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 * http://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 package org.apache.commons.math4.legacy.analysis.function;
19
20 import java.util.Arrays;
21
22 import org.apache.commons.math4.legacy.analysis.ParametricUnivariateFunction;
23 import org.apache.commons.math4.legacy.analysis.differentiation.DerivativeStructure;
24 import org.apache.commons.math4.legacy.analysis.differentiation.UnivariateDifferentiableFunction;
25 import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
26 import org.apache.commons.math4.legacy.exception.NotStrictlyPositiveException;
27 import org.apache.commons.math4.legacy.exception.NullArgumentException;
28 import org.apache.commons.math4.core.jdkmath.JdkMath;
29 import org.apache.commons.numbers.core.Precision;
30
31 /**
32 * <a href="http://en.wikipedia.org/wiki/Gaussian_function">
33 * Gaussian</a> function.
34 *
35 * @since 3.0
36 */
37 public class Gaussian implements UnivariateDifferentiableFunction {
38 /** Mean. */
39 private final double mean;
40 /** Inverse of the standard deviation. */
41 private final double is;
42 /** Inverse of twice the square of the standard deviation. */
43 private final double i2s2;
44 /** Normalization factor. */
45 private final double norm;
46
47 /**
48 * Gaussian with given normalization factor, mean and standard deviation.
49 *
50 * @param norm Normalization factor.
51 * @param mean Mean.
52 * @param sigma Standard deviation.
53 * @throws NotStrictlyPositiveException if {@code sigma <= 0}.
54 */
55 public Gaussian(double norm,
56 double mean,
57 double sigma)
58 throws NotStrictlyPositiveException {
59 if (sigma <= 0) {
60 throw new NotStrictlyPositiveException(sigma);
61 }
62
63 this.norm = norm;
64 this.mean = mean;
65 this.is = 1 / sigma;
66 this.i2s2 = 0.5 * is * is;
67 }
68
69 /**
70 * Normalized gaussian with given mean and standard deviation.
71 *
72 * @param mean Mean.
73 * @param sigma Standard deviation.
74 * @throws NotStrictlyPositiveException if {@code sigma <= 0}.
75 */
76 public Gaussian(double mean,
77 double sigma)
78 throws NotStrictlyPositiveException {
79 this(1 / (sigma * JdkMath.sqrt(2 * Math.PI)), mean, sigma);
80 }
81
82 /**
83 * Normalized gaussian with zero mean and unit standard deviation.
84 */
85 public Gaussian() {
86 this(0, 1);
87 }
88
89 /** {@inheritDoc} */
90 @Override
91 public double value(double x) {
92 return value(x - mean, norm, i2s2);
93 }
94
95 /**
96 * Parametric function where the input array contains the parameters of
97 * the Gaussian. Ordered as follows:
98 * <ul>
99 * <li>Norm</li>
100 * <li>Mean</li>
101 * <li>Standard deviation</li>
102 * </ul>
103 */
104 public static class Parametric implements ParametricUnivariateFunction {
105 /**
106 * Computes the value of the Gaussian at {@code x}.
107 *
108 * @param x Value for which the function must be computed.
109 * @param param Values of norm, mean and standard deviation.
110 * @return the value of the function.
111 * @throws NullArgumentException if {@code param} is {@code null}.
112 * @throws DimensionMismatchException if the size of {@code param} is
113 * not 3.
114 * @throws NotStrictlyPositiveException if {@code param[2]} is negative.
115 */
116 @Override
117 public double value(double x, double ... param)
118 throws NullArgumentException,
119 DimensionMismatchException,
120 NotStrictlyPositiveException {
121 validateParameters(param);
122
123 final double diff = x - param[1];
124 final double i2s2 = 1 / (2 * param[2] * param[2]);
125 return Gaussian.value(diff, param[0], i2s2);
126 }
127
128 /**
129 * Computes the value of the gradient at {@code x}.
130 * The components of the gradient vector are the partial
131 * derivatives of the function with respect to each of the
132 * <em>parameters</em> (norm, mean and standard deviation).
133 *
134 * @param x Value at which the gradient must be computed.
