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 package org.apache.commons.math.estimation;
18
19 import java.io.Serializable;
20 import java.util.Arrays;
21
22
23 /**
24 * This class solves a least squares problem.
25 *
26 * <p>This implementation <em>should</em> work even for over-determined systems
27 * (i.e. systems having more variables than equations). Over-determined systems
28 * are solved by ignoring the variables which have the smallest impact according
29 * to their jacobian column norm. Only the rank of the matrix and some loop bounds
30 * are changed to implement this.</p>
31 *
32 * <p>The resolution engine is a simple translation of the MINPACK <a
33 * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
34 * changes. The changes include the over-determined resolution and the Q.R.
35 * decomposition which has been rewritten following the algorithm described in the
36 * P. Lascaux and R. Theodor book <i>Analyse numérique matricielle
37 * appliquée à l'art de l'ingénieur</i>, Masson 1986.</p>
38 * <p>The authors of the original fortran version are:
39 * <ul>
40 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
41 * <li>Burton S. Garbow</li>
42 * <li>Kenneth E. Hillstrom</li>
43 * <li>Jorge J. More</li>
44 * </ul>
45 * The redistribution policy for MINPACK is available <a
46 * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
47 * is reproduced below.</p>
48 *
49 * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
50 * <tr><td>
51 * Minpack Copyright Notice (1999) University of Chicago.
52 * All rights reserved
53 * </td></tr>
54 * <tr><td>
55 * Redistribution and use in source and binary forms, with or without
56 * modification, are permitted provided that the following conditions
57 * are met:
58 * <ol>
59 * <li>Redistributions of source code must retain the above copyright
60 * notice, this list of conditions and the following disclaimer.</li>
61 * <li>Redistributions in binary form must reproduce the above
62 * copyright notice, this list of conditions and the following
63 * disclaimer in the documentation and/or other materials provided
64 * with the distribution.</li>
65 * <li>The end-user documentation included with the redistribution, if any,
66 * must include the following acknowledgment:
67 * <code>This product includes software developed by the University of
68 * Chicago, as Operator of Argonne National Laboratory.</code>
69 * Alternately, this acknowledgment may appear in the software itself,
70 * if and wherever such third-party acknowledgments normally appear.</li>
71 * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
72 * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
73 * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
74 * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
75 * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
76 * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
77 * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
78 * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
79 * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
80 * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
81 * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
82 * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
83 * BE CORRECTED.</strong></li>
84 * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
85 * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
86 * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
87 * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
88 * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
89 * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
90 * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
91 * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
92 * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
93 * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
94 * <ol></td></tr>
95 * </table>
96
97 * @version $Revision: 825919 $ $Date: 2009-10-16 10:51:55 -0400 (Fri, 16 Oct 2009) $
98 * @since 1.2
99 * @deprecated as of 2.0, everything in package org.apache.commons.math.estimation has
100 * been deprecated and replaced by package org.apache.commons.math.optimization.general
101 *
102 */
103 @Deprecated
104 public class LevenbergMarquardtEstimator extends AbstractEstimator implements Serializable {
105
106 /** Serializable version identifier */
107 private static final long serialVersionUID = -5705952631533171019L;
108
109 /** Number of solved variables. */
110 private int solvedCols;
111
112 /** Diagonal elements of the R matrix in the Q.R. decomposition. */
113 private double[] diagR;
114
115 /** Norms of the columns of the jacobian matrix. */
116 private double[] jacNorm;
117
118 /** Coefficients of the Householder transforms vectors. */
119 private double[] beta;
120
121 /** Columns permutation array. */
122 private int[] permutation;
123
124 /** Rank of the jacobian matrix. */
125 private int rank;
126
127 /** Levenberg-Marquardt parameter. */
128 private double lmPar;
129
130 /** Parameters evolution direction associated with lmPar. */
131 private double[] lmDir;
132
133 /** Positive input variable used in determining the initial step bound. */
134 private double initialStepBoundFactor;
135
136 /** Desired relative error in the sum of squares. */
137 private double costRelativeTolerance;
138
139 /** Desired relative error in the approximate solution parameters. */
140 private double parRelativeTolerance;
141
142 /** Desired max cosine on the orthogonality between the function vector
143 * and the columns of the jacobian. */
144 private double orthoTolerance;
145
146 /**
147 * Build an estimator for least squares problems.
