Curve fitting with two variables

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Kishor Kumar Johari 2021-12-27

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评论： William Rose 2022-1-7

Hi community!

I have y = f(x), and z = f(x,y), now i want to fit z vs y and need final graph like z vs y. In general, it can be simply done by replecing x by its y equivalent in z but here it is not possible since x-variable is in the integral.

Thank you for your help!

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John D'Errico 2021-12-28

编辑：John D'Errico 2021-12-28

在 MATLAB Online 中打开

Note that at least one of these integrals has a direct special function form. For example:

syms x t

int(t/(1+exp(t-x)),t,0,inf)

ans =

Where polylog is the polylogarithm function.

help polylog
--- help for double/polylog ---

 POLYLOG  polylogarithm function.
    POLYLOG(0,X) = X./(1-X)
    POLYLOG(1,X) = -log(1-X)
    POLYLOG(N+1,X) = int(polylog(N,t)/t,t,0,X) for 0 < X < 1 and any N; or for N >= 1 and any X

    Documentation for double/polylog
       doc double/polylog

    Other functions named polylog

       single/polylog    sym/polylog

I'm reminded of the joke about a famous visiting professor in mathematics and physics, who when asked to give a talk to the students, accepts. He stands at a blackboard (do they even have blackboards anymore?) and covers it with long arcane messes of symbols, filled with various special functions, etc. After a while, the silnece of the students gets to him, so he turns around, and asks the students to raise their hand if they have never seen a Bessel function. One student finally timidly raises their hand, as if to concede they have never seen Bessel functions. The professor nods his head, then turns back to point at the blackboard, with only the statement "There's one now." Then he continues writing on the blackboard.

So, well, there's a polylogarithm.

Kishor Kumar Johari 2022-1-7

@William Rose Thank you so much for the MATLAB code!

I'll try to fit my data using the provided code.

William Rose 2022-1-7

@Kishor Kumar Johari, You're welcome. Good luck with your work!

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Answer 1

William Rose 2021-12-28

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https://ww2.mathworks.cn/matlabcentral/answers/1618240-curve-fitting-with-two-variables#answer_863820

编辑：William Rose 2021-12-28

在 MATLAB Online 中打开

@Kishor Kumar Johari,

Here is a sample solution. It uses calcZ(), which I posted previously, and it uses the simulated data file which I posted previously. The main program is fitKKJ.m. It calls fmincon() and it passes function sseZY() to fmincon(). Function sseZY is attached. There are comments in fitKKJ.m and in sseZY.m which explain how each works.

Here is the console output when I run the script:

fitKKJ

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in

feasible directions, to within the value of the optimality tolerance,

and constraints are satisfied to within the value of the constraint tolerance.

Best fit values: a=0.771, x=0.047.

The simulated data (ten y,z pairs) was generated with a=1, x=0, and Gaussian noise was added.

The script also plots the measured (or simulated, in this case) data and the best-fit approximation. See plot below.

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Kishor Kumar Johari 2022-1-7

@William Rose First of all thank you so much for the efforts you made!

I think there is some confusion about 'x'. It is not a constant of fit, 'a' is only the constant of fit.

'x' is the variable on which 'y' as well as 'z' depends. Here, I would like to explain the problem in detail for more clarity.

There are total four parameters 'x', 'y', 'z', and 'a' are to play. Experimetally, 'y' and 'z' are measured, and 'x' can easily be determined using the equation (2) for each value of 'z' which is corressponding to 'y' . Here, it is to be noted that ultimately we get a dataset for all of three 'x', 'y', 'z'. Now, it is simple when we have to calculate 'a' for only a single set of data, for example we have an experimetally measured z = 0.002 corressponding to y = 3.5, then 'x' can easily be calculated using 'z' like, x = 1.2 (taken arbitrary number for example, not exactly calculated), then using the values of 'z' and 'x', 'a' can simply be calculated.

But the problem arises, when we have to fit a data set with n number of data points, to calculate 'a' as constant of fit.

Now, the problem statement be like we have one input parameter 'x' for generating theoretical data for the fit, and the fit plot should appear 'z' vs 'y'.

Here, it is to be noted that the meaning of the range of 'x' can arbitrarily be given means that an arbitrary data set can be given as input to generate data for 'y' and 'z'.

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Curve fitting with two variables

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Curve fitting with two variables

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