Raymond | dee0849 | 2015-04-02 10:43:13 -0700 | [diff] [blame] | 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 | import org.apache.commons.math.exception.util.LocalizedFormats; |
| 23 | import org.apache.commons.math.util.FastMath; |
| 24 | |
| 25 | |
| 26 | /** |
| 27 | * This class solves a least squares problem. |
| 28 | * |
| 29 | * <p>This implementation <em>should</em> work even for over-determined systems |
| 30 | * (i.e. systems having more variables than equations). Over-determined systems |
| 31 | * are solved by ignoring the variables which have the smallest impact according |
| 32 | * to their jacobian column norm. Only the rank of the matrix and some loop bounds |
| 33 | * are changed to implement this.</p> |
| 34 | * |
| 35 | * <p>The resolution engine is a simple translation of the MINPACK <a |
| 36 | * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor |
| 37 | * changes. The changes include the over-determined resolution and the Q.R. |
| 38 | * decomposition which has been rewritten following the algorithm described in the |
| 39 | * P. Lascaux and R. Theodor book <i>Analyse numérique matricielle |
| 40 | * appliquée à l'art de l'ingénieur</i>, Masson 1986.</p> |
| 41 | * <p>The authors of the original fortran version are: |
| 42 | * <ul> |
| 43 | * <li>Argonne National Laboratory. MINPACK project. March 1980</li> |
| 44 | * <li>Burton S. Garbow</li> |
| 45 | * <li>Kenneth E. Hillstrom</li> |
| 46 | * <li>Jorge J. More</li> |
| 47 | * </ul> |
| 48 | * The redistribution policy for MINPACK is available <a |
| 49 | * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it |
| 50 | * is reproduced below.</p> |
| 51 | * |
| 52 | * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0"> |
| 53 | * <tr><td> |
| 54 | * Minpack Copyright Notice (1999) University of Chicago. |
| 55 | * All rights reserved |
| 56 | * </td></tr> |
| 57 | * <tr><td> |
| 58 | * Redistribution and use in source and binary forms, with or without |
| 59 | * modification, are permitted provided that the following conditions |
| 60 | * are met: |
| 61 | * <ol> |
| 62 | * <li>Redistributions of source code must retain the above copyright |
| 63 | * notice, this list of conditions and the following disclaimer.</li> |
| 64 | * <li>Redistributions in binary form must reproduce the above |
| 65 | * copyright notice, this list of conditions and the following |
| 66 | * disclaimer in the documentation and/or other materials provided |
| 67 | * with the distribution.</li> |
| 68 | * <li>The end-user documentation included with the redistribution, if any, |
| 69 | * must include the following acknowledgment: |
| 70 | * <code>This product includes software developed by the University of |
| 71 | * Chicago, as Operator of Argonne National Laboratory.</code> |
| 72 | * Alternately, this acknowledgment may appear in the software itself, |
| 73 | * if and wherever such third-party acknowledgments normally appear.</li> |
| 74 | * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS" |
| 75 | * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE |
| 76 | * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND |
| 77 | * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR |
| 78 | * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES |
| 79 | * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE |
| 80 | * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY |
| 81 | * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR |
| 82 | * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF |
| 83 | * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4) |
| 84 | * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION |
| 85 | * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL |
| 86 | * BE CORRECTED.</strong></li> |
| 87 | * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT |
| 88 | * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF |
| 89 | * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT, |
| 90 | * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF |
| 91 | * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF |
| 92 | * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER |
| 93 | * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT |
| 94 | * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE, |
