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


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

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

1 Answer 1


For completion's sake (and to help improve the stats of this site, ha) I have to wonder if this paper wouldn't also answer your question?

ABSTRACT: We discuss the choice of polynomial basis for approximation of uncertainty propagation through complex simulation models with capability to output derivative information. Our work is part of a larger research effort in uncertainty quantification using sampling methods augmented with derivative information. The approach has new challenges compared with standard polynomial regression. In particular, we show that a tensor product multivariate orthogonal polynomial basis of an arbitrary degree may no longer be constructed. We provide sufficient conditions for an orthonormal set of this type to exist, a basis for the space it spans. We demonstrate the benefits of the basis in the propagation of material uncertainties through a simplified model of heat transport in a nuclear reactor core. Compared with the tensor product Hermite polynomial basis, the orthogonal basis results in a better numerical conditioning of the regression procedure, a modest improvement in approximation error when basis polynomials are chosen a priori, and a significant improvement when basis polynomials are chosen adaptively, using a stepwise fitting procedure.

Otherwise, the tensor-product basis of one-dimensional polynomials is not only the appropriate technique, but also the only one I can find for this.


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