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1048 lines
40 KiB
JavaScript
1048 lines
40 KiB
JavaScript
/**
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* This class solves a least-squares problem using the Levenberg-Marquardt algorithm.
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*
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* <p>This implementation <em>should</em> work even for over-determined systems
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* (i.e. systems having more point than equations). Over-determined systems
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* are solved by ignoring the point which have the smallest impact according
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* to their jacobian column norm. Only the rank of the matrix and some loop bounds
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* are changed to implement this.</p>
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*
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* <p>The resolution engine is a simple translation of the MINPACK <a
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* href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
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* changes. The changes include the over-determined resolution, the use of
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* inherited convergence checker and the Q.R. decomposition which has been
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* rewritten following the algorithm described in the
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* P. Lascaux and R. Theodor book <i>Analyse numérique matricielle
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* appliquée à l'art de l'ingénieur</i>, Masson 1986.</p>
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* <p>The authors of the original fortran version are:
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* <ul>
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* <li>Argonne National Laboratory. MINPACK project. March 1980</li>
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* <li>Burton S. Garbow</li>
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* <li>Kenneth E. Hillstrom</li>
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* <li>Jorge J. More</li>
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* </ul>
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* The redistribution policy for MINPACK is available <a
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* href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
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* is reproduced below.</p>
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*
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* <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
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* <tr><td>
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* Minpack Copyright Notice (1999) University of Chicago.
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* All rights reserved
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* </td></tr>
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* <tr><td>
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* <ol>
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* <li>Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.</li>
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* <li>Redistributions in binary form must reproduce the above
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* copyright notice, this list of conditions and the following
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* disclaimer in the documentation and/or other materials provided
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* with the distribution.</li>
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* <li>The end-user documentation included with the redistribution, if any,
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* must include the following acknowledgment:
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* <code>This product includes software developed by the University of
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* Chicago, as Operator of Argonne National Laboratory.</code>
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* Alternately, this acknowledgment may appear in the software itself,
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* if and wherever such third-party acknowledgments normally appear.</li>
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* <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
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* WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
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* UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
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* THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
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* IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
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* OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
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* OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
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* OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
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* USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
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* THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
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* DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
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* UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
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* BE CORRECTED.</strong></li>
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* <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
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* HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
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* ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
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* INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
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* ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
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* PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
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* SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
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* (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
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* EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
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* POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
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* <ol></td></tr>
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* </table>
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*
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* @version $Id: LevenbergMarquardtOptimizer.java 1416643 2012-12-03 19:37:14Z tn $
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* @constructor
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*/
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export default function LMOptimizer(startPoint, target, model, jacobian) {
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this.startPoint = startPoint;
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this.target = target;
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this.evalCount = 0;
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this.evalMaximalCount = 100000;
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this.model = model;
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this.jacobian = jacobian;
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this.identity = function(size) {
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var out = [];
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for (var row = 0; row < size; ++row) {
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out.push([]);
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for (var col = 0; col < size; ++col) {
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out[row].push( row === col ? 1 : 0);
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}
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}
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return out;
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}
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/** Square-root of the weight matrix. */
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this.weightMatrixSqrt = this.identity(target.length);//TMath.identity(new TMath.Matrix(target.length, target.length)); //TODO:
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this.weightMatrix = this.identity(target.length);
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/** Cost value (square root of the sum of the residuals). */
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this.cost = null;
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/** Number of solved point. */
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this.solvedCols = null;
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/** Diagonal elements of the R matrix in the Q.R. decomposition. */
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this.diagR = null;
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/** Norms of the columns of the jacobian matrix. */
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this.jacNorm = null;
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/** Coefficients of the Householder transforms vectors. */
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this.beta = null;
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/** Columns permutation array. */
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this.permutation = null;
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/** Rank of the jacobian matrix. */
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this.rank = null;
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/** Levenberg-Marquardt parameter. */
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this.lmPar = null;
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/** Parameters evolution direction associated with lmPar. */
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this.lmDir = null;
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/** Positive input variable used in determining the initial step bound. */
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this.initialStepBoundFactor = null;
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/** Desired relative error in the sum of squares. */
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this.costRelativeTolerance = null;
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/** Desired relative error in the approximate solution parameters. */
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this.parRelativeTolerance = null;
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/** Desired max cosine on the orthogonality between the function vector
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* and the columns of the jacobian. */
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this.orthoTolerance = null;
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/** Threshold for QR ranking. */
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this.qrRankingThreshold = null;
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/** Weighted residuals. */
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this.weightedResidual = null;
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/** Weighted Jacobian. */
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this.weightedJacobian = null;
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this.checker = null;
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function arr(size) {
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var out = [];
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out.length = size;
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for (var i = 0; i < size; ++i) {
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out[i] = 0;
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}
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return out;
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}
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function Arrays_fill(a, fromIndex, toIndex,val) {
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for (var i = fromIndex; i < toIndex; i++)
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a[i] = val;
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}
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// var SAFE_MIN = Number.MIN_VALUE; //FIXME!!!!
