Fitting 4 parameters with constraints

I am trying to fit a function to a data set in order to fit 4 parameters. I have been using lsqcurvefit because it is the only way that I know how to do a good easy fitting. But the problem is that I need to add constraints to my parameters. This is not possible using lsqcurvefit. I wonder if there is a function or a manual way of doing this.

where A1 D1 A2 D2 are unknown cosntants.

The constraints I need to have is that D1 and D2 are less than 0.001 (D1 and D2 < 0.0008). Any solution that includes either D1 or D2 greater than that is not of interest.

A solution that lsqcurvefit is finding is

A1 = 0.364

D1 = 0.001338

A2 = 0.610

D2 = 0.000110

*****Plot as a semilogY ****

D1 in this case is greater than my desired constraint. I would like for it to skip it and keep iterating.

I will include my data below if anybody could help me I appreciate it!

clear; clc; clf; close all;

D_0 = [0.5 0.00001 0.5 0.00001]; %initial guess if necessary for A1 D1 A2 D2 respectively

xdata = [9.85 32.05 66.83 114.44 174.55 247.57 333.00 431.44 542.19 666.04 802.12 951.39 1113.38 1287.47 1474.87 1674.29 1887.11 2600.14 2863.78 3139.17 3428.23 4043.41 4369.45 4709.33 5425.99 5802.67 6193.38 6595.39];

ydata = [1 0.9569 0.9528 0.8894 0.8387 0.8995 0.7911 0.773 0.7523 0.7155 0.7086 0.6478 0.6269 0.6175 0.574 0.551 0.4991 0.4559 0.4449 0.4314 0.4212 0.407 0.3856 0.3511 0.3526 0.303 0.3148 0.2912];

Answer 1

[Star Strider]

‘But the problem is that I need to add constraints to my parameters. This is not possible using lsqcurvefit.’

Yes, it is, however only constraint bounds.

objfcn = @(b,x) b(1).*exp(-b(2).*x) + b(3).*exp(-b(4).*x);
D_0 = [0.5 0.00001 0.5 0.00001];
xdata = [9.85	32.05	66.83	114.44	174.55	247.57	333.00	431.44	542.19	666.04	802.12	951.39	1113.38	1287.47	1474.87	1674.29	1887.11	2600.14	2863.78	3139.17	3428.23	4043.41	4369.45	4709.33	5425.99	5802.67	6193.38	6595.39];
ydata = [1	0.9569	0.9528	0.8894	0.8387	0.8995	0.7911	0.773	0.7523	0.7155	0.7086	0.6478	0.6269	0.6175	0.574	0.551	0.4991	0.4559	0.4449	0.4314	0.4212	0.407	0.3856	0.3511	0.3526	0.303	0.3148	0.2912];
[B,rsdnrm] = lsqcurvefit(objfcn, D_0, xdata, ydata, -Inf(1,4), [Inf 0.001 Inf 0.001])

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

B = 1×4

-18.9893 0.0001 19.7668 0.0001

rsdnrm = 0.1898

xv = linspace(min(xdata), max(xdata));

figure
plot(xdata, ydata, 'pg')
hold on
plot(xv, objfcn(B,xv), '-r')
hold off
grid
xlabel('x')
ylabel('y')

.

[Alfredo Scigliani]

Thank you for your input! Nonetheless, I am trying to fit a multiexponential fucntion, what the fitting is doing is finding a solution as a monoexponential function. If both of the D parameters are the same, then you can add the two terms together and it will just be 1 exponential. D1 and D2 must be different from eachother as well.

Try plotting is as semilogy, if multiple exponentials are in play, the line will be curved.

[Alex Sha]

if want D1 and D2 are all less than 0.001:

Sum Squared Error (SSE): 0.0116703699647364
Root of Mean Square Error (RMSE): 0.0204156539770838
Correlation Coef. (R): 0.995666057914124
R-Square: 0.991350898882251

Parameter	Best Estimate
----------	-------------
a1	0.428678645876139
a2	0.534815808010459
d1	0.001
d2	8.58505177096869E-5

while, if want D1 and D2 are all less than 0.0008:

Sum Squared Error (SSE): 0.0140886399832964
Root of Mean Square Error (RMSE): 0.0224313555918754
Correlation Coef. (R): 0.994765584731217
R-Square: 0.989558568565641

Parameter	Best Estimate
----------	-------------
a1	0.495993548397224
a2	0.458843635970715
d1	0.0008
d2	5.99343414652501E-5