/* * 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. * *

This implementation should 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.

* *

The resolution engine is a simple translation of the MINPACK lmder 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 Analyse numérique matricielle * appliquée à l'art de l'ingénieur, Masson 1986.

The authors of the original fortran version are: *

Argonne National Laboratory. MINPACK project. March 1980
Burton S. Garbow
Kenneth E. Hillstrom
Jorge J. More

* The redistribution policy for MINPACK is available here, for convenience, it * is reproduced below.

* * * * *

* Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: *

Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer.
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.
The end-user documentation included with the redistribution, if any, * must include the following acknowledgment: * This product includes software developed by the University of * Chicago, as Operator of Argonne National Laboratory. * Alternately, this acknowledgment may appear in the software itself, * if and wherever such third-party acknowledgments normally appear.
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* @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. *

The default values for the algorithm settings are: *

{@link #setInitialStepBoundFactor initial step bound factor}: 100.0
{@link #setMaxCostEval maximal cost evaluations}: 1000
{@link #setCostRelativeTolerance cost relative tolerance}: 1.0e-10
{@link #setParRelativeTolerance parameters relative tolerance}: 1.0e-10
{@link #setOrthoTolerance orthogonality tolerance}: 1.0e-10

*/ 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. *

The algorithm used is a modified Levenberg-Marquardt one, based * on the MINPACK lmder * 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.

The authors of the original fortran function are:

Argonne National Laboratory. MINPACK project. March 1980
Burton S. Garbow
Kenneth E. Hillstrom
Jorge J. More

Luc Maisonobe did the Java translation.

* * @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. *

This implementation is a translation in Java of the MINPACK * lmpar * routine.

This method sets the lmPar and lmDir attributes.

The authors of the original fortran function are:

Argonne National Laboratory. MINPACK project. March 1980
Burton S. Garbow
Kenneth E. Hillstrom
Jorge J. More

Luc Maisonobe did the Java translation.

* * @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. *

This implementation is a translation in Java of the MINPACK * qrsolv * routine.

This method sets the lmDir and lmDiag attributes.

The authors of the original fortran function are:

Argonne National Laboratory. MINPACK project. March 1980
Burton S. Garbow
Kenneth E. Hillstrom
Jorge J. More

Luc Maisonobe did the Java translation.

* * @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. *

As suggested in the P. Lascaux and R. Theodor book * Analyse numérique matricielle appliquée à * l'art de l'ingénieur (Masson, 1986), instead of representing * the Householder transforms with u_k unit vectors such that: *

   * H_k = I - 2u_k.u_k^t
   *

* we use _k non-unit vectors such that: *

   * H_k = I - beta_kv_k.v_k^t
   *

* where v_k = a_k - alpha_k e_k. * The beta_k coefficients are provided upon exit as recomputing * them from the v_k vectors would be costly.

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.

* @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; } } } }