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<?php
// Levenberg-Marquardt in PHP
// http://www.idiom.com/~zilla/Computer/Javanumeric/LM.java
class LevenbergMarquardt {
/**
* Calculate the current sum-squared-error
*
* Chi-squared is the distribution of squared Gaussian errors,
* thus the name.
*
* @param double[][] $x
* @param double[] $a
* @param double[] $y,
* @param double[] $s,
* @param object $f
*/
function chiSquared($x, $a, $y, $s, $f) {
$npts = count($y);
$sum = 0.0;
for ($i = 0; $i < $npts; ++$i) {
$d = $y[$i] - $f->val($x[$i], $a);
$d = $d / $s[$i];
$sum = $sum + ($d*$d);
}
return $sum;
} // function chiSquared()
/**
* Minimize E = sum {(y[k] - f(x[k],a)) / s[k]}^2
* The individual errors are optionally scaled by s[k].
* Note that LMfunc implements the value and gradient of f(x,a),
* NOT the value and gradient of E with respect to a!
*
* @param x array of domain points, each may be multidimensional
* @param y corresponding array of values
* @param a the parameters/state of the model
* @param vary false to indicate the corresponding a[k] is to be held fixed
* @param s2 sigma^2 for point i
* @param lambda blend between steepest descent (lambda high) and
* jump to bottom of quadratic (lambda zero).
* Start with 0.001.
* @param termepsilon termination accuracy (0.01)
* @param maxiter stop and return after this many iterations if not done
* @param verbose set to zero (no prints), 1, 2
*
* @return the new lambda for future iterations.
* Can use this and maxiter to interleave the LM descent with some other
* task, setting maxiter to something small.
*/
function solve($x, $a, $y, $s, $vary, $f, $lambda, $termepsilon, $maxiter, $verbose) {
$npts = count($y);
$nparm = count($a);
if ($verbose > 0) {
print("solve x[".count($x)."][".count($x[0])."]");
print(" a[".count($a)."]");
println(" y[".count(length)."]");
}
$e0 = $this->chiSquared($x, $a, $y, $s, $f);
//double lambda = 0.001;
$done = false;
// g = gradient, H = hessian, d = step to minimum
// H d = -g, solve for d
$H = array();
$g = array();
//double[] d = new double[nparm];
$oos2 = array();
for($i = 0; $i < $npts; ++$i) {
$oos2[$i] = 1./($s[$i]*$s[$i]);
}
$iter = 0;
$term = 0; // termination count test
do {
++$iter;
// hessian approximation
for( $r = 0; $r < $nparm; ++$r) {
for( $c = 0; $c < $nparm; ++$c) {
for( $i = 0; $i < $npts; ++$i) {
if ($i == 0) $H[$r][$c] = 0.;
$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( $r = 0; $r < $nparm; ++$r)
$H[$r][$r] *= (1. + $lambda);
// gradient
for( $r = 0; $r < $nparm; ++$r) {
for( $i = 0; $i < $npts; ++$i) {
if ($i == 0) $g[$r] = 0.;
$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( $r = 0; $r < $nparm; ++$r) {
$g[$r] = -0.5 * $g[$r];
for( $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;
} // function solve()
} // class LevenbergMarquardt
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