135 * @param param Values of norm, mean and standard deviation.
136 * @return the gradient vector at {@code x}.
137 * @throws NullArgumentException if {@code param} is {@code null}.
138 * @throws DimensionMismatchException if the size of {@code param} is
139 * not 3.
140 * @throws NotStrictlyPositiveException if {@code param[2]} is negative.
141 */
142 @Override
143 public double[] gradient(double x, double ... param)
144 throws NullArgumentException,
145 DimensionMismatchException,
146 NotStrictlyPositiveException {
147 validateParameters(param);
148
149 final double norm = param[0];
150 final double diff = x - param[1];
151 final double sigma = param[2];
152 final double i2s2 = 1 / (2 * sigma * sigma);
153
154 final double n = Gaussian.value(diff, 1, i2s2);
155 final double m = norm * n * 2 * i2s2 * diff;
156 final double s = m * diff / sigma;
157
158 return new double[] { n, m, s };
159 }
160
161 /**
162 * Validates parameters to ensure they are appropriate for the evaluation of
163 * the {@link #value(double,double[])} and {@link #gradient(double,double[])}
164 * methods.
165 *
166 * @param param Values of norm, mean and standard deviation.
167 * @throws NullArgumentException if {@code param} is {@code null}.
168 * @throws DimensionMismatchException if the size of {@code param} is
169 * not 3.
170 * @throws NotStrictlyPositiveException if {@code param[2]} is negative.
171 */
172 private void validateParameters(double[] param)
173 throws NullArgumentException,
174 DimensionMismatchException,
175 NotStrictlyPositiveException {
176 if (param == null) {
177 throw new NullArgumentException();
178 }
179 if (param.length != 3) {
180 throw new DimensionMismatchException(param.length, 3);
181 }
182 if (param[2] <= 0) {
183 throw new NotStrictlyPositiveException(param[2]);
184 }
185 }
186 }
187
188 /**
189 * @param xMinusMean {@code x - mean}.
190 * @param norm Normalization factor.
191 * @param i2s2 Inverse of twice the square of the standard deviation.
192 * @return the value of the Gaussian at {@code x}.
193 */
194 private static double value(double xMinusMean,
195 double norm,
196 double i2s2) {
197 return norm * JdkMath.exp(-xMinusMean * xMinusMean * i2s2);
198 }
199
200 /** {@inheritDoc}
201 * @since 3.1
202 */
203 @Override
204 public DerivativeStructure value(final DerivativeStructure t)
205 throws DimensionMismatchException {
206
207 final double u = is * (t.getValue() - mean);
208 double[] f = new double[t.getOrder() + 1];
209
210 // the nth order derivative of the Gaussian has the form:
211 // dn(g(x)/dxn = (norm / s^n) P_n(u) exp(-u^2/2) with u=(x-m)/s
212 // where P_n(u) is a degree n polynomial with same parity as n
213 // P_0(u) = 1, P_1(u) = -u, P_2(u) = u^2 - 1, P_3(u) = -u^3 + 3 u...
214 // the general recurrence relation for P_n is:
215 // P_n(u) = P_(n-1)'(u) - u P_(n-1)(u)
216 // as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
217 final double[] p = new double[f.length];
218 p[0] = 1;
219 final double u2 = u * u;
220 double coeff = norm * JdkMath.exp(-0.5 * u2);
221 if (coeff <= Precision.SAFE_MIN) {
222 Arrays.fill(f, 0.0);
223 } else {
224 f[0] = coeff;
225 for (int n = 1; n < f.length; ++n) {
226
227 // update and evaluate polynomial P_n(x)
228 double v = 0;
229 p[n] = -p[n - 1];
230 for (int k = n; k >= 0; k -= 2) {
231 v = v * u2 + p[k];
232 if (k > 2) {
233 p[k - 2] = (k - 1) * p[k - 1] - p[k - 3];
234 } else if (k == 2) {
235 p[0] = p[1];
236 }
237 }
238 if ((n & 0x1) == 1) {
239 v *= u;
240 }
241
242 coeff *= is;
243 f[n] = coeff * v;
244 }
245 }
246
247 return t.compose(f);
248 }
249 }