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diff --git a/src/main/java/org/apache/commons/math/estimation/LevenbergMarquardtEstimator.java b/src/main/java/org/apache/commons/math/estimation/LevenbergMarquardtEstimator.java new file mode 100644 index 0000000..78a4661 --- /dev/null +++ b/src/main/java/org/apache/commons/math/estimation/LevenbergMarquardtEstimator.java @@ -0,0 +1,897 @@ +/* + * Licensed to the Apache Software Foundation (ASF) under one or more + * contributor license agreements. See the NOTICE file distributed with + * this work for additional information regarding copyright ownership. + * The ASF licenses this file to You under the Apache License, Version 2.0 + * (the "License"); you may not use this file except in compliance with + * the License. You may obtain a copy of the License at + * + * http://www.apache.org/licenses/LICENSE-2.0 + * + * Unless required by applicable law or agreed to in writing, software + * distributed under the License is distributed on an "AS IS" BASIS, + * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + * See the License for the specific language governing permissions and + * limitations under the License. + */ +package org.apache.commons.math.estimation; + +import java.io.Serializable; +import java.util.Arrays; + +import org.apache.commons.math.exception.util.LocalizedFormats; +import org.apache.commons.math.util.FastMath; + + +/** + * This class solves a least squares problem. + * + * <p>This implementation <em>should</em> work even for over-determined systems + * (i.e. systems having more variables than equations). Over-determined systems + * are solved by ignoring the variables which have the smallest impact according + * to their jacobian column norm. Only the rank of the matrix and some loop bounds + * are changed to implement this.</p> + * + * <p>The resolution engine is a simple translation of the MINPACK <a + * href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor + * changes. The changes include the over-determined resolution and the Q.R. + * decomposition which has been rewritten following the algorithm described in the + * P. Lascaux and R. Theodor book <i>Analyse numérique matricielle + * appliquée à l'art de l'ingénieur</i>, Masson 1986.</p> + * <p>The authors of the original fortran version are: + * <ul> + * <li>Argonne National Laboratory. MINPACK project. March 1980</li> + * <li>Burton S. Garbow</li> + * <li>Kenneth E. Hillstrom</li> + * <li>Jorge J. More</li> + * </ul> + * The redistribution policy for MINPACK is available <a + * href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it + * is reproduced below.</p> + * + * <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0"> + * <tr><td> + * Minpack Copyright Notice (1999) University of Chicago. + * All rights reserved + * </td></tr> + * <tr><td> + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * <ol> + * <li>Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer.</li> + * <li>Redistributions in binary form must reproduce the above + * copyright notice, this list of conditions and the following + * disclaimer in the documentation and/or other materials provided + * with the distribution.</li> + * <li>The end-user documentation included with the redistribution, if any, + * must include the following acknowledgment: + * <code>This product includes software developed by the University of + * Chicago, as Operator of Argonne National Laboratory.</code> + * Alternately, this acknowledgment may appear in the software itself, + * if and wherever such third-party acknowledgments normally appear.</li> + * <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS" + * WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE + * UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND + * THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES + * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE + * OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY + * OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR + * USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF + * THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4) + * DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION + * UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL + * BE CORRECTED.