LSQcurvefit does not yield same result for comparable data sets

I have used the following code to fit my data. Key is that the fit captures the peak at the start of the curve.

For a lot of data sets, the method works (example are x_good,y_good).

For two sets this method does not work adequate enough (x5, y5 and x10, y10)

Anyone who can see if I am doing something wrong? Or give a different method which works better?

clear all
clc
close all

% Working data set
x_good = [0 0.0025	0.020666667	0.0915	0.129	0.169916667	0.204	0.2415	0.299833333	0.359833333	0.409833333	0.459833333	0.589833333	0.674833333];
y_good = [0.000001 -1.791689626	-2.085814283 -1.254192484	-1.130978553	-0.852663995	-0.650156083	-0.605732792	-0.552092535	-0.466707459	-0.381595899	-0.341840176	-0.112360934	-0.107089368];

%Not working data set
x5 = [0 0.0025	0.020666667	0.0915	0.129	0.169916667	0.204	0.299833333	0.359833333	0.509833333	0.589833333	0.674833333];
y5 = [0.000001	-1.512917459	-1.397221246	-0.861543826	-0.678138048	-0.538930943	-0.440640054	-0.253865251	-0.192352332	0.001791084	0.078138742	0.156925367];

%Not working data set
x10 = [0 0.0025	0.020666667	0.0915	0.129	0.169916667	0.204	0.2415	0.299833333	0.359833333	0.409833333	0.459833333	0.589833333	0.674833333];
y10 = [0.000001 -0.909146404	-1.389416499	-0.736431181	-0.767464076	-0.430784784	-0.298350016	-0.477736703	-0.174485909	-0.10975744	-0.038531763	0.009471926	0.133414928	0.178482903];

% LSQ
t = x5;   
y =y10;
xspace = linspace(t(1), t(end), 1000);


options = optimoptions('lsqcurvefit', 'MaxFunctionEvaluations', 100e3)

% fit function y = c(1)*exp(-lam(1)*t) + c(2)*exp(-lam(2)*t)
F = @(x,t)(x(1)*exp(-x(2)*t) + x(3)*exp(-x(4)*t));
x4 = [1 1 1 0];
[x,resnorm,~,~,output] = lsqcurvefit(F,x4,t,y, [], [], options)

figure
plot(t,y,'ro')
title("Least squared method ")
hold on
plot(t,F(x,t), 'r')
hold on 
plot(xspace, F(x, xspace), '--r')
set(gca, 'YDir','reverse')
legend("data points", "LSQ, resnorm: " + resnorm, "LSQ continous")

Answer 1

[Torsten]

Use

x4 = [-2.1782 5.0283 2.1782 720.8491];

instead of

x4 = [1 1 1 0];

as initial guess for the parameters.

[Rebecca Belmer]

Thank you! It works!

Can you tell me how I can best guess x4? Or how you found these values?

[Torsten]

From the fit results of y_good against x_good :-)