148 * <p>The default values for the algorithm settings are:
149 * <ul>
150 * <li>{@link #setInitialStepBoundFactor initial step bound factor}: 100.0</li>
151 * <li>{@link #setMaxCostEval maximal cost evaluations}: 1000</li>
152 * <li>{@link #setCostRelativeTolerance cost relative tolerance}: 1.0e-10</li>
153 * <li>{@link #setParRelativeTolerance parameters relative tolerance}: 1.0e-10</li>
154 * <li>{@link #setOrthoTolerance orthogonality tolerance}: 1.0e-10</li>
155 * </ul>
156 * </p>
157 */
158 public LevenbergMarquardtEstimator() {
159
160 // set up the superclass with a default max cost evaluations setting
161 setMaxCostEval(1000);
162
163 // default values for the tuning parameters
164 setInitialStepBoundFactor(100.0);
165 setCostRelativeTolerance(1.0e-10);
166 setParRelativeTolerance(1.0e-10);
167 setOrthoTolerance(1.0e-10);
168
169 }
170
171 /**
172 * Set the positive input variable used in determining the initial step bound.
173 * This bound is set to the product of initialStepBoundFactor and the euclidean norm of diag*x if nonzero,
174 * or else to initialStepBoundFactor itself. In most cases factor should lie
175 * in the interval (0.1, 100.0). 100.0 is a generally recommended value
176 *
177 * @param initialStepBoundFactor initial step bound factor
178 * @see #estimate
179 */
180 public void setInitialStepBoundFactor(double initialStepBoundFactor) {
181 this.initialStepBoundFactor = initialStepBoundFactor;
182 }
183
184 /**
185 * Set the desired relative error in the sum of squares.
186 *
187 * @param costRelativeTolerance desired relative error in the sum of squares
188 * @see #estimate
189 */
190 public void setCostRelativeTolerance(double costRelativeTolerance) {
191 this.costRelativeTolerance = costRelativeTolerance;
192 }
193
194 /**
195 * Set the desired relative error in the approximate solution parameters.
196 *
197 * @param parRelativeTolerance desired relative error
198 * in the approximate solution parameters
199 * @see #estimate
200 */
201 public void setParRelativeTolerance(double parRelativeTolerance) {
202 this.parRelativeTolerance = parRelativeTolerance;
203 }
204
205 /**
206 * Set the desired max cosine on the orthogonality.
207 *
208 * @param orthoTolerance desired max cosine on the orthogonality
209 * between the function vector and the columns of the jacobian
210 * @see #estimate
211 */
212 public void setOrthoTolerance(double orthoTolerance) {
213 this.orthoTolerance = orthoTolerance;
214 }
215
216 /**
217 * Solve an estimation problem using the Levenberg-Marquardt algorithm.
218 * <p>The algorithm used is a modified Levenberg-Marquardt one, based
219 * on the MINPACK <a href="http://www.netlib.org/minpack/lmder.f">lmder</a>
220 * routine. The algorithm settings must have been set up before this method
221 * is called with the {@link #setInitialStepBoundFactor},
222 * {@link #setMaxCostEval}, {@link #setCostRelativeTolerance},
223 * {@link #setParRelativeTolerance} and {@link #setOrthoTolerance} methods.
