunction {
    042        /** Mean. */
    043        private final double mean;
    044        /** Inverse of the standard deviation. */
    045        private final double is;
    046        /** Inverse of twice the square of the standard deviation. */
    047        private final double i2s2;
    048        /** Normalization factor. */
    049        private final double norm;
    050
    051        /**
    052         * Gaussian with given normalization factor, mean and standard deviation.
    053         *
    054         * @param norm Normalization factor.
    055         * @param mean Mean.
    056         * @param sigma Standard deviation.
    057         * @throws NotStrictlyPositiveException if {@code sigma <= 0}.
    058         */
    059        public Gaussian(double norm,
    060                        double mean,
    061                        double sigma)
    062            throws NotStrictlyPositiveException {
    063            if (sigma <= 0) {
    064                throw new NotStrictlyPositiveException(sigma);
    065            }
    066
    067            this.norm = norm;
    068            this.mean = mean;
    069            this.is   = 1 / sigma;
    070            this.i2s2 = 0.5 * is * is;
    071        }
    072
    073        /**
    074         * Normalized gaussian with given mean and standard deviation.
    075         *
    076         * @param mean Mean.
    077         * @param sigma Standard deviation.
    078         * @throws NotStrictlyPositiveException if {@code sigma <= 0}.
    079         */
    080        public Gaussian(double mean,
    081                        double sigma)
    082            throws NotStrictlyPositiveException {
    083            this(1 / (sigma * FastMath.sqrt(2 * Math.PI)), mean, sigma);
    084        }
    085
    086        /**
    087         * Normalized gaussian with zero mean and unit standard deviation.
    088         */
    089        public Gaussian() {
    090            this(0, 1);
    091        }
    092
    093        /** {@inheritDoc} */
    094        public double value(double x) {
    095            return value(x - mean, norm, i2s2);
    096        }
    097
    098        /** {@inheritDoc}
    099         * @deprecated as of 3.1, replaced by {@link #value(DerivativeStructure)}
    100         */
    101        @Deprecated
    102        public UnivariateFunction derivative() {
    103            return FunctionUtils.toDifferentiableUnivariateFunction(this).derivative();
    104        }
    105
    106        /**
    107         * Parametric function where the input array contains the parameters of
    108         * the Gaussian, ordered as follows:
    109         * <ul>
    110         *  <li>Norm</li>
    111         *  <li>Mean</li>
    112         *  <li>Standard deviation</li>
    113         * </ul>
    114         */
    115        public static class Parametric implements ParametricUnivariateFunction {
    116            /**
    117             * Computes the value of the Gaussian at {@code x}.
    118             *
    119             * @param x Value for which the function must be computed.
    120             * @param param Values of norm, mean and standard deviation.
    121             * @return the value of the function.
    122             * @throws NullArgumentException if {@code param} is {@code null}.
    123             * @throws DimensionMismatchException if the size of {@code param} is
    124             * not 3.
    125             * @throws NotStrictlyPositiveException if {@code param[2]} is negative.
    126             */
    127            public double value(double x, double ... param)
    128                throws NullArgumentException,
    129                       DimensionMismatchException,
    130                       NotStrictlyPositiveException {
    131                validateParameters(param);
    132
    133                final double diff = x - param[1];
    134                final double i2s2 = 1 / (2 * param[2] * param[2]);
    135                return Gaussian.value(diff, param[0], i2s2);
    136            }
    137
    138            /**
    139             * Computes the value of the gradient at {@code x}.
    140             * The components of the gradient vector are the partial
    141             * derivatives of the function with respect to each of the
    142             * <em>parameters</em> (norm, mean and standard deviation).
    143             *
    144             * @param x Value at which the gradient must be computed.
    145             * @param param Values of norm, mean and standard deviation.
    146             * @return the gradient vector at {@code x}.
    147             * @throws NullArgumentException if {@code param} is {@code null}.
    148             * @throws DimensionMismatchException if the size of {@code param} is
    149             * not 3.
    150             * @throws NotStrictlyPositiveException if {@code param[2]} is negative.