</strong></li> + * <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT + * HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF + * ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT, + * INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF + * ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF + * PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER + * SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT + * (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE, + * EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE + * POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li> + * <ol></td></tr> + * </table> + + * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 août 2010) $ + * @since 1.2 + * @deprecated as of 2.0, everything in package org.apache.commons.math.estimation has + * been deprecated and replaced by package org.apache.commons.math.optimization.general + * + */ +@Deprecated +public class LevenbergMarquardtEstimator extends AbstractEstimator implements Serializable { + + /** Serializable version identifier */ + private static final long serialVersionUID = -5705952631533171019L; + + /** Number of solved variables. */ + private int solvedCols; + + /** Diagonal elements of the R matrix in the Q.R. decomposition. */ + private double[] diagR; + + /** Norms of the columns of the jacobian matrix. */ + private double[] jacNorm; + + /** Coefficients of the Householder transforms vectors. */ + private double[] beta; + + /** Columns permutation array. */ + private int[] permutation; + + /** Rank of the jacobian matrix. */ + private int rank; + + /** Levenberg-Marquardt parameter. */ + private double lmPar; + + /** Parameters evolution direction associated with lmPar. */ + private double[] lmDir; + + /** Positive input variable used in determining the initial step bound. */ + private double initialStepBoundFactor; + + /** Desired relative error in the sum of squares. */ + private double costRelativeTolerance; + + /** Desired relative error in the approximate solution parameters. */ + private double parRelativeTolerance; + + /** Desired max cosine on the orthogonality between the function vector + * and the columns of the jacobian. */ + private double orthoTolerance; + + /** + * Build an estimator for least squares problems. + * <p>The default values for the algorithm settings are: + * <ul> + * <li>{@link #setInitialStepBoundFactor initial step bound factor}: 100.0</li> + * <li>{@link #setMaxCostEval maximal cost evaluations}: 1000</li> + * <li>{@link #setCostRelativeTolerance cost relative tolerance}: 1.0e-10</li> + * <li>{@link #setParRelativeTolerance parameters relative tolerance}: 1.0e-10</li> + * <li>{@link #setOrthoTolerance orthogonality tolerance}: 1.0e-10</li> + * </ul> + * </p> + */ + public LevenbergMarquardtEstimator() { + + // set up the superclass with a default max cost evaluations setting + setMaxCostEval(1000); + + // default values for the tuning parameters + setInitialStepBoundFactor(100.0); + setCostRelativeTolerance(1.0e-10); + setParRelativeTolerance(1.0e-10); + setOrthoTolerance(1.0e-10); + + } + + /** + * Set the positive input variable used in determining the initial step bound. + * This bound is set to the product of initialStepBoundFactor and the euclidean norm of diag*x if nonzero, + * or else to initialStepBoundFactor itself. In most cases factor should lie + * in the interval (0.1, 100.0). 