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 * Licensed to the Apache Software Foundation (ASF) under one or more
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 * 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
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package org.apache.commons.math3.analysis.function;

import java.util.Arrays;

import org.apache.commons.math3.analysis.FunctionUtils;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.analysis.DifferentiableUnivariateFunction;
import org.apache.commons.math3.analysis.ParametricUnivariateFunction;
import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableFunction;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.util.FastMath;


 Sigmoid function. It is the inverse of the logit function. A more flexible version, the generalised logistic, is implemented by the Logistic class. 
Since:
3.0/**
 * <a href="http://en.wikipedia.org/wiki/Sigmoid_function">
 *  Sigmoid</a> function.
 * It is the inverse of the {@link Logit logit} function.
 * A more flexible version, the generalised logistic, is implemented
 * by the {@link Logistic} class.
 *
 * @since 3.0
 */
public class Sigmoid implements UnivariateDifferentiableFunction, DifferentiableUnivariateFunction {
    
Lower asymptote. /** Lower asymptote. */
    private final double lo;
    
Higher asymptote. /** Higher asymptote. */
    private final double hi;

    
Usual sigmoid function, where the lower asymptote is 0 and the higher
asymptote is 1.
/**
     * Usual sigmoid function, where the lower asymptote is 0 and the higher
     * asymptote is 1.
     */
    public Sigmoid() {
        this(0, 1);
    }

    
Sigmoid function.
Params:
lo – Lower asymptote.
hi – Higher asymptote./**
     * Sigmoid function.
     *
     * @param lo Lower asymptote.
     * @param hi Higher asymptote.
     */
    public Sigmoid(double lo,
                   double hi) {
        this.lo = lo;
        this.hi = hi;
    }

    
{@inheritDoc}
Deprecated:
as of 3.1, replaced by value(DerivativeStructure)/** {@inheritDoc}
     * @deprecated as of 3.1, replaced by {@link #value(DerivativeStructure)}
     */
    @Deprecated
    public UnivariateFunction derivative() {
        return FunctionUtils.toDifferentiableUnivariateFunction(this).derivative();
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public double value(double x) {
        return value(x, lo, hi);
    }

    
Parametric function where the input array contains the parameters of the sigmoid function, ordered as follows: 

 
Lower asymptote
 
Higher asymptote

/**
     * Parametric function where the input array contains the parameters of
     * the {@link Sigmoid#Sigmoid(double,double) sigmoid function}, ordered
     * as follows:
     * <ul>
     *  <li>Lower asymptote</li>
     *  <li>Higher asymptote</li>
     * </ul>
     */
    public static class Parametric implements ParametricUnivariateFunction {
        
Computes the value of the sigmoid at x. 
Params:
x – Value for which the function must be computed.
param – Values of lower asymptote and higher asymptote.
Throws:
NullArgumentException – if param is null.
DimensionMismatchException – if the size of param is not 2.
Returns:
the value of the function./**
         * Computes the value of the sigmoid at {@code x}.
         *
         * @param x Value for which the function must be computed.
         * @param param Values of lower asymptote and higher asymptote.
         * @return the value of the function.
         * @throws NullArgumentException if {@code param} is {@code null}.
         * @throws DimensionMismatchException if the size of {@code param} is
         * not 2.
         */
        public double value(double x, double ... param)
            throws NullArgumentException,
                   DimensionMismatchException {
            validateParameters(param);
            return Sigmoid.value(x, param[0], param[1]);
        }

