/*
    * 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.math4.legacy.analysis.function;
  
  import java.util.Arrays;
  
  import org.apache.commons.math4.legacy.analysis.ParametricUnivariateFunction;
  import org.apache.commons.math4.legacy.analysis.differentiation.DerivativeStructure;
  import org.apache.commons.math4.legacy.analysis.differentiation.UnivariateDifferentiableFunction;
  import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
  import org.apache.commons.math4.legacy.exception.NullArgumentException;
  import org.apache.commons.math4.core.jdkmath.JdkMath;
  
  /**
   * <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 {
      /** Lower asymptote. */
      private final double lo;
      /** Higher asymptote. */
      private final double hi;
  
      /**
       * Usual sigmoid function, where the lower asymptote is 0 and the higher
       * asymptote is 1.
       */
      public Sigmoid() {
          this(0, 1);
      }
  
      /**
       * Sigmoid function.
       *
       * @param lo Lower asymptote.
       * @param hi Higher asymptote.
       */
      public Sigmoid(double lo,
                     double hi) {
          this.lo = lo;
          this.hi = hi;
      }
  
      /** {@inheritDoc} */
      @Override
      public double value(double x) {
          return value(x, lo, hi);
      }
  
      /**
       * 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 {@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.
           */
          @Override
          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 {@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.
          */
         @Override
         public double[] gradient(double x, double ... param)
             throws NullArgumentException,
                    DimensionMismatchException {
             validateParameters(param);
 
             final double invExp1 = 1 / (1 + JdkMath.exp(-x));
 
             return new double[] { 1 - invExp1, invExp1 };
         }
 
         /**
          * 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);
             }
         }
     }
 
     /**
      * @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 + JdkMath.exp(-x));
     }
 
     /** {@inheritDoc}
      * @since 3.1
      */
     @Override
     public DerivativeStructure value(final DerivativeStructure t)
         throws DimensionMismatchException {
 
         double[] f = new double[t.getOrder() + 1];
         final double exp = JdkMath.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);
     }
 }