100.0 is a generally recommended value + * + * @param initialStepBoundFactor initial step bound factor + * @see #estimate + */ + public void setInitialStepBoundFactor(double initialStepBoundFactor) { + this.initialStepBoundFactor = initialStepBoundFactor; + } + + /** + * Set the desired relative error in the sum of squares. + * + * @param costRelativeTolerance desired relative error in the sum of squares + * @see #estimate + */ + public void setCostRelativeTolerance(double costRelativeTolerance) { + this.costRelativeTolerance = costRelativeTolerance; + } + + /** + * Set the desired relative error in the approximate solution parameters. + * + * @param parRelativeTolerance desired relative error + * in the approximate solution parameters + * @see #estimate + */ + public void setParRelativeTolerance(double parRelativeTolerance) { + this.parRelativeTolerance = parRelativeTolerance; + } + + /** + * Set the desired max cosine on the orthogonality. + * + * @param orthoTolerance desired max cosine on the orthogonality + * between the function vector and the columns of the jacobian + * @see #estimate + */ + public void setOrthoTolerance(double orthoTolerance) { + this.orthoTolerance = orthoTolerance; + } + + /** + * Solve an estimation problem using the Levenberg-Marquardt algorithm. + * <p>The algorithm used is a modified Levenberg-Marquardt one, based + * on the MINPACK <a href="http://www.netlib.org/minpack/lmder.f">lmder</a> + * routine. The algorithm settings must have been set up before this method + * is called with the {@link #setInitialStepBoundFactor}, + * {@link #setMaxCostEval}, {@link #setCostRelativeTolerance}, + * {@link #setParRelativeTolerance} and {@link #setOrthoTolerance} methods. + * If these methods have not been called, the default values set up by the + * {@link #LevenbergMarquardtEstimator() constructor} will be used.</p> + * <p>The authors of the original fortran function are:</p> + * <ul> + * <li>Argonne National Laboratory. MINPACK project. March 1980</li> + * <li>Burton S. Garbow</li> + * <li>Kenneth E. Hillstrom</li> + * <li>Jorge J. More</li> + * </ul> + * <p>Luc Maisonobe did the Java translation.</p> + * + * @param problem estimation problem to solve + * @exception EstimationException if convergence cannot be + * reached with the specified algorithm settings or if there are more variables + * than equations + * @see #setInitialStepBoundFactor + * @see #setCostRelativeTolerance + * @see #setParRelativeTolerance + * @see #setOrthoTolerance + */ + @Override + public void estimate(EstimationProblem problem) + throws EstimationException { + + initializeEstimate(problem); + + // arrays shared with the other private methods + solvedCols = FastMath.min(rows, cols); + diagR = new double[cols]; + jacNorm = new double[cols]; + beta = new double[cols]; + permutation = new int[cols]; + lmDir = new double[cols]; + + // local variables + double delta = 0; + double xNorm = 0; + double[] diag = new double[cols]; + double[] oldX = new double[cols]; + double[] oldRes = new double[rows]; + double[] work1 = new double[cols]; + double[] work2 = new double[cols]; + double[] work3 = new double[cols]; + + // evaluate the function at the starting point and calculate its norm + updateResidualsAndCost(); + + // outer loop + lmPar = 0; + boolean firstIteration = true; + while (true) { + + // compute the Q.R. decomposition of the jacobian matrix + updateJacobian(); + qrDecomposition(); + + // compute Qt.res + qTy(residuals); + + // now we don't need Q anymore, + // so let jacobian contain the R matrix with its diagonal elements + for (int k = 0; k < solvedCols; ++k) { + int pk = permutation[k]; + jacobian[k * cols + pk] = diagR[pk]; + } + + if (firstIteration) { + + // scale the variables according to the norms of the columns + // of the initial jacobian + xNorm = 0; + for (int k = 0; k < cols; ++k) { + double dk = jacNorm[k]; + if (dk == 0) { + dk = 1.0; + } + double xk = dk * parameters[k].getEstimate(); + xNorm += xk * xk; + diag[k] = dk; + } + xNorm = FastMath.sqrt(xNorm); + + // initialize the step bound delta + delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm); + + } + + // check orthogonality between function vector and jacobian columns + double maxCosine = 0; + if (cost != 0) { + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + double s = jacNorm[pj]; + if (s != 0) { + double sum = 