Advanced Data Fitting - More than Simple Regressions

[HumanistEngineer]

Hello All,
I want to improve myself in data fitting in order to derive new equations for the data from experimental results and/or performance analysis. I am an engineering researcher and since I found some out-of-world formulations derived from performance data, I need to learn this advanced data fitting (or whatever its name is) discipline.
Please guide me to the exact field name and/or books to achieve this skill. I guess that this is not just Math but also Physics so that one could put the relationship between the input data and/or dimensionless quantities.
For example: How come one could derive this formulation for the correction factor data (LMTD heat exchanger) as shown below:
Here is the formulation/expression:

For the graph:

Thank you in advance.

 

[FactChecker]

I think these equations must come from expertise in the physics of the subject matter rather than from a curve-fitting approach.

 

[Stephen Tashi]

I want to improve myself in data fitting in order to derive new equations for the data from experimental results and/or performance analysis.

Are you trying to derive new equations? Or are you trying to fit a family of equations that is defined by some constant parameters to data by estimating the values of the constant parameters?

 

[HumanistEngineer]

Are you trying to derive new equations? Or are you trying to fit a family of equations that is defined by some constant parameters to data by estimating the values of the constant parameters?

For now, I want to fit a family of equations (at least I can find the most close function form so that I can run an optimization to find the coefficients of the function form). I have been reading a lot in most of the web forums, if I understand correctly, there is a way to find equation patterns of which some suggests working with slopes and finding the coefficients by use of that (while keeping the other input parameters constant) or something similar as this approach. I am totally new so can't interpret what is told in most.
Why I need this is to integrate the heat exchanger model in my simulations but not limited to this.
Long way but the best is to spend time again what was teached in Thermodynamics, Heat Transfer, Fluid Mechanics,... and this time paying attention about the theoretical basis of the formulations derived. But still a detailed reference book is my need to understand how such relation between input and/or dimensionless parameters and the results (math part) are found by some (as the LMTD correction factor example could be made). Also, I need a function for e(ffectiveness)-NTU method.
Thank you.

 

[FactChecker]

For now, I want to fit a family of equations (at least I can find the most close function form so that I can run an optimization to find the coefficients of the function form).

This is still confusing to me. I think of a "family of equations" as functions that only differ by some parameters. So "fitting" the family of equations would mean running an optimization to find the coefficients (parameters) of the function form. But you seem to have something different in mind since you talk about doing the parameter optimization after fitting the family. Could you explain a little more what you mean by "fitting a family of equations"?

 

[HumanistEngineer]

This is still confusing to me. I think of a "family of equations" as functions that only differ by some parameters. So "fitting" the family of equations would mean running an optimization to find the coefficients (parameters) of the function form. But you seem to have something different in mind since you talk about doing the parameter optimization after fitting the family. Could you explain a little more what you mean by "fitting a family of equations"?

Let me explain by my words. I want to derive/define a sole function form of an exponential equation i.e. a exp (b x + c) + d and run an optimization that minimizes the error between calculated and the real data by finding the coefficients i.e. a, b, c, d (at last the optimization can result for some of these coefficients in zero so I will remove that part considering multiplication i.e. b). Function form is not straightforward to find since two original input parameters P and R affect the correction factor (each line in the above graph is different values of R while all lines change with changing P value). This is my quick solution.

My curiosity is about the equation formation as in the first question comment of mine. That is out-of-world for me that someone could define the expressions (mid parameter X ) and, by use of X, R, and P, another complex equation formation that finds the correction factor (Y-Axis). Such complex relationing, I want to learn that but don't know which discipline involves such high relationing/regression.

Thank you.

 

[FactChecker]

Are you saying that you already know exp (b x + c) + d or that the first step is to find it among many other alternatives? If the later is the case, then I think that you are trying to use statistics to replace subject matter expertise. I don't recommend that. The equations in your original post are very complicated. They were not found with statistics. There is some theoretical basis for them.

 

[HumanistEngineer]

Are you saying that you already know exp (b x + c) + d or that the first step is to find it among many other alternatives? If the later is the case, then I think that you are trying to use statistics to replace subject matter expertise. I don't recommend that. The equations in your original post are very complicated. They were not found with statistics. There is some theoretical basis for them.

Quick solution to integrate this correction factor in my large simulation can be ok with the most close function form i.e. a exp (b x + c) + d. But since I need to publish after some time I need to learn to drive the real expression or something close to that. I am sure that there must be a field in Math to find such relations between the input, mid-calculation datas and the result. If someone will guide me, I will buy the books for this field.