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 package org.apache.commons.math.estimation;
018
019 import java.io.Serializable;
020 import java.util.Arrays;
021
022
023 /**
024 * This class solves a least squares problem.
025 *
026 * <p>This implementation <em>should</em> work even for over-determined systems
027 * (i.e. systems having more variables than equations). Over-determined systems
028 * are solved by ignoring the variables which have the smallest impact according
029 * to their jacobian column norm. Only the rank of the matrix and some loop bounds
030 * are changed to implement this.</p>
031 *
032 * <p>The resolution engine is a simple translation of the MINPACK <a
033 * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
034 * changes. The changes include the over-determined resolution and the Q.R.
035 * decomposition which has been rewritten following the algorithm described in the
036 * P. Lascaux and R. Theodor book <i>Analyse numérique matricielle
037 * appliquée à l'art de l'ingénieur</i>, Masson 1986. The
038 * redistribution policy for MINPACK is available <a
039 * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
040 * is reproduced below.</p>
041 *
042 * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
043 * <tr><td>
044 * Minpack Copyright Notice (1999) University of Chicago.
045 * All rights reserved
046 * </td></tr>
047 * <tr><td>
048 * Redistribution and use in source and binary forms, with or without
049 * modification, are permitted provided that the following conditions
050 * are met:
051 * <ol>
052 * <li>Redistributions of source code must retain the above copyright
053 * notice, this list of conditions and the following disclaimer.</li>
054 * <li>Redistributions in binary form must reproduce the above
055 * copyright notice, this list of conditions and the following
056 * disclaimer in the documentation and/or other materials provided
057 * with the distribution.</li>
058 * <li>The end-user documentation included with the redistribution, if any,
059 * must include the following acknowledgment:
060 * <code>This product includes software developed by the University of
061 * Chicago, as Operator of Argonne National Laboratory.</code>
062 * Alternately, this acknowledgment may appear in the software itself,
063 * if and wherever such third-party acknowledgments normally appear.</li>
064 * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
065 * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
066 * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
067 * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
068 * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
069 * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
070 * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
071 * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
072 * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
073 * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
074 * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
075 * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
076 * BE CORRECTED.</strong></li>
077 * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
078 * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
079 * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
080 * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
081 * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
082 * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
083 * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
084 * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
085 * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
086 * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
087 * <ol></td></tr>
088 * </table>
089
090 * @author Argonne National Laboratory. MINPACK project. March 1980 (original fortran)
091 * @author Burton S. Garbow (original fortran)
092 * @author Kenneth E. Hillstrom (original fortran)
093 * @author Jorge J. More (original fortran)
094
095 * @version $Revision: 762087 $ $Date: 2009-04-05 10:20:18 -0400 (Sun, 05 Apr 2009) $
096 * @since 1.2
097 * @deprecated as of 2.0, everything in package org.apache.commons.math.estimation has
098 * been deprecated and replaced by package org.apache.commons.math.optimization.general
099 *
100 */
101 @Deprecated
102 public class LevenbergMarquardtEstimator extends AbstractEstimator implements Serializable {
103
104 /**
105 * Build an estimator for least squares problems.
106 * <p>The default values for the algorithm settings are:
107 * <ul>
108 * <li>{@link #setInitialStepBoundFactor initial step bound factor}: 100.0</li>
109 * <li>{@link #setMaxCostEval maximal cost evaluations}: 1000</li>
110 * <li>{@link #setCostRelativeTolerance cost relative tolerance}: 1.0e-10</li>
111 * <li>{@link #setParRelativeTolerance parameters relative tolerance}: 1.0e-10</li>
112 * <li>{@link #setOrthoTolerance orthogonality tolerance}: 1.0e-10</li>
113 * </ul>
114 * </p>
115 */
116 public LevenbergMarquardtEstimator() {
117
118 // set up the superclass with a default max cost evaluations setting
119 setMaxCostEval(1000);
120
121 // default values for the tuning parameters
122 setInitialStepBoundFactor(100.0);
123 setCostRelativeTolerance(1.0e-10);
124 setParRelativeTolerance(1.0e-10);
125 setOrthoTolerance(1.0e-10);
126
127 }
128
129 /**
130 * Set the positive input variable used in determining the initial step bound.
