# Multivariate orthogonal polynomial regression?

As a means of motivating the question, consider a regresison problem where we seek to estimate $Y$ using observed variables $\{ a, b \}$

When doing multivariate polynomial regresison, I try to find the optimal paramitization of the function

$$f(y)=c_{1}a+c_{2}b+c_{3}a^{2}+c_{4}ab+c_{5}b^{2}+\cdots$$

which best fit the data in a least squared sense.

The problem with this, however, is that the parameters $c_i$ are not independent. Is there a way to do the regression on a different set of "basis" vectors which are orthogonal? Doing this has many obvious advantages

1) the coefficients are no longer correlated. 2) the values of the $c_i$'s themselves no longer depend on the degree of the coefficients. 3) This also has the computational advantage of being able to drop the higher order terms for a coarser but still accurate approximation to the data.

This is easily achieved in the single variable case using orthogonal polynomials, using a well studied set such as the Chebyshev Polynomials. It's not obvious however (to me anyway) how to generalize this! It occured to me that I could mutiply chebyshev polynomials pairwise, but I'm not sure if that is the mathematically correct thing to do.

Your help is appreciated

• How about the tensor-product basis of your one-dimensional polynomials? This sounds like what you were alluding to and they will be orthogonal. Jun 21, 2011 at 12:10
• I think that is a satisfactory answer as a quesiton :) Jun 21, 2011 at 23:50
• Did you get anywhere with this? I am also looking for a solution to multivariate regression using orthogonal polynomials. Thank you Nov 1, 2018 at 9:55
• Please also see the referenced paper in the answer to another question which is specific to Chebyshev polynomials: stats.stackexchange.com/a/434939/274906 Oct 11, 2021 at 5:32