Representing a given curve as the sum of an exponential + gaussian curve

Hi,

I have a given curve y(x) which behaves like an exponential curve until a point say x=.25 and then it follows a Gaussian distribution until the end (say x=8.25).

How do I use cfit(or any other method) to fit data into my curve?

My course of action was that split the range of x into 2 parts - part 1- x upto which there is exponential behavior and part 2 - later part of x for which there is a Gaussian nature.

So i found the coefficients for each case respectively(part 1 and part 2 using standard model fits like 'exp1' and 'Gauss1') i.e ( a and b for a*exp(b*x) and a1,b1,c1 for a1*exp(-((x-b1)/c1)^2) ) for different ranges of x (according to the behavior) .

And then used these coefficients in fitting in the cfit tool for my general model a*exp(b*x)+ a1*exp(-((x-b1)/c1)^2) for all range of x . But this does not give me the correct answer(which is obvious).

Basically i need to have a curve which can represent both exponential behavior as well as Gaussian nature using a single equation .How do i do this?

Thank you.

Answer 1

[Matt J]

Your model would really be something like this

 a*exp(b*x).*(x<=K)+ a1*exp(-((x-b1)/c1)^2).*(x>K)

where K is the transition point separating the two ranges. I think you would have to curve fit each piece separately as you've been doing, but to evaluate the curve using a single expression, you would use the above.

[Thejas]

I don't understand a particular feature about MATLAB .... It tries to find coefficients for the curve fitting of standard models but fails to do so when we have our custom model even when the model is not complicated ?

Is there a way in this case to find the coefficients somehow? i.e. a,a1,b,b1,c1

[Matt J]

Well, you have the pieces to build what you want. Suppose you write functions

 [error,a,b] = Fit1(x,y)

 [error, a1,b1,c1] = Fit2(x,y)

that return parameter estimates and errors using the model from each piece. I'm pretty sure cfit lets you build such functions. Then you could write a 3rd function

    function [error,a,a1,b,b1,c1]=Fit3(K,x,y)

       I=(x<=K);
       [error1,a,b] =Fit1(x(I),y(I))
       [error2, a1,b1,c1] = Fit2(x(~I),y(~I));

       error=error1+error2;

and finally

 Kfit=fminsearch(@(K)Fit3(K,x,y), Kguess);

 [~,a,a1,b,b1,c1]=Fit3(Kfit,x,y);

[Thejas]

Thank you Matt :) That was good