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1   /*
2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
4    * this work for additional information regarding copyright ownership.
5    * The ASF licenses this file to You under the Apache License, Version 2.0
6    * (the "License"); you may not use this file except in compliance with
7    * the License.  You may obtain a copy of the License at
8    *
9    *      https://www.apache.org/licenses/LICENSE-2.0
10   *
11   * Unless required by applicable law or agreed to in writing, software
12   * distributed under the License is distributed on an "AS IS" BASIS,
13   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17  
18  /*
19   * This is not the original file distributed by the Apache Software Foundation
20   * It has been modified by the Hipparchus project
21   */
22  
23  package org.hipparchus.analysis.function;
24  
25  import java.util.Arrays;
26  
27  import org.hipparchus.analysis.ParametricUnivariateFunction;
28  import org.hipparchus.analysis.differentiation.Derivative;
29  import org.hipparchus.analysis.differentiation.UnivariateDifferentiableFunction;
30  import org.hipparchus.exception.LocalizedCoreFormats;
31  import org.hipparchus.exception.MathIllegalArgumentException;
32  import org.hipparchus.exception.NullArgumentException;
33  import org.hipparchus.util.FastMath;
34  import org.hipparchus.util.MathUtils;
35  import org.hipparchus.util.Precision;
36  
37  /**
38   * <a href="http://en.wikipedia.org/wiki/Gaussian_function">
39   *  Gaussian</a> function.
40   *
41   */
42  public class Gaussian implements UnivariateDifferentiableFunction {
43      /** Mean. */
44      private final double mean;
45      /** Inverse of the standard deviation. */
46      private final double is;
47      /** Inverse of twice the square of the standard deviation. */
48      private final double i2s2;
49      /** Normalization factor. */
50      private final double norm;
51  
52      /**
53       * Gaussian with given normalization factor, mean and standard deviation.
54       *
55       * @param norm Normalization factor.
56       * @param mean Mean.
57       * @param sigma Standard deviation.
58       * @throws MathIllegalArgumentException if {@code sigma <= 0}.
59       */
60      public Gaussian(double norm,
61                      double mean,
62                      double sigma)
63          throws MathIllegalArgumentException {
64          if (sigma <= 0) {
65              throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL_BOUND_EXCLUDED,
66                                                     sigma, 0);
67          }
68  
69          this.norm = norm;
70          this.mean = mean;
71          this.is   = 1 / sigma;
72          this.i2s2 = 0.5 * is * is;
73      }
74  
75      /**
76       * Normalized gaussian with given mean and standard deviation.
77       *
78       * @param mean Mean.
79       * @param sigma Standard deviation.
80       * @throws MathIllegalArgumentException if {@code sigma <= 0}.
81       */
82      public Gaussian(double mean,
83                      double sigma)
84          throws MathIllegalArgumentException {
85          this(1 / (sigma * FastMath.sqrt(2 * Math.PI)), mean, sigma);
86      }
87  
88      /**
89       * Normalized gaussian with zero mean and unit standard deviation.
90       */
91      public Gaussian() {
92          this(0, 1);
93      }
94  
95      /** {@inheritDoc} */
96      @Override
97      public double value(double x) {
98          return value(x - mean, norm, i2s2);
99      }
100 
101     /**
102      * Parametric function where the input array contains the parameters of
103      * the Gaussian, ordered as follows:
104      * <ul>
105      *  <li>Norm</li>
106      *  <li>Mean</li>
107      *  <li>Standard deviation</li>
108      * </ul>
109      */
110     public static class Parametric implements ParametricUnivariateFunction {
111 
112         /** Empty constructor.
113          * <p>
114          * This constructor is not strictly necessary, but it prevents spurious
115          * javadoc warnings with JDK 18 and later.
116          * </p>
117          * @since 3.0
118          */
119         public Parametric() { // NOPMD - unnecessary constructor added intentionally to make javadoc happy
120             // nothing to do
121         }
122 
123         /**
124          * Computes the value of the Gaussian at {@code x}.
125          *
126          * @param x Value for which the function must be computed.
127          * @param param Values of norm, mean and standard deviation.
128          * @return the value of the function.
129          * @throws NullArgumentException if {@code param} is {@code null}.
130          * @throws MathIllegalArgumentException if the size of {@code param} is
131          * not 3.
132          * @throws MathIllegalArgumentException if {@code param[2]} is negative.
