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Nonlinear fitting: how do I split the linear and the nonlinear problems?

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I am fitting a function to some data I simulated. I managed to get intelligent constraints that help the fit quite a bit, even with a lot of noise.
This is the function and as you can see, c(1) and c(2) are linear, while lam(1), lam(2), lam(3) and lam(4) are nonlinear. I am following the procedure explained here ( to split linear and nonlinear parameters.
% Create a function that computes the value of the response at times t when the parameters are c and lam
diffun = ((c(1)) .* ((1 - exp(-t / lam(1))) .* exp(-t / lam(2))) * (Vm - (-70)) + ...
((c(2)) .* ((1 - exp(-t / lam(3))) .* exp(-t / lam(4))) * -30));
This is the code that I came up with, but for some reason it's not working. To generate the data:
function [EPSC, IPSC, CPSC, t] = generate_current(G_max_chl, G_max_glu, EGlu, EChl, Vm, tau_rise_In, tau_decay_In, tau_rise_Ex, tau_decay_Ex,tmax)
dt = 0.1; % time step duration (ms)
t = 0:dt:tmax-dt;
% Compute compound current
IPSC = ((G_max_chl) .* ((1 - exp(-t / tau_rise_In)) .* exp(-t / tau_decay_In)) * (Vm - EChl));
EPSC = ((G_max_glu) .* ((1 - exp(-t / tau_rise_Ex)) .* exp(-t / tau_decay_Ex)) * (Vm - EGlu));
To fit the function:
% Simulated data
[EPSC,IPSC,CPSC,t] = generate_current(80,15,0,-70,-30,0.44,15,0.73,3,120);
ydata = awgn(CPSC,25,'measured'); % Add white noise
% Values
Vm = -30;
% Initial values for fitting
gmc = 40; gmg = 20; tde = 0.2; tdi = 8; tre = 1.56; tri = 3;
% Objective function
c = optimvar('c',2); % Linear parameters
lam = optimvar('lam',4); % Nonlinear parameters
% Bounds
c.LowerBound = [0, 0];
c.UpperBound = [200, 200];
lam.LowerBound = [0.16,7.4,1.1,2.6];
lam.UpperBound = [0.29,8.4,2.3,3.3];
x0.c = [gmc,gmg]; % Starting values
x0.lam = [tri,tdi,tre,tde]; % Starting values
% Create a function that computes the value of the response at times t when the parameters are c and lam
diffun = ((c(1)) .* ((1 - exp(-t / lam(1))) .* exp(-t / lam(2))) * (Vm - (-70)) + ...
((c(2)) .* ((1 - exp(-t / lam(3))) .* exp(-t / lam(4))) * -30));
%Solve the problem using solve starting from initial point x02
x02.lam = x0.lam;
%To do so, first convert the fitvector function to an optimization expression using fcn2optimexpr.
F2 = fcn2optimexpr(@(x) fitvector(x,t,ydata),lam,'OutputSize',[length(t),1]);
% Create a new optimization problem with objective as the sum of squared differences between the converted fitvector function and the data y
ssqprob2 = optimproblem('Objective',sum((F2' - ydata).^2));
[sol2,fval2,exitflag2,output2] = solve(ssqprob2,x02)
% Plot
resp = evaluate(diffun,sol2);
hold on
hold off
The error is:
Solving problem using lsqnonlin.
Error using optim.problemdef.OptimizationProblem/solve
Matrix dimensions must agree.
Error in SplittFit (line 37)
[sol2,fval2,exitflag2,output2] = solve(ssqprob2,x02)
Caused by:
Failure in initial objective function evaluation. LSQNONLIN cannot continue.
Not sure why.

Accepted Answer

Alan Weiss
Alan Weiss on 25 Mar 2021
You should follow the example more closely. In the example the lambda variables only are declared to be optimization variables; the c variables are not, and are computed by backslash for given values of the lambda variables in the fitvector function.
For your case you will need to update the fitvector function from the example to handle a 4-D lambda vector that differs from your 4-D function because you have some (1-exp(-t/lambda)) terms, not just exp(-t/lambda). You need to write out the linear equations in c and solve those equations in the fitvector function.
Good luck,
Alan Weiss
MATLAB mathematical toolbox documentation
Yanxin Liu
Yanxin Liu on 25 Aug 2021
I was also studying the Matlab example you pasted link. From my understanding, the equation you want to fit to must have the form y = c(1)*exp(-lam(1)*t) + ... + c(n)*exp(-lam(n)*t). That's say, each columne of matrix A can only represent one nonlinear parameter (lambda). Then the linear parameters c can be derived from backslash by trying different pairs of lambdas. In your problem, the lam1, lam2, also lam3, lam4 are entangled together.
Alan Weiss
Alan Weiss on 25 Aug 2021
The response function can have any form. For complicated examples, see Fit ODE, Problem-Based and Fit an Ordinary Differential Equation (ODE), which have objective functions given by the solution of ODEs.
By no means are you required to split the response into a linear and a nonlinear part; when you can, then you can use the trick of fitting the linear part after fitting the nonlinear part.
Alan Weiss
MATLAB mathematical toolbox documentation

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