001 /*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements. See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License. You may obtain a copy of the License at
008 *
009 * http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017
018 package org.apache.commons.math.optimization.fitting;
019
020 import org.apache.commons.math.optimization.DifferentiableMultivariateVectorialOptimizer;
021 import org.apache.commons.math.analysis.function.HarmonicOscillator;
022 import org.apache.commons.math.exception.ZeroException;
023 import org.apache.commons.math.exception.NumberIsTooSmallException;
024 import org.apache.commons.math.exception.util.LocalizedFormats;
025 import org.apache.commons.math.util.FastMath;
026
027 /**
028 * Class that implements a curve fitting specialized for sinusoids.
029 *
030 * Harmonic fitting is a very simple case of curve fitting. The
031 * estimated coefficients are the amplitude a, the pulsation ω and
032 * the phase φ: <code>f (t) = a cos (ω t + φ)</code>. They are
033 * searched by a least square estimator initialized with a rough guess
034 * based on integrals.
035 *
036 * @version $Id: HarmonicFitter.java 1131229 2011-06-03 20:49:25Z luc $
037 * @since 2.0
038 */
039 public class HarmonicFitter extends CurveFitter {
040 /**
041 * Simple constructor.
042 * @param optimizer Optimizer to use for the fitting.
043 */
044 public HarmonicFitter(final DifferentiableMultivariateVectorialOptimizer optimizer) {
045 super(optimizer);
046 }
047
048 /**
049 * Fit an harmonic function to the observed points.
050 *
051 * @param initialGuess First guess values in the following order:
052 * <ul>
053 * <li>Amplitude</li>
054 * <li>Angular frequency</li>
055 * <li>Phase</li>
056 * </ul>
057 * @return the parameters of the harmonic function that best fits the
058 * observed points (in the same order as above).
059 */
060 public double[] fit(double[] initialGuess) {
061 return fit(new HarmonicOscillator.Parametric(), initialGuess);
062 }
063
064 /**
065 * Fit an harmonic function to the observed points.
066 * An initial guess will be automatically computed.
067 *
068 * @return the parameters of the harmonic function that best fits the
069 * observed points (see the other {@link #fit(double[]) fit} method.
070 * @throws NumberIsTooSmallException if the sample is too short for the
071 * the first guess to be computed.
072 * @throws ZeroException if the first guess cannot be computed because
073 * the abscissa range is zero.
074 */
075 public double[] fit() {
076 return fit((new ParameterGuesser(getObservations())).guess());
077 }
078
079 /**
080 * This class guesses harmonic coefficients from a sample.
081 * <p>The algorithm used to guess the coefficients is as follows:</p>
082 *
083 * <p>We know f (t) at some sampling points t<sub>i</sub> and want to find a,
084 * ω and φ such that f (t) = a cos (ω t + φ).
085 * </p>
086 *
087 * <p>From the analytical expression, we can compute two primitives :
088 * <pre>
089 * If2 (t) = ∫ f<sup>2</sup> = a<sup>2</sup> × [t + S (t)] / 2
090 * If'2 (t) = ∫ f'<sup>2</sup> = a<sup>2</sup> ω<sup>2</sup> × [t - S (t)] / 2
091 * where S (t) = sin (2 (ω t + φ)) / (2 ω)
092 * </pre>
093 * </p>
094 *
095 * <p>We can remove S between these expressions :
096 * <pre>
097 * If'2 (t) = a<sup>2</sup> ω<sup>2</sup> t - ω<sup>2</sup> If2 (t)
098 * </pre>
099 * </p>
100 *
101 * <p>The preceding expression shows that If'2 (t) is a linear
102 * combination of both t and If2 (t): If'2 (t) = A × t + B × If2 (t)
103 * </p>
104 *
105 * <p>From the primitive, we can deduce the same form for definite
106 * integrals between t<sub>1</sub> and t<sub>i</sub> for each t<sub>i</sub> :
107 * <pre>
108 * If2 (t<sub>i</sub>) - If2 (t<sub>1</sub>) = A × (t<sub>i</sub> - t<sub>1</sub>) + B × (If2 (t<sub>i</sub>) - If2 (t<sub>1</sub>))
109 * </pre>
110 * </p>
111 *
112 * <p>We can find the coefficients A and B that best fit the sample
113 * to this linear expression by computing the definite integrals for
114 * each sample points.