224 * If these methods have not been called, the default values set up by the
225 * {@link #LevenbergMarquardtEstimator() constructor} will be used.</p>
226 * <p>The authors of the original fortran function are:</p>
227 * <ul>
228 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
229 * <li>Burton S. Garbow</li>
230 * <li>Kenneth E. Hillstrom</li>
231 * <li>Jorge J. More</li>
232 * </ul>
233 * <p>Luc Maisonobe did the Java translation.</p>
234 *
235 * @param problem estimation problem to solve
236 * @exception EstimationException if convergence cannot be
237 * reached with the specified algorithm settings or if there are more variables
238 * than equations
239 * @see #setInitialStepBoundFactor
240 * @see #setCostRelativeTolerance
241 * @see #setParRelativeTolerance
242 * @see #setOrthoTolerance
243 */
244 @Override
245 public void estimate(EstimationProblem problem)
246 throws EstimationException {
247
248 initializeEstimate(problem);
249
250 // arrays shared with the other private methods
251 solvedCols = Math.min(rows, cols);
252 diagR = new double[cols];
253 jacNorm = new double[cols];
254 beta = new double[cols];
255 permutation = new int[cols];
256 lmDir = new double[cols];
257
258 // local variables
259 double delta = 0;
260 double xNorm = 0;
261 double[] diag = new double[cols];
262 double[] oldX = new double[cols];
263 double[] oldRes = new double[rows];
264 double[] work1 = new double[cols];
265 double[] work2 = new double[cols];
266 double[] work3 = new double[cols];
267
268 // evaluate the function at the starting point and calculate its norm
269 updateResidualsAndCost();
270
271 // outer loop
272 lmPar = 0;
273 boolean firstIteration = true;
274 while (true) {
275
276 // compute the Q.R. decomposition of the jacobian matrix
277 updateJacobian();
278 qrDecomposition();
279
280 // compute Qt.res
281 qTy(residuals);
282
283 // now we don't need Q anymore,
284 // so let jacobian contain the R matrix with its diagonal elements
285 for (int k = 0; k < solvedCols; ++k) {
286 int pk = permutation[k];
287 jacobian[k * cols + pk] = diagR[pk];
288 }
289
290 if (firstIteration) {
291
292 // scale the variables according to the norms of the columns
293 // of the initial jacobian
294 xNorm = 0;
295 for (int k = 0; k < cols; ++k) {
296 double dk = jacNorm[k];
297 if (dk == 0) {
298 dk = 1.0;
299 }
300 double xk = dk * parameters[k].getEstimate();
301 xNorm += xk * xk;
302 diag[k] = dk;
303 }
304 xNorm = Math.sqrt(xNorm);
305
306 // initialize the step bound delta
307 delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
308
309 }
310
311 // check orthogonality between function vector and jacobian columns
312 double maxCosine = 0;
313 if (cost != 0) {
314 for (int j = 0; j < solvedCols; ++j) {
315 int pj = permutation[j];
316 double s = jacNorm[pj];
317 if (s != 0) {
318 double sum = 0;
319 int index = pj;
320 for (int i = 0; i <= j; ++i) {
321 sum += jacobian[index] * residuals[i];
322 index += cols;
323 }
324 maxCosine = Math.max(maxCosine, Math.abs(sum) / (s * cost));
325 }
326 }
327 }
328 if (maxCosine <= orthoTolerance) {
329 return;
330 }
331
332 // rescale if necessary
333 for (int j = 0; j < cols; ++j) {
334 diag[j] = Math.max(diag[j], jacNorm[j]);
335 }
336
337 // inner loop
338 for (double ratio = 0; ratio < 1.0e-4;) {
339
340 // save the state
341 for (int j = 0; j < solvedCols; ++j) {
342 int pj = permutation[j];
343 oldX[pj] = parameters[pj].getEstimate();
344 }
345 double previousCost = cost;
346 double[] tmpVec = residuals;
347 residuals = oldRes;
348 oldRes = tmpVec;
349
350 // determine the Levenberg-Marquardt parameter
351 determineLMParameter(oldRes, delta, diag, work1, work2, work3);
352
353 // compute the new point and the norm of the evolution direction
354 double lmNorm = 0;
355 for (int j = 0; j < solvedCols; ++j) {
356 int pj = permutation[j];
357 lmDir[pj] = -lmDir[pj];
358 parameters[pj].setEstimate(oldX[pj] + lmDir[pj]);
359 double s = diag[pj] * lmDir[pj];
360 lmNorm += s * s;
361 }
362 lmNorm = Math.sqrt(lmNorm);
363
364 // on the first iteration, adjust the initial step bound.