131 * This bound is set to the product of initialStepBoundFactor and the euclidean norm of diag*x if nonzero,
132 * or else to initialStepBoundFactor itself. In most cases factor should lie
133 * in the interval (0.1, 100.0). 100.0 is a generally recommended value
134 *
135 * @param initialStepBoundFactor initial step bound factor
136 * @see #estimate
137 */
138 public void setInitialStepBoundFactor(double initialStepBoundFactor) {
139 this.initialStepBoundFactor = initialStepBoundFactor;
140 }
141
142 /**
143 * Set the desired relative error in the sum of squares.
144 *
145 * @param costRelativeTolerance desired relative error in the sum of squares
146 * @see #estimate
147 */
148 public void setCostRelativeTolerance(double costRelativeTolerance) {
149 this.costRelativeTolerance = costRelativeTolerance;
150 }
151
152 /**
153 * Set the desired relative error in the approximate solution parameters.
154 *
155 * @param parRelativeTolerance desired relative error
156 * in the approximate solution parameters
157 * @see #estimate
158 */
159 public void setParRelativeTolerance(double parRelativeTolerance) {
160 this.parRelativeTolerance = parRelativeTolerance;
161 }
162
163 /**
164 * Set the desired max cosine on the orthogonality.
165 *
166 * @param orthoTolerance desired max cosine on the orthogonality
167 * between the function vector and the columns of the jacobian
168 * @see #estimate
169 */
170 public void setOrthoTolerance(double orthoTolerance) {
171 this.orthoTolerance = orthoTolerance;
172 }
173
174 /**
175 * Solve an estimation problem using the Levenberg-Marquardt algorithm.
176 * <p>The algorithm used is a modified Levenberg-Marquardt one, based
177 * on the MINPACK <a href="http://www.netlib.org/minpack/lmder.f">lmder</a>
178 * routine. The algorithm settings must have been set up before this method
179 * is called with the {@link #setInitialStepBoundFactor},
180 * {@link #setMaxCostEval}, {@link #setCostRelativeTolerance},
181 * {@link #setParRelativeTolerance} and {@link #setOrthoTolerance} methods.
182 * If these methods have not been called, the default values set up by the
183 * {@link #LevenbergMarquardtEstimator() constructor} will be used.</p>
184 * <p>The authors of the original fortran function are:</p>
185 * <ul>
186 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
187 * <li>Burton S. Garbow</li>
188 * <li>Kenneth E. Hillstrom</li>
189 * <li>Jorge J. More</li>
190 * </ul>
191 * <p>Luc Maisonobe did the Java translation.</p>
192 *
193 * @param problem estimation problem to solve
194 * @exception EstimationException if convergence cannot be
195 * reached with the specified algorithm settings or if there are more variables
196 * than equations
197 * @see #setInitialStepBoundFactor
198 * @see #setCostRelativeTolerance
199 * @see #setParRelativeTolerance
200 * @see #setOrthoTolerance
201 */
202 @Override
203 public void estimate(EstimationProblem problem)
204 throws EstimationException {
205
206 initializeEstimate(problem);
207
208 // arrays shared with the other private methods
209 solvedCols = Math.min(rows, cols);
210 diagR = new double[cols];
211 jacNorm = new double[cols];
212 beta = new double[cols];
213 permutation = new int[cols];
214 lmDir = new double[cols];
215
216 // local variables
217 double delta = 0, xNorm = 0;
218 double[] diag = new double[cols];
219 double[] oldX = new double[cols];
220 double[] oldRes = new double[rows];
221 double[] work1 = new double[cols];
222 double[] work2 = new double[cols];
223 double[] work3 = new double[cols];
224
225 // evaluate the function at the starting point and calculate its norm
226 updateResidualsAndCost();
227
228 // outer loop
229 lmPar = 0;
230 boolean firstIteration = true;
231 while (true) {
232
233 // compute the Q.R. decomposition of the jacobian matrix
