## Advanced Regression - Deriving New Equations

## Advanced Regression - Deriving New Equations

(OP)

Hello,

During my tasks in different disciplines, I am reaching some new performance measure as according to the topic studied. I already used multi-regression in some and, if that was not enough to fit my data, I used interpolation method 'The Vandermonde Method' and lately discovered the powerful griddata function by Matlab.

I need to derive new expression/equations for the data I reach each time (like regression coefficients so that I can use them in my model). My question arises since I have no idea how the old scientists could find the advanced expressions for their data (Please check the formulations below). For example, how one could put the relation for the correction factor as a function of R and P as given below. As you can see, the relations of R and P with correction factor is not straightforward to be understood. Normally the curve of correction factor seems to be power-like regression for each P with changing R values. But the formulation below is totally out-of-world.

I believe that there is a topic in mathematics like regression, interpolation etc. for such expression deriving. Checked google but couldn't find anything. Any help, guidance can be rewarding.

In Example:

Nusselt correlations

or

Correction Factor - LMTD Theory(most inspiring one) :)

During my tasks in different disciplines, I am reaching some new performance measure as according to the topic studied. I already used multi-regression in some and, if that was not enough to fit my data, I used interpolation method 'The Vandermonde Method' and lately discovered the powerful griddata function by Matlab.

I need to derive new expression/equations for the data I reach each time (like regression coefficients so that I can use them in my model). My question arises since I have no idea how the old scientists could find the advanced expressions for their data (Please check the formulations below). For example, how one could put the relation for the correction factor as a function of R and P as given below. As you can see, the relations of R and P with correction factor is not straightforward to be understood. Normally the curve of correction factor seems to be power-like regression for each P with changing R values. But the formulation below is totally out-of-world.

I believe that there is a topic in mathematics like regression, interpolation etc. for such expression deriving. Checked google but couldn't find anything. Any help, guidance can be rewarding.

In Example:

Nusselt correlations

or

Correction Factor - LMTD Theory(most inspiring one) :)

“In life, the truest guide is science” – Mustafa Kemal Atatürk

## RE: Advanced Regression - Deriving New Equations

Knowing what the relationship should be from theory, then tweeking it.

another day in paradise, or is paradise one day closer ?

## RE: Advanced Regression - Deriving New Equations

## RE: Advanced Regression - Deriving New Equations

Regards.

“In life, the truest guide is science” – Mustafa Kemal Atatürk

## RE: Advanced Regression - Deriving New Equations

## RE: Advanced Regression - Deriving New Equations

If you start off with nothing but a bunch of data points and derive an expression to represent them, you'll generally wind up with a fairly simple expression= and the more scattered the points are, the simpler the expression.

If the problem at hand has an approximate solution, the expression may come from that.

For example, the "exact" solution to the idealized problem may be a power series or double power series, and taking the first one or two terms of that may be used as the basis of the approximation.

If the problem is non-dimensionalized, that affects the form of the solution.

If a curve is graphed on log paper, those exponents are just multipliers on the graph.

## RE: Advanced Regression - Deriving New Equations

Dik

## RE: Advanced Regression - Deriving New Equations

https://www.mathworks.com/help/curvefit/csaps.html

## RE: Advanced Regression - Deriving New Equations

It will be rewarding to give the correction factor curve, its expression being the formulation that I did submit in my first post (but giving here as well).

They seem to have a characteristic of an exponential function for each curve (considering some coefficients being negative). So I tried to fit the data for my custom function a exp (b x + c) + d for each line in the graph (by use of Matlab optimization tool) and obtained (for two lines) that only the coefficients b and c change with changing lines. It results something but I didn't get satisfied with that. The custom function was generated by use of info given at Nancy Marcus - Graphs of Exponential Functions.

“In life, the truest guide is science” – Mustafa Kemal Atatürk

## RE: Advanced Regression - Deriving New Equations

Then you need to have a cost function that grows bigger as a function of error. Usually the cost function is a sum of errors between the actual and estimated values for each parameter/coefficient you want to identify. A good algorithm to start with is BFGS. Matlab should have a routine that uses the BFGS algorithm. A simpler algorithm is Levenberg-Marquardt. Don't try to understand the math behind these algorithms. Just look in Matlab do see how Matlab implements these algorithms.

If you want to get a gut feel for what is going on then look at this first

https://www.youtube.com/edit?o=U&video_id=qzr6...

You will see that I too lay it out like a grid or map to show the basics.

However, when identifying many coefficients the grid isn't practical and the more advanced algorithms must be use.

https://www.youtube.com/edit?o=U&video_id=lerm...

These algorithms are the same ones control guys use to find a model for the system they are trying to control which is basically what you are trying to do.

Peter Nachtwey

Delta Computer Systems

http://www.deltamotion.com

http://forum.deltamotion.com/