datafit

Parameter identification based on measured data

Calling Sequence

[p,err]=datafit([imp,] G [,DG],Z [,W],[contr],p0,[algo],[df0,[mem]],
[work],[stop],['in'])

Arguments

:imp scalar argument used to set the trace mode. imp=0 nothing

(execpt errors) is reported, imp=1 initial and final reports, imp=2 adds a report per iteration, imp>2 add reports on linear search. Warning, most of these reports are written on the Scilab standard output.

: :G function descriptor (e=G(p,z), e: ne x 1, p: np x 1, z: nz x 1) : :DG partial of G wrt p function descriptor (optional; S=DG(p,z), S:

ne x np)

: :Z matrix [z_1,z_2,...z_n] where z_i (nz x 1) is the ith measurement : :W weighting matrix of size ne x ne (optional; defaut no

ponderation)

: :contr ‘b’,binf,bsup with binf and bsup real vectors with same

dimension as p0. binf and bsup are lower and upper bounds on p.

: :p0 initial guess (size np x 1) : :algo ‘qn’ or ‘gc’ or ‘nd’ . This string stands for quasi-

Newton (default), conjugate gradient or non-differentiable respectively. Note that ‘nd’ does not accept bounds on x ).

: :df0 real scalar. Guessed decreasing of f at first iteration. (

df0=1 is the default value).

: :mem : integer, number of variables used to approximate the Hessian,

( algo=’gc’ or ‘nd’). Default value is around 6.

: :stop sequence of optional parameters controlling the convergence of the algorithm. stop= ‘ar’,nap, [iter [,epsg [,epsf [,epsx]]]]

“ar” reserved keyword for stopping rule selection defined as follows:  

: :nap maximum number of calls to fun allowed. : :iter maximum number of iterations allowed. : :epsg threshold on gradient norm. : :epsf threshold controlling decreasing of f : :epsx threshold controlling variation of x. This vector (possibly

matrix) of same size as x0 can be used to scale x.

:

: :”in” reserved keyword for initialization of parameters used when

fun in given as a Fortran routine (see below).

: :p Column vector, optimal solution found : :err scalar, least square error. :

Description

datafit is used for fitting data to a model. For a given function G(p,z), this function finds the best vector of parameters p for approximating G(p,z_i)=0 for a set of measurement vectors z_i. Vector p is found by minimizing G(p,z_1)’WG(p,z_1)+G(p,z_2)’WG(p,z_2)+...+G(p,z_n)’WG(p,z_n)

datafit is an improved version of fit_dat.

Examples

//generate the data
function y=FF(x, p)
  y=p(1)*(x-p(2))+p(3)*x.*x
endfunction
X=[];Y=[];
pg=[34;12;14] //parameter used to generate data
for x=0:.1:3
  Y=[Y,FF(x,pg)+100*(`rand`_()-.5)];
  X=[X,x];
end
Z=[Y;X];

//The criterion function
function e=G(p, z),
  y=z(1),x=z(2);
  e=y-FF(x,p),
endfunction

//Solve the problem
p0=[3;5;10]
[p,err]=datafit(G,Z,p0);

`scf`_(0);`clf`_()
`plot2d`_(X,FF(X,pg),5) //the curve without noise
`plot2d`_(X,Y,-1)  // the noisy data
`plot2d`_(X,FF(X,p),12) //the solution

//the gradient of the criterion function
function s=DG(p, z),
  a=p(1),b=p(2),c=p(3),y=z(1),x=z(2),
  s=-[x-b,-a,x*x]
endfunction

[p,err]=datafit(G,DG,Z,p0);
`scf`_(1);
`clf`_()
`plot2d`_(X,FF(X,pg),5) //the curve without noise
`plot2d`_(X,Y,-1)  // the noisy data
`plot2d`_(X,FF(X,p),12) //the solution

// Add some bounds on the estimate of the parameters
// We want positive estimation (the result will not change)
[p,err]=datafit(G,DG,Z,'b',[0;0;0],[%inf;%inf;%inf],p0,algo='gc');
`scf`_(1);
`clf`_()
`plot2d`_(X,FF(X,pg),5) //the curve without noise
`plot2d`_(X,Y,-1)  // the noisy data
`plot2d`_(X,FF(X,p),12) //the solution

See Also

• lsqrsolve minimize the sum of the squares of nonlinear functions, levenberg-marquardt algorithm
• optim non-linear optimization routine
• leastsq Solves non-linear least squares problems