0; + int index = pj; + for (int i = 0; i <= j; ++i) { + sum += jacobian[index] * residuals[i]; + index += cols; + } + maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * cost)); + } + } + } + if (maxCosine <= orthoTolerance) { + return; + } + + // rescale if necessary + for (int j = 0; j < cols; ++j) { + diag[j] = FastMath.max(diag[j], jacNorm[j]); + } + + // inner loop + for (double ratio = 0; ratio < 1.0e-4;) { + + // save the state + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + oldX[pj] = parameters[pj].getEstimate(); + } + double previousCost = cost; + double[] tmpVec = residuals; + residuals = oldRes; + oldRes = tmpVec; + + // determine the Levenberg-Marquardt parameter + determineLMParameter(oldRes, delta, diag, work1, work2, work3); + + // compute the new point and the norm of the evolution direction + double lmNorm = 0; + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + lmDir[pj] = -lmDir[pj]; + parameters[pj].setEstimate(oldX[pj] + lmDir[pj]); + double s = diag[pj] * lmDir[pj]; + lmNorm += s * s; + } + lmNorm = FastMath.sqrt(lmNorm); + + // on the first iteration, adjust the initial step bound. + if (firstIteration) { + delta = FastMath.min(delta, lmNorm); + } + + // evaluate the function at x + p and calculate its norm + updateResidualsAndCost(); + + // compute the scaled actual reduction + double actRed = -1.0; + if (0.1 * cost < previousCost) { + double r = cost / previousCost; + actRed = 1.0 - r * r; + } + + // compute the scaled predicted reduction + // and the scaled directional derivative + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + double dirJ = lmDir[pj]; + work1[j] = 0; + int index = pj; + for (int i = 0; i <= j; ++i) { + work1[i] += jacobian[index] * dirJ; + index += cols; + } + } + double coeff1 = 0; + for (int j = 0; j < solvedCols; ++j) { + coeff1 += work1[j] * work1[j]; + } + double pc2 = previousCost * previousCost; + coeff1 = coeff1 / pc2; + double coeff2 = lmPar * lmNorm * lmNorm / pc2; + double preRed = coeff1 + 2 * coeff2; + double dirDer = -(coeff1 + coeff2); + + // ratio of the actual to the predicted reduction + ratio = (preRed == 0) ? 0 : (actRed / preRed); + + // update the step bound + if (ratio <= 0.25) { + double tmp = + (actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5; + if ((0.1 * cost >= previousCost) || (tmp < 0.1)) { + tmp = 0.1; + } + delta = tmp * FastMath.min(delta, 10.0 * lmNorm); + lmPar /= tmp; + } else if ((lmPar == 0) || (ratio >= 0.75)) { + delta = 2 * lmNorm; + lmPar *= 0.5; + } + + // test for successful iteration. + if (ratio >= 1.0e-4) { + // successful iteration, update the norm + firstIteration = false; + xNorm = 0; + for (int k = 0; k < cols; ++k) { + double xK = diag[k] * parameters[k].getEstimate(); + xNorm += xK * xK; + } + xNorm = FastMath.sqrt(xNorm); + } else { + // failed iteration, reset the previous values + cost = previousCost; + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + parameters[pj].setEstimate(oldX[pj]); + } + tmpVec = residuals; + residuals = oldRes; + oldRes = tmpVec; + } + + // tests for convergence. + if (((FastMath.abs(actRed) <= costRelativeTolerance) && + (preRed <= costRelativeTolerance) && + (ratio <= 2.0)) || + (delta <= parRelativeTolerance * xNorm)) { + return; + } + + // tests for termination and stringent tolerances + // (2.2204e-16 is the machine epsilon for IEEE754) + if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) { + throw new EstimationException("cost relative tolerance is too small ({0})," + + " no further reduction in the" + + " sum of squares is possible", + costRelativeTolerance); + } else if (delta <= 2.2204e-16 * xNorm) { + throw new EstimationException("parameters relative tolerance is too small" + + " ({0}), no further improvement in" + + " the approximate solution is possible", + parRelativeTolerance); + } else if (maxCosine <= 2.2204e-16) { + throw new EstimationException("orthogonality tolerance is too small ({0})," + + " solution is orthogonal to the jacobian", + orthoTolerance); + } + + } + + } + + } + + /** + * Determine the Levenberg-Marquardt parameter. + * <p>This implementation is a translation in Java of the MINPACK + * <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a> + * routine.