365 if (firstIteration) {
366 delta = Math.min(delta, lmNorm);
367 }
368
369 // evaluate the function at x + p and calculate its norm
370 updateResidualsAndCost();
371
372 // compute the scaled actual reduction
373 double actRed = -1.0;
374 if (0.1 * cost < previousCost) {
375 double r = cost / previousCost;
376 actRed = 1.0 - r * r;
377 }
378
379 // compute the scaled predicted reduction
380 // and the scaled directional derivative
381 for (int j = 0; j < solvedCols; ++j) {
382 int pj = permutation[j];
383 double dirJ = lmDir[pj];
384 work1[j] = 0;
385 int index = pj;
386 for (int i = 0; i <= j; ++i) {
387 work1[i] += jacobian[index] * dirJ;
388 index += cols;
389 }
390 }
391 double coeff1 = 0;
392 for (int j = 0; j < solvedCols; ++j) {
393 coeff1 += work1[j] * work1[j];
394 }
395 double pc2 = previousCost * previousCost;
396 coeff1 = coeff1 / pc2;
397 double coeff2 = lmPar * lmNorm * lmNorm / pc2;
398 double preRed = coeff1 + 2 * coeff2;
399 double dirDer = -(coeff1 + coeff2);
400
401 // ratio of the actual to the predicted reduction
402 ratio = (preRed == 0) ? 0 : (actRed / preRed);
403
404 // update the step bound
405 if (ratio <= 0.25) {
406 double tmp =
407 (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
408 if ((0.1 * cost >= previousCost) || (tmp < 0.1)) {
409 tmp = 0.1;
410 }
411 delta = tmp * Math.min(delta, 10.0 * lmNorm);
412 lmPar /= tmp;
413 } else if ((lmPar == 0) || (ratio >= 0.75)) {
414 delta = 2 * lmNorm;
415 lmPar *= 0.5;
416 }
417
418 // test for successful iteration.
419 if (ratio >= 1.0e-4) {
420 // successful iteration, update the norm
421 firstIteration = false;
422 xNorm = 0;
423 for (int k = 0; k < cols; ++k) {
424 double xK = diag[k] * parameters[k].getEstimate();
425 xNorm += xK * xK;
426 }
427 xNorm = Math.sqrt(xNorm);
428 } else {
429 // failed iteration, reset the previous values
430 cost = previousCost;
431 for (int j = 0; j < solvedCols; ++j) {
432 int pj = permutation[j];
433 parameters[pj].setEstimate(oldX[pj]);
434 }
435 tmpVec = residuals;
436 residuals = oldRes;
437 oldRes = tmpVec;
438 }
439
440 // tests for convergence.
441 if (((Math.abs(actRed) <= costRelativeTolerance) &&
442 (preRed <= costRelativeTolerance) &&
443 (ratio <= 2.0)) ||
444 (delta <= parRelativeTolerance * xNorm)) {
445 return;
446 }
447
448 // tests for termination and stringent tolerances
449 // (2.2204e-16 is the machine epsilon for IEEE754)
450 if ((Math.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
451 throw new EstimationException("cost relative tolerance is too small ({0})," +
452 " no further reduction in the" +
453 " sum of squares is possible",
454 costRelativeTolerance);
455 } else if (delta <= 2.2204e-16 * xNorm) {
456 throw new EstimationException("parameters relative tolerance is too small" +
457 " ({0}), no further improvement in" +
458 " the approximate solution is possible",
459 parRelativeTolerance);
460 } else if (maxCosine <= 2.2204e-16) {
461 throw new EstimationException("orthogonality tolerance is too small ({0})," +
462 " solution is orthogonal to the jacobian",
463 orthoTolerance);
464 }
465
466 }
467
468 }
469
470 }
471
472 /**
473 * Determine the Levenberg-Marquardt parameter.