115 * </p>
116 *
117 * <p>For a bilinear expression z (x<sub>i</sub>, y<sub>i</sub>) = A × x<sub>i</sub> + B × y<sub>i</sub>, the
118 * coefficients A and B that minimize a least square criterion
119 * ∑ (z<sub>i</sub> - z (x<sub>i</sub>, y<sub>i</sub>))<sup>2</sup> are given by these expressions:</p>
120 * <pre>
121 *
122 * ∑y<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
123 * A = ------------------------
124 * ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>y<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>y<sub>i</sub>
125 *
126 * ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub>
127 * B = ------------------------
128 * ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>y<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>y<sub>i</sub>
129 * </pre>
130 * </p>
131 *
132 *
133 * <p>In fact, we can assume both a and ω are positive and
134 * compute them directly, knowing that A = a<sup>2</sup> ω<sup>2</sup> and that
135 * B = - ω<sup>2</sup>. The complete algorithm is therefore:</p>
136 * <pre>
137 *
138 * for each t<sub>i</sub> from t<sub>1</sub> to t<sub>n-1</sub>, compute:
139 * f (t<sub>i</sub>)
140 * f' (t<sub>i</sub>) = (f (t<sub>i+1</sub>) - f(t<sub>i-1</sub>)) / (t<sub>i+1</sub> - t<sub>i-1</sub>)
141 * x<sub>i</sub> = t<sub>i</sub> - t<sub>1</sub>
142 * y<sub>i</sub> = ∫ f<sup>2</sup> from t<sub>1</sub> to t<sub>i</sub>
143 * z<sub>i</sub> = ∫ f'<sup>2</sup> from t<sub>1</sub> to t<sub>i</sub>
144 * update the sums ∑x<sub>i</sub>x<sub>i</sub>, ∑y<sub>i</sub>y<sub>i</sub>, ∑x<sub>i</sub>y<sub>i</sub>, ∑x<sub>i</sub>z<sub>i</sub> and ∑y<sub>i</sub>z<sub>i</sub>
145 * end for
146 *
147 * |--------------------------
148 * \ | ∑y<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
149 * a = \ | ------------------------
150 * \| ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
151 *
152 *
153 * |--------------------------
154 * \ | ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>z<sub>i</sub> - ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>z<sub>i</sub>
155 * ω = \ | ------------------------
156 * \| ∑x<sub>i</sub>x<sub>i</sub> ∑y<sub>i</sub>y<sub>i</sub> - ∑x<sub>i</sub>y<sub>i</sub> ∑x<sub>i</sub>y<sub>i</sub>
157 *
158 * </pre>
159 * </p>
160 *
161 * <p>Once we know ω, we can compute:
162 * <pre>
163 * fc = ω f (t) cos (ω t) - f' (t) sin (ω t)
164 * fs = ω f (t) sin (ω t) + f' (t) cos (ω t)
165 * </pre>
166 * </p>
167 *
168 * <p>It appears that <code>fc = a ω cos (φ)</code> and
169 * <code>fs = -a ω sin (φ)</code>, so we can use these
170 * expressions to compute φ. The best estimate over the sample is
171 * given by averaging these expressions.
172 * </p>
173 *
174 * <p>Since integrals and means are involved in the preceding
175 * estimations, these operations run in O(n) time, where n is the
176 * number of measurements.</p>
177 */
178 public static class ParameterGuesser {
179 /** Sampled observations. */
180 private final WeightedObservedPoint[] observations;
181 /** Amplitude. */
182 private double a;
183 /** Angular frequency. */
184 private double omega;
185 /** Phase. */
186 private double phi;
187
188 /**
189 * Simple constructor.
190 * @param observations sampled observations
191 * @throws NumberIsTooSmallException if the sample is too short or if
192 * the first guess cannot be computed.
193 */
194 public ParameterGuesser(WeightedObservedPoint[] observations) {
195 if (observations.length < 4) {
196 throw new NumberIsTooSmallException(LocalizedFormats.INSUFFICIENT_OBSERVED_POINTS_IN_SAMPLE,
197 observations.length, 4, true);
198 }
199
200 this.observations = observations.clone();
201 }
202
203 /**
204 * Estimate a first guess of the coefficients.
205 *
206 * @return the guessed coefficients, in the following order:
207 * <ul>
208 * <li>Amplitude</li>
209 * <li>Angular frequency</li>
210 * <li>Phase</li>
211 * </ul>
212 */
213 public double[] guess() {
214 sortObservations();
215 guessAOmega();
216 guessPhi();
217 return new double[] { a, omega, phi };
218 }
219
220 /**
221 * Sort the observations with respect to the abscissa.
222 */
223 private void sortObservations() {
224 // Since the samples are almost always already sorted, this
225 // method is implemented as an insertion sort that reorders the
226 // elements in place. Insertion sort is very efficient in this case.