</p> + * <p>This method sets the lmPar and lmDir attributes.</p> + * <p>The authors of the original fortran function are:</p> + * <ul> + * <li>Argonne National Laboratory. MINPACK project. March 1980</li> + * <li>Burton S. Garbow</li> + * <li>Kenneth E. Hillstrom</li> + * <li>Jorge J. More</li> + * </ul> + * <p>Luc Maisonobe did the Java translation.</p> + * + * @param qy array containing qTy + * @param delta upper bound on the euclidean norm of diagR * lmDir + * @param diag diagonal matrix + * @param work1 work array + * @param work2 work array + * @param work3 work array + */ + private void determineLMParameter(double[] qy, double delta, double[] diag, + double[] work1, double[] work2, double[] work3) { + + // compute and store in x the gauss-newton direction, if the + // jacobian is rank-deficient, obtain a least squares solution + for (int j = 0; j < rank; ++j) { + lmDir[permutation[j]] = qy[j]; + } + for (int j = rank; j < cols; ++j) { + lmDir[permutation[j]] = 0; + } + for (int k = rank - 1; k >= 0; --k) { + int pk = permutation[k]; + double ypk = lmDir[pk] / diagR[pk]; + int index = pk; + for (int i = 0; i < k; ++i) { + lmDir[permutation[i]] -= ypk * jacobian[index]; + index += cols; + } + lmDir[pk] = ypk; + } + + // evaluate the function at the origin, and test + // for acceptance of the Gauss-Newton direction + double dxNorm = 0; + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + double s = diag[pj] * lmDir[pj]; + work1[pj] = s; + dxNorm += s * s; + } + dxNorm = FastMath.sqrt(dxNorm); + double fp = dxNorm - delta; + if (fp <= 0.1 * delta) { + lmPar = 0; + return; + } + + // if the jacobian is not rank deficient, the Newton step provides + // a lower bound, parl, for the zero of the function, + // otherwise set this bound to zero + double sum2; + double parl = 0; + if (rank == solvedCols) { + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + work1[pj] *= diag[pj] / dxNorm; + } + sum2 = 0; + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + double sum = 0; + int index = pj; + for (int i = 0; i < j; ++i) { + sum += jacobian[index] * work1[permutation[i]]; + index += cols; + } + double s = (work1[pj] - sum) / diagR[pj]; + work1[pj] = s; + sum2 += s * s; + } + parl = fp / (delta * sum2); + } + + // calculate an upper bound, paru, for the zero of the function + sum2 = 0; + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + double sum = 0; + int index = pj; + for (int i = 0; i <= j; ++i) { + sum += jacobian[index] * qy[i]; + index += cols; + } + sum /= diag[pj]; + sum2 += sum * sum; + } + double gNorm = FastMath.sqrt(sum2); + double paru = gNorm / delta; + if (paru == 0) { + // 2.2251e-308 is the smallest positive real for IEE754 + paru = 2.2251e-308 / FastMath.min(delta, 0.1); + } + + // if the input par lies outside of the interval (parl,paru), + // set par to the closer endpoint + lmPar = FastMath.min(paru, FastMath.max(lmPar, parl)); + if (lmPar == 0) { + lmPar = gNorm / dxNorm; + } + + for (int countdown = 10; countdown >= 0; --countdown) { + + // evaluate the function at the current value of lmPar + if (lmPar == 0) { + lmPar = FastMath.max(2.2251e-308, 0.001 * paru); + } + double sPar = FastMath.sqrt(lmPar); + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + work1[pj] = sPar * diag[pj]; + } + determineLMDirection(qy, work1, work2, work3); + + dxNorm = 0; + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + double s = diag[pj] * lmDir[pj]; + work3[pj] = s; + dxNorm += s * s; + } + dxNorm = FastMath.sqrt(dxNorm); + double previousFP = fp; + fp = dxNorm - delta; + + // if the function is small enough, accept the current value + // of lmPar, also test for the exceptional cases where parl is zero + if ((FastMath.abs(fp) <= 0.1 * delta) || + ((parl == 0) && (fp <= previousFP) && (previousFP < 0))) { + return; + } + + // compute