    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.NullArgumentException;
    029    import org.apache.commons.math3.exception.DimensionMismatchException;
    030    import org.apache.commons.math3.util.FastMath;
    031
    032    /**
    033     * <a href="http://en.wikipedia.org/wiki/Sigmoid_function">
    034     *  Sigmoid</a> function.
    035     * It is the inverse of the {@link Logit logit} function.
    036     * A more flexible version, the generalised logistic, is implemented
    037     * by the {@link Logistic} class.
    038     *
    039     * @since 3.0
    040     * @version $Id: Sigmoid.java 1391927 2012-09-30 00:03:30Z erans $
    041     */
    042    public class Sigmoid implements UnivariateDifferentiableFunction, DifferentiableUnivariateFunction {
    043        /** Lower asymptote. */
    044        private final double lo;
    045        /** Higher asymptote. */
    046        private final double hi;
    047
    048        /**
    049         * Usual sigmoid function, where the lower asymptote is 0 and the higher
    050         * asymptote is 1.
    051         */
    052        public Sigmoid() {
    053            this(0, 1);
    054        }
    055
    056        /**
    057         * Sigmoid function.
    058         *
    059         * @param lo Lower asymptote.
    060         * @param hi Higher asymptote.
    061         */
    062        public Sigmoid(double lo,
    063                       double hi) {
    064            this.lo = lo;
    065            this.hi = hi;
    066        }
    067
    068        /** {@inheritDoc}
    069         * @deprecated as of 3.1, replaced by {@link #value(DerivativeStructure)}
    070         */
    071        @Deprecated
    072        public UnivariateFunction derivative() {
    073            return FunctionUtils.toDifferentiableUnivariateFunction(this).derivative();
    074        }
    075
    076        /** {@inheritDoc} */
    077        public double value(double x) {
    078            return value(x, lo, hi);
    079        }
    080
    081        /**
    082         * Parametric function where the input array contains the parameters of
    083         * the logit function, ordered as follows:
    084         * <ul>
    085         *  <li>Lower asymptote</li>
    086         *  <li>Higher asymptote</li>
    087         * </ul>
    088         */
    089        public static class Parametric implements ParametricUnivariateFunction {
    090            /**
    091             * Computes the value of the sigmoid at {@code x}.
    092             *
    093             * @param x Value for which the function must be computed.
    094             * @param param Values of lower asymptote and higher asymptote.
    095             * @return the value of the function.
    096             * @throws NullArgumentException if {@code param} is {@code null}.
    097             * @throws DimensionMismatchException if the size of {@code param} is
    098             * not 2.
    099             */
    100            public double value(double x, double ... param)
    101                throws NullArgumentException,
    102                       DimensionMismatchException {
    103                validateParameters(param);
    104                return Sigmoid.value(x, param[0], param[1]);
    105            }
    106
    107            /**
    108             * Computes the value of the gradient at {@code x}.
    109             * The components of the gradient vector are the partial
    110             * derivatives of the function with respect to each of the
    111             * <em>parameters</em> (lower asymptote and higher asymptote).
    112             *
    113             * @param x Value at which the gradient must be computed.
    114             * @param param Values for lower asymptote and higher asymptote.
    115             * @return the gradient vector at {@code x}.
    116             * @throws NullArgumentException if {@code param} is {@code null}.
    117             * @throws DimensionMismatchException if the size of {@code param} is
    118             * not 2.
    119             */
    120            public double[] gradient(double x, double ... param)
    121                throws NullArgumentException,
    122                       DimensionMismatchException {
    123                validateParameters(param);
    124
    125                final double invExp1 = 1 / (1 + FastMath.exp(-x));
    126
    127                return new double[] { 1 - invExp1, invExp1 };
    128            }
    129
    130            /**
    131             * Validates parameters to ensure they are appropriate for the evaluation of
    132             * the {@link #value(double,double[])} and {@link #gradient(double,double[])}
    133             * methods.
    134             *
    135             * @param param Values for lower and higher asymptotes.
    136             * @throws NullArgumentException if {@code param} is {@code null}.
    137             * @throws DimensionMismatchException if the size of {@code param} is
    138             * not 2.
    139             */
    140            private void validateParameters(double[] param)
    141                throws NullArgumentException,
    142                       DimensionMismatchException {
    143                if (param == null) {
    144                    throw new NullArgumentException();
    145                }
    146                if (param.length != 2) {
    147                    throw new DimensionMismatchException(param.length, 2);
    148                }
    149            }
    150        }
    151
    152        /**
    153         * @param x Value at which to compute the sigmoid.
    154         * @param lo Lower asymptote.
    155         * @param hi Higher asymptote.
    156         * @return the value of the sigmoid function at {@code x}.
    157         */
    158        private static double value(double x,
    159                                    double lo,
    160                                    double hi) {
    161            return lo + (hi - lo) / (1 + FastMath.exp(-x));
    162        }
    163
    164        /** {@inheritDoc}
    165         * @since 3.1
    166         */
    167        public DerivativeStructure value(final DerivativeStructure t) {
    168
    169            double[] f = new double[t.getOrder() + 1];
    170            final double exp = FastMath.exp(-t.getValue());
    171            if (Double.isInfinite(exp)) {
    172
    173                // special handling near lower boundary, to avoid NaN
    174                f[0] = lo;
    175                Arrays.fill(f, 1, f.length, 0.0);
    176
    177            } else {
    178
    179                // the nth order derivative of sigmoid has the form:
    180                // dn(sigmoid(x)/dxn = P_n(exp(-x)) / (1+exp(-x))^(n+1)
    181                // where P_n(t) is a degree n polynomial with normalized higher term
    182                // P_0(t) = 1, P_1(t) = t, P_2(t) = t^2 - t, P_3(t) = t^3 - 4 t^2 + t...
    183                // the general recurrence relation for P_n is:
    184                // P_n(x) = n t P_(n-1)(t) - t (1 + t) P_(n-1)'(t)
    185                final double[] p = new double[f.length];
    186
    187                final double inv   = 1 / (1 + exp);
    188                double coeff = hi - lo;
    189                for (int n = 0; n < f.length; ++n) {
    190
    191                    // update and evaluate polynomial P_n(t)
    192                    double v = 0;
    193                    p[n] = 1;
    194                    for (int k = n; k >= 0; --k) {
    195                        v = v * exp + p[k];
    196                        if (k > 1) {
    197                            p[k - 1] = (n - k + 2) * p[k - 2] - (k - 1) * p[k - 1];
    198                        } else {
    199                            p[0] = 0;
    200                        }
    201                    }
    202
    203                    coeff *= inv;
    204                    f[n]   = coeff * v;
    205
    206                }
    207
    208                // fix function value
    209                f[0] += lo;
    210
    211            }
    212
    213            return t.compose(f);
    214
    215        }
    216
    217    }