| 95 | * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE |
| 96 | * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li> |
| 97 | * <ol></td></tr> |
| 98 | * </table> |
| 99 | |
| 100 | * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 août 2010) $ |
| 101 | * @since 1.2 |
| 102 | * @deprecated as of 2.0, everything in package org.apache.commons.math.estimation has |
| 103 | * been deprecated and replaced by package org.apache.commons.math.optimization.general |
| 104 | * |
| 105 | */ |
| 106 | @Deprecated |
| 107 | public class LevenbergMarquardtEstimator extends AbstractEstimator implements Serializable { |
| 108 | |
| 109 | /** Serializable version identifier */ |
| 110 | private static final long serialVersionUID = -5705952631533171019L; |
| 111 | |
| 112 | /** Number of solved variables. */ |
| 113 | private int solvedCols; |
| 114 | |
| 115 | /** Diagonal elements of the R matrix in the Q.R. decomposition. */ |
| 116 | private double[] diagR; |
| 117 | |
| 118 | /** Norms of the columns of the jacobian matrix. */ |
| 119 | private double[] jacNorm; |
| 120 | |
| 121 | /** Coefficients of the Householder transforms vectors. */ |
| 122 | private double[] beta; |
| 123 | |
| 124 | /** Columns permutation array. */ |
| 125 | private int[] permutation; |
| 126 | |
| 127 | /** Rank of the jacobian matrix. */ |
| 128 | private int rank; |
| 129 | |
| 130 | /** Levenberg-Marquardt parameter. */ |
| 131 | private double lmPar; |
| 132 | |
| 133 | /** Parameters evolution direction associated with lmPar. */ |
| 134 | private double[] lmDir; |
| 135 | |
| 136 | /** Positive input variable used in determining the initial step bound. */ |
| 137 | private double initialStepBoundFactor; |
| 138 | |
| 139 | /** Desired relative error in the sum of squares. */ |
| 140 | private double costRelativeTolerance; |
| 141 | |
| 142 | /** Desired relative error in the approximate solution parameters. */ |
| 143 | private double parRelativeTolerance; |
| 144 | |
| 145 | /** Desired max cosine on the orthogonality between the function vector |
| 146 | * and the columns of the jacobian. */ |
| 147 | private double orthoTolerance; |
| 148 | |
| 149 | /** |
| 150 | * Build an estimator for least squares problems. |
| 151 | * <p>The default values for the algorithm settings are: |
| 152 | * <ul> |
| 153 | * <li>{@link #setInitialStepBoundFactor initial step bound factor}: 100.0</li> |
| 154 | * <li>{@link #setMaxCostEval maximal cost evaluations}: 1000</li> |
| 155 | * <li>{@link #setCostRelativeTolerance cost relative tolerance}: 1.0e-10</li> |
| 156 | * <li>{@link #setParRelativeTolerance parameters relative tolerance}: 1.0e-10</li> |
| 157 | * <li>{@link #setOrthoTolerance orthogonality tolerance}: 1.0e-10</li> |
| 158 | * </ul> |
| 159 | * </p> |
| 160 | */ |
| 161 | public LevenbergMarquardtEstimator() { |
| 162 | |
| 163 | // set up the superclass with a default max cost evaluations setting |
| 164 | setMaxCostEval(1000); |
| 165 | |
| 166 | // default values for the tuning parameters |
| 167 | setInitialStepBoundFactor(100.0); |
| 168 | setCostRelativeTolerance(1.0e-10); |
| 169 | setParRelativeTolerance(1.0e-10); |
| 170 | setOrthoTolerance(1.0e-10); |
| 171 | |
| 172 | } |
| 173 | |
| 174 | /** |
| 175 | * Set the positive input variable used in determining the initial step bound. |
| 176 | * This bound is set to the product of initialStepBoundFactor and the euclidean norm of diag*x if nonzero, |
| 177 | * or else to initialStepBoundFactor itself. In most cases factor should lie |
| 178 | * in the interval (0.1, 100.0). 100.0 is a generally recommended value |
| 179 | * |
| 180 | * @param initialStepBoundFactor initial step bound factor |
| 181 | * @see #estimate |
| 182 | */ |
| 183 | public void setInitialStepBoundFactor(double initialStepBoundFactor) { |
| 184 | this.initialStepBoundFactor = initialStepBoundFactor; |
| 185 | } |
| 186 | |
| 187 | /** |
| 188 | * Set the desired relative error in the sum of squares. |
| 189 | * |
| 190 | * @param costRelativeTolerance desired relative error in the sum of squares |
| 191 | * @see #estimate |
| 192 | */ |
| 193 | public void setCostRelativeTolerance(double costRelativeTolerance) { |
| 194 | this.costRelativeTolerance = costRelativeTolerance; |
