Levenberg Marquardt curve-fitting
minimize sum of weighted squared residuals. Javascript version of matlab library from Henri Gavin.
See example for usage
--------- INPUT VARIABLES -----------
func = function of n independent variables, 't', and m parameters, 'p',
returning the simulated model: y_hat = func(t,p,c)
p = n-vector of initial guess of parameter values
t = m-vectors or matrix of independent variables (used as arg to func)
y_dat = m-vectors or matrix of data to be fit by func(t,p)
weight = weighting vector for least squares fit ( weight >= 0 ) ...
inverse of the standard measurement errors
Default: sqrt(d.o.f. / ( y_dat' * y_dat ))
dp = fractional increment of 'p' for numerical derivatives
dp(j)>0 central differences calculated
dp(j)<0 one sided 'backwards' differences calculated
dp(j)=0 sets corresponding partials to zero; i.e. holds p(j) fixed
Default: 0.001;
p_min = n-vector of lower bounds for parameter values
p_max = n-vector of upper bounds for parameter values
c = an optional matrix of values passed to func(t,p,c)
opts = vector of algorithmic parameters
parameter defaults meaning
opts(1) = prnt 3 >1 intermediate results; >2 plots
opts(2) = MaxIter 10*Npar maximum number of iterations
opts(3) = epsilon_1 1e-3 convergence tolerance for gradient
opts(4) = epsilon_2 1e-3 convergence tolerance for parameters
opts(5) = epsilon_3 1e-3 convergence tolerance for Chi-square
opts(6) = epsilon_4 1e-2 determines acceptance of a L-M step
opts(7) = lambda_0 1e-2 initial value of L-M paramter
opts(8) = lambda_UP_fac 11 factor for increasing lambda
opts(9) = lambda_DN_fac 9 factor for decreasing lambda
opts(10) = Update_Type 1 1: Levenberg-Marquardt lambda update
2: Quadratic update
3: Nielsen's lambda update equations
##---------- OUTPUT VARIABLES -----------
p = least-squares optimal estimate of the parameter values
X2 = Chi squared criteria
Henri Gavin, Dept. Civil & Environ. Engineering, Duke Univ. 22 Sep 2013
modified from: [http://octave.sourceforge.net/optim/function/leasqr.html]
using references by
Press, et al., Numerical Recipes, Cambridge Univ. Press, 1992, Chapter 15.
Sam Roweis [http://www.cs.toronto.edu/~roweis/notes/lm.pdf]
Manolis Lourakis [http://www.ics.forth.gr/~lourakis/levmar/levmar.pdf]
Hans Nielson [http://www2.imm.dtu.dk/~hbn/publ/TR9905.ps]
Mathworks optimization toolbox reference manual
K. Madsen, H.B., Nielsen, and O. Tingleff
[http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3215/pdf/imm3215.pdf]