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var SAFE_MIN = 1e-30; //FIXME!!!!
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/**
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* Build an optimizer for least squares problems with default values
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* for all the tuning parameters (see the {@link
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* #LevenbergMarquardtOptimizer(double,double,double,double,double)
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* other contructor}.
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* The default values for the algorithm settings are:
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* <ul>
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* <li>Initial step bound factor: 100</li>
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* <li>Cost relative tolerance: 1e-10</li>
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* <li>Parameters relative tolerance: 1e-10</li>
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* <li>Orthogonality tolerance: 1e-10</li>
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* <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
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* </ul>
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*/
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this.init = function() {
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this.init1(100, 1e-10, 1e-10, 1e-10, SAFE_MIN);
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}
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/**
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* Build an optimizer for least squares problems with default values
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* for some of the tuning parameters (see the {@link
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* #LevenbergMarquardtOptimizer(double,double,double,double,double)
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* other contructor}.
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* The default values for the algorithm settings are:
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* <ul>
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* <li>Initial step bound factor}: 100</li>
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* <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li>
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* </ul>
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*
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* @param costRelativeTolerance Desired relative error in the sum of
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* squares.
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* @param parRelativeTolerance Desired relative error in the approximate
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* solution parameters.
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* @param orthoTolerance Desired max cosine on the orthogonality between
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* the function vector and the columns of the Jacobian.
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*/
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this.init0 = function(costRelativeTolerance,
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parRelativeTolerance,
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orthoTolerance) {
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this.init1(100, costRelativeTolerance, parRelativeTolerance, orthoTolerance,
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SAFE_MIN);
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}
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/**
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* The arguments control the behaviour of the default convergence checking
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* procedure.
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* Additional criteria can defined through the setting of a {@link
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* ConvergenceChecker}.
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*
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* @param initialStepBoundFactor Positive input variable used in
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* determining the initial step bound. This bound is set to the
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* product of initialStepBoundFactor and the euclidean norm of
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* {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
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* itself. In most cases factor should lie in the interval
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* {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
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* @param costRelativeTolerance Desired relative error in the sum of
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* squares.
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* @param parRelativeTolerance Desired relative error in the approximate
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* solution parameters.
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* @param orthoTolerance Desired max cosine on the orthogonality between
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* the function vector and the columns of the Jacobian.
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* @param threshold Desired threshold for QR ranking. If the squared norm
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* of a column vector is smaller or equal to this threshold during QR
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* decomposition, it is considered to be a zero vector and hence the rank
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* of the matrix is reduced.
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*/
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this.init1 = function(initialStepBoundFactor,
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costRelativeTolerance,
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parRelativeTolerance,
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orthoTolerance,
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threshold) {
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this.initialStepBoundFactor = initialStepBoundFactor;
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this.costRelativeTolerance = costRelativeTolerance;
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this.parRelativeTolerance = parRelativeTolerance;
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this.orthoTolerance = orthoTolerance;
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this.qrRankingThreshold = threshold;
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}
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/** {@inheritDoc} */
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this.doOptimize = function() {
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var nR = this.target.length; // Number of observed data.
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var currentPoint = this.startPoint;
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var nC = currentPoint.length; // Number of parameters.
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// arrays shared with the other private methods
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this.solvedCols = Math.min(nR, nC);
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this.diagR = arr(nC);
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this.jacNorm = arr(nC);
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this.beta = arr(nC);
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this.permutation = arr(nC);
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this.lmDir = arr(nC);
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// local point
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var delta = 0;
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var xNorm = 0;
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var diag = arr(nC);
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var oldX = arr(nC);
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var oldRes = arr(nR);
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var oldObj = arr(nR);
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var qtf = arr(nR);
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var work1 = arr(nC);
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var work2 = arr(nC);
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var work3 = arr(nC);
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var weightMatrixSqrt = this.getWeightSquareRoot();
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// Evaluate the function at the starting point and calculate its norm.