| 195 | } |
| 196 | |
| 197 | /** |
| 198 | * Set the desired relative error in the approximate solution parameters. |
| 199 | * |
| 200 | * @param parRelativeTolerance desired relative error |
| 201 | * in the approximate solution parameters |
| 202 | * @see #estimate |
| 203 | */ |
| 204 | public void setParRelativeTolerance(double parRelativeTolerance) { |
| 205 | this.parRelativeTolerance = parRelativeTolerance; |
| 206 | } |
| 207 | |
| 208 | /** |
| 209 | * Set the desired max cosine on the orthogonality. |
| 210 | * |
| 211 | * @param orthoTolerance desired max cosine on the orthogonality |
| 212 | * between the function vector and the columns of the jacobian |
| 213 | * @see #estimate |
| 214 | */ |
| 215 | public void setOrthoTolerance(double orthoTolerance) { |
| 216 | this.orthoTolerance = orthoTolerance; |
| 217 | } |
| 218 | |
| 219 | /** |
| 220 | * Solve an estimation problem using the Levenberg-Marquardt algorithm. |
| 221 | * <p>The algorithm used is a modified Levenberg-Marquardt one, based |
| 222 | * on the MINPACK <a href="http://www.netlib.org/minpack/lmder.f">lmder</a> |
| 223 | * routine. The algorithm settings must have been set up before this method |
| 224 | * is called with the {@link #setInitialStepBoundFactor}, |
| 225 | * {@link #setMaxCostEval}, {@link #setCostRelativeTolerance}, |
| 226 | * {@link #setParRelativeTolerance} and {@link #setOrthoTolerance} methods. |
| 227 | * If these methods have not been called, the default values set up by the |
| 228 | * {@link #LevenbergMarquardtEstimator() constructor} will be used.</p> |
| 229 | * <p>The authors of the original fortran function are:</p> |
| 230 | * <ul> |
| 231 | * <li>Argonne National Laboratory. MINPACK project. March 1980</li> |
| 232 | * <li>Burton S. Garbow</li> |
| 233 | * <li>Kenneth E. Hillstrom</li> |
| 234 | * <li>Jorge J. More</li> |
| 235 | * </ul> |
| 236 | * <p>Luc Maisonobe did the Java translation.</p> |
| 237 | * |
| 238 | * @param problem estimation problem to solve |
| 239 | * @exception EstimationException if convergence cannot be |
| 240 | * reached with the specified algorithm settings or if there are more variables |
| 241 | * than equations |
| 242 | * @see #setInitialStepBoundFactor |
| 243 | * @see #setCostRelativeTolerance |
| 244 | * @see #setParRelativeTolerance |
| 245 | * @see #setOrthoTolerance |
| 246 | */ |
| 247 | @Override |
| 248 | public void estimate(EstimationProblem problem) |
| 249 | throws EstimationException { |
| 250 | |
| 251 | initializeEstimate(problem); |
| 252 | |
| 253 | // arrays shared with the other private methods |
| 254 | solvedCols = FastMath.min(rows, cols); |
| 255 | diagR = new double[cols]; |
| 256 | jacNorm = new double[cols]; |
| 257 | beta = new double[cols]; |
| 258 | permutation = new int[cols]; |
| 259 | lmDir = new double[cols]; |
| 260 | |
| 261 | // local variables |
| 262 | double delta = 0; |
| 263 | double xNorm = 0; |
| 264 | double[] diag = new double[cols]; |
| 265 | double[] oldX = new double[cols]; |
| 266 | double[] oldRes = new double[rows]; |
| 267 | double[] work1 = new double[cols]; |
| 268 | double[] work2 = new double[cols]; |
| 269 | double[] work3 = new double[cols]; |
| 270 | |
| 271 | // evaluate the function at the starting point and calculate its norm |
| 272 | updateResidualsAndCost(); |
| 273 | |
| 274 | // outer loop |
| 275 | lmPar = 0; |
| 276 | boolean firstIteration = true; |
| 277 | while (true) { |
| 278 | |
| 279 | // compute the Q.R. decomposition of the jacobian matrix |
| 280 | updateJacobian(); |
| 281 | qrDecomposition(); |
| 282 | |
| 283 | // compute Qt.res |
| 284 | qTy(residuals); |
| 285 | |
| 286 | // now we don't need Q anymore, |
| 287 | // so let jacobian contain the R matrix with its diagonal elements |
| 288 | for (int k = 0; k < solvedCols; ++k) { |
| 289 | int pk = permutation[k]; |
| 290 | jacobian[k * cols + pk] = diagR[pk]; |
| 291 | } |
| 292 | |
| 293 | if (firstIteration) { |
| 294 | |
| 295 | // scale the variables according to the norms of the columns |
| 296 | // of the initial jacobian |
| 297 | xNorm = 0; |
| 298 | for (int k = 0; k < cols; ++k) { |
| 299 | double dk = jacNorm[k]; |
| 300 | if (dk == 0) { |
| 301 | dk = 1.0; |
| 302 | } |
| 303 | double xk = dk * parameters[k].getEstimate(); |
