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 }