        
Computes the value of the gradient at x. The components of the gradient vector are the partial derivatives of the function with respect to each of the parameters (lower asymptote and higher asymptote).
Params:
x – Value at which the gradient must be computed.
param – Values for lower asymptote and higher asymptote.
Throws:
NullArgumentException – if param is null.
DimensionMismatchException – if the size of param is not 2.
Returns:
the gradient vector at x./**
         * Computes the value of the gradient at {@code x}.
         * The components of the gradient vector are the partial
         * derivatives of the function with respect to each of the
         * <em>parameters</em> (lower asymptote and higher asymptote).
         *
         * @param x Value at which the gradient must be computed.
         * @param param Values for lower asymptote and higher asymptote.
         * @return the gradient vector at {@code x}.
         * @throws NullArgumentException if {@code param} is {@code null}.
         * @throws DimensionMismatchException if the size of {@code param} is
         * not 2.
         */
        public double[] gradient(double x, double ... param)
            throws NullArgumentException,
                   DimensionMismatchException {
            validateParameters(param);

            final double invExp1 = 1 / (1 + FastMath.exp(-x));

            return new double[] { 1 - invExp1, invExp1 };
        }

        
Validates parameters to ensure they are appropriate for the evaluation of the value(double, double[]) and gradient(double, double[]) methods. 
Params:
param – Values for lower and higher asymptotes.
Throws:
NullArgumentException – if param is null.
DimensionMismatchException – if the size of param is not 2./**
         * Validates parameters to ensure they are appropriate for the evaluation of
         * the {@link #value(double,double[])} and {@link #gradient(double,double[])}
         * methods.
         *
         * @param param Values for lower and higher asymptotes.
         * @throws NullArgumentException if {@code param} is {@code null}.
         * @throws DimensionMismatchException if the size of {@code param} is
         * not 2.
         */
        private void validateParameters(double[] param)
            throws NullArgumentException,
                   DimensionMismatchException {
            if (param == null) {
                throw new NullArgumentException();
            }
            if (param.length != 2) {
                throw new DimensionMismatchException(param.length, 2);
            }
        }
    }

    
Params:
x – Value at which to compute the sigmoid.
lo – Lower asymptote.
hi – Higher asymptote.
Returns:
the value of the sigmoid function at x./**
     * @param x Value at which to compute the sigmoid.
     * @param lo Lower asymptote.
     * @param hi Higher asymptote.
     * @return the value of the sigmoid function at {@code x}.
     */
    private static double value(double x,
                                double lo,
                                double hi) {
        return lo + (hi - lo) / (1 + FastMath.exp(-x));
    }

    
{@inheritDoc}
Since:
3.1/** {@inheritDoc}
     * @since 3.1
     */
    public DerivativeStructure value(final DerivativeStructure t)
        throws DimensionMismatchException {

        double[] f = new double[t.getOrder() + 1];
        final double exp = FastMath.exp(-t.getValue());
        if (Double.isInfinite(exp)) {

            // special handling near lower boundary, to avoid NaN
            f[0] = lo;
            Arrays.fill(f, 1, f.length, 0.0);

        } else {

            // the nth order derivative of sigmoid has the form:
            // dn(sigmoid(x)/dxn = P_n(exp(-x)) / (1+exp(-x))^(n+1)
            // where P_n(t) is a degree n polynomial with normalized higher term
            // P_0(t) = 1, P_1(t) = t, P_2(t) = t^2 - t, P_3(t) = t^3 - 4 t^2 + t...
            // the general recurrence relation for P_n is:
            // P_n(x) = n t P_(n-1)(t) - t (1 + t) P_(n-1)'(t)
            final double[] p = new double[f.length];

            final double inv   = 1 / (1 + exp);
            double coeff = hi - lo;
            for (int n = 0; n < f.length; ++n) {

                // update and evaluate polynomial P_n(t)
                double v = 0;
                p[n] = 1;
                for (int k = n; k >= 0; --k) {
                    v = v * exp + p[k];
                    if (k > 1) {
                        p[k - 1] = (n - k + 2) * p[k - 2] - (k - 1) * p[k - 1];
                    } else {
                        p[0] = 0;
                    }
                }

                coeff *= inv;
                f[n]   = coeff * v;

            }

            // fix function value
            f[0] += lo;

        }

        return t.compose(f);

    }

}