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var currentObjective = this.computeObjectiveValue(currentPoint);
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var currentResiduals = this.computeResiduals(currentObjective);
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var current = [currentPoint, currentObjective];
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var currentCost = this.computeCost(currentResiduals);
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// Outer loop.
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this.lmPar = 0;
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var firstIteration = true;
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var iter = 0;
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while (true) {
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++iter;
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var previous = current;
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// QR decomposition of the jacobian matrix
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this.qrDecomposition(this.computeWeightedJacobian(currentPoint));
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this.weightedResidual = this.operate(weightMatrixSqrt, currentResiduals);
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for (var i = 0; i < nR; i++) {
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qtf[i] = this.weightedResidual[i];
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}
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// compute Qt.res
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this.qTy(qtf);
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// now we don't need Q anymore,
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// so let jacobian contain the R matrix with its diagonal elements
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for (var k = 0; k < this.solvedCols; ++k) {
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var pk = this.permutation[k];
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this.weightedJacobian[k][pk] = this.diagR[pk];
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}
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if (firstIteration) {
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// scale the point according to the norms of the columns
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// of the initial jacobian
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xNorm = 0;
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for (var k = 0; k < nC; ++k) {
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var dk = this.jacNorm[k];
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if (dk == 0) {
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dk = 1.0;
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}
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var xk = dk * currentPoint[k];
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xNorm += xk * xk;
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diag[k] = dk;
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}
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xNorm = Math.sqrt(xNorm);
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// initialize the step bound delta
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delta = (xNorm == 0) ? this.initialStepBoundFactor : (this.initialStepBoundFactor * xNorm);
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}
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// check orthogonality between function vector and jacobian columns
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var maxCosine = 0;
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if (currentCost != 0) {
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for (var j = 0; j < this.solvedCols; ++j) {
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var pj = this.permutation[j];
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var s = this.jacNorm[pj];
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if (s != 0) {
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var sum = 0;
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for (var i = 0; i <= j; ++i) {
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sum += this.weightedJacobian[i][pj] * qtf[i];
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}
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maxCosine = Math.max(maxCosine, Math.abs(sum) / (s * currentCost));
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}
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}
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}
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if (maxCosine <= this.orthoTolerance) {
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// Convergence has been reached.
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this.setCost(currentCost);
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return current;
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}
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// rescale if necessary
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for (var j = 0; j < nC; ++j) {
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diag[j] = Math.max(diag[j], this.jacNorm[j]);
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}
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// Inner loop.
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for (var ratio = 0; ratio < 1.0e-4;) {
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// save the state
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for (var j = 0; j < this.solvedCols; ++j) {
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var pj = this.permutation[j];
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oldX[pj] = currentPoint[pj];
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}
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var previousCost = currentCost;
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var tmpVec = this.weightedResidual;
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this.weightedResidual = oldRes;
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oldRes = tmpVec;
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tmpVec = currentObjective;
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currentObjective = oldObj;
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oldObj = tmpVec;
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// determine the Levenberg-Marquardt parameter
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this.determineLMParameter(qtf, delta, diag, work1, work2, work3);
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// compute the new point and the norm of the evolution direction
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var lmNorm = 0;
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for (var j = 0; j < this.solvedCols; ++j) {
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var pj = this.permutation[j];
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this.lmDir[pj] = -this.lmDir[pj];
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currentPoint[pj] = oldX[pj] + this.lmDir[pj];
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var s = diag[pj] * this.lmDir[pj];
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lmNorm += s * s;
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}
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lmNorm = Math.sqrt(lmNorm);
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// on the first iteration, adjust the initial step bound.
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if (firstIteration) {
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delta = Math.min(delta, lmNorm);
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}
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// Evaluate the function at x + p and calculate its norm.