474 * <p>This implementation is a translation in Java of the MINPACK
475 * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
476 * routine.</p>
477 * <p>This method sets the lmPar and lmDir attributes.</p>
478 * <p>The authors of the original fortran function are:</p>
479 * <ul>
480 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
481 * <li>Burton S. Garbow</li>
482 * <li>Kenneth E. Hillstrom</li>
483 * <li>Jorge J. More</li>
484 * </ul>
485 * <p>Luc Maisonobe did the Java translation.</p>
486 *
487 * @param qy array containing qTy
488 * @param delta upper bound on the euclidean norm of diagR * lmDir
489 * @param diag diagonal matrix
490 * @param work1 work array
491 * @param work2 work array
492 * @param work3 work array
493 */
494 private void determineLMParameter(double[] qy, double delta, double[] diag,
495 double[] work1, double[] work2, double[] work3) {
496
497 // compute and store in x the gauss-newton direction, if the
498 // jacobian is rank-deficient, obtain a least squares solution
499 for (int j = 0; j < rank; ++j) {
500 lmDir[permutation[j]] = qy[j];
501 }
502 for (int j = rank; j < cols; ++j) {
503 lmDir[permutation[j]] = 0;
504 }
505 for (int k = rank - 1; k >= 0; --k) {
506 int pk = permutation[k];
507 double ypk = lmDir[pk] / diagR[pk];
508 int index = pk;
509 for (int i = 0; i < k; ++i) {
510 lmDir[permutation[i]] -= ypk * jacobian[index];
511 index += cols;
512 }
513 lmDir[pk] = ypk;
514 }
515
516 // evaluate the function at the origin, and test
517 // for acceptance of the Gauss-Newton direction
518 double dxNorm = 0;
519 for (int j = 0; j < solvedCols; ++j) {
520 int pj = permutation[j];
521 double s = diag[pj] * lmDir[pj];
522 work1[pj] = s;
523 dxNorm += s * s;
524 }
525 dxNorm = Math.sqrt(dxNorm);
526 double fp = dxNorm - delta;
527 if (fp <= 0.1 * delta) {
528 lmPar = 0;
529 return;
530 }
531
532 // if the jacobian is not rank deficient, the Newton step provides
533 // a lower bound, parl, for the zero of the function,
534 // otherwise set this bound to zero
535 double sum2;
536 double parl = 0;
537 if (rank == solvedCols) {
538 for (int j = 0; j < solvedCols; ++j) {
539 int pj = permutation[j];
540 work1[pj] *= diag[pj] / dxNorm;
541 }
542 sum2 = 0;
543 for (int j = 0; j < solvedCols; ++j) {
544 int pj = permutation[j];
545 double sum = 0;
546 int index = pj;
547 for (int i = 0; i < j; ++i) {
548 sum += jacobian[index] * work1[permutation[i]];
549 index += cols;
550 }
551 double s = (work1[pj] - sum) / diagR[pj];
552 work1[pj] = s;
553 sum2 += s * s;
554 }
555 parl = fp / (delta * sum2);
556 }
557
558 // calculate an upper bound, paru, for the zero of the function
559 sum2 = 0;
560 for (int j = 0; j < solvedCols; ++j) {
561 int pj = permutation[j];
562 double sum = 0;
563 int index = pj;
564 for (int i = 0; i <= j; ++i) {
565 sum += jacobian[index] * qy[i];
566 index += cols;
567 }
568 sum /= diag[pj];
569 sum2 += sum * sum;
570 }
571 double gNorm = Math.sqrt(sum2);
572 double paru = gNorm / delta;
573 if (paru == 0) {
574 // 2.2251e-308 is the smallest positive real for IEE754
575 paru = 2.2251e-308 / Math.min(delta, 0.1);
576 }
577
578 // if the input par lies outside of the interval (parl,paru),
579 // set par to the closer endpoint
580 lmPar = Math.min(paru, Math.max(lmPar, parl));
581 if (lmPar == 0) {
582 lmPar = gNorm / dxNorm;
583 }
584
585 for (int countdown = 10; countdown >= 0; --countdown) {
586
587 // evaluate the function at the current value of lmPar