| 304 | xNorm += xk * xk; |
| 305 | diag[k] = dk; |
| 306 | } |
| 307 | xNorm = FastMath.sqrt(xNorm); |
| 308 | |
| 309 | // initialize the step bound delta |
| 310 | delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm); |
| 311 | |
| 312 | } |
| 313 | |
| 314 | // check orthogonality between function vector and jacobian columns |
| 315 | double maxCosine = 0; |
| 316 | if (cost != 0) { |
| 317 | for (int j = 0; j < solvedCols; ++j) { |
| 318 | int pj = permutation[j]; |
| 319 | double s = jacNorm[pj]; |
| 320 | if (s != 0) { |
| 321 | double sum = 0; |
| 322 | int index = pj; |
| 323 | for (int i = 0; i <= j; ++i) { |
| 324 | sum += jacobian[index] * residuals[i]; |
| 325 | index += cols; |
| 326 | } |
| 327 | maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * cost)); |
| 328 | } |
| 329 | } |
| 330 | } |
| 331 | if (maxCosine <= orthoTolerance) { |
| 332 | return; |
| 333 | } |
| 334 | |
| 335 | // rescale if necessary |
| 336 | for (int j = 0; j < cols; ++j) { |
| 337 | diag[j] = FastMath.max(diag[j], jacNorm[j]); |
| 338 | } |
| 339 | |
| 340 | // inner loop |
| 341 | for (double ratio = 0; ratio < 1.0e-4;) { |
| 342 | |
| 343 | // save the state |
| 344 | for (int j = 0; j < solvedCols; ++j) { |
| 345 | int pj = permutation[j]; |
| 346 | oldX[pj] = parameters[pj].getEstimate(); |
| 347 | } |
| 348 | double previousCost = cost; |
| 349 | double[] tmpVec = residuals; |
| 350 | residuals = oldRes; |
| 351 | oldRes = tmpVec; |
| 352 | |
| 353 | // determine the Levenberg-Marquardt parameter |
| 354 | determineLMParameter(oldRes, delta, diag, work1, work2, work3); |
| 355 | |
| 356 | // compute the new point and the norm of the evolution direction |
| 357 | double lmNorm = 0; |
| 358 | for (int j = 0; j < solvedCols; ++j) { |
| 359 | int pj = permutation[j]; |
| 360 | lmDir[pj] = -lmDir[pj]; |
| 361 | parameters[pj].setEstimate(oldX[pj] + lmDir[pj]); |
| 362 | double s = diag[pj] * lmDir[pj]; |
| 363 | lmNorm += s * s; |
| 364 | } |
| 365 | lmNorm = FastMath.sqrt(lmNorm); |
| 366 | |
| 367 | // on the first iteration, adjust the initial step bound. |
| 368 | if (firstIteration) { |
| 369 | delta = FastMath.min(delta, lmNorm); |
| 370 | } |
| 371 | |
| 372 | // evaluate the function at x + p and calculate its norm |
| 373 | updateResidualsAndCost(); |
| 374 | |
| 375 | // compute the scaled actual reduction |
| 376 | double actRed = -1.0; |
| 377 | if (0.1 * cost < previousCost) { |
| 378 | double r = cost / previousCost; |
| 379 | actRed = 1.0 - r * r; |
| 380 | } |
| 381 | |
| 382 | // compute the scaled predicted reduction |
| 383 | // and the scaled directional derivative |
| 384 | for (int j = 0; j < solvedCols; ++j) { |
| 385 | int pj = permutation[j]; |
| 386 | double dirJ = lmDir[pj]; |
| 387 | work1[j] = 0; |
| 388 | int index = pj; |
| 389 | for (int i = 0; i <= j; ++i) { |
| 390 | work1[i] += jacobian[index] * dirJ; |
| 391 | index += cols; |
| 392 | } |
| 393 | } |
| 394 | double coeff1 = 0; |
| 395 | for (int j = 0; j < solvedCols; ++j) { |
| 396 | coeff1 += work1[j] * work1[j]; |
| 397 | } |
| 398 | double pc2 = previousCost * previousCost; |
| 399 | coeff1 = coeff1 / pc2; |
| 400 | double coeff2 = lmPar * lmNorm * lmNorm / pc2; |
| 401 | double preRed = coeff1 + 2 * coeff2; |
| 402 | double dirDer = -(coeff1 + coeff2); |
| 403 | |
| 404 | // ratio of the actual to the predicted reduction |
| 405 | ratio = (preRed == 0) ? 0 : (actRed / preRed); |
| 406 | |
| 407 | // update the step bound |
| 408 | if (ratio <= 0.25) { |
| 409 | double tmp = |
| 410 | (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5; |
| 411 | if ((0.1 * cost >= previousCost) || (tmp < 0.1)) { |
| 412 | tmp = 0.1; |
| 413 | } |
| 414 | delta = tmp * FastMath.min(delta, 10.0 * lmNorm); |
| 415 | lmPar /= tmp; |
| 416 | } else if ((lmPar == 0) || (ratio >= 0.75)) { |
| 417 | delta = 2 * lmNorm; |
| 418 | lmPar *= 0.5; |
| 419 | } |
| 420 | |
| 421 | // test for successful iteration. |
| 422 | if (ratio >= 1.0e-4) { |
| 423 | // successful iteration, update the norm |
| 424 | firstIteration = false; |
| 425 | xNorm = 0; |
| 426 | for (int k = 0; k < cols; ++k) { |
| 427 | double xK = diag[k] * parameters[k].getEstimate(); |
| 428 | xNorm += xK * xK; |
| 429 | } |
| 430 | xNorm = FastMath.sqrt(xNorm); |
