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 018package org.apache.commons.math3.optimization.fitting; 019 020import org.apache.commons.math3.optimization.DifferentiableMultivariateVectorOptimizer; 021import org.apache.commons.math3.analysis.function.HarmonicOscillator; 022import org.apache.commons.math3.exception.ZeroException; 023import org.apache.commons.math3.exception.NumberIsTooSmallException; 024import org.apache.commons.math3.exception.MathIllegalStateException; 025import org.apache.commons.math3.exception.util.LocalizedFormats; 026import org.apache.commons.math3.util.FastMath; 027 028/** 029 * Class that implements a curve fitting specialized for sinusoids. 030 * 031 * Harmonic fitting is a very simple case of curve fitting. The 032 * estimated coefficients are the amplitude a, the pulsation ω and 033 * the phase φ: <code>f (t) = a cos (ω t + φ)</code>. They are 034 * searched by a least square estimator initialized with a rough guess 035 * based on integrals. 036 * 037 * @deprecated As of 3.1 (to be removed in 4.0). 038 * @since 2.0 039 */ 040@Deprecated 041public class HarmonicFitter extends CurveFitter<HarmonicOscillator.Parametric> { 042 /** 043 * Simple constructor. 044 * @param optimizer Optimizer to use for the fitting. 045 */ 046 public HarmonicFitter(final DifferentiableMultivariateVectorOptimizer optimizer) { 047 super(optimizer); 048 } 049 050 /** 051 * Fit an harmonic function to the observed points. 052 * 053 * @param initialGuess First guess values in the following order: 054 * <ul> 055 * <li>Amplitude</li> 056 * <li>Angular frequency</li> 057 * <li>Phase</li> 058 * </ul> 059 * @return the parameters of the harmonic function that best fits the 060 * observed points (in the same order as above). 061 */ 062 public double[] fit(double[] initialGuess) { 063 return fit(new HarmonicOscillator.Parametric(), initialGuess); 064 } 065 066 /** 067 * Fit an harmonic function to the observed points. 068 * An initial guess will be automatically computed. 069 * 070 * @return the parameters of the harmonic function that best fits the 071 * observed points (see the other {@link #fit(double[]) fit} method. 072 * @throws NumberIsTooSmallException if the sample is too short for the 073 * the first guess to be computed. 074 * @throws ZeroException if the first guess cannot be computed because 075 * the abscissa range is zero. 076 */ 077 public double[] fit() { 078 return fit((new ParameterGuesser(getObservations())).guess()); 079 } 080 081 /** 082 * This class guesses harmonic coefficients from a sample. 083 * <p>The algorithm used to guess the coefficients is as follows:</p> 084 * 085 * <p>We know f (t) at some sampling points t<sub>i</sub> and want to find a, 086 * ω and φ such that f (t) = a cos (ω t + φ). 087 * </p> 088 * 089 * <p>From the analytical expression, we can compute two primitives : 090 * <pre> 091 * If2 (t) = ∫ f<sup>2</sup> = a<sup>2</sup> × [t + S (t)] / 2 092 * If'2 (t) = ∫ f'<sup>2</sup> = a<sup>2</sup> ω<sup>2</sup> × [t - S (t)] / 2 093 * where S (t) = sin (2 (ω t + φ)) / (2 ω) 094 * </pre> 095 * </p> 096 * 097 * <p>We can remove S between these expressions : 098 * <pre> 099 * If'2 (t) = a<sup>2</sup> ω<sup>2</sup> t - ω<sup>2</sup> If2 (t) 100 * </pre> 101 * </p> 102 * 103 * <p>The preceding expression shows that If'2 (t) is a linear 104 * combination of both t and If2 (t): If'2 (t) = A × t + B × If2 (t) 105 * </p> 106 * 107 * <p>From the primitive, we can deduce the same form for definite 108 * integrals between t<sub>1</sub> and t<sub>i</sub> for each t<sub>i</sub> : 109 * <pre> 110 * 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>)) 111 * </pre> 112 * </p> 113 * 114 * <p>We can find the coefficients A and B that best fit the sample 115 * to this linear expression by computing the definite integrals for 116 * each sample points. 