Fitting ad hoc function to data using MatLab

2012 7 15
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Dear all,
I have the following data:
x=[0 0 0 11 14 14 19 19 26 37 47 47 53 53 63 74 81 81 86 86 89 100 100 100];
y=[33 33 27 32 32 26 43 38 42 42 45 40 51 43 73 61 52 67 69 79 89 60 86 75];
I have plotted these data using
plot (x, y,'o')
Now I would like to fit the function of the form:
f=(p^(gamma))/(p^gamma+(1-p))^(1/gamma)
to the data. p(i) is equal to x(i). gamma is a variable.
Basically,
(1) I need to find gamma which produces the best fit of the function f (specified above) to the data;
(2) I need to plot f against the data (on the same graph).
I am sure this can be done but for some reason cannot come up with a good algorithm. Could you please suggest some solutions?
Thank you,
Ganna

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the cyclist
the cyclist 2012-7-15
编辑：the cyclist 2012-7-15

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If you have the Statistics Toolbox, you can use the function nlinfit() to do this.
Here is a simple example of its use:
function [] = nlinfitExample()
% Here is an example of using nlinfit(). For simplicity, none of
% of the fitted parameters are actually nonlinear!
% Define the data to be fit
x=(0:1:10)'; % Explanatory variable
y = 5 + 3*x + 7*x.^2; % Response variable (if response were perfect)
y = y + 2*randn((size(x)));% Add some noise to response variable
% Define function that will be used to fit data
% (F is a vector of fitting parameters)
f = @(F,x) F(1) + F(2).*x + F(3).*x.^2;
F_fitted = nlinfit(x,y,f,[1 1 1]);
% Display fitted coefficients
disp(['F = ',num2str(F_fitted)])
% Plot the data and fit
figure(1)
plot(x,y,'*',x,f(F_fitted,x),'g');
legend('data','fit')
end

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Ganna
Ganna 2012-7-15
Thank you so much for the quick reply. The problem is that my fitting parameter (gamma) is a scalar and what i need to find out is the y=f(x gamma). In your example, it is y = 5 + 3*x + 7*x.^2. My problem is that I know the function I am fitting, but need to estimate gamma first. I have tried to use a vector of values
y=[33 33 27 32 32 26 43 38 42 42 45 40 51 43 73 61 52 67 69 79 89 60 86 75];
instead of y=f(x) as in your example
but I can't make it work.
Basically, I have two vectors: one is x another is y. My y=(x^(gamma))/(x^gamma+(1-x))^(1/gamma) where gamma is a scalar.
I need to find gamma.
Sorry, this is probably completely stupid but for some reason I can't make it work.
Thank again,
Ganna
the cyclist
the cyclist 2012-7-16
编辑：the cyclist 2012-7-16
You have not fully understood my example. Specifically, you have confused the few lines of code that I used to generate some random data (e.g. y = 5 + ...), with the part of the code that actually does the fitting.
So, in the section where I comment "% Define the data to be fit", you already have that. Those are the (x,y) data you already have.
Where I have written
f = @(F,x) F(1) + F(2).*x + F(3).*x.^2;
that is where you will define your own functional form. Your gamma will take the place of my F (and you will not need three parameters like I have in my example). So, I think it will be
f = @(Gamma,x) x.^Gamma % etc ...
Then where I have
F_fitted = nlinfit(x,y,f,[1 1 1]);
you will have the same thing, except you will substitute just one starting guess for gamma (instead of three starting guesses for my three-element vector F.)
I hope that helps.

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