588 if (lmPar == 0) {
589 lmPar = Math.max(2.2251e-308, 0.001 * paru);
590 }
591 double sPar = Math.sqrt(lmPar);
592 for (int j = 0; j < solvedCols; ++j) {
593 int pj = permutation[j];
594 work1[pj] = sPar * diag[pj];
595 }
596 determineLMDirection(qy, work1, work2, work3);
597
598 dxNorm = 0;
599 for (int j = 0; j < solvedCols; ++j) {
600 int pj = permutation[j];
601 double s = diag[pj] * lmDir[pj];
602 work3[pj] = s;
603 dxNorm += s * s;
604 }
605 dxNorm = Math.sqrt(dxNorm);
606 double previousFP = fp;
607 fp = dxNorm - delta;
608
609 // if the function is small enough, accept the current value
610 // of lmPar, also test for the exceptional cases where parl is zero
611 if ((Math.abs(fp) <= 0.1 * delta) ||
612 ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
613 return;
614 }
615
616 // compute the Newton correction
617 for (int j = 0; j < solvedCols; ++j) {
618 int pj = permutation[j];
619 work1[pj] = work3[pj] * diag[pj] / dxNorm;
620 }
621 for (int j = 0; j < solvedCols; ++j) {
622 int pj = permutation[j];
623 work1[pj] /= work2[j];
624 double tmp = work1[pj];
625 for (int i = j + 1; i < solvedCols; ++i) {
626 work1[permutation[i]] -= jacobian[i * cols + pj] * tmp;
627 }
628 }
629 sum2 = 0;
630 for (int j = 0; j < solvedCols; ++j) {
631 double s = work1[permutation[j]];
632 sum2 += s * s;
633 }
634 double correction = fp / (delta * sum2);
635
636 // depending on the sign of the function, update parl or paru.
637 if (fp > 0) {
638 parl = Math.max(parl, lmPar);
639 } else if (fp < 0) {
640 paru = Math.min(paru, lmPar);
641 }
642
643 // compute an improved estimate for lmPar
644 lmPar = Math.max(parl, lmPar + correction);
645
646 }
647 }
648
649 /**
650 * Solve a*x = b and d*x = 0 in the least squares sense.
651 * <p>This implementation is a translation in Java of the MINPACK
652 * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
653 * routine.</p>
654 * <p>This method sets the lmDir and lmDiag attributes.</p>
655 * <p>The authors of the original fortran function are:</p>
656 * <ul>
657 * <li>Argonne National Laboratory. MINPACK project. March 1980</li>
658 * <li>Burton S. Garbow</li>
659 * <li>Kenneth E. Hillstrom</li>
660 * <li>Jorge J. More</li>
661 * </ul>
662 * <p>Luc Maisonobe did the Java translation.</p>
663 *
664 * @param qy array containing qTy
665 * @param diag diagonal matrix
666 * @param lmDiag diagonal elements associated with lmDir
667 * @param work work array
668 */
669 private void determineLMDirection(double[] qy, double[] diag,
670 double[] lmDiag, double[] work) {
671
672 // copy R and Qty to preserve input and initialize s
673 // in particular, save the diagonal elements of R in lmDir
674 for (int j = 0; j < solvedCols; ++j) {
675 int pj = permutation[j];
676 for (int i = j + 1; i < solvedCols; ++i) {
677 jacobian[i * cols + pj] = jacobian[j * cols + permutation[i]];
678 }
679 lmDir[j] = diagR[pj];
680 work[j] = qy[j];
681 }
682
683 // eliminate the diagonal matrix d using a Givens rotation
684 for (int j = 0; j < solvedCols; ++j) {
685
686 // prepare the row of d to be eliminated, locating the
687 // diagonal element using p from the Q.R. factorization
688 int pj = permutation[j];
689 double dpj = diag[pj];
690 if (dpj != 0) {
691 Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
692 }
693 lmDiag[j] = dpj;
694
695 // the transformations to eliminate the row of d
696 // modify only a single element of Qty
697 // beyond the first n, which is initially zero.