| 431 | } else { |
| 432 | // failed iteration, reset the previous values |
| 433 | cost = previousCost; |
| 434 | for (int j = 0; j < solvedCols; ++j) { |
| 435 | int pj = permutation[j]; |
| 436 | parameters[pj].setEstimate(oldX[pj]); |
| 437 | } |
| 438 | tmpVec = residuals; |
| 439 | residuals = oldRes; |
| 440 | oldRes = tmpVec; |
| 441 | } |
| 442 | |
| 443 | // tests for convergence. |
| 444 | if (((FastMath.abs(actRed) <= costRelativeTolerance) && |
| 445 | (preRed <= costRelativeTolerance) && |
| 446 | (ratio <= 2.0)) || |
| 447 | (delta <= parRelativeTolerance * xNorm)) { |
| 448 | return; |
| 449 | } |
| 450 | |
| 451 | // tests for termination and stringent tolerances |
| 452 | // (2.2204e-16 is the machine epsilon for IEEE754) |
| 453 | if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) { |
| 454 | throw new EstimationException("cost relative tolerance is too small ({0})," + |
| 455 | " no further reduction in the" + |
| 456 | " sum of squares is possible", |
| 457 | costRelativeTolerance); |
| 458 | } else if (delta <= 2.2204e-16 * xNorm) { |
| 459 | throw new EstimationException("parameters relative tolerance is too small" + |
| 460 | " ({0}), no further improvement in" + |
| 461 | " the approximate solution is possible", |
| 462 | parRelativeTolerance); |
| 463 | } else if (maxCosine <= 2.2204e-16) { |
| 464 | throw new EstimationException("orthogonality tolerance is too small ({0})," + |
| 465 | " solution is orthogonal to the jacobian", |
| 466 | orthoTolerance); |
| 467 | } |
| 468 | |
| 469 | } |
| 470 | |
| 471 | } |
| 472 | |
| 473 | } |
| 474 | |
| 475 | /** |
| 476 | * Determine the Levenberg-Marquardt parameter. |
| 477 | * <p>This implementation is a translation in Java of the MINPACK |
| 478 | * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a> |
| 479 | * routine.</p> |
| 480 | * <p>This method sets the lmPar and lmDir attributes.</p> |
| 481 | * <p>The authors of the original fortran function are:</p> |
| 482 | * <ul> |
| 483 | * <li>Argonne National Laboratory. MINPACK project. March 1980</li> |
| 484 | * <li>Burton S. Garbow</li> |
| 485 | * <li>Kenneth E. Hillstrom</li> |
| 486 | * <li>Jorge J. More</li> |
| 487 | * </ul> |
| 488 | * <p>Luc Maisonobe did the Java translation.</p> |
| 489 | * |
| 490 | * @param qy array containing qTy |
| 491 | * @param delta upper bound on the euclidean norm of diagR * lmDir |
| 492 | * @param diag diagonal matrix |
| 493 | * @param work1 work array |
| 494 | * @param work2 work array |
| 495 | * @param work3 work array |
| 496 | */ |
| 497 | private void determineLMParameter(double[] qy, double delta, double[] diag, |
| 498 | double[] work1, double[] work2, double[] work3) { |
| 499 | |
| 500 | // compute and store in x the gauss-newton direction, if the |
| 501 | // jacobian is rank-deficient, obtain a least squares solution |
| 502 | for (int j = 0; j < rank; ++j) { |
| 503 | lmDir[permutation[j]] = qy[j]; |
| 504 | } |
| 505 | for (int j = rank; j < cols; ++j) { |
| 506 | lmDir[permutation[j]] = 0; |
| 507 | } |
| 508 | for (int k = rank - 1; k >= 0; --k) { |
| 509 | int pk = permutation[k]; |
| 510 | double ypk = lmDir[pk] / diagR[pk]; |
| 511 | int index = pk; |
| 512 | for (int i = 0; i < k; ++i) { |
| 513 | lmDir[permutation[i]] -= ypk * jacobian[index]; |
| 514 | index += cols; |
| 515 | } |
| 516 | lmDir[pk] = ypk; |
| 517 | } |
| 518 | |
| 519 | // evaluate the function at the origin, and test |
| 520 | // for acceptance of the Gauss-Newton direction |
| 521 | double dxNorm = 0; |
| 522 | for (int j = 0; j < solvedCols; ++j) { |
| 523 | int pj = permutation[j]; |
| 524 | double s = diag[pj] * lmDir[pj]; |
| 525 | work1[pj] = s; |
| 526 | dxNorm += s * s; |
| 527 | } |
| 528 | dxNorm = FastMath.sqrt(dxNorm); |
| 529 | double fp = dxNorm - delta; |
| 530 | if (fp <= 0.1 * delta) { |
| 531 | lmPar = 0; |
| 532 | return; |
| 533 | } |
| 534 | |
| 535 | // if the jacobian is not rank deficient, the Newton step provides |
| 536 | // a lower bound, parl, for the zero of the function, |
| 537 | // otherwise set this bound to zero |
| 538 | double sum2; |
| 539 | double parl = 0; |
| 540 | if (rank == solvedCols) { |
| 541 | for (int j = 0; j < solvedCols; ++j) { |
| 542 | int pj = permutation[j]; |
| 543 | work1[pj] *= diag[pj] / dxNorm; |
