Using LINEST to derive a multiple polynomial equation

**Richatom** · 12-07-2014, 04:16 AM

Hello,

I am using LINEST to derive a formula for y from an array of known y values and two input parameters, x1 and x2.

Since y varies non-linearly with both x1 and x2, I have added two more input parameters x1^2 and x2^2.

I am getting fairly good correlation, but I am wondering if I can further improve the correlation by adding more input parameters such as x1^3, x2^3, or even x1*x2?

**MrShorty** · 12-07-2014, 11:14 AM

The short answer to your question is, yes. Adding another parameter to your regression will almost universally "improve" the fit. And you can continue adding parameters until you reach the point where Excel's (or whatever program you are using for the regression) algorithm rejects additional parameters or becomes numerically unstable, or until you fit the data exactly.

The longer and harder part of your question is how to truly justify these additional parameters. I am not familiar with them, but I understand that there are statistical tests out there that can be used when deciding if an additional parameter truly improves the fit or not -- is truly "significant" in the regression. For a longer, harder answer to your question, you might search out these tests and study what is meant when a parameter is truly significant and when it is not.

Another thing I find when I start to get tempted to add cubic and quartic terms to a regression is to wonder if a generic polynomial is the best equation to use for the regression. This further broadens the question beyond "Excel" and into the expected mathematical models that would be expected to represent your data. In the end, your question is not a trivial question, and there is much for you to consider.

**Richatom** · 12-08-2014, 09:09 AM

Thank you for your help. I experimented with adding a few variables and as you mentioned it does not make much difference. What was particularly surprising is that when I tried using powers of the variables that one would anticipate as being close approximation to the physical model (in this case x1^2, x2^0.5), it made little or no difference compared to using simply x1^2, x1^3, x2^2,x2^3 etc.