Fit Gaussian into symbolic expression

I have a symbolic expression which contains very complex terms. I wish to fit this expression into a Gaussian Distribution.

The fit function takes only arrays as arguments. So i tried to find the data points of my symbolic expression using subs, but is there a better way to fit the expression directly into a gaussian?

EDIT: Not only Gaussian but solution for any other fittype such as 'rat', 'poly', etc, is happily accepted.

[Walter Roberson]

is it a fixed Gaussian that you are trying to fit? Each different Gaussian would be best fit by a different set of parameters .

To confirm: you have a symbolic expression in several variables and a fixed Gaussian in one of the variables and you want to fit the remaining variables in the expression ?

[Dhritishmam Sarmah]

I want my final output to be symbolic, but the fit method doesn't return symbolic. so was wondering if i can get symbolic result directly.

[Walter Roberson]

To confirm: you have a complicated symbolic expression in several variables and a particular model (Gaussian, rat, poly, etc.) with fixed parameters in one of the variables, and you want to fit the remaining variables in the complicated symbolic expression ?

So minimize

sum( (Complicated_Expression(x, expression_parameter1, expression_parameter2, expression_parameter3, etc) - ParticularModelWithNoParameters(x))^2 )

and you are looking for expression_parameter1, expression_parameter2, expression_parameter3, etc that minimizes that residue? And you are hoping for closed form expressions for the expression_parameters, but none-the-less representing numeric values? Or you are just hoping for symbolic results because you want numeric results but in higher precision than double precision?

Or when you say symbolic, do you mean minimize

 sum( (Complicated_Expression(x, expression_parameter1, expression_parameter2, expression_parameter3, etc) - ParticularModelWithParameters(x, model_parameter1, model_parameter2, etc))^2 )

where model_parameter1, model_parameter2, etc are symbolic variables, and you are hoping for symbolic expressions for expression_parameter1, expression_parameter2, expression_parameter3, etc in terms of model_parameter1, model_parameter2, etc ?

If you are hoping for closed form expressions, whether for you are using ParticularModelWithNoParameters (and so looking for closed form representation of numeric values) or for ParticularModelWithParameters (and so looking for closed form solutions involving symbolic variables), then you will not find those for a Gaussian model. You might be able to find such for a polynomial model... but probably not.

Realistically, the best you are likely to be able to achieve is for

 sum( (Complicated_Expression(x, expression_parameter1, expression_parameter2, expression_parameter3, etc) - ParticularModelWithNoParameters(x))^2 )

searching for symbolic results because you want numeric results but in higher precision than double precision. And even then, Gaussians are notoriously difficult to fit properly unless you manage to hit a good starting point.

[Dhritishmam Sarmah]

Let me show, what i have been doing until now;

syms s
H % my symbolic expression. I know the terms in H since i solved it on pen and paper.
% also used fplot to see its plot.

x= -5:1/100:5;
y = double(subs(H, s, x)); %time consuming step;
f1 = fit(x', y', 'rat44');      % example fittype, can also have selfdefinded type
% f1 = 
%      General model Rat44:
%      f1(x) = (p1*x^4 + p2*x^3 + p3*x^2 + p4*x + p5) /
%                (x^4 + q1*x^3 + q2*x^2 + q3*x + q4)
H_fit = (f1.p1*s^4 + f1.p2*s^3 + f1.p3*s^2 + f1.p4*s + f1.p5)/ ...
        (s^4 + f1.q1*s^3 + f1.q2*s^2 + f1.q3*s + f1.q4);

For different fittypes i have to write the last line separately for different expression. Is there a better way?

[Walter Roberson]

extract the model formula from the fit object and str2sym it. extract the parameter names and values and subs()

However in practice you might be able to just feval() the fit object at particular locations or plot the fit object .