package org.spaceroots.mantissa.estimation;
import java.io.Serializable;
import java.util.Arrays;
/** 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. This feature has undergone only basic testing
* for now and should still be considered experimental.
* 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
* redistribution policy for MINPACK is available here, for convenience, it
* is reproduced below.
*
*
* Minpack Copyright Notice (1999) University of Chicago.
* All rights reserved
*
* 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.
* - 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.
* - 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.
*
* @author Argonne National Laboratory. MINPACK project. March 1980 (original fortran)
* @author Burton S. Garbow (original fortran)
* @author Kenneth E. Hillstrom (original fortran)
* @author Jorge J. More (original fortran)
* @author Luc Maisonobe (Java translation)
*/
public class LevenbergMarquardtEstimator implements Serializable, Estimator {
/** 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() {
// default values for the tuning parameters
setInitialStepBoundFactor(100.0);
setMaxCostEval(1000);
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 maximal number of cost evaluations.
* @param maxCostEval maximal number of cost evaluations
* @see #estimate
*/
public void setMaxCostEval(int maxCostEval) {
this.maxCostEval = maxCostEval;
}
/** 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;
}
/** Get the number of cost evaluations.
* @return number of cost evaluations
* */
public int getCostEvaluations() {
return costEvaluations;
}
/** Get the number of jacobian evaluations.
* @return number of jacobian evaluations
* */
public int getJacobianEvaluations() {
return jacobianEvaluations;
}
/** Update the jacobian matrix.
*/
private void updateJacobian() {
++jacobianEvaluations;
Arrays.fill(jacobian, 0);
for (int i = 0, index = 0; i < rows; i++) {
WeightedMeasurement wm = measurements[i];
double factor = -Math.sqrt(wm.getWeight());
for (int j = 0; j < cols; ++j) {
jacobian[index++] = factor * wm.getPartial(parameters[j]);
}
}
}
/** Update the residuals array and cost function value.
*/
private void updateResidualsAndCost() {
++costEvaluations;
cost = 0;
for (int i = 0, index = 0; i < rows; i++, index += cols) {
WeightedMeasurement wm = measurements[i];
double residual = wm.getResidual();
residuals[i] = Math.sqrt(wm.getWeight()) * residual;
cost += wm.getWeight() * residual * residual;
}
cost = Math.sqrt(cost);
}
/** Get the Root Mean Square value.
* Get the Root Mean Square value, i.e. the root of the arithmetic
* mean of the square of all weighted residuals. This is related to the
* criterion that is minimized by the estimator as follows: if
* c if the criterion, and n is the number of
* measurements, then the RMS is sqrt (c/n).
* @param problem estimation problem
* @return RMS value
*/
public double getRMS(EstimationProblem problem) {
WeightedMeasurement[] wm = problem.getMeasurements();
double criterion = 0;
for (int i = 0; i < wm.length; ++i) {
double residual = wm[i].getResidual();
criterion += wm[i].getWeight() * residual * residual;
}
return Math.sqrt(criterion / wm.length);
}
/** 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 #setMaxCostEval
* @see #setCostRelativeTolerance
* @see #setParRelativeTolerance
* @see #setOrthoTolerance
*/
public void estimate(EstimationProblem problem)
throws EstimationException {
// retrieve the equations and the parameters
measurements = problem.getMeasurements();
parameters = problem.getUnboundParameters();
// arrays shared with the other private methods
rows = measurements.length;
cols = parameters.length;
solvedCols = Math.min(rows, cols);
jacobian = new double[rows * cols];
diagR = new double[cols];
jacNorm = new double[cols];
beta = new double[cols];
permutation = new int[cols];
lmDir = new double[cols];
residuals = new double[rows];
// local variables
double delta = 0, 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;
costEvaluations = 0;
jacobianEvaluations = 0;
boolean firstIteration = true;
while (costEvaluations < maxCostEval) {
// 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 = Math.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;
for (int i = 0, index = pj; i <= j; ++i, index += cols) {
sum += jacobian[index] * residuals[i];
}
maxCosine = Math.max(maxCosine, Math.abs(sum) / (s * cost));
}
}
}
if (maxCosine <= orthoTolerance) {
return;
}
// rescale if necessary
for (int j = 0; j < cols; ++j) {
diag[j] = Math.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 = Math.sqrt(lmNorm);
// on the first iteration, adjust the initial step bound.