the Newton correction + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + work1[pj] = work3[pj] * diag[pj] / dxNorm; + } + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + work1[pj] /= work2[j]; + double tmp = work1[pj]; + for (int i = j + 1; i < solvedCols; ++i) { + work1[permutation[i]] -= jacobian[i * cols + pj] * tmp; + } + } + sum2 = 0; + for (int j = 0; j < solvedCols; ++j) { + double s = work1[permutation[j]]; + sum2 += s * s; + } + double correction = fp / (delta * sum2); + + // depending on the sign of the function, update parl or paru. + if (fp > 0) { + parl = FastMath.max(parl, lmPar); + } else if (fp < 0) { + paru = FastMath.min(paru, lmPar); + } + + // compute an improved estimate for lmPar + lmPar = FastMath.max(parl, lmPar + correction); + + } + } + + /** + * Solve a*x = b and d*x = 0 in the least squares sense. + * <p>This implementation is a translation in Java of the MINPACK + * <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a> + * routine.</p> + * <p>This method sets the lmDir and lmDiag attributes.</p> + * <p>The authors of the original fortran function are:</p> + * <ul> + * <li>Argonne National Laboratory. MINPACK project. March 1980</li> + * <li>Burton S. Garbow</li> + * <li>Kenneth E. Hillstrom</li> + * <li>Jorge J. More</li> + * </ul> + * <p>Luc Maisonobe did the Java translation.</p> + * + * @param qy array containing qTy + * @param diag diagonal matrix + * @param lmDiag diagonal elements associated with lmDir + * @param work work array + */ + private void determineLMDirection(double[] qy, double[] diag, + double[] lmDiag, double[] work) { + + // copy R and Qty to preserve input and initialize s + // in particular, save the diagonal elements of R in lmDir + for (int j = 0; j < solvedCols; ++j) { + int pj = permutation[j]; + for (int i = j + 1; i < solvedCols; ++i) { + jacobian[i * cols + pj] = jacobian[j * cols + permutation[i]]; + } + lmDir[j] = diagR[pj]; + work[j] = qy[j]; + } + + // eliminate the diagonal matrix d using a Givens rotation + for (int j = 0; j < solvedCols; ++j) { + + // prepare the row of d to be eliminated, locating the + // diagonal element using p from the Q.R. factorization + int pj = permutation[j]; + double dpj = diag[pj]; + if (dpj != 0) { + Arrays.fill(lmDiag, j + 1, lmDiag.length, 0); + } + lmDiag[j] = dpj; + + // the transformations to eliminate the row of d + // modify only a single element of Qty + // beyond the first n, which is initially zero. + double qtbpj = 0; + for (int k = j; k < solvedCols; ++k) { + int pk = permutation[k]; + + // determine a Givens rotation which eliminates the + // appropriate element in the current row of d + if (lmDiag[k] != 0) { + + final double sin; + final double cos; + double rkk = jacobian[k * cols + pk]; + if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) { + final double cotan = rkk / lmDiag[k]; + sin = 1.0 / FastMath.sqrt(1.0 + cotan * cotan); + cos = sin * cotan; + } else { + final double tan = lmDiag[k] / rkk; + cos = 1.0 / FastMath.sqrt(1.0 + tan * tan); + sin = cos * tan; + } + + // compute the modified diagonal element of R and + // the modified element of (Qty,0) + jacobian[k * cols + pk] = cos * rkk + sin * lmDiag[k]; + final double temp = cos * work[k] + sin * qtbpj; + qtbpj = -sin * work[k] + cos * qtbpj; + work[k] = temp; + + // accumulate the tranformation in the row of s + for (int i = k + 1; i < solvedCols; ++i) { + double rik = jacobian[i * cols + pk]; + final double temp2 = cos * rik + sin * lmDiag[i]; + lmDiag[i] = -sin * rik + cos * lmDiag[i]; + jacobian[i * cols + pk] = temp2; + } + + } + } + + // store the diagonal element of s and restore + // the corresponding diagonal element of R + int index = j * cols + permutation[j]; + lmDiag[j] = jacobian[index]; + jacobian[index] = lmDir[j]; + + } + + // solve the triangular system for z, if the system is + // singular, then obtain a least squares solution + int nSing = solvedCols; + for (int j = 0; j < solvedCols; ++j) { + if ((lmDiag[j] == 0) && (nSing == solvedCols)) { + nSing = j; + } + if (nSing < solvedCols) { + work[j] = 0; + } + } + if (nSing > 0) { + for (int j = nSing - 1; j >= 0; --j) { + int pj = permutation[j]; + double sum = 0; + for (int i = j + 1; i < nSing; ++i) { + sum += jacobian[i * cols + pj] * work[i]; + } + work[j] = (work[j] - sum) / lmDiag[j]; + } + } + + // permute the components of z back to components of lmDir + for (int j = 0; j < lmDir.length; ++j) { + lmDir[permutation[j]] = work[j]; + } + + } + + /** + * Decompose a matrix A as A.P = Q.R using Householder transforms. + * <p>As suggested in the P. Lascaux and R. Theodor book + * <i>Analyse numérique matricielle appliquée à + * l'art de l'ingénieur</i> (Masson, 1986), instead of representing + * the Householder transforms with u<sub>k</sub> unit vectors such that: + * <pre> + * H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup> + * </pre> + * we use <sub>k</sub> non-unit vectors such that: + * <pre> + * H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup> + * </pre> + * where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>. + * The beta<sub>k</sub> coefficients are provided upon exit as recomputing + * them from the v<sub>k</sub> vectors would be costly.</p> + * <p>This decomposition handles rank deficient cases since the tranformations + * are performed in non-increasing columns norms order thanks to columns + * pivoting. The diagonal elements of the R matrix are therefore also in + * non-increasing absolute values order.</p> + * @exception EstimationException if the decomposition cannot be performed + */ + private void qrDecomposition() throws EstimationException { + + // initializations + for (int k = 0; k < cols; ++k) { + permutation[k] = k; + double norm2 = 0; + for (int index = k; index < jacobian.length; index += cols) { + double akk = jacobian[index]; + norm2 += akk * akk; + } + jacNorm[k] = FastMath.sqrt(norm2); + } + + // transform the matrix column after column + for (int k = 0; k < cols; ++k) { + + // select the column with the greatest norm on active components + int nextColumn = -1; + double ak2 = Double.NEGATIVE_INFINITY; + for (int i = k; i < cols; ++i) { + double norm2 = 0; + int iDiag = k * cols + permutation[i]; + for (int index = iDiag; index < jacobian.length; index += cols) { + double aki = jacobian[index]; + norm2 += aki * aki; + } + if (Double.isInfinite(norm2) || Double.isNaN(norm2)) { + throw new EstimationException( + LocalizedFormats.UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN, + rows, cols); + } + if (norm2 > ak2) { + nextColumn = i; + ak2 = norm2; + } + } + if (ak2 == 0) { + rank = k; + return; + } + int pk = permutation[nextColumn]; + permutation[nextColumn] = permutation[k]; + permutation[k] = pk; + + // choose alpha such that Hk.u = alpha ek + int kDiag = k * cols + pk; + double akk = jacobian[kDiag]; + double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2); + double betak = 1.0 / (ak2 - akk * alpha); + beta[pk] = betak; + + // transform the current column + diagR[pk] = alpha; + jacobian[kDiag] -= alpha; + + // transform the remaining columns + for (int dk = cols - 1 - k; dk > 0; --dk) { + int dkp = permutation[k + dk] - pk; + double gamma = 0; + for (int index = kDiag; index < jacobian.length; index += cols) { + gamma += jacobian[index] * jacobian[index + dkp]; + } + gamma *= betak; + for (int index = kDiag; index < jacobian.length; index += cols) { + jacobian[index + dkp] -= gamma * jacobian[index]; + } + } + + } + + rank = solvedCols; + + } + + /** + * Compute the product Qt.y for some Q.R. decomposition. + * + * @param y vector to multiply (will be overwritten with the result) + */ + private void qTy(double[] y) { + for (int k = 0; k < cols; ++k) { + int pk = permutation[k]; + int kDiag = k * cols + pk; + double gamma = 0; + int index = kDiag; + for (int i = k; i < rows; ++i) { + gamma += jacobian[index] * y[i]; + index += cols; + } + gamma *= beta[pk]; + index = kDiag; + for (int i = k; i < rows; ++i) { + y[i] -= gamma * jacobian[index]; + index += cols; + } + } + } + +} |