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currentObjective = this.computeObjectiveValue(currentPoint);
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currentResiduals = this.computeResiduals(currentObjective);
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current = [currentPoint, currentObjective];
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currentCost = this.computeCost(currentResiduals);
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// compute the scaled actual reduction
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var actRed = -1.0;
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if (0.1 * currentCost < previousCost) {
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var r = currentCost / previousCost;
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actRed = 1.0 - r * r;
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}
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// compute the scaled predicted reduction
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// and the scaled directional derivative
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for (var j = 0; j < this.solvedCols; ++j) {
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var pj = this.permutation[j];
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var dirJ = this.lmDir[pj];
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work1[j] = 0;
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for (var i = 0; i <= j; ++i) {
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work1[i] += this.weightedJacobian[i][pj] * dirJ;
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}
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}
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var coeff1 = 0;
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for (var j = 0; j < this.solvedCols; ++j) {
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coeff1 += work1[j] * work1[j];
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}
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var pc2 = previousCost * previousCost;
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coeff1 = coeff1 / pc2;
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var coeff2 = this.lmPar * lmNorm * lmNorm / pc2;
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var preRed = coeff1 + 2 * coeff2;
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var dirDer = -(coeff1 + coeff2);
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// ratio of the actual to the predicted reduction
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ratio = (preRed == 0) ? 0 : (actRed / preRed);
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// update the step bound
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if (ratio <= 0.25) {
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var tmp =
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(actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
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if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) {
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tmp = 0.1;
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}
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delta = tmp * Math.min(delta, 10.0 * lmNorm);
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this.lmPar /= tmp;
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} else if ((this.lmPar == 0) || (ratio >= 0.75)) {
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delta = 2 * lmNorm;
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this.lmPar *= 0.5;
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}
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// test for successful iteration.
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if (ratio >= 1.0e-4) {
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// successful iteration, update the norm
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firstIteration = false;
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xNorm = 0;
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for (var k = 0; k < nC; ++k) {
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var xK = diag[k] * currentPoint[k];
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xNorm += xK * xK;
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}
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xNorm = Math.sqrt(xNorm);
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// tests for convergence.
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if (this.checker != null) {
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// we use the vectorial convergence checker
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if (this.checker.call(iter, previous, current)) {
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this.setCost(currentCost);
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return current;
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}
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}
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} else {
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// failed iteration, reset the previous values
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currentCost = previousCost;
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for (var j = 0; j < this.solvedCols; ++j) {
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var pj = this.permutation[j];
