lm.java

Java实现的各种数学算法

// levenberg-marquardt in java 
//
// To use this, implement the functions in the LMfunc interface.
//
// This library uses simple matrix routines from the JAMA java matrix package,
// which is in the public domain.  Reference:
//    http://math.nist.gov/javanumerics/jama/
// (JAMA has a matrix object class.  An earlier library JNL, which is no longer
// available, represented matrices as low-level arrays.  Several years 
// ago the performance of JNL matrix code was better than that of JAMA,
// though improvements in java compilers may have fixed this by now.)
//
// One further recommendation would be to use an inverse based
// on Choleski decomposition, which is easy to implement and
// suitable for the symmetric inverse required here.  There is a choleski
// routine at idiom.com/~zilla.
//
// If you make an improved version, please consider adding your
// name to it ("modified by ...") and send it back to me
// (and put it on the web).
//
// ----------------------------------------------------------------
// 
// This library is free software; you can redistribute it and/or
// modify it under the terms of the GNU Library General Public
// License as published by the Free Software Foundation; either
// version 2 of the License, or (at your option) any later version.
// 
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Library General Public License for more details.
// 
// You should have received a copy of the GNU Library General Public
// License along with this library; if not, write to the
// Free Software Foundation, Inc., 59 Temple Place - Suite 330,
// Boston, MA  02111-1307, USA.
//
// initial author contact info:  
// jplewis  www.idiom.com/~zilla  zilla # computer.org,   #=at
//
// Improvements by:
// dscherba  www.ncsa.uiuc.edu/~dscherba  
// Jonathan Jackson   j.jackson # ucl.ac.uk


package ZS.Solve;

// see comment above
import Jama.*;

/**
 * Levenberg-Marquardt, implemented from the general description
 * in Numerical Recipes (NR), then tweaked slightly to mostly
 * match the results of their code.
 * Use for nonlinear least squares assuming Gaussian errors.
 *
 * TODO this holds some parameters fixed by simply not updating them.
 * this may be ok if the number if fixed parameters is small,
 * but if the number of varying parameters is larger it would
 * be more efficient to make a smaller hessian involving only
 * the variables.
 *
 * The NR code assumes a statistical context, e.g. returns
 * covariance of parameter errors; we do not do this.
 */
public final class LM
{

  /**
   * calculate the current sum-squared-error
   * (Chi-squared is the distribution of squared Gaussian errors,
   * thus the name)
   */
  static double chiSquared(double[][] x, double[] a, double[] y, double[] s, 
			   LMfunc f)
  {
    int npts = y.length;
    double sum = 0.;

    for( int i = 0; i < npts; i++ ) {
      double d = y[i] - f.val(x[i], a);
      d = d / s[i];
      sum = sum + (d*d);
    }

    return sum;
  } //chiSquared


  /**
   * Minimize E = sum {(y[k] - f(x[k],a)) / s[k]}^2
   * The individual errors are optionally scaled by s[k].
   * Note that LMfunc implements the value and gradient of f(x,a),
   * NOT the value and gradient of E with respect to a!
   * 
   * @param x array of domain points, each may be multidimensional
   * @param y corresponding array of values
   * @param a the parameters/state of the model
   * @param vary false to indicate the corresponding a[k] is to be held fixed
   * @param s2 sigma^2 for point i
   * @param lambda blend between steepest descent (lambda high) and
   *	jump to bottom of quadratic (lambda zero).
   * 	Start with 0.001.
   * @param termepsilon termination accuracy (0.01)
   * @param maxiter	stop and return after this many iterations if not done
   * @param verbose	set to zero (no prints), 1, 2
   *
   * @return the new lambda for future iterations.
   *  Can use this and maxiter to interleave the LM descent with some other
   *  task, setting maxiter to something small.
   */
  public static double solve(double[][] x, double[] a, double[] y, double[] s,
			     boolean[] vary, LMfunc f,
			     double lambda, double termepsilon, int maxiter,
			     int verbose)
    throws Exception
  {
    int npts = y.length;
    int nparm = a.length;
    assert s.length == npts;
    assert x.length == npts;
    if (verbose > 0) {
      System.out.print("solve x["+x.length+"]["+x[0].length+"]" );
      System.out.print(" a["+a.length+"]");
      System.out.println(" y["+y.length+"]");
    }

    double e0 = chiSquared(x, a, y, s, f);
    //double lambda = 0.001;
    boolean done = false;

    // g = gradient, H = hessian, d = step to minimum
    // H d = -g, solve for d
    double[][] H = new double[nparm][nparm];
    double[] g = new double[nparm];
    //double[] d = new double[nparm];

    double[] oos2 = new double[s.length];
    for( int i = 0; i < npts; i++ )  oos2[i] = 1./(s[i]*s[i]);

    int iter = 0;
    int term = 0;	// termination count test

    do {
      ++iter;

