How to use multistart with 'fit' function

Hi There,

i want to use 'Multistart' to find the best solution for a bi-exponential decay function that i am fitting to my data with bounds.

Below is the code i am using.

% bi-exponential decay model   


fparam = fittype(@(a,b,c,d,x)(a)*exp(-(1/b)*x)+abs(1-a)*(exp(-(1/c)*x))+d);   

 lb1 = [0,0,0,0];
 ub1 = [1,30,200,80];
 b0 = [0.3,0.2,20,10];
 x = [3.39,8.59,13.8,19,24.2,29.4,34.6,39.81,45.0,50.21];
 y = [1,0.2905,0.0894,0.0838,0.1173,0.1006,0.0782,0.0894,0.1061,0.0726]
 
  opts = fitoptions('Display','Off','Method','NonlinearLeastSquares','Normalize','Off',...
                           'Startpoint',b0, 'Robust','On','Lower',lb1,'Upper',ub1,...
                           'TolFun',1e-3); 
                       
[estTmp,Goft,Out] = fit(x,y,fparam,opts);

Since i need to start somewhere i have randomly chosen 'b0' values. I want to avoid local minima, so i want to use 'Multistart' function for optimization and find the best solution within the 'bounds' (here given by 'lb1', 'ub1').

Can someone please help me bridge 'Multistart' with 'fit' ?

P.S. I know someone might suggest to use 'lsqnonlin' instead of 'fit'. The reason i am not using it is becasue i need the output 'stats' (e.g. goodness of fit and R^2 values) which i might have to compute manually in 'lsqnonlin'. So please suggest me only the ways to combine 'Multistart' and 'fit'

Answer 1

[Alan Weiss]

As documented, the only local solvers available for MultiStart are fmincon, fminunc, lsqcurvefit, and lsqnonlin. Sorry, no other local solvers are supported.

Alan Weiss

MATLAB mathematical toolbox documentation

[roni zidane]

Hi Alan,

That's unfortunate. In that case, Would you have any solution about how to calculate Goodness of Fit, and R^2 from either 'lsqnolin' or 'lsqcurvefit' ?

The reason i avoid using either 'lsqnonlin' or 'lsqcurvefit' are the following:

1) i want to use 'Robust' option as mentioned in my 'opts' variable in my original post.

Also when i try to find best fit solution with 'lsqnonlin', I get fig.1 result, where the instead of starting to fit from datapoint 1 , The fitting start from 1.4 or so (Though i have provided 'lb' and 'ub'), and it is far from the ideal solution.

So ,

2) my question will be, how do i make sure that the fitting starts at datapoint 1 with 'lsqnonlin' ? 3) How do i calculate goodness of Fit and R^2 from 'lsqnonlin' result?

[Alan Weiss]

As far as getting the curve to start from level 1 at x = 3.39, I suggest that you shift all x values by 3.39 (so x = 0 corresponds to your first data point) and then set d = 0 in your formula (fit parameters a, b, and c, but do not allow a nonzero value of d). Then your curve will start at the height 1 at x = 0.

To calculate the goodness of fit, see this old documentation example.

I am not sure that calculating R^2 makes much sense in nonlinear fitting.

Alan Weiss

MATLAB mathematical toolbox documentation

Answer 2

[Alex Sha]

Multi-solutions (Parameter a is different):

1:

Root of Mean Square Error (RMSE): 0.109662308225183

Sum of Squared Residual: 0.12025821845275

Correlation Coef. (R): 0.948796774493673

R-Square: 0.900215319289597

Parameter Best Estimate

---------- -------------

a 0.057712785437929

b 7.20470183282193

c 7.20470183925339

d 0.0814111332963631

2:

Root of Mean Square Error (RMSE): 0.109662308225183

Sum of Squared Residual: 0.12025821845275

Correlation Coef. (R): 0.948796774685829

R-Square: 0.900215319654232

Parameter Best Estimate

---------- -------------

a 0.0035886762500919

b 7.20470208037725

c 7.20470182707246

d 0.0814111343856455

3:

Sum of Squared Residual: 0.12025821845275

Correlation Coef. (R): 0.948796775443418

R-Square: 0.900215321091827

Parameter Best Estimate

---------- -------------

a 0.176245451532707

b 7.20470187824529

c 7.20470176505555

d 0.0814111355698764