698 double qtbpj = 0;
699 for (int k = j; k < solvedCols; ++k) {
700 int pk = permutation[k];
701
702 // determine a Givens rotation which eliminates the
703 // appropriate element in the current row of d
704 if (lmDiag[k] != 0) {
705
706 final double sin;
707 final double cos;
708 double rkk = jacobian[k * cols + pk];
709 if (Math.abs(rkk) < Math.abs(lmDiag[k])) {
710 final double cotan = rkk / lmDiag[k];
711 sin = 1.0 / Math.sqrt(1.0 + cotan * cotan);
712 cos = sin * cotan;
713 } else {
714 final double tan = lmDiag[k] / rkk;
715 cos = 1.0 / Math.sqrt(1.0 + tan * tan);
716 sin = cos * tan;
717 }
718
719 // compute the modified diagonal element of R and
720 // the modified element of (Qty,0)
721 jacobian[k * cols + pk] = cos * rkk + sin * lmDiag[k];
722 final double temp = cos * work[k] + sin * qtbpj;
723 qtbpj = -sin * work[k] + cos * qtbpj;
724 work[k] = temp;
725
726 // accumulate the tranformation in the row of s
727 for (int i = k + 1; i < solvedCols; ++i) {
728 double rik = jacobian[i * cols + pk];
729 final double temp2 = cos * rik + sin * lmDiag[i];
730 lmDiag[i] = -sin * rik + cos * lmDiag[i];
731 jacobian[i * cols + pk] = temp2;
732 }
733
734 }
735 }
736
737 // store the diagonal element of s and restore
738 // the corresponding diagonal element of R
739 int index = j * cols + permutation[j];
740 lmDiag[j] = jacobian[index];
741 jacobian[index] = lmDir[j];
742
743 }
744
745 // solve the triangular system for z, if the system is
746 // singular, then obtain a least squares solution
747 int nSing = solvedCols;
748 for (int j = 0; j < solvedCols; ++j) {
749 if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
750 nSing = j;
751 }
752 if (nSing < solvedCols) {
753 work[j] = 0;
754 }
755 }
756 if (nSing > 0) {
757 for (int j = nSing - 1; j >= 0; --j) {
758 int pj = permutation[j];
759 double sum = 0;
760 for (int i = j + 1; i < nSing; ++i) {
761 sum += jacobian[i * cols + pj] * work[i];
762 }
763 work[j] = (work[j] - sum) / lmDiag[j];
764 }
765 }
766
767 // permute the components of z back to components of lmDir
768 for (int j = 0; j < lmDir.length; ++j) {
769 lmDir[permutation[j]] = work[j];
770 }
771
772 }
773
774 /**
775 * Decompose a matrix A as A.P = Q.R using Householder transforms.