133          */
134         @Override
135         public double value(double x, double ... param)
136             throws MathIllegalArgumentException, NullArgumentException {
137             validateParameters(param);
138 
139             final double diff = x - param[1];
140             final double i2s2 = 1 / (2 * param[2] * param[2]);
141             return Gaussian.value(diff, param[0], i2s2);
142         }
143 
144         /**
145          * Computes the value of the gradient at {@code x}.
146          * The components of the gradient vector are the partial
147          * derivatives of the function with respect to each of the
148          * <em>parameters</em> (norm, mean and standard deviation).
149          *
150          * @param x Value at which the gradient must be computed.
151          * @param param Values of norm, mean and standard deviation.
152          * @return the gradient vector at {@code x}.
153          * @throws NullArgumentException if {@code param} is {@code null}.
154          * @throws MathIllegalArgumentException if the size of {@code param} is
155          * not 3.
156          * @throws MathIllegalArgumentException if {@code param[2]} is negative.
157          */
158         @Override
159         public double[] gradient(double x, double ... param)
160             throws MathIllegalArgumentException, NullArgumentException {
161             validateParameters(param);
162 
163             final double norm = param[0];
164             final double diff = x - param[1];
165             final double sigma = param[2];
166             final double i2s2 = 1 / (2 * sigma * sigma);
167 
168             final double n = Gaussian.value(diff, 1, i2s2);
169             final double m = norm * n * 2 * i2s2 * diff;
170             final double s = m * diff / sigma;
171 
172             return new double[] { n, m, s };
173         }
174 
175         /**
176          * Validates parameters to ensure they are appropriate for the evaluation of
177          * the {@link #value(double,double[])} and {@link #gradient(double,double[])}
178          * methods.
179          *
180          * @param param Values of norm, mean and standard deviation.
181          * @throws NullArgumentException if {@code param} is {@code null}.
182          * @throws MathIllegalArgumentException if the size of {@code param} is
183          * not 3.
184          * @throws MathIllegalArgumentException if {@code param[2]} is negative.
185          */
186         private void validateParameters(double[] param)
187             throws MathIllegalArgumentException, NullArgumentException {
188             if (param == null) {
189                 throw new NullArgumentException();
190             }
191             MathUtils.checkDimension(param.length, 3);
192             if (param[2] <= 0) {
193                 throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL_BOUND_EXCLUDED,
194                                                        param[2], 0);
195             }
196         }
197     }
198 
199     /**
200      * @param xMinusMean {@code x - mean}.
201      * @param norm Normalization factor.
202      * @param i2s2 Inverse of twice the square of the standard deviation.
203      * @return the value of the Gaussian at {@code x}.
204      */
205     private static double value(double xMinusMean,
206                                 double norm,
207                                 double i2s2) {
208         return norm * FastMath.exp(-xMinusMean * xMinusMean * i2s2);
209     }
210 
211     /** {@inheritDoc}
212      */
213     @Override
214     public <T extends Derivative<T>> T value(T t)
215         throws MathIllegalArgumentException {
216 
217         final double u = is * (t.getValue() - mean);
218         double[] f = new double[t.getOrder() + 1];
219 
220         // the nth order derivative of the Gaussian has the form:
221         // dn(g(x)/dxn = (norm / s^n) P_n(u) exp(-u^2/2) with u=(x-m)/s
222         // where P_n(u) is a degree n polynomial with same parity as n
223         // P_0(u) = 1, P_1(u) = -u, P_2(u) = u^2 - 1, P_3(u) = -u^3 + 3 u...
224         // the general recurrence relation for P_n is:
225         // P_n(u) = P_(n-1)'(u) - u P_(n-1)(u)
226         // as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
227         final double[] p = new double[f.length];
228         p[0] = 1;
229         final double u2 = u * u;
230         double coeff = norm * FastMath.exp(-0.5 * u2);
231         if (coeff <= Precision.SAFE_MIN) {
232             Arrays.fill(f, 0.0);
233         } else {
234             f[0] = coeff;
235             for (int n = 1; n < f.length; ++n) {
236 
237                 // update and evaluate polynomial P_n(x)
238                 double v = 0;
239                 p[n] = -p[n - 1];
240                 for (int k = n; k >= 0; k -= 2) {
241                     v = v * u2 + p[k];
242                     if (k > 2) {
243                         p[k - 2] = (k - 1) * p[k - 1] - p[k - 3];
244                     } else if (k == 2) {
245                         p[0] = p[1];
246                     }
247                 }
248                 if ((n & 0x1) == 1) {
249                     v *= u;
250                 }
251 
252                 coeff *= is;
253                 f[n] = coeff * v;
254 
255             }
256         }
257 
258         return t.compose(f);
259 
260     }
261 
262 }