234 updateJacobian();
235 qrDecomposition();
236
237 // compute Qt.res
238 qTy(residuals);
239
240 // now we don't need Q anymore,
241 // so let jacobian contain the R matrix with its diagonal elements
242 for (int k = 0; k < solvedCols; ++k) {
243 int pk = permutation[k];
244 jacobian[k * cols + pk] = diagR[pk];
245 }
246
247 if (firstIteration) {
248
249 // scale the variables according to the norms of the columns
250 // of the initial jacobian
251 xNorm = 0;
252 for (int k = 0; k < cols; ++k) {
253 double dk = jacNorm[k];
254 if (dk == 0) {
255 dk = 1.0;
256 }
257 double xk = dk * parameters[k].getEstimate();
258 xNorm += xk * xk;
259 diag[k] = dk;
260 }
261 xNorm = Math.sqrt(xNorm);
262
263 // initialize the step bound delta
264 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
265
266 }
267
268 // check orthogonality between function vector and jacobian columns
269 double maxCosine = 0;
270 if (cost != 0) {
271 for (int j = 0; j < solvedCols; ++j) {
272 int pj = permutation[j];
273 double s = jacNorm[pj];
274 if (s != 0) {
275 double sum = 0;
276 for (int i = 0, index = pj; i <= j; ++i, index += cols) {
277 sum += jacobian[index] * residuals[i];
278 }
279 maxCosine = Math.max(maxCosine, Math.abs(sum) / (s * cost));
280 }
281 }
282 }
283 if (maxCosine <= orthoTolerance) {
284 return;
285 }
286
287 // rescale if necessary
288 for (int j = 0; j < cols; ++j) {
289 diag[j] = Math.max(diag[j], jacNorm[j]);
290 }
291
292 // inner loop
293 for (double ratio = 0; ratio < 1.0e-4;) {
294
295 // save the state
296 for (int j = 0; j < solvedCols; ++j) {
297 int pj = permutation[j];
298 oldX[pj] = parameters[pj].getEstimate();
299 }
300 double previousCost = cost;
301 double[] tmpVec = residuals;
302 residuals = oldRes;
303 oldRes = tmpVec;
304
305 // determine the Levenberg-Marquardt parameter
306 determineLMParameter(oldRes, delta, diag, work1, work2, work3);
307
308 // compute the new point and the norm of the evolution direction
309 double lmNorm = 0;
310 for (int j = 0; j < solvedCols; ++j) {
311 int pj = permutation[j];
312 lmDir[pj] = -lmDir[pj];
313 parameters[pj].setEstimate(oldX[pj] + lmDir[pj]);
314 double s = diag[pj] * lmDir[pj];
315 lmNorm += s * s;
316 }
317 lmNorm = Math.sqrt(lmNorm);
318
319 // on the first iteration, adjust the initial step bound.
320 if (firstIteration) {
321 delta = Math.min(delta, lmNorm);
322 }
323
324 // evaluate the function at x + p and calculate its norm
325 updateResidualsAndCost();
326
327 // compute the scaled actual reduction
328 double actRed = -1.0;
329 if (0.1 * cost < previousCost) {
330 double r = cost / previousCost;
331 actRed = 1.0 - r * r;
332 }
333
334 // compute the scaled predicted reduction
335 // and the scaled directional derivative
336 for (int j = 0; j < solvedCols; ++j) {
337 int pj = permutation[j];
338 double dirJ = lmDir[pj];
339 work1[j] = 0;
340 for (int i = 0, index = pj; i <= j; ++i, index += cols) {
341 work1[i] += jacobian[index] * dirJ;
342 }
343 }
344 double coeff1 = 0;
345 for (int j = 0; j < solvedCols; ++j) {
346 coeff1 += work1[j] * work1[j];
347 }
348 double pc2 = previousCost * previousCost;
349 coeff1 = coeff1 / pc2;
350 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
351 double preRed = coeff1 + 2 * coeff2;
352 double dirDer = -(coeff1 + coeff2);
353
354 // ratio of the actual to the predicted reduction
355 ratio = (preRed == 0) ? 0 : (actRed / preRed);
356
357 // update the step bound
358 if (ratio <= 0.25) {
359 double tmp =
360 (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
361 if ((0.1 * cost >= previousCost) || (tmp < 0.1)) {
362 tmp = 0.1;
363 }
364 delta = tmp * Math.min(delta, 10.0 * lmNorm);
365 lmPar /= tmp;
366 } else if ((lmPar == 0) || (ratio >= 0.75)) {
367 delta = 2 * lmNorm;
368 lmPar *= 0.5;
369 }
370
371 // test for successful iteration.