      // hessian approximation
      for( int r = 0; r < nparm; r++ ) {
	for( int c = 0; c < nparm; c++ ) {
	  for( int i = 0; i < npts; i++ ) {
	    if (i == 0) H[r][c] = 0.;
	    double[] xi = x[i];
	    H[r][c] += (oos2[i] * f.grad(xi, a, r) * f.grad(xi, a, c));
	  }  //npts
	} //c
      } //r

      // boost diagonal towards gradient descent
      for( int r = 0; r < nparm; r++ )
	H[r][r] *= (1. + lambda);

      // gradient
      for( int r = 0; r < nparm; r++ ) {
	for( int i = 0; i < npts; i++ ) {
	  if (i == 0) g[r] = 0.;
	  double[] xi = x[i];
	  g[r] += (oos2[i] * (y[i]-f.val(xi,a)) * f.grad(xi, a, r));
	}
      } //npts

      // scale (for consistency with NR, not necessary)
      if (false) {
	for( int r = 0; r < nparm; r++ ) {
	  g[r] = -0.5 * g[r];
	  for( int c = 0; c < nparm; c++ ) {
	    H[r][c] *= 0.5;
	  }
	}
      }

      // solve H d = -g, evaluate error at new location
      //double[] d = DoubleMatrix.solve(H, g);
      double[] d = (new Matrix(H)).lu().solve(new Matrix(g, nparm)).getRowPackedCopy();
      //double[] na = DoubleVector.add(a, d);
      double[] na = (new Matrix(a, nparm)).plus(new Matrix(d, nparm)).getRowPackedCopy();
      double e1 = chiSquared(x, na, y, s, f);

      if (verbose > 0) {
	System.out.println("\n\niteration "+iter+" lambda = "+lambda);
	System.out.print("a = ");
        (new Matrix(a, nparm)).print(10, 2);
	if (verbose > 1) {
          System.out.print("H = "); 
          (new Matrix(H)).print(10, 2);
          System.out.print("g = "); 
          (new Matrix(g, nparm)).print(10, 2);
          System.out.print("d = "); 
          (new Matrix(d, nparm)).print(10, 2);
	}
	System.out.print("e0 = " + e0 + ": ");
	System.out.print("moved from ");
        (new Matrix(a, nparm)).print(10, 2);
	System.out.print("e1 = " + e1 + ": ");
	if (e1 < e0) {
	  System.out.print("to ");
          (new Matrix(na, nparm)).print(10, 2);
	}
	else {
	  System.out.println("move rejected");
	}
      }

      // termination test (slightly different than NR)
      if (Math.abs(e1-e0) > termepsilon) {
	term = 0;
      }
      else {
	term++;
	if (term == 4) {
	  System.out.println("terminating after " + iter + " iterations");
	  done = true;
	}
      }
      if (iter >= maxiter) done = true;

      // in the C++ version, found that changing this to e1 >= e0
      // was not a good idea.  See comment there.
      //
      if (e1 > e0 || Double.isNaN(e1)) { // new location worse than before
	lambda *= 10.;
      }
      else {		// new location better, accept new parameters
	lambda *= 0.1;
	e0 = e1;
	// simply assigning a = na will not get results copied back to caller
	for( int i = 0; i < nparm; i++ ) {
	  if (vary[i]) a[i] = na[i];
	}
      }

    } while(!done);

    return lambda;
  } //solve

  //----------------------------------------------------------------

  /**
   * solve for phase, amplitude and frequency of a sinusoid
   */
  static class LMSineTest implements LMfunc
  {
    static final int	PHASE = 0;
    static final int	AMP = 1;
    static final int	FREQ = 2;

    public double[] initial()
    {
      double[] a = new double[3];
      a[PHASE] = 0.;
      a[AMP] = 1.;
      a[FREQ] = 1.;
      return a;
    } //initial

    public double val(double[] x, double[] a)
    {
      return a[AMP] * Math.sin(a[FREQ]*x[0] + a[PHASE]);
    } //val

    public double grad(double[] x, double[] a, int a_k)
    {
      if (a_k == AMP)
	return Math.sin(a[FREQ]*x[0] + a[PHASE]);

      else if (a_k == FREQ)
	return a[AMP] * Math.cos(a[FREQ]*x[0] + a[PHASE]) * x[0];

      else if (a_k == PHASE)
	return a[AMP] * Math.cos(a[FREQ]*x[0] + a[PHASE]);

      else {
	assert false;
	return 0.;
      }
    } //grad


    public Object[] testdata() {
      double[] a = new double[3];
      a[PHASE] = 0.111;
      a[AMP] = 1.222;
      a[FREQ] = 1.333;

      int npts = 10;
      double[][] x = new double[npts][1];
      double[] y = new double[npts];