if (firstIteration) {
delta = Math.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;
for (int i = 0, index = pj; i <= j; ++i, index += cols) {
work1[i] += jacobian[index] * dirJ;
}
}
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 * Math.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 = Math.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 (((Math.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 (costEvaluations >= maxCostEval) {
break;
}
if ((Math.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",
new String[] {
Double.toString(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",
new String[] {
Double.toString(parRelativeTolerance)
});
} else if (maxCosine <= 2.2204e-16) {
throw new EstimationException("orthogonality tolerance is too small ({0}),"
+ " solution is orthogonal to the jacobian",
new String[] {
Double.toString(orthoTolerance)
});
}
}
}
throw new EstimationException("maximal number of evaluations exceeded ({0})",
new String[] {
Integer.toString(maxCostEval)
});
}
/** 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];
for (int i = 0, index = pk; i < k; ++i, index += cols) {
lmDir[permutation[i]] -= ypk * jacobian[index];
}
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 = Math.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, 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;
for (int i = 0, index = pj; i < j; ++i, index += cols) {
sum += jacobian[index] * work1[permutation[i]];
}
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;
for (int i = 0, index = pj; i <= j; ++i, index += cols) {
sum += jacobian[index] * qy[i];
}
sum /= diag[pj];
sum2 += sum * sum;
}
double gNorm = Math.sqrt(sum2);
double paru = gNorm / delta;
if (paru == 0) {
// 2.2251e-308 is the smallest positive real for IEE754
paru = 2.2251e-308 / Math.min(delta, 0.1);
}
// if the input par lies outside of the interval (parl,paru),
// set par to the closer endpoint
lmPar = Math.min(paru, Math.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 = Math.max(2.2251e-308, 0.001 * paru);
}
double sPar = Math.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 = Math.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 ((Math.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 = Math.max(parl, lmPar);
} else if (fp < 0) {
paru = Math.min(paru, lmPar);
}
// compute an improved estimate for lmPar
lmPar = Math.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) {
double sin, cos;
double rkk = jacobian[k * cols + pk];
if (Math.abs(rkk) < Math.abs(lmDiag[k])) {
double cotan = rkk / lmDiag[k];
sin = 1.0 / Math.sqrt(1.0 + cotan * cotan);
cos = sin * cotan;
} else {
double tan = lmDiag[k] / rkk;
cos = 1.0 / Math.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];
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];
temp = cos * rik + sin * lmDiag[i];
lmDiag[i] = -sin * rik + cos * lmDiag[i];
jacobian[i * cols + pk] = temp;
}
}
}
// 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 uk unit vectors such that:
*
* Hk = I - 2uk.ukt
*
* we use k non-unit vectors such that:
*
* Hk = I - betakvk.vkt
*
* where vk = ak - alphak ek.
* The betak coefficients are provided upon exit as recomputing
* them from the vk 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.
*/
private void qrDecomposition() {
// 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] = Math.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 (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) ? -Math.sqrt(ak2) : Math.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;
for (int i = k, index = kDiag; i < rows; ++i, index += cols) {
gamma += jacobian[index] * y[i];
}
gamma *= beta[pk];
for (int i = k, index = kDiag; i < rows; ++i, index += cols) {
y[i] -= gamma * jacobian[index];
}
}
}
/** Array of measurements. */
private WeightedMeasurement[] measurements;
/** Array of parameters. */
private EstimatedParameter[] parameters;
/** Jacobian matrix.
* Depending on the computation phase, this matrix is either in
* canonical form (just after the calls to updateJacobian) or in
* Q.R. decomposed form (after calls to qrDecomposition)
*/
private double[] jacobian;
/** Number of columns of the jacobian matrix. */
private int cols;
/** Number of solved variables. */
private int solvedCols;
/** Number of rows of the jacobian matrix. */
private int rows;
/** 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;
/** Residuals array.
* Depending on the computation phase, this array is either in
* canonical form (just after the calls to updateResiduals) or in
* premultiplied by Qt form (just after calls to qTy)
*/
private double[] residuals;
/** Cost value (square root of the sum of the residuals). */
private double cost;
/** Positive input variable used in determining the initial step bound. */
private double initialStepBoundFactor;
/** Maximal number of cost evaluations. */
private int maxCostEval;
/** Number of cost evaluations. */
private int costEvaluations;
/** Number of jacobian evaluations. */
private int jacobianEvaluations;
/** 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;
private static final long serialVersionUID = 5387476316105068340L;
}
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