372 if (ratio >= 1.0e-4) {
373 // successful iteration, update the norm
374 firstIteration = false;
375 xNorm = 0;
376 for (int k = 0; k < cols; ++k) {
377 double xK = diag[k] * parameters[k].getEstimate();
378 xNorm += xK * xK;
379 }
380 xNorm = Math.sqrt(xNorm);
381 } else {
382 // failed iteration, reset the previous values
383 cost = previousCost;
384 for (int j = 0; j < solvedCols; ++j) {
385 int pj = permutation[j];
386 parameters[pj].setEstimate(oldX[pj]);
387 }
388 tmpVec = residuals;
389 residuals = oldRes;
390 oldRes = tmpVec;
391 }
392
393 // tests for convergence.
394 if (((Math.abs(actRed) <= costRelativeTolerance) &&
395 (preRed <= costRelativeTolerance) &&
396 (ratio <= 2.0)) ||
397 (delta <= parRelativeTolerance * xNorm)) {
398 return;
399 }
400
401 // tests for termination and stringent tolerances
402 // (2.2204e-16 is the machine epsilon for IEEE754)
403 if ((Math.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
404 throw new EstimationException("cost relative tolerance is too small ({0})," +
405 " no further reduction in the" +
406 " sum of squares is possible",
407 costRelativeTolerance);
408 } else if (delta <= 2.2204e-16 * xNorm) {
409 throw new EstimationException("parameters relative tolerance is too small" +
410 " ({0}), no further improvement in" +
411 " the approximate solution is possible",
412 parRelativeTolerance);
413 } else if (maxCosine <= 2.2204e-16) {
414 throw new EstimationException("orthogonality tolerance is too small ({0})," +
415 " solution is orthogonal to the jacobian",
416 orthoTolerance);
417 }
418
419 }
420
421 }
422
423 }
424
425 /**
426 * Determine the Levenberg-Marquardt parameter.
427 * <p>This implementation is a translation in Java of the MINPACK
428 * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
429 * routine.</p>
430 * <p>This method sets the lmPar and lmDir attributes.</p>
431 * <p>The authors of the original fortran function are:</p>
432 * <ul>
433 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
434 * <li>Burton S. Garbow</li>
435 * <li>Kenneth E. Hillstrom</li>
436 * <li>Jorge J. More</li>
437 * </ul>
438 * <p>Luc Maisonobe did the Java translation.</p>
439 *
440 * @param qy array containing qTy
441 * @param delta upper bound on the euclidean norm of diagR * lmDir
442 * @param diag diagonal matrix
443 * @param work1 work array
444 * @param work2 work array
445 * @param work3 work array
446 */
447 private void determineLMParameter(double[] qy, double delta, double[] diag,
448 double[] work1, double[] work2, double[] work3) {
449
450 // compute and store in x the gauss-newton direction, if the
451 // jacobian is rank-deficient, obtain a least squares solution
452 for (int j = 0; j < rank; ++j) {
453 lmDir[permutation[j]] = qy[j];
454 }
455 for (int j = rank; j < cols; ++j) {
456 lmDir[permutation[j]] = 0;
457 }
458 for (int k = rank - 1; k >= 0; --k) {
459 int pk = permutation[k];
460 double ypk = lmDir[pk] / diagR[pk];
461 for (int i = 0, index = pk; i < k; ++i, index += cols) {
462 lmDir[permutation[i]] -= ypk * jacobian[index];
463 }
464 lmDir[pk] = ypk;
465 }
466
467 // evaluate the function at the origin, and test
468 // for acceptance of the Gauss-Newton direction
469 double dxNorm = 0;
470 for (int j = 0; j < solvedCols; ++j) {
471 int pj = permutation[j];
472 double s = diag[pj] * lmDir[pj];
473 work1[pj] = s;
474 dxNorm += s * s;
475 }
476 dxNorm = Math.sqrt(dxNorm);
477 double fp = dxNorm - delta;
478 if (fp <= 0.1 * delta) {
479 lmPar = 0;
480 return;
481 }
482
483 // if the jacobian is not rank deficient, the Newton step provides
484 // a lower bound, parl, for the zero of the function,
485 // otherwise set this bound to zero
486 double sum2, parl = 0;
487 if (rank == solvedCols) {
488 for (int j = 0; j < solvedCols; ++j) {
489 int pj = permutation[j];
490 work1[pj] *= diag[pj] / dxNorm;
491 }
492 sum2 = 0;
493 for (int j = 0; j < solvedCols; ++j) {
494 int pj = permutation[j];
495 double sum = 0;
496 for (int i = 0, index = pj; i < j; ++i, index += cols) {
497 sum += jacobian[index] * work1[permutation[i]];
498 }
499 double s = (work1[pj] - sum) / diagR[pj];
500 work1[pj] = s;
501 sum2 += s * s;
502 }
503 parl = fp / (delta * sum2);
504 }
505
506 // calculate an upper bound, paru, for the zero of the function
