How to fit a custom equation?

My equation is y=a(1-exp(-b(c+x)) x=[0,80,100,120,150] y=[2195,4265,4824,5143.5,5329] When I am solving it in matlab, I am not getting a proper fit in addition, sse=6.5196e+05 and r square=0.899. Although the r square value is acceptable, the sse is too high. Therefore kindly help to get minimum sse. Further I have tried in curve fitting tool but I got same thing.

Answer 1

[Star Strider]

I get good results with this:

yf = @(b,x) b(1).*(1-exp(-b(2)*(b(3)+x)));

B0 = [5000; 0.01; 10];
[Bm,normresm] = fminsearch(@(b) norm(y - yf(b,x)), B0);
SSE = sum((y - yf(Bm,x)).^2)

Bm =
          6677.76372320411
        0.0084077646869843
          47.1622210493944

normresm =
          195.173589996072

SSE =
          38092.7302319547

[madhuri dubey]

The value of sse is too high. Is it possible to optimize it?

[madhuri dubey]

Further, I would like to know how did you fix the initial values of B0?

[Star Strider]

The SSE value is about as good as it can be with my code. It is about 6% of the value you initially reported.

I experimented with the initial parameter estimates. It is clear that ‘b(1)’ is the maximum value of ‘y’. The truly important parameter in your model is ‘b(2)’. An appropriate initial estimate for it here is ‘x(end)/y(end)’, reflecting my initial estimate for it as 0.01. If that is close to ideal, the other parameters will be easily estimated.

I would have used the Global Optimization Toolbox ga (genetic algorithm) to find the best parameter set if my observational guesses for it had not worked.

[madhuri dubey]

Thank You so much. If I want to fit quadratic form of equation y=a(x+b)^2+c(x+b)+d, where a,b,c & d are constant and I want to know the starting value of these constant to initiate curve fitting or to get the best result of the curve fit. Can you elaborate on how to use Global Optimization Toolbox ga (genetic algorithm) for this purpose? I have tried but I am unable to find a solution.

[Star Strider]

As always, my pleasure.

I would be tempted to use polyfit to get initial estimates of ‘a’, ‘c’, and ‘d’ (estimated as [-0.09, 34, 2200] when I did it) , then let your nonlinear parameter estimation routine (similar to my code) estimate them and ‘b’ (that I would initially estimate as 10).

I usually create my own initial population for ga. I would be tempted here to use a matrix of 500 individuals, defined as:

init_pop = randi(5000, 500, 4);

using the appropriate options function (linked to in the See Also section in the ga documentation) to define it as such. The ga function is efficient, however since it has to search the entire parameter space, it will take time for it to converge.

Note that you have 5 data pairs and you are now estimating 4 parameters.

[madhuri dubey]

When I use polyfit to get initial estimates of ‘a’, ‘c’, and ‘d’ , I got [-0.0001, 0.0346, 2.1807]. Why there is difference in constant values for the same data.

[Star Strider]

There isn’t. You’re ignoring the constant multiplication factor 1.0E+03. The full result:

p =

     1.0e+03 *

    -0.000088159928538   0.034559355475118   2.180742845451099

Answer 2

[Image Analyst]

For what it's worth, I used fitnlm() (Fit a non-linear model) because that's the function I'm more familiar with. You can see it gives the same results as Star's method in the image below. I'm attaching the full demo to determine the coefficients and plot the figure.

Answer 3

[Alex Sha]

If don't care the type of fitting function, try the below function, much simple but with much better result:

y = b1+b2*x^1.5+b3*x^3;

Root of Mean Square Error (RMSE): 18.0563068929128
Sum of Squared Residual: 1630.15109305524
Correlation Coef. (R): 0.999873897456616
R-Square: 0.999747810815083
Adjusted R-Square: 0.999495621630167
Determination Coef. (DC): 0.999747810815083
Chi-Square: 0.165495427247465
F-Statistic: 3964.27710070773

Parameter	Best Estimate
----------	-------------
b1	2195.84396843687
b2	3.66989234203779
b3	-0.00107101963512847