Find the parameters of integration

Hello everyone, I have a question that hasn't been able to find it on my own though, so any help is greatly appreciated.

I try to find the parameters a, b, c, d and e which satisfy below equations.

Where T_1~T_5 and q_1~q_5 are input parameters which already know.

Are there any functions that can solve these kinds of problem in MATLAB?

Thanks for any helpful ideas.

[Walter Roberson]

Is f known?

Do you need closed form solutions, or are numeric solutions okay?

[Kim Jeong Min]

Hello, Mr. Roberson

The specific form of function f is:

The original purpose was to obtain an analytic function of the integral, but since the analytic integration is impossible in the case of the function, I want to obtain each parameter through numerical integration.

[Walter Roberson]

do you have the curve fitting toolbox?

[Kim Jeong Min]

I do not have "Curve Fitting Toolbox" right now, but I am willing to purchase if it is necessary to solve the problem.

[Walter Roberson]

It is not necessary -- it just makes it easier, probably.

[Kim Jeong Min]

If the problem is very easy to solve using "Curve Fitting Toolbox" (or the problem is too complicated without "Curve Fitting Toolbox"). Please tell me the solution with "Curve Fitting Toolbox".

Answer 1

[Ameer Hamza]

Try this. This requires optimization toolbox. This uses the sum of squared error from your equations as an objective function.

T0 = 1;
Tx = [2; 3; 4; 5; 6]; % from T1 to T5
qx = [2; 3; 4; 5; 6];

f = @(a,b,c,d,e, T) (a+b*T+c*T.^2).^(1./(d+e*T));
f_int = @(a,b,c,d,e, T0, T1) integral(@(T) f(a,b,c,d,e,T), T0, T1);
obj_fun = @(a,b,c,d,e) sum((arrayfun(@(T0, T1) f_int(a,b,c,d,e, T0, T1), T0*ones(size(Tx)), Tx)-qx).^2);
sol = fmincon(@(x) obj_fun(x(1),x(2),x(3),x(4),x(5)), [1;1;1;1;1], [], []);

The sol contain value of a, b, c, d and e in order.

[Walter Roberson]

Unfortunately my tests indicate that fmincon does not do a good job on this; it gets caught in local minima. The curve fitting toolbox does a lot better, at least for a first pass value.

[Ameer Hamza]

Maybe some other definition of objective function will produce better estimation.

[Kim Jeong Min]

What is the physical meaning of objective function? Is it the sum of squares? Is "fmincon" try to find the parameters which have a minimum sum of squares?

[Walter Roberson]

The problem is finding good initial conditions for fmincon as it is strictly a local minimizer.

Yes, that objective function is the sum of squares of differences between the modeled outcome and the known outcome; if you had a perfect fit then the objective function would be 0. This is a standard technique.

[Walter Roberson]

Could you post some sample T and q values for us to test with?

[Kim Jeong Min]

Thank you for using so much personal time to solve this problem.

The conditions are:

Sorry for the mistake I made (missing exponential) for the specific form of function.

[Ameer Hamza]

As Walter pointed out, the difficulty is finding a suitable starting point for optimization. After the introduction of exp() term, it is even more difficult. The new integrand is very sensitive to the value of b and c, for the given value of integral limits. I don't have curve fitting toolbox but maybe it is able to estimate parameters for this new integrand.