507 sum2 = 0;
508 for (int j = 0; j < solvedCols; ++j) {
509 int pj = permutation[j];
510 double sum = 0;
511 for (int i = 0, index = pj; i <= j; ++i, index += cols) {
512 sum += jacobian[index] * qy[i];
513 }
514 sum /= diag[pj];
515 sum2 += sum * sum;
516 }
517 double gNorm = Math.sqrt(sum2);
518 double paru = gNorm / delta;
519 if (paru == 0) {
520 // 2.2251e-308 is the smallest positive real for IEE754
521 paru = 2.2251e-308 / Math.min(delta, 0.1);
522 }
523
524 // if the input par lies outside of the interval (parl,paru),
525 // set par to the closer endpoint
526 lmPar = Math.min(paru, Math.max(lmPar, parl));
527 if (lmPar == 0) {
528 lmPar = gNorm / dxNorm;
529 }
530
531 for (int countdown = 10; countdown >= 0; --countdown) {
532
533 // evaluate the function at the current value of lmPar
534 if (lmPar == 0) {
535 lmPar = Math.max(2.2251e-308, 0.001 * paru);
536 }
537 double sPar = Math.sqrt(lmPar);
538 for (int j = 0; j < solvedCols; ++j) {
539 int pj = permutation[j];
540 work1[pj] = sPar * diag[pj];
541 }
542 determineLMDirection(qy, work1, work2, work3);
543
544 dxNorm = 0;
545 for (int j = 0; j < solvedCols; ++j) {
546 int pj = permutation[j];
547 double s = diag[pj] * lmDir[pj];
548 work3[pj] = s;
549 dxNorm += s * s;
550 }
551 dxNorm = Math.sqrt(dxNorm);
552 double previousFP = fp;
553 fp = dxNorm - delta;
554
555 // if the function is small enough, accept the current value
556 // of lmPar, also test for the exceptional cases where parl is zero
557 if ((Math.abs(fp) <= 0.1 * delta) ||
558 ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
559 return;
560 }
561
562 // compute the Newton correction
563 for (int j = 0; j < solvedCols; ++j) {
564 int pj = permutation[j];
565 work1[pj] = work3[pj] * diag[pj] / dxNorm;
566 }
567 for (int j = 0; j < solvedCols; ++j) {
568 int pj = permutation[j];
569 work1[pj] /= work2[j];
570 double tmp = work1[pj];
571 for (int i = j + 1; i < solvedCols; ++i) {
572 work1[permutation[i]] -= jacobian[i * cols + pj] * tmp;
573 }
574 }
575 sum2 = 0;
576 for (int j = 0; j < solvedCols; ++j) {
577 double s = work1[permutation[j]];
578 sum2 += s * s;
579 }
580 double correction = fp / (delta * sum2);
581
582 // depending on the sign of the function, update parl or paru.
583 if (fp > 0) {
584 parl = Math.max(parl, lmPar);
585 } else if (fp < 0) {
586 paru = Math.min(paru, lmPar);
587 }
588
589 // compute an improved estimate for lmPar
590 lmPar = Math.max(parl, lmPar + correction);
591
592 }
593 }
594
595 /**
596 * Solve a*x = b and d*x = 0 in the least squares sense.
597 * <p>This implementation is a translation in Java of the MINPACK
598 * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
599 * routine.</p>
600 * <p>This method sets the lmDir and lmDiag attributes.</p>
601 * <p>The authors of the original fortran function are:</p>
602 * <ul>
603 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
604 * <li>Burton S. Garbow</li>
605 * <li>Kenneth E. Hillstrom</li>
606 * <li>Jorge J. More</li>
607 * </ul>
608 * <p>Luc Maisonobe did the Java translation.</p>
609 *
610 * @param qy array containing qTy
611 * @param diag diagonal matrix
612 * @param lmDiag diagonal elements associated with lmDir
613 * @param work work array
614 */
615 private void determineLMDirection(double[] qy, double[] diag,
616 double[] lmDiag, double[] work) {
617
618 // copy R and Qty to preserve input and initialize s
619 // in particular, save the diagonal elements of R in lmDir
620 for (int j = 0; j < solvedCols; ++j) {
621 int pj = permutation[j];
622 for (int i = j + 1; i < solvedCols; ++i) {
623 jacobian[i * cols + pj] = jacobian[j * cols + permutation[i]];
624 }
625 lmDir[j] = diagR[pj];
626 work[j] = qy[j];
627 }
628
629 // eliminate the diagonal matrix d using a Givens rotation
630 for (int j = 0; j < solvedCols; ++j) {
631
632 // prepare the row of d to be eliminated, locating the
633 // diagonal element using p from the Q.R. factorization
634 int pj = permutation[j];
635 double dpj = diag[pj];
636 if (dpj != 0) {
637 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
638 }
639 lmDiag[j] = dpj;
640
641 // the transformations to eliminate the row of d
642 // modify only a single element of Qty
643 // beyond the first n, which is initially zero.