776 * <p>As suggested in the P. Lascaux and R. Theodor book
777 * <i>Analyse numérique matricielle appliquée à
778 * l'art de l'ingénieur</i> (Masson, 1986), instead of representing
779 * the Householder transforms with u<sub>k</sub> unit vectors such that:
780 * <pre>
781 * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
782 * </pre>
783 * we use <sub>k</sub> non-unit vectors such that:
784 * <pre>
785 * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
786 * </pre>
787 * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
788 * The beta<sub>k</sub> coefficients are provided upon exit as recomputing
789 * them from the v<sub>k</sub> vectors would be costly.</p>
790 * <p>This decomposition handles rank deficient cases since the tranformations
791 * are performed in non-increasing columns norms order thanks to columns
792 * pivoting. The diagonal elements of the R matrix are therefore also in
793 * non-increasing absolute values order.</p>
794 * @exception EstimationException if the decomposition cannot be performed
795 */
796 private void qrDecomposition() throws EstimationException {
797
798 // initializations
799 for (int k = 0; k < cols; ++k) {
800 permutation[k] = k;
801 double norm2 = 0;
802 for (int index = k; index < jacobian.length; index += cols) {
803 double akk = jacobian[index];
804 norm2 += akk * akk;
805 }
806 jacNorm[k] = Math.sqrt(norm2);
807 }
808
809 // transform the matrix column after column
810 for (int k = 0; k < cols; ++k) {
811
812 // select the column with the greatest norm on active components
813 int nextColumn = -1;
814 double ak2 = Double.NEGATIVE_INFINITY;
815 for (int i = k; i < cols; ++i) {
816 double norm2 = 0;
817 int iDiag = k * cols + permutation[i];
818 for (int index = iDiag; index < jacobian.length; index += cols) {
819 double aki = jacobian[index];
820 norm2 += aki * aki;
821 }
822 if (Double.isInfinite(norm2) || Double.isNaN(norm2)) {
823 throw new EstimationException(
824 "unable to perform Q.R decomposition on the {0}x{1} jacobian matrix",
825 rows, cols);
826 }
827 if (norm2 > ak2) {
828 nextColumn = i;
829 ak2 = norm2;
830 }
831 }
832 if (ak2 == 0) {
833 rank = k;
834 return;
835 }
836 int pk = permutation[nextColumn];
837 permutation[nextColumn] = permutation[k];
838 permutation[k] = pk;
839
840 // choose alpha such that Hk.u = alpha ek
841 int kDiag = k * cols + pk;
842 double akk = jacobian[kDiag];
843 double alpha = (akk > 0) ? -Math.sqrt(ak2) : Math.sqrt(ak2);
844 double betak = 1.0 / (ak2 - akk * alpha);
845 beta[pk] = betak;
846
847 // transform the current column
848 diagR[pk] = alpha;
849 jacobian[kDiag] -= alpha;
850
851 // transform the remaining columns
852 for (int dk = cols - 1 - k; dk > 0; --dk) {
853 int dkp = permutation[k + dk] - pk;
854 double gamma = 0;
855 for (int index = kDiag; index < jacobian.length; index += cols) {
856 gamma += jacobian[index] * jacobian[index + dkp];
857 }
858 gamma *= betak;
859 for (int index = kDiag; index < jacobian.length; index += cols) {
860 jacobian[index + dkp] -= gamma * jacobian[index];
861 }
862 }
863
864 }
865
866 rank = solvedCols;
867
868 }
869
870 /**
871 * Compute the product Qt.y for some Q.R. decomposition.
872 *
873 * @param y vector to multiply (will be overwritten with the result)
874 */
875 private void qTy(double[] y) {
876 for (int k = 0; k < cols; ++k) {
877 int pk = permutation[k];
878 int kDiag = k * cols + pk;
879 double gamma = 0;
880 int index = kDiag;
881 for (int i = k; i < rows; ++i) {
882 gamma += jacobian[index] * y[i];
883 index += cols;
884 }
885 gamma *= beta[pk];
886 index = kDiag;
887 for (int i = k; i < rows; ++i) {
888 y[i] -= gamma * jacobian[index];
889 index += cols;
890 }
891 }
892 }
893
894 }