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currentPoint[pj] = oldX[pj];
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}
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tmpVec = this.weightedResidual;
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this.weightedResidual = oldRes;
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oldRes = tmpVec;
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tmpVec = currentObjective;
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currentObjective = oldObj;
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oldObj = tmpVec;
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// Reset "current" to previous values.
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current = [currentPoint, currentObjective];
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}
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// Default convergence criteria.
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if ((Math.abs(actRed) <= this.costRelativeTolerance &&
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preRed <= this.costRelativeTolerance &&
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ratio <= 2.0) ||
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delta <= this.parRelativeTolerance * xNorm) {
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this.setCost(currentCost);
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return current;
|
|
}
|
|
|
|
// tests for termination and stringent tolerances
|
|
// (2.2204e-16 is the machine epsilon for IEEE754)
|
|
if ((Math.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
|
|
throw "TOO_SMALL_COST_RELATIVE_TOLERANCE: " + this.costRelativeTolerance;
|
|
} else if (delta <= 2.2204e-16 * xNorm) {
|
|
throw "TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE: " + this.parRelativeTolerance;
|
|
} else if (maxCosine <= 2.2204e-16) {
|
|
throw "TOO_SMALL_ORTHOGONALITY_TOLERANCE: " + this.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
|
|
*/
|
|
this.determineLMParameter = function(qy, delta, diag,
|
|
work1, work2, work3) {
|
|
var nC = this.weightedJacobian[0].length;
|
|
|
|
// compute and store in x the gauss-newton direction, if the
|
|
// jacobian is rank-deficient, obtain a least squares solution
|
|
for (var j = 0; j < this.rank; ++j) {
|
|
this.lmDir[this.permutation[j]] = qy[j];
|
|
}
|
|
for (var j = this.rank; j < nC; ++j) {
|
|
this.lmDir[this.permutation[j]] = 0;
|
|
}
|
|
for (var k = this.rank - 1; k >= 0; --k) {
|
|
var pk = this.permutation[k];
|
|
var ypk = this.lmDir[pk] / this.diagR[pk];
|
|
for (var i = 0; i < k; ++i) {
|
|
this.lmDir[this.permutation[i]] -= ypk * this.weightedJacobian[i][pk];
|
|
}
|
|
this.lmDir[pk] = ypk;
|
|
}
|
|
|
|
// evaluate the function at the origin, and test
|
|
// for acceptance of the Gauss-Newton direction
|
|
var dxNorm = 0;
|
|
for (var j = 0; j < this.solvedCols; ++j) {
|
|
var pj = this.permutation[j];
|
|
var s = diag[pj] * this.lmDir[pj];
|
|
work1[pj] = s;
|
|
dxNorm += s * s;
|
|
}
|
|
dxNorm = Math.sqrt(dxNorm);
|
|
var fp = dxNorm - delta;
|
|
if (fp <= 0.1 * delta) {
|
|
this.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
|
|
var sum2;
|
|
var parl = 0;
|
|
if (this.rank == this.solvedCols) {
|
|
for (var j = 0; j < this.solvedCols; ++j) {
|
|
var pj = this.permutation[j];
|
|
work1[pj] *= diag[pj] / dxNorm;
|
|
}
|
|
sum2 = 0;
|
|
for (var j = 0; j < this.solvedCols; ++j) {
|
|
var pj = this.permutation[j];
|
|
var sum = 0;
|
|
for (var i = 0; i < j; ++i) {
|
|
sum += this.weightedJacobian[i][pj] * work1[this.permutation[i]];
|
|
}
|
|
var s = (work1[pj] - sum) / this.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 (var j = 0; j < this.solvedCols; ++j) {
|
|
var pj = this.permutation[j];
|
|
var sum = 0;
|
|
for (var i = 0; i <= j; ++i) {
|
|
sum += this.weightedJacobian[i][pj] * qy[i];
|
|
}
|
|
sum /= diag[pj];
|
|
sum2 += sum * sum;
|
|
}
|
|
var gNorm = Math.sqrt(sum2);
|
|
var 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
|
|
this.lmPar = Math.min(paru, Math.max(this.lmPar, parl));
|
|
if (this.lmPar == 0) {
|
|
this.lmPar = gNorm / dxNorm;
|
|
}
|
|
|
|
for (var countdown = 10; countdown >= 0; --countdown) {
|
|
|
|
// evaluate the function at the current value of lmPar
|
|
if (this.lmPar == 0) {
|
|
this.lmPar = Math.max(2.2251e-308, 0.001 * paru);
|
|
}
|
|
var sPar = Math.sqrt(this.lmPar);
|
|
for (var j = 0; j < this.solvedCols; ++j) {
|
|
var pj = this.permutation[j];
|
|
work1[pj] = sPar * diag[pj];
|
|
}
|
|
this.determineLMDirection(qy, work1, work2, work3);
|
|
|
|
dxNorm = 0;
|
|
for (var j = 0; j < this.solvedCols; ++j) {
|
|
var pj = this.permutation[j];
|
|
var s = diag[pj] * this.lmDir[pj];
|
|
work3[pj] = s;
|
|
dxNorm += s * s;
|
|
}
|
|
dxNorm = Math.sqrt(dxNorm);
|
|
var 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 (var j = 0; j < this.solvedCols; ++j) {
|
|
var pj = this.permutation[j];
|
|
work1[pj] = work3[pj] * diag[pj] / dxNorm;
|
|
}
|
|
for (var j = 0; j < this.solvedCols; ++j) {
|
|
var pj = this.permutation[j];
|
|
work1[pj] /= work2[j];
|
|
var tmp = work1[pj];
|
|
for (var i = j + 1; i < this.solvedCols; ++i) {
|
|
work1[this.permutation[i]] -= this.weightedJacobian[i][pj] * tmp;
|
|
}
|
|
}
|
|
sum2 = 0;
|
|
for (var j = 0; j < this.solvedCols; ++j) {
|
|
var s = work1[this.permutation[j]];
|
|
sum2 += s * s;
|
|
}
|
|
var correction = fp / (delta * sum2);
|
|
|
|
// depending on the sign of the function, update parl or paru.