117 * </p> 118 * 119 * <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 120 * coefficients A and B that minimize a least square criterion 121 * ∑ (z<sub>i</sub> - z (x<sub>i</sub>, y<sub>i</sub>))<sup>2</sup> are given by these expressions:</p> 122 * <pre> 123 * 124 * ∑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> 125 * A = ------------------------ 126 * ∑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> 127 * 128 * ∑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> 129 * B = ------------------------ 130 * ∑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> 131 * </pre> 132 * </p> 133 * 134 * 135 * <p>In fact, we can assume both a and ω are positive and 136 * compute them directly, knowing that A = a<sup>2</sup> ω<sup>2</sup> and that 137 * B = - ω<sup>2</sup>. The complete algorithm is therefore:</p> 138 * <pre> 139 * 140 * for each t<sub>i</sub> from t<sub>1</sub> to t<sub>n-1</sub>, compute: 141 * f (t<sub>i</sub>) 142 * 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>) 143 * x<sub>i</sub> = t<sub>i</sub> - t<sub>1</sub> 144 * y<sub>i</sub> = ∫ f<sup>2</sup> from t<sub>1</sub> to t<sub>i</sub> 145 * z<sub>i</sub> = ∫ f'<sup>2</sup> from t<sub>1</sub> to t<sub>i</sub> 146 * 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> 147 * end for 148 * 149 * |-------------------------- 150 * \ | ∑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> 151 * a = \ | ------------------------ 152 * \| ∑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> 153 * 154 * 155 * |-------------------------- 156 * \ | ∑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> 157 * ω = \ | ------------------------ 158 * \| ∑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> 159 * 160 * </pre> 161 * </p> 162 * 163 * <p>Once we know ω, we can compute: 164 * <pre> 165 * fc = ω f (t) cos (ω t) - f' (t) sin (ω t) 166 * fs = ω f (t) sin (ω t) + f' (t) cos (ω t) 167 * </pre> 168 * </p> 169 * 170 * <p>It appears that <code>fc = a ω cos (φ)</code> and 171 * <code>fs = -a ω sin (φ)</code>, so we can use these 172 * expressions to compute φ. The best estimate over the sample is 173 * given by averaging these expressions. 174 * </p> 175 * 176 * <p>Since integrals and means are involved in the preceding 177 * estimations, these operations run in O(n) time, where n is the 178 * number of measurements.</p> 179 */ 180 public static class ParameterGuesser { 181 /** Amplitude. */ 182 private final double a; 183 /** Angular frequency. */ 184 private final double omega; 185 /** Phase. */ 186 private final double phi; 187 188 /** 189 * Simple constructor. 190 * 191 * @param observations Sampled observations. 192 * @throws NumberIsTooSmallException if the sample is too short. 193 * @throws ZeroException if the abscissa range is zero. 194 * @throws MathIllegalStateException when the guessing procedure cannot 195 * produce sensible results. 196 */ 197 public ParameterGuesser(WeightedObservedPoint[] observations) { 198 if (observations.length < 4) { 199 throw new NumberIsTooSmallException(LocalizedFormats.INSUFFICIENT_OBSERVED_POINTS_IN_SAMPLE, 200 observations.length, 4, true); 201 } 202 203 final WeightedObservedPoint[] sorted = sortObservations(observations); 204 205 final double aOmega[] = guessAOmega(sorted); 206 a = aOmega[0]; 207 omega = aOmega[1]; 208 209 phi = guessPhi(sorted); 210 } 211 212 /** 213 * Gets an estimation of the parameters. 214 * 215 * @return the guessed parameters, in the following order: 216 * <ul> 217 * <li>Amplitude</li> 218 * <li>Angular frequency</li> 219 * <li>Phase</li> 220 * </ul> 221 */ 222 public double[] guess() { 223 return new double[] { a, omega, phi }; 224 } 225 226 /** 227 * Sort the observations with respect to the abscissa. 228 * 229 * @param unsorted Input observations. 230 * @return the input observations, sorted. 231 */ 232 private WeightedObservedPoint[] sortObservations(WeightedObservedPoint[] unsorted) { 233 final WeightedObservedPoint[] observations = unsorted.clone(); 234 235 // Since the samples are almost always already sorted, this 236 // method is implemented as an insertion sort that reorders the 237 // elements in place. Insertion sort is very efficient in this case. 238 WeightedObservedPoint curr = observations[0]; 239 for (int j = 1; j < observations.length; ++j) { 240 WeightedObservedPoint prec = curr; 241 curr = observations[j]; 242 if (curr.getX() < prec.getX()) { 243 // the current element should be inserted closer to the beginning 244 int i = j - 1; 245 WeightedObservedPoint mI = observations[i]; 246 while ((i >= 0) && (curr.getX() < mI.getX())) { 247 observations[i + 1] = mI; 248 if (i-- != 0) { 249 mI = observations[i]; 250 } 251 } 252 observations[i + 1] = curr; 253 curr = observations[j]; 254 } 255 } 256 257 return observations; 258 } 259 260 /** 261 * Estimate a first guess of the amplitude and angular frequency. 