| 544 | } |
| 545 | sum2 = 0; |
| 546 | for (int j = 0; j < solvedCols; ++j) { |
| 547 | int pj = permutation[j]; |
| 548 | double sum = 0; |
| 549 | int index = pj; |
| 550 | for (int i = 0; i < j; ++i) { |
| 551 | sum += jacobian[index] * work1[permutation[i]]; |
| 552 | index += cols; |
| 553 | } |
| 554 | double s = (work1[pj] - sum) / diagR[pj]; |
| 555 | work1[pj] = s; |
| 556 | sum2 += s * s; |
| 557 | } |
| 558 | parl = fp / (delta * sum2); |
| 559 | } |
| 560 | |
| 561 | // calculate an upper bound, paru, for the zero of the function |
| 562 | sum2 = 0; |
| 563 | for (int j = 0; j < solvedCols; ++j) { |
| 564 | int pj = permutation[j]; |
| 565 | double sum = 0; |
| 566 | int index = pj; |
| 567 | for (int i = 0; i <= j; ++i) { |
| 568 | sum += jacobian[index] * qy[i]; |
| 569 | index += cols; |
| 570 | } |
| 571 | sum /= diag[pj]; |
| 572 | sum2 += sum * sum; |
| 573 | } |
| 574 | double gNorm = FastMath.sqrt(sum2); |
| 575 | double paru = gNorm / delta; |
| 576 | if (paru == 0) { |
| 577 | // 2.2251e-308 is the smallest positive real for IEE754 |
| 578 | paru = 2.2251e-308 / FastMath.min(delta, 0.1); |
| 579 | } |
| 580 | |
| 581 | // if the input par lies outside of the interval (parl,paru), |
| 582 | // set par to the closer endpoint |
| 583 | lmPar = FastMath.min(paru, FastMath.max(lmPar, parl)); |
| 584 | if (lmPar == 0) { |
| 585 | lmPar = gNorm / dxNorm; |
| 586 | } |
| 587 | |
| 588 | for (int countdown = 10; countdown >= 0; --countdown) { |
| 589 | |
| 590 | // evaluate the function at the current value of lmPar |
| 591 | if (lmPar == 0) { |
| 592 | lmPar = FastMath.max(2.2251e-308, 0.001 * paru); |
| 593 | } |
| 594 | double sPar = FastMath.sqrt(lmPar); |
| 595 | for (int j = 0; j < solvedCols; ++j) { |
| 596 | int pj = permutation[j]; |
| 597 | work1[pj] = sPar * diag[pj]; |
| 598 | } |
| 599 | determineLMDirection(qy, work1, work2, work3); |
| 600 | |
| 601 | dxNorm = 0; |
| 602 | for (int j = 0; j < solvedCols; ++j) { |
| 603 | int pj = permutation[j]; |
| 604 | double s = diag[pj] * lmDir[pj]; |
| 605 | work3[pj] = s; |
| 606 | dxNorm += s * s; |
| 607 | } |
| 608 | dxNorm = FastMath.sqrt(dxNorm); |
| 609 | double previousFP = fp; |
| 610 | fp = dxNorm - delta; |
| 611 | |
| 612 | // if the function is small enough, accept the current value |
| 613 | // of lmPar, also test for the exceptional cases where parl is zero |
| 614 | if ((FastMath.abs(fp) <= 0.1 * delta) || |
| 615 | ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) { |
| 616 | return; |
| 617 | } |
| 618 | |
| 619 | // compute the Newton correction |
| 620 | for (int j = 0; j < solvedCols; ++j) { |
| 621 | int pj = permutation[j]; |
| 622 | work1[pj] = work3[pj] * diag[pj] / dxNorm; |
| 623 | } |
| 624 | for (int j = 0; j < solvedCols; ++j) { |
| 625 | int pj = permutation[j]; |
| 626 | work1[pj] /= work2[j]; |
| 627 | double tmp = work1[pj]; |
| 628 | for (int i = j + 1; i < solvedCols; ++i) { |
| 629 | work1[permutation[i]] -= jacobian[i * cols + pj] * tmp; |
| 630 | } |
| 631 | } |
| 632 | sum2 = 0; |
| 633 | for (int j = 0; j < solvedCols; ++j) { |
| 634 | double s = work1[permutation[j]]; |
| 635 | sum2 += s * s; |
| 636 | } |
| 637 | double correction = fp / (delta * sum2); |
| 638 | |
| 639 | // depending on the sign of the function, update parl or paru. |
| 640 | if (fp > 0) { |
| 641 | parl = FastMath.max(parl, lmPar); |
| 642 | } else if (fp < 0) { |
| 643 | paru = FastMath.min(paru, lmPar); |
| 644 | } |
| 645 | |
| 646 | // compute an improved estimate for lmPar |
| 647 | lmPar = FastMath.max(parl, lmPar + correction); |
| 648 | |
| 649 | } |
| 650 | } |
| 651 | |
| 652 | /** |
| 653 | * Solve a*x = b and d*x = 0 in the least squares sense. |
| 654 | * <p>This implementation is a translation in Java of the MINPACK |
| 655 | * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a> |
| 656 | * routine.</p> |
| 657 | * <p>This method sets the lmDir and lmDiag attributes.</p> |
| 658 | * <p>The authors of the original fortran function are:</p> |
| 659 | * <ul> |
| 660 | * <li>Argonne National Laboratory. MINPACK project. March 1980</li> |
| 661 | * <li>Burton S. Garbow</li> |
| 662 | * <li>Kenneth E. Hillstrom</li> |
| 663 | * <li>Jorge J. More</li> |
| 664 | * </ul> |
