Non linear fit extremely bad : what am I doing wrong ?

Hello everybody,

I've spent my evening trying to understand why the nonlinear fit was so bad with my data and I couldn't find why.

I tried many different options, like the 'fit' function, 'fminsearsh', the fitting curve tool, etc...

For some reasons, the fit is OK only if my starting points are close from the real coeff at 0.1% ...

What am I doing wrong ? 1.1 isn't THAT far from 1. (same problem without the constant 'a')

Thank you for your help,

close all
clear all

rng('default')
t = 0:0.02:10; x = t.*sin(2*pi*1*t) + 0.1*randn(1, length(t)); % x*sin(x) with noise
x = x'; t = t';

% fitting part
x0 = [0 1.1];
fitfun = fittype( @(a,b,x) a+x.*sin(2*pi*b*x) );
[fitted_curve,gof] = fit(t,x,fitfun,'StartPoint',x0)

fitted_curve = 
     General model:
     fitted_curve(x) = a+x.*sin(2*pi*b*x)
     Coefficients (with 95% confidence bounds):
       a =     -0.1064  (-0.5048, 0.292)
       b =       1.117  (1.115, 1.119)

gof = struct with fields:
           sse: 1.0263e+04
       rsquare: -0.2337
           dfe: 499
    adjrsquare: -0.2362
          rmse: 4.5352

Answer 1

[Matt J]

There are ways to derive an accurate x0 more systematically, e.g.,

rng('default')
t = 0:0.02:10; x = t.*sin(2*pi*1*t) + 0.1*randn(1, length(t)); % x*sin(x) with noise
x = x'; t = t';


%Coarse fit
a0=mean(x);
I=t>=1;
z=(x(I)-a0)./t(I); %Ideally, z would be purely sinusoidal
cfit=fit(t(I),z,'fourier1');

% Fine fit
x0 = [a0 cfit.w/2/pi];
fitfun = fittype( @(a,b,x) a+x.*sin(2*pi*b*x) );
[cfit,gof] = fit(t,x,fitfun,'StartPoint',x0),

cfit = 
     General model:
     cfit(x) = a+x.*sin(2*pi*b*x)
     Coefficients (with 95% confidence bounds):
       a =   -0.002197  (-0.01101, 0.006614)
       b =           1  (0.9999, 1)

gof = struct with fields:
           sse: 5.0280
       rsquare: 0.9994
           dfe: 499
    adjrsquare: 0.9994
          rmse: 0.1004

plot(cfit,t,x)

[Nassim Mhammedi]

Indeed I should've known that the search for starting points is part of the work.

Thank you for your answer and the elegant code!