262 * This method assumes that the {@link #sortObservations(WeightedObservedPoint[])} method 263 * has been called previously. 264 * 265 * @param observations Observations, sorted w.r.t. abscissa. 266 * @throws ZeroException if the abscissa range is zero. 267 * @throws MathIllegalStateException when the guessing procedure cannot 268 * produce sensible results. 269 * @return the guessed amplitude (at index 0) and circular frequency 270 * (at index 1). 271 */ 272 private double[] guessAOmega(WeightedObservedPoint[] observations) { 273 final double[] aOmega = new double[2]; 274 275 // initialize the sums for the linear model between the two integrals 276 double sx2 = 0; 277 double sy2 = 0; 278 double sxy = 0; 279 double sxz = 0; 280 double syz = 0; 281 282 double currentX = observations[0].getX(); 283 double currentY = observations[0].getY(); 284 double f2Integral = 0; 285 double fPrime2Integral = 0; 286 final double startX = currentX; 287 for (int i = 1; i < observations.length; ++i) { 288 // one step forward 289 final double previousX = currentX; 290 final double previousY = currentY; 291 currentX = observations[i].getX(); 292 currentY = observations[i].getY(); 293 294 // update the integrals of f<sup>2</sup> and f'<sup>2</sup> 295 // considering a linear model for f (and therefore constant f') 296 final double dx = currentX - previousX; 297 final double dy = currentY - previousY; 298 final double f2StepIntegral = 299 dx * (previousY * previousY + previousY * currentY + currentY * currentY) / 3; 300 final double fPrime2StepIntegral = dy * dy / dx; 301 302 final double x = currentX - startX; 303 f2Integral += f2StepIntegral; 304 fPrime2Integral += fPrime2StepIntegral; 305 306 sx2 += x * x; 307 sy2 += f2Integral * f2Integral; 308 sxy += x * f2Integral; 309 sxz += x * fPrime2Integral; 310 syz += f2Integral * fPrime2Integral; 311 } 312 313 // compute the amplitude and pulsation coefficients 314 double c1 = sy2 * sxz - sxy * syz; 315 double c2 = sxy * sxz - sx2 * syz; 316 double c3 = sx2 * sy2 - sxy * sxy; 317 if ((c1 / c2 < 0) || (c2 / c3 < 0)) { 318 final int last = observations.length - 1; 319 // Range of the observations, assuming that the 320 // observations are sorted. 321 final double xRange = observations[last].getX() - observations[0].getX(); 322 if (xRange == 0) { 323 throw new ZeroException(); 324 } 325 aOmega[1] = 2 * Math.PI / xRange; 326 327 double yMin = Double.POSITIVE_INFINITY; 328 double yMax = Double.NEGATIVE_INFINITY; 329 for (int i = 1; i < observations.length; ++i) { 330 final double y = observations[i].getY(); 331 if (y < yMin) { 332 yMin = y; 333 } 334 if (y > yMax) { 335 yMax = y; 336 } 337 } 338 aOmega[0] = 0.5 * (yMax - yMin); 339 } else { 340 if (c2 == 0) { 341 // In some ill-conditioned cases (cf. MATH-844), the guesser 342 // procedure cannot produce sensible results. 343 throw new MathIllegalStateException(LocalizedFormats.ZERO_DENOMINATOR); 344 } 345 346 aOmega[0] = FastMath.sqrt(c1 / c2); 347 aOmega[1] = FastMath.sqrt(c2 / c3); 348 } 349 350 return aOmega; 351 } 352 353 /** 354 * Estimate a first guess of the phase. 355 * 356 * @param observations Observations, sorted w.r.t. abscissa. 357 * @return the guessed phase. 358 */ 359 private double guessPhi(WeightedObservedPoint[] observations) { 360 // initialize the means 361 double fcMean = 0; 362 double fsMean = 0; 363 364 double currentX = observations[0].getX(); 365 double currentY = observations[0].getY(); 366 for (int i = 1; i < observations.length; ++i) { 367 // one step forward 368 final double previousX = currentX; 369 final double previousY = currentY; 370 currentX = observations[i].getX(); 371 currentY = observations[i].getY(); 372 final double currentYPrime = (currentY - previousY) / (currentX - previousX); 373 374 double omegaX = omega * currentX; 375 double cosine = FastMath.cos(omegaX); 376 double sine = FastMath.sin(omegaX); 377 fcMean += omega * currentY * cosine - currentYPrime * sine; 378 fsMean += omega * currentY * sine + currentYPrime * cosine; 379 } 380 381 return FastMath.atan2(-fsMean, fcMean); 382 } 383 } 384}