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

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Samuele Bolotta on 25 Mar 2021
Commented: Alan Weiss on 30 Mar 2021
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 (https://it.mathworks.com/help/optim/ug/nonlinear-data-fitting-problem-based-example.html#NonlinearDataFittingProblemBasedExample-4) 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));
CPSC = IPSC + EPSC;
end
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
plot(t,resp)
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.

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
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Alan Weiss on 30 Mar 2021
I don't see the code for SplittFit. So I cannot see where there is a problem. I can assure you that the "MathWorks example" runs without error. Is it possible that you are running an old version of MATLAB, one that does not support least-squares fitting in the problem-based approach?
Alan Weiss
MATLAB mathematical toolbox documentation