|
|
if (fp > 0) {
|
|
parl = Math.max(parl, this.lmPar);
|
|
} else if (fp < 0) {
|
|
paru = Math.min(paru, this.lmPar);
|
|
}
|
|
|
|
// compute an improved estimate for lmPar
|
|
this.lmPar = Math.max(parl, this.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
|
|
*/
|
|
this.determineLMDirection = function(qy, diag, lmDiag, work) {
|
|
|
|
// copy R and Qty to preserve input and initialize s
|
|
// in particular, save the diagonal elements of R in lmDir
|
|
for (var j = 0; j < this.solvedCols; ++j) {
|
|
var pj = this.permutation[j];
|
|
for (var i = j + 1; i < this.solvedCols; ++i) {
|
|
this.weightedJacobian[i][pj] = this.weightedJacobian[j][this.permutation[i]];
|
|
}
|
|
this.lmDir[j] = this.diagR[pj];
|
|
work[j] = qy[j];
|
|
}
|
|
|
|
// eliminate the diagonal matrix d using a Givens rotation
|
|
for (var j = 0; j < this.solvedCols; ++j) {
|
|
|
|
// prepare the row of d to be eliminated, locating the
|
|
// diagonal element using p from the Q.R. factorization
|
|
var pj = this.permutation[j];
|
|
var 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.
|
|
var qtbpj = 0;
|
|
for (var k = j; k < this.solvedCols; ++k) {
|
|
var pk = this.permutation[k];
|
|
|
|
// determine a Givens rotation which eliminates the
|
|
// appropriate element in the current row of d
|
|
if (lmDiag[k] != 0) {
|
|
|
|
var sin;
|
|
var cos;
|
|
var rkk = this.weightedJacobian[k][pk];
|
|
if (Math.abs(rkk) < Math.abs(lmDiag[k])) {
|
|
var cotan = rkk / lmDiag[k];
|
|
sin = 1.0 / Math.sqrt(1.0 + cotan * cotan);
|
|
cos = sin * cotan;
|
|
} else {
|
|
var 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)
|
|
this.weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
|
|
var temp = cos * work[k] + sin * qtbpj;
|
|
qtbpj = -sin * work[k] + cos * qtbpj;
|
|
work[k] = temp;
|
|
|
|
// accumulate the tranformation in the row of s
|
|
for (var i = k + 1; i < this.solvedCols; ++i) {
|
|
var rik = this.weightedJacobian[i][pk];
|
|
var temp2 = cos * rik + sin * lmDiag[i];
|
|
lmDiag[i] = -sin * rik + cos * lmDiag[i];
|
|
this.weightedJacobian[i][pk] = temp2;
|
|
}
|
|
}
|
|
}
|
|
|
|
// store the diagonal element of s and restore
|
|
// the corresponding diagonal element of R
|
|
lmDiag[j] = this.weightedJacobian[j][this.permutation[j]];
|
|
this.weightedJacobian[j][this.permutation[j]] = this.lmDir[j];
|
|
}
|
|
|
|
// solve the triangular system for z, if the system is
|
|
// singular, then obtain a least squares solution
|
|
var nSing = this.solvedCols;
|
|
for (var j = 0; j < this.solvedCols; ++j) {
|
|
if ((lmDiag[j] == 0) && (nSing == this.solvedCols)) {
|
|
nSing = j;
|
|
}
|
|
if (nSing < this.solvedCols) {
|
|
work[j] = 0;
|
|
}
|
|
}
|
|
if (nSing > 0) {
|
|
for (var j = nSing - 1; j >= 0; --j) {
|
|
var pj = this.permutation[j];
|
|
var sum = 0;
|
|
for (var i = j + 1; i < nSing; ++i) {
|
|
sum += this.weightedJacobian[i][pj] * work[i];
|
|
}
|
|
work[j] = (work[j] - sum) / lmDiag[j];
|
|
}
|
|
}
|
|
|
|
// permute the components of z back to components of lmDir
|
|
for (var j = 0; j < this.lmDir.length; ++j) {
|
|
this.lmDir[this.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>
|
|
*
|
|
* @param jacobian Weighted Jacobian matrix at the current point.
|
|
* @exception ConvergenceException if the decomposition cannot be performed
|
|
*/
|
|
this.qrDecomposition = function(jacobian) {
|
|
// Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
|
|
// hence the multiplication by -1.