644 double qtbpj = 0;
645 for (int k = j; k < solvedCols; ++k) {
646 int pk = permutation[k];
647
648 // determine a Givens rotation which eliminates the
649 // appropriate element in the current row of d
650 if (lmDiag[k] != 0) {
651
652 double sin, cos;
653 double rkk = jacobian[k * cols + pk];
654 if (Math.abs(rkk) < Math.abs(lmDiag[k])) {
655 double cotan = rkk / lmDiag[k];
656 sin = 1.0 / Math.sqrt(1.0 + cotan * cotan);
657 cos = sin * cotan;
658 } else {
659 double tan = lmDiag[k] / rkk;
660 cos = 1.0 / Math.sqrt(1.0 + tan * tan);
661 sin = cos * tan;
662 }
663
664 // compute the modified diagonal element of R and
665 // the modified element of (Qty,0)
666 jacobian[k * cols + pk] = cos * rkk + sin * lmDiag[k];
667 double temp = cos * work[k] + sin * qtbpj;
668 qtbpj = -sin * work[k] + cos * qtbpj;
669 work[k] = temp;
670
671 // accumulate the tranformation in the row of s
672 for (int i = k + 1; i < solvedCols; ++i) {
673 double rik = jacobian[i * cols + pk];
674 temp = cos * rik + sin * lmDiag[i];
675 lmDiag[i] = -sin * rik + cos * lmDiag[i];
676 jacobian[i * cols + pk] = temp;
677 }
678
679 }
680 }
681
682 // store the diagonal element of s and restore
683 // the corresponding diagonal element of R
684 int index = j * cols + permutation[j];
685 lmDiag[j] = jacobian[index];
686 jacobian[index] = lmDir[j];
687
688 }
689
690 // solve the triangular system for z, if the system is
691 // singular, then obtain a least squares solution
692 int nSing = solvedCols;
693 for (int j = 0; j < solvedCols; ++j) {
694 if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
695 nSing = j;
696 }
697 if (nSing < solvedCols) {
698 work[j] = 0;
699 }
700 }
701 if (nSing > 0) {
702 for (int j = nSing - 1; j >= 0; --j) {
703 int pj = permutation[j];
704 double sum = 0;
705 for (int i = j + 1; i < nSing; ++i) {
706 sum += jacobian[i * cols + pj] * work[i];
707 }
708 work[j] = (work[j] - sum) / lmDiag[j];
709 }
710 }
711
712 // permute the components of z back to components of lmDir
713 for (int j = 0; j < lmDir.length; ++j) {
714 lmDir[permutation[j]] = work[j];
715 }
716
717 }
718
719 /**
720 * Decompose a matrix A as A.P = Q.R using Householder transforms.
