Least square curve fit

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anala reddy
anala reddy on 25 Aug 2017
Answered: Alex Sha on 9 Nov 2019
For function like y = a*(x-b)^c, how can I use the least square curve fit feature to find out the coefficients a, b and c? But If i use the custom equation in cftool it reports " Complex value computed by model function, fitting cannot continue. Try using or tightening upper and lower bounds on coefficients".

Answers (2)

Star Strider
Star Strider on 25 Aug 2017
I do not have the Curve Fitting Toolbox, so I cannot provide an exact example. However the problem is obvious — the complex values result from (x-b)<0, and -Inf at x=b. You need to constrain ‘b’ so that (x-b)>0. This requires b<x, so constrain ‘b’ to be less than ‘min(x)-1E-8’ (with the ‘1E-8’ preventing (x-b)=0). Set that as the upper bound of ‘b’.
Star Strider
Star Strider on 27 Aug 2017
This works. It does not (in my opinion) produce a good fit, and only ‘a(1)’ is significantly different from zero. (Parameter confidence intervals that include zero are not needed in the model.) A logistic model might be a better fit, if it describes the system that produced your data.
data = load('nmoschar.txt');
x = data(:,1);
y = data(:,2);
predicted = @(a,x) a(1).*((x-a(2)).^a(3));
ub_a2 = min(x)-1E-8;
a0 = [sqrt(eps), max(x), 1.4];
[ahat,resnorm,residual,exitflag,output,lambda,jacobian] = lsqcurvefit(predicted,a0,x,y, [0,0,0]-1E-8, [Inf,ub_a2,Inf]);
ci = nlparci(ahat,residual,'jacobian',real(jacobian));
plot(x, y, 'pg')
hold on
plot(x, predicted(ahat,x), '-r')
hold off
It is necessary to constrain the parameters to avoid complex coefficients.
The nlparci function requires the Statistics and Machine Learning Toolbox.

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Alex Sha
Alex Sha on 9 Nov 2019
The function of lsqcurvefit use local optimization algorithm, so it is too weak on fault tolerance, try to use global optimization algorithms, easy to get proper result:
Root of Mean Square Error (RMSE): 1.89485474476935E-5
Sum of Squared Residual: 3.94952195415243E-9
Correlation Coef. (R): 0.99509458980833
R-Square: 0.990213242665809
Adjusted R-Square: 0.987766553332261
Determination Coef. (DC): 0.98908887287217
Chi-Square: 4.92331654510374E-5
F-Statistic: 348.803598712296
Parameter Best Estimate
---------- -------------
a 0.000558941792124118
b -9.85792530954279E-16
c 3.84941267720528

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