|
|
this.weightedJacobian = this.scalarMultiply(jacobian, -1);
|
|
|
|
var nR = this.weightedJacobian.length;
|
|
var nC = this.weightedJacobian[0].length;
|
|
|
|
// initializations
|
|
for (var k = 0; k < nC; ++k) {
|
|
this.permutation[k] = k;
|
|
var norm2 = 0;
|
|
for (var i = 0; i < nR; ++i) {
|
|
var akk = this.weightedJacobian[i][k];
|
|
norm2 += akk * akk;
|
|
}
|
|
this.jacNorm[k] = Math.sqrt(norm2);
|
|
}
|
|
|
|
// transform the matrix column after column
|
|
for (var k = 0; k < nC; ++k) {
|
|
|
|
// select the column with the greatest norm on active components
|
|
var nextColumn = -1;
|
|
var ak2 = Number.NEGATIVE_INFINITY;
|
|
for (var i = k; i < nC; ++i) {
|
|
var norm2 = 0;
|
|
for (var j = k; j < nR; ++j) {
|
|
var aki = this.weightedJacobian[j][this.permutation[i]];
|
|
norm2 += aki * aki;
|
|
}
|
|
if (!isFinite(norm2)) {
|
|
throw "UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN";
|
|
}
|
|
if (norm2 > ak2) {
|
|
nextColumn = i;
|
|
ak2 = norm2;
|
|
}
|
|
}
|
|
if (ak2 <= this.qrRankingThreshold) {
|
|
this.rank = k;
|
|
return;
|
|
}
|
|
var pk = this.permutation[nextColumn];
|
|
this.permutation[nextColumn] = this.permutation[k];
|
|
this.permutation[k] = pk;
|
|
|
|
// choose alpha such that Hk.u = alpha ek
|
|
var akk = this.weightedJacobian[k][pk];
|
|
var alpha = (akk > 0) ? -Math.sqrt(ak2) : Math.sqrt(ak2);
|
|
var betak = 1.0 / (ak2 - akk * alpha);
|
|
this.beta[pk] = betak;
|
|
|
|
// transform the current column
|
|
this.diagR[pk] = alpha;
|
|
this.weightedJacobian[k][pk] -= alpha;
|
|
|
|
// transform the remaining columns
|
|
for (var dk = nC - 1 - k; dk > 0; --dk) {
|
|
var gamma = 0;
|
|
for (var j = k; j < nR; ++j) {
|
|
gamma += this.weightedJacobian[j][pk] * this.weightedJacobian[j][this.permutation[k + dk]];
|
|
}
|
|
gamma *= betak;
|
|
for (var j = k; j < nR; ++j) {
|
|
this.weightedJacobian[j][this.permutation[k + dk]] -= gamma * this.weightedJacobian[j][pk];
|
|
}
|
|
}
|
|
}
|
|
this.rank = this.solvedCols;
|
|
}
|
|
|
|
/**
|
|
* Compute the product Qt.y for some Q.R. decomposition.
|
|
*
|
|
* @param y vector to multiply (will be overwritten with the result)
|
|
*/
|
|
this.qTy = function(y) {
|
|
var nR = this.weightedJacobian.length;
|
|
var nC = this.weightedJacobian[0].length;
|
|
|
|
for (var k = 0; k < nC; ++k) {
|
|
var pk = this.permutation[k];
|
|
var gamma = 0;
|
|
for (var i = k; i < nR; ++i) {
|
|
gamma += this.weightedJacobian[i][pk] * y[i];
|
|
}
|
|
gamma *= this.beta[pk];
|
|
for (var i = k; i < nR; ++i) {
|
|
y[i] -= gamma * this.weightedJacobian[i][pk];
|
|
}
|
|
}
|
|
}
|
|
|
|
/**
|
|
* Computes the weighted Jacobian matrix.
|
|
*
|
|
* @param params Model parameters at which to compute the Jacobian.
|
|
* @return the weighted Jacobian: W<sup>1/2</sup> J.
|
|
* @throws DimensionMismatchException if the Jacobian dimension does not
|
|
* match problem dimension.
|
|
*/
|
|
this.computeWeightedJacobian = function(params) {
|
|
// return this.weightMatrixSqrt.multiply(this.jacobian(params));
|
|
|
|
//TODO: since weighted matrix is always identity return jacobian itself
|
|
return this.jacobian(params);
|
|
}
|
|
|
|
this.scalarMultiply = function(m, s) {
|
|
var rowCount = m.length;
|
|
var columnCount = m[0].length;
|
|
var out = [];
|
|
for (var row = 0; row < rowCount; ++row) {
|
|
out.push([]);
|
|
for (var col = 0; col < columnCount; ++col) {
|
|
out[row].push(m[row][col] * s);
|
|
}
|
|
}
|
|
|
|
return out;
|
|
}
|
|
|
|
this.operate = function(m, v) {
|
|
var nRows = m.length;
|
|
var nCols = m[0].length;
|
|
if (v.length != nCols) {
|
|
throw "DimensionMismatchException: " + v.length + "!=" + nCols;
|
|
}
|
|
var out = [];
|
|
for (var row = 0; row < nRows; row++) {
|
|
var dataRow = m[row];
|
|
var sum = 0;
|
|
for (var i = 0; i < nCols; i++) {
|
|
sum += dataRow[i] * v[i];
|
|
}
|
|
out[row] = sum;
|
|
}
|
|
return out;
|
|
}
|
|
|
|
/**
|
|
* Computes the cost.
