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-   -   Polynomial regression (http://book.caltech.edu/bookforum/showthread.php?t=1628)

Daniel 09-18-2012 11:59 PM

Polynomial regression
 
I'm curious about using polynomial regression with multiple features. I understand how to use polynomial hypotheses with univariate regression, but I'm unsure how to extend this to multiple features. Let's say I want to use a cubic polynomial. Do I introduce three terms in the hypothesis for each feature x^1 and x^2 and x^3 ?? So in effect, I'm tripling the number of features (or terms in the hypothesis equation)?

Thank you.

Daniel

magdon 09-19-2012 05:49 PM

Re: Polynomial regression
 
You can certainly implement your suggestion, but it is technically not the 3rd order polynomial transform, although it is a form of polynomial transform. You would introduce every term that is at most 3rd order to get the full 3rd order polynomial transform. Here are those terms:

1

x_1,\ x_2,\ x_3

x_1^2,\ x_1x_2,\ x_1x_3,\ x_2^2,\ x_2x_3,\ x_2^3

x_1^3,\ x_1^2x_2,\ x_1^2x_3,\ x_1x_2^2,\ x_1x_3^2,\ x_1x_2x_3,\ x_2^3,\ x_2^2x_3,\ x_2x_3^2,\ x_3^3

If you have d variables and you want the Q-order polynomial transform, there are quick algorithms to generate these terms. As you can see, the number of terms rises very quickly. In this example you have done much worse than tripling. In general, for the Q-order polynomial transform with d variables, the number of terms is

\left(\begin{array}{c}d+Q\\Q\end{array}\right)=\frac{(d+Q)!}{d!Q!}.

Quote:

Originally Posted by Daniel (Post 5567)
I'm curious about using polynomial regression with multiple features. I understand how to use polynomial hypotheses with univariate regression, but I'm unsure how to extend this to multiple features. Let's say I want to use a cubic polynomial. Do I introduce three terms in the hypothesis for each feature x^1 and x^2 and x^3 ?? So in effect, I'm tripling the number of features (or terms in the hypothesis equation)?

Thank you.

Daniel


Daniel 09-27-2012 10:00 PM

Re: Polynomial regression
 
Dear Dr. Magdon,

Thank you so much for the informative reply. Of course, your solution makes perfect sense. It just wasn't obvious to me. I shall experiment with this.

Cheers,

Daniel


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