| 665 | * <p>Luc Maisonobe did the Java translation.</p> |
| 666 | * |
| 667 | * @param qy array containing qTy |
| 668 | * @param diag diagonal matrix |
| 669 | * @param lmDiag diagonal elements associated with lmDir |
| 670 | * @param work work array |
| 671 | */ |
| 672 | private void determineLMDirection(double[] qy, double[] diag, |
| 673 | double[] lmDiag, double[] work) { |
| 674 | |
| 675 | // copy R and Qty to preserve input and initialize s |
| 676 | // in particular, save the diagonal elements of R in lmDir |
| 677 | for (int j = 0; j < solvedCols; ++j) { |
| 678 | int pj = permutation[j]; |
| 679 | for (int i = j + 1; i < solvedCols; ++i) { |
| 680 | jacobian[i * cols + pj] = jacobian[j * cols + permutation[i]]; |
| 681 | } |
| 682 | lmDir[j] = diagR[pj]; |
| 683 | work[j] = qy[j]; |
| 684 | } |
| 685 | |
| 686 | // eliminate the diagonal matrix d using a Givens rotation |
| 687 | for (int j = 0; j < solvedCols; ++j) { |
| 688 | |
| 689 | // prepare the row of d to be eliminated, locating the |
| 690 | // diagonal element using p from the Q.R. factorization |
| 691 | int pj = permutation[j]; |
| 692 | double dpj = diag[pj]; |
| 693 | if (dpj != 0) { |
| 694 | Arrays.fill(lmDiag, j + 1, lmDiag.length, 0); |
| 695 | } |
| 696 | lmDiag[j] = dpj; |
| 697 | |
| 698 | // the transformations to eliminate the row of d |
| 699 | // modify only a single element of Qty |
| 700 | // beyond the first n, which is initially zero. |
| 701 | double qtbpj = 0; |
| 702 | for (int k = j; k < solvedCols; ++k) { |
| 703 | int pk = permutation[k]; |
| 704 | |
| 705 | // determine a Givens rotation which eliminates the |
| 706 | // appropriate element in the current row of d |
| 707 | if (lmDiag[k] != 0) { |
| 708 | |
| 709 | final double sin; |
| 710 | final double cos; |
| 711 | double rkk = jacobian[k * cols + pk]; |
| 712 | if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) { |
| 713 | final double cotan = rkk / lmDiag[k]; |
| 714 | sin = 1.0 / FastMath.sqrt(1.0 + cotan * cotan); |
| 715 | cos = sin * cotan; |
| 716 | } else { |
| 717 | final double tan = lmDiag[k] / rkk; |
| 718 | cos = 1.0 / FastMath.sqrt(1.0 + tan * tan); |
| 719 | sin = cos * tan; |
| 720 | } |
| 721 | |
| 722 | // compute the modified diagonal element of R and |
| 723 | // the modified element of (Qty,0) |
| 724 | jacobian[k * cols + pk] = cos * rkk + sin * lmDiag[k]; |
| 725 | final double temp = cos * work[k] + sin * qtbpj; |
| 726 | qtbpj = -sin * work[k] + cos * qtbpj; |
| 727 | work[k] = temp; |
| 728 | |
| 729 | // accumulate the tranformation in the row of s |
| 730 | for (int i = k + 1; i < solvedCols; ++i) { |
| 731 | double rik = jacobian[i * cols + pk]; |
| 732 | final double temp2 = cos * rik + sin * lmDiag[i]; |
| 733 | lmDiag[i] = -sin * rik + cos * lmDiag[i]; |
| 734 | jacobian[i * cols + pk] = temp2; |
| 735 | } |
| 736 | |
| 737 | } |
| 738 | } |
| 739 | |
| 740 | // store the diagonal element of s and restore |
| 741 | // the corresponding diagonal element of R |
| 742 | int index = j * cols + permutation[j]; |
| 743 | lmDiag[j] = jacobian[index]; |
| 744 | jacobian[index] = lmDir[j]; |
| 745 | |
| 746 | } |
| 747 | |
| 748 | // solve the triangular system for z, if the system is |
| 749 | // singular, then obtain a least squares solution |
| 750 | int nSing = solvedCols; |
| 751 | for (int j = 0; j < solvedCols; ++j) { |
| 752 | if ((lmDiag[j] == 0) && (nSing == solvedCols)) { |
| 753 | nSing = j; |
| 754 | } |
| 755 | if (nSing < solvedCols) { |
| 756 | work[j] = 0; |
| 757 | } |
| 758 | } |
| 759 | if (nSing > 0) { |
| 760 | for (int j = nSing - 1; j >= 0; --j) { |
| 761 | int pj = permutation[j]; |
| 762 | double sum = 0; |
| 763 | for (int i = j + 1; i < nSing; ++i) { |
| 764 | sum += jacobian[i * cols + pj] * work[i]; |
| 765 | } |
| 766 | work[j] = (work[j] - sum) / lmDiag[j]; |
| 767 | } |
| 768 | } |
| 769 | |
| 770 | // permute the components of z back to components of lmDir |
| 771 | for (int j = 0; j < lmDir.length; ++j) { |
| 772 | lmDir[permutation[j]] = work[j]; |
| 773 | } |
| 774 | |
| 775 | } |
| 776 | |
| 777 | /** |
| 778 | * Decompose a matrix A as A.P = Q.R using Householder transforms. |
| 779 | * <p>As suggested in the P. Lascaux and R. Theodor book |
| 780 | * <i>Analyse numérique matricielle appliquée à |