|
|
*
|
|
* @param residuals Residuals.
|
|
* @return the cost.
|
|
* @see #computeResiduals(double[])
|
|
*/
|
|
this.computeCost = function(residuals) {
|
|
return Math.sqrt(this.dotProduct( residuals, this.operate(this.getWeight(), residuals)));
|
|
}
|
|
|
|
|
|
this.dotProduct = function(v1, v2) {
|
|
var dot = 0;
|
|
for (var i = 0; i < v1.length; i++) {
|
|
dot += v1[i] * v2[i];
|
|
}
|
|
return dot;
|
|
}
|
|
|
|
/**
|
|
* Gets the root-mean-square (RMS) value.
|
|
*
|
|
* The RMS the root of the arithmetic mean of the square of all weighted
|
|
* residuals.
|
|
* This is related to the criterion that is minimized by the optimizer
|
|
* as follows: If <em>c</em> if the criterion, and <em>n</em> is the
|
|
* number of measurements, then the RMS is <em>sqrt (c/n)</em>.
|
|
*
|
|
* @return the RMS value.
|
|
*/
|
|
this.getRMS = function() {
|
|
return Math.sqrt(this.getChiSquare() / this.target.length);
|
|
}
|
|
|
|
/**
|
|
* Get a Chi-Square-like value assuming the N residuals follow N
|
|
* distinct normal distributions centered on 0 and whose variances are
|
|
* the reciprocal of the weights.
|
|
* @return chi-square value
|
|
*/
|
|
this.getChiSquare = function() {
|
|
return this.cost * this.cost;
|
|
}
|
|
|
|
/**
|
|
* Gets the square-root of the weight matrix.
|
|
*
|
|
* @return the square-root of the weight matrix.
|
|
*/
|
|
this.getWeightSquareRoot = function() {
|
|
return this.weightMatrixSqrt;//.copy(); FIXME for now it's always identity
|
|
}
|
|
|
|
this.getWeight = function() {
|
|
return this.weightMatrix;//.copy(); FIXME for now it's always identity
|
|
}
|
|
|
|
/**
|
|
* Sets the cost.
|
|
*
|
|
* @param cost Cost value.
|
|
*/
|
|
this.setCost = function(cost) {
|
|
this.cost = cost;
|
|
}
|
|
|
|
/**
|
|
* Computes the residuals.
|
|
* The residual is the difference between the observed (target)
|
|
* values and the model (objective function) value.
|
|
* There is one residual for each element of the vector-valued
|
|
* function.
|
|
*
|
|
* @param objectiveValue Value of the the objective function. This is
|
|
* the value returned from a call to
|
|
* {@link #computeObjectiveValue(double[]) computeObjectiveValue}
|
|
* (whose array argument contains the model parameters).
|
|
* @return the residuals.
|
|
* @throws DimensionMismatchException if {@code params} has a wrong
|
|
* length.
|
|
*/
|
|
this.computeResiduals = function(objectiveValue) {
|
|
var target = this.target;
|
|
if (objectiveValue.length != target.length) {
|
|
throw "DimensionMismatchException: " + target.length + " != " + objectiveValue.length;
|
|
}
|
|
|
|
var residuals = arr(target.length);
|
|
for (var i = 0; i < target.length; i++) {
|
|
residuals[i] = target[i] - objectiveValue[i];
|
|
}
|
|
|
|
return residuals;
|
|
}
|
|
|
|
this.computeObjectiveValue = function(params) {
|
|
if (++this.evalCount > this.evalMaximalCount) {
|
|
throw "TOO MANY FUNCTION EVALUATION"
|
|
}
|
|
return this.model(params);
|
|
}
|
|
|
|
}
|