Hi there, I am trying to find an expression for the parameter 'alpha' as a function of x (the length): alpha = f(x). Means that alpha must be estimated as a profile, rather than a data point. My alpha is dependent on 4 experimental and independent variables (Temperature T, Pressure P, mass flow rate m and ratio R) which each of them individually affect this profile.
I tried to use a 3rd order polynom for its description. Alpha = A x^3 + B x ^2+ C x + D.
In order to take the influence of these 4 variables on alpha values, I have written the 4 factors of A, B, C and D as:
A = a(1) * T ^ a(2) + a(3) * P ^ a(4) + a(5) * m ^ a(6) + a(7) * ratio ^ a(8) + a(9)
B = b(1) * T ^ b(2) + b(3) * P ^ b(4) + b(5) * m ^ b(6) + b(7) * ratio ^ b(8) + b(9)
C = c(1) * T ^ c(2) + c(3) * P ^ c(4) + c(5) * m ^ c(6) + c(7) * ratio ^ c(8) + c(9)
D = d(1) * T ^ d(2) + d(3) * P ^ d(4) + d(5) * m ^ d(6) + d(7) * ratio ^ d(8) + d(9)
I try to fit my experimental data with lsqnonlin, levenberg-marquardt algorithm and find the above mentioned 36 variables, so that I can have an expression for A, B, C and D, to finally predict alpha. But my problem is that: 1st: I think it’s very inefficient to use 36 parameters for a function, but I just have no better idea!! 2nd: the solver does converge and I find my 36 values (local minimum possible). But when I calculate alpha the value it gives me is going below zero (which is wrong).. and also is not really correct..
Instead I would like to use a Gaussian function to do the fit, since it describes the decaying trend of alpha better (in the attachment) and doesnt go to below zero either, but I cannot find what is the corresponding function for 'polyfit' for a Gaussian fit, and if it helps to reduce the number of the parameters and improve the accuracy..
I would greatly appreciate any comments or hints from both mathematical and coding point of view