721 * <p>As suggested in the P. Lascaux and R. Theodor book
722 * <i>Analyse numérique matricielle appliquée à
723 * l'art de l'ingénieur</i> (Masson, 1986), instead of representing
724 * the Householder transforms with u<sub>k</sub> unit vectors such that:
725 * <pre>
726 * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
727 * </pre>
728 * we use <sub>k</sub> non-unit vectors such that:
729 * <pre>
730 * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
731 * </pre>
732 * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
733 * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
734 * them from the v<sub>k</sub> vectors would be costly.</p>
735 * <p>This decomposition handles rank deficient cases since the tranformations
736 * are performed in non-increasing columns norms order thanks to columns
737 * pivoting. The diagonal elements of the R matrix are therefore also in
738 * non-increasing absolute values order.</p>
739 * @exception EstimationException if the decomposition cannot be performed
740 */
741 private void qrDecomposition() throws EstimationException {
742
743 // initializations
744 for (int k = 0; k < cols; ++k) {
745 permutation[k] = k;
746 double norm2 = 0;
747 for (int index = k; index < jacobian.length; index += cols) {
748 double akk = jacobian[index];
749 norm2 += akk * akk;
750 }
751 jacNorm[k] = Math.sqrt(norm2);
752 }
753
754 // transform the matrix column after column
755 for (int k = 0; k < cols; ++k) {
756
757 // select the column with the greatest norm on active components
758 int nextColumn = -1;
759 double ak2 = Double.NEGATIVE_INFINITY;
760 for (int i = k; i < cols; ++i) {
761 double norm2 = 0;
762 int iDiag = k * cols + permutation[i];
763 for (int index = iDiag; index < jacobian.length; index += cols) {
764 double aki = jacobian[index];
765 norm2 += aki * aki;
766 }
767 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
768 throw new EstimationException(
769 "unable to perform Q.R decomposition on the {0}x{1} jacobian matrix",
770 rows, cols);
771 }
772 if (norm2 > ak2) {
773 nextColumn = i;
774 ak2 = norm2;
775 }
776 }
777 if (ak2 == 0) {
778 rank = k;
779 return;
780 }
781 int pk = permutation[nextColumn];
782 permutation[nextColumn] = permutation[k];
783 permutation[k] = pk;
784
785 // choose alpha such that Hk.u = alpha ek
786 int kDiag = k * cols + pk;
787 double akk = jacobian[kDiag];
788 double alpha = (akk > 0) ? -Math.sqrt(ak2) : Math.sqrt(ak2);
789 double betak = 1.0 / (ak2 - akk * alpha);
790 beta[pk] = betak;
791
792 // transform the current column
793 diagR[pk] = alpha;
794 jacobian[kDiag] -= alpha;
795
796 // transform the remaining columns
797 for (int dk = cols - 1 - k; dk > 0; --dk) {
798 int dkp = permutation[k + dk] - pk;
799 double gamma = 0;
800 for (int index = kDiag; index < jacobian.length; index += cols) {
801 gamma += jacobian[index] * jacobian[index + dkp];
802 }
803 gamma *= betak;
804 for (int index = kDiag; index < jacobian.length; index += cols) {
805 jacobian[index + dkp] -= gamma * jacobian[index];
806 }
807 }
808
809 }
810
811 rank = solvedCols;
812
813 }
814
815 /**
816 * Compute the product Qt.y for some Q.R. decomposition.
817 *
818 * @param y vector to multiply (will be overwritten with the result)
819 */
820 private void qTy(double[] y) {
821 for (int k = 0; k < cols; ++k) {
822 int pk = permutation[k];
823 int kDiag = k * cols + pk;
824 double gamma = 0;
825 for (int i = k, index = kDiag; i < rows; ++i, index += cols) {
826 gamma += jacobian[index] * y[i];
827 }
828 gamma *= beta[pk];
829 for (int i = k, index = kDiag; i < rows; ++i, index += cols) {
830 y[i] -= gamma * jacobian[index];
831 }
832 }
833 }
834
835 /** Number of solved variables. */
836 private int solvedCols;
837
838 /** Diagonal elements of the R matrix in the Q.R. decomposition. */
839 private double[] diagR;
840
841 /** Norms of the columns of the jacobian matrix. */
842 private double[] jacNorm;
843
844 /** Coefficients of the Householder transforms vectors. */
845 private double[] beta;
846
847 /** Columns permutation array. */
848 private int[] permutation;
849
850 /** Rank of the jacobian matrix. */
851 private int rank;
852
853 /** Levenberg-Marquardt parameter. */
854 private double lmPar;
855
856 /** Parameters evolution direction associated with lmPar. */
857 private double[] lmDir;
858
859 /** Positive input variable used in determining the initial step bound. */
860 private double initialStepBoundFactor;
861
862 /** Desired relative error in the sum of squares. */
863 private double costRelativeTolerance;
864
865 /** Desired relative error in the approximate solution parameters. */
866 private double parRelativeTolerance;
867
868 /** Desired max cosine on the orthogonality between the function vector
869 * and the columns of the jacobian. */
870 private double orthoTolerance;
871
872 /** Serializable version identifier */
873 private static final long serialVersionUID = -5705952631533171019L;
874
875 }