| 781 | * l'art de l'ingénieur</i> (Masson, 1986), instead of representing |
| 782 | * the Householder transforms with u<sub>k</sub> unit vectors such that: |
| 783 | * <pre> |
| 784 | * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup> |
| 785 | * </pre> |
| 786 | * we use <sub>k</sub> non-unit vectors such that: |
| 787 | * <pre> |
| 788 | * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup> |
| 789 | * </pre> |
| 790 | * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>. |
| 791 | * The beta<sub>k</sub> coefficients are provided upon exit as recomputing |
| 792 | * them from the v<sub>k</sub> vectors would be costly.</p> |
| 793 | * <p>This decomposition handles rank deficient cases since the tranformations |
| 794 | * are performed in non-increasing columns norms order thanks to columns |
| 795 | * pivoting. The diagonal elements of the R matrix are therefore also in |
| 796 | * non-increasing absolute values order.</p> |
| 797 | * @exception EstimationException if the decomposition cannot be performed |
| 798 | */ |
| 799 | private void qrDecomposition() throws EstimationException { |
| 800 | |
| 801 | // initializations |
| 802 | for (int k = 0; k < cols; ++k) { |
| 803 | permutation[k] = k; |
| 804 | double norm2 = 0; |
| 805 | for (int index = k; index < jacobian.length; index += cols) { |
| 806 | double akk = jacobian[index]; |
| 807 | norm2 += akk * akk; |
| 808 | } |
| 809 | jacNorm[k] = FastMath.sqrt(norm2); |
| 810 | } |
| 811 | |
| 812 | // transform the matrix column after column |
| 813 | for (int k = 0; k < cols; ++k) { |
| 814 | |
| 815 | // select the column with the greatest norm on active components |
| 816 | int nextColumn = -1; |
| 817 | double ak2 = Double.NEGATIVE_INFINITY; |
| 818 | for (int i = k; i < cols; ++i) { |
| 819 | double norm2 = 0; |
| 820 | int iDiag = k * cols + permutation[i]; |
| 821 | for (int index = iDiag; index < jacobian.length; index += cols) { |
| 822 | double aki = jacobian[index]; |
| 823 | norm2 += aki * aki; |
| 824 | } |
| 825 | if (Double.isInfinite(norm2) || Double.isNaN(norm2)) { |
| 826 | throw new EstimationException( |
| 827 | LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN, |
| 828 | rows, cols); |
| 829 | } |
| 830 | if (norm2 > ak2) { |
| 831 | nextColumn = i; |
| 832 | ak2 = norm2; |
| 833 | } |
| 834 | } |
| 835 | if (ak2 == 0) { |
| 836 | rank = k; |
| 837 | return; |
| 838 | } |
| 839 | int pk = permutation[nextColumn]; |
| 840 | permutation[nextColumn] = permutation[k]; |
| 841 | permutation[k] = pk; |
| 842 | |
| 843 | // choose alpha such that Hk.u = alpha ek |
| 844 | int kDiag = k * cols + pk; |
| 845 | double akk = jacobian[kDiag]; |
| 846 | double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2); |
| 847 | double betak = 1.0 / (ak2 - akk * alpha); |
| 848 | beta[pk] = betak; |
| 849 | |
| 850 | // transform the current column |
| 851 | diagR[pk] = alpha; |
| 852 | jacobian[kDiag] -= alpha; |
| 853 | |
| 854 | // transform the remaining columns |
| 855 | for (int dk = cols - 1 - k; dk > 0; --dk) { |
| 856 | int dkp = permutation[k + dk] - pk; |
| 857 | double gamma = 0; |
| 858 | for (int index = kDiag; index < jacobian.length; index += cols) { |
| 859 | gamma += jacobian[index] * jacobian[index + dkp]; |
| 860 | } |
| 861 | gamma *= betak; |
| 862 | for (int index = kDiag; index < jacobian.length; index += cols) { |
| 863 | jacobian[index + dkp] -= gamma * jacobian[index]; |
| 864 | } |
| 865 | } |
| 866 | |
| 867 | } |
| 868 | |
| 869 | rank = solvedCols; |
| 870 | |
| 871 | } |
| 872 | |
| 873 | /** |
| 874 | * Compute the product Qt.y for some Q.R. decomposition. |
| 875 | * |
| 876 | * @param y vector to multiply (will be overwritten with the result) |
| 877 | */ |
| 878 | private void qTy(double[] y) { |
| 879 | for (int k = 0; k < cols; ++k) { |
| 880 | int pk = permutation[k]; |
| 881 | int kDiag = k * cols + pk; |
| 882 | double gamma = 0; |
| 883 | int index = kDiag; |
| 884 | for (int i = k; i < rows; ++i) { |
| 885 | gamma += jacobian[index] * y[i]; |
| 886 | index += cols; |
| 887 | } |
| 888 | gamma *= beta[pk]; |
| 889 | index = kDiag; |
| 890 | for (int i = k; i < rows; ++i) { |
| 891 | y[i] -= gamma * jacobian[index]; |
| 892 | index += cols; |
| 893 | } |
| 894 | } |
| 895 | } |
| 896 | |
| 897 | } |