    151             */
    152            public double[] gradient(double x, double ... param)
    153                throws NullArgumentException,
    154                       DimensionMismatchException,
    155                       NotStrictlyPositiveException {
    156                validateParameters(param);
    157
    158                final double norm = param[0];
    159                final double diff = x - param[1];
    160                final double sigma = param[2];
    161                final double i2s2 = 1 / (2 * sigma * sigma);
    162
    163                final double n = Gaussian.value(diff, 1, i2s2);
    164                final double m = norm * n * 2 * i2s2 * diff;
    165                final double s = m * diff / sigma;
    166
    167                return new double[] { n, m, s };
    168            }
    169
    170            /**
    171             * Validates parameters to ensure they are appropriate for the evaluation of
    172             * the {@link #value(double,double[])} and {@link #gradient(double,double[])}
    173             * methods.
    174             *
    175             * @param param Values of norm, mean and standard deviation.
    176             * @throws NullArgumentException if {@code param} is {@code null}.
    177             * @throws DimensionMismatchException if the size of {@code param} is
    178             * not 3.
    179             * @throws NotStrictlyPositiveException if {@code param[2]} is negative.
    180             */
    181            private void validateParameters(double[] param)
    182                throws NullArgumentException,
    183                       DimensionMismatchException,
    184                       NotStrictlyPositiveException {
    185                if (param == null) {
    186                    throw new NullArgumentException();
    187                }
    188                if (param.length != 3) {
    189                    throw new DimensionMismatchException(param.length, 3);
    190                }
    191                if (param[2] <= 0) {
    192                    throw new NotStrictlyPositiveException(param[2]);
    193                }
    194            }
    195        }
    196
    197        /**
    198         * @param xMinusMean {@code x - mean}.
    199         * @param norm Normalization factor.
    200         * @param i2s2 Inverse of twice the square of the standard deviation.
    201         * @return the value of the Gaussian at {@code x}.
    202         */
    203        private static double value(double xMinusMean,
    204                                    double norm,
    205                                    double i2s2) {
    206            return norm * FastMath.exp(-xMinusMean * xMinusMean * i2s2);
    207        }
    208
    209        /** {@inheritDoc}
    210         * @since 3.1
    211         */
    212        public DerivativeStructure value(final DerivativeStructure t) {
    213
    214            final double u = is * (t.getValue() - mean);
    215            double[] f = new double[t.getOrder() + 1];
    216
    217            // the nth order derivative of the Gaussian has the form:
    218            // dn(g(x)/dxn = (norm / s^n) P_n(u) exp(-u^2/2) with u=(x-m)/s
    219            // where P_n(u) is a degree n polynomial with same parity as n
    220            // P_0(u) = 1, P_1(u) = -u, P_2(u) = u^2 - 1, P_3(u) = -u^3 + 3 u...
    221            // the general recurrence relation for P_n is:
    222            // P_n(u) = P_(n-1)'(u) - u P_(n-1)(u)
    223            // as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
    224            final double[] p = new double[f.length];
    225            p[0] = 1;
    226            final double u2 = u * u;
    227            double coeff = norm * FastMath.exp(-0.5 * u2);
    228            if (coeff <= Precision.SAFE_MIN) {
    229                Arrays.fill(f, 0.0);
    230            } else {
    231                f[0] = coeff;
    232                for (int n = 1; n < f.length; ++n) {
    233
    234                    // update and evaluate polynomial P_n(x)
    235                    double v = 0;
    236                    p[n] = -p[n - 1];
    237                    for (int k = n; k >= 0; k -= 2) {
    238                        v = v * u2 + p[k];
    239                        if (k > 2) {
    240                            p[k - 2] = (k - 1) * p[k - 1] - p[k - 3];
    241                        } else if (k == 2) {
    242                            p[0] = p[1];
    243                        }
    244                    }
    245                    if ((n & 0x1) == 1) {
    246                        v *= u;
    247                    }
    248
    249                    coeff *= is;
    250                    f[n] = coeff * v;
    251
    252                }
    253            }
    254
    255            return t.compose(f);
    256
    257        }
    258
    259    }