227 WeightedObservedPoint curr = observations[0];
228 for (int j = 1; j < observations.length; ++j) {
229 WeightedObservedPoint prec = curr;
230 curr = observations[j];
231 if (curr.getX() < prec.getX()) {
232 // the current element should be inserted closer to the beginning
233 int i = j - 1;
234 WeightedObservedPoint mI = observations[i];
235 while ((i >= 0) && (curr.getX() < mI.getX())) {
236 observations[i + 1] = mI;
237 if (i-- != 0) {
238 mI = observations[i];
239 }
240 }
241 observations[i + 1] = curr;
242 curr = observations[j];
243 }
244 }
245 }
246
247 /**
248 * Estimate a first guess of the amplitude and angular frequency.
249 * This method assumes that the {@link #sortObservations()} method
250 * has been called previously.
251 *
252 * @throws ZeroException if the abscissa range is zero.
253 */
254 private void guessAOmega() {
255 // initialize the sums for the linear model between the two integrals
256 double sx2 = 0;
257 double sy2 = 0;
258 double sxy = 0;
259 double sxz = 0;
260 double syz = 0;
261
262 double currentX = observations[0].getX();
263 double currentY = observations[0].getY();
264 double f2Integral = 0;
265 double fPrime2Integral = 0;
266 final double startX = currentX;
267 for (int i = 1; i < observations.length; ++i) {
268 // one step forward
269 final double previousX = currentX;
270 final double previousY = currentY;
271 currentX = observations[i].getX();
272 currentY = observations[i].getY();
273
274 // update the integrals of f<sup>2</sup> and f'<sup>2</sup>
275 // considering a linear model for f (and therefore constant f')
276 final double dx = currentX - previousX;
277 final double dy = currentY - previousY;
278 final double f2StepIntegral =
279 dx * (previousY * previousY + previousY * currentY + currentY * currentY) / 3;
280 final double fPrime2StepIntegral = dy * dy / dx;
281
282 final double x = currentX - startX;
283 f2Integral += f2StepIntegral;
284 fPrime2Integral += fPrime2StepIntegral;
285
286 sx2 += x * x;
287 sy2 += f2Integral * f2Integral;
288 sxy += x * f2Integral;
289 sxz += x * fPrime2Integral;
290 syz += f2Integral * fPrime2Integral;
291 }
292
293 // compute the amplitude and pulsation coefficients
294 double c1 = sy2 * sxz - sxy * syz;
295 double c2 = sxy * sxz - sx2 * syz;
296 double c3 = sx2 * sy2 - sxy * sxy;
297 if ((c1 / c2 < 0) || (c2 / c3 < 0)) {
298 final int last = observations.length - 1;
299 // Range of the observations, assuming that the
300 // observations are sorted.
301 final double xRange = observations[last].getX() - observations[0].getX();
302 if (xRange == 0) {
303 throw new ZeroException();
304 }
305 omega = 2 * Math.PI / xRange;
306
307 double yMin = Double.POSITIVE_INFINITY;
308 double yMax = Double.NEGATIVE_INFINITY;
309 for (int i = 1; i < observations.length; ++i) {
310 final double y = observations[i].getY();
311 if (y < yMin) {
312 yMin = y;
313 }
314 if (y > yMax) {
315 yMax = y;
316 }
317 }
318 a = 0.5 * (yMax - yMin);
319 } else {
320 a = FastMath.sqrt(c1 / c2);
321 omega = FastMath.sqrt(c2 / c3);
322 }
323 }
324
325 /**
326 * Estimate a first guess of the phase.
327 */
328 private void guessPhi() {
329 // initialize the means
330 double fcMean = 0;
331 double fsMean = 0;
332
333 double currentX = observations[0].getX();
334 double currentY = observations[0].getY();
335 for (int i = 1; i < observations.length; ++i) {
336 // one step forward
337 final double previousX = currentX;
338 final double previousY = currentY;
339 currentX = observations[i].getX();
340 currentY = observations[i].getY();
341 final double currentYPrime = (currentY - previousY) / (currentX - previousX);
342
343 double omegaX = omega * currentX;
344 double cosine = FastMath.cos(omegaX);
345 double sine = FastMath.sin(omegaX);
346 fcMean += omega * currentY * cosine - currentYPrime * sine;
347 fsMean += omega * currentY * sine + currentYPrime * cosine;
348 }
349
350 phi = FastMath.atan2(-fsMean, fcMean);
351 }
352 }
353 }