On Discrete Least Squares Polynomial Fit, Linear Spaces and Data Classification

where n P is the set of polynomials of degree at most n . The motivation for this short note comes from a mistake in the proof of Theorem 1 in [5] and explained in the Remark 2 below. The goal of this paper is to clarify the dimension of some vector spaces encountered in solving this problem, establish a property useful for proving the existence of a WLSE for exponential models , and suggest a way to classify data using the best polynomial fits. For a standard presentation of the theory related to best (polynomial) least squares fit see [1, 3, 7, 8, . The best polynomial fit problem can be solved by considering an orthogonal projection onto n P or, equivalently, by considering an orthogonal projection onto a subspace of m IR . In Section 2 we briefly review the solution of the problem in n P and specify the dimension of subspaces of polynomials. In the first part of the Section 3 we consider the subspaces of m IR that play a role in solving the problem in m IR . In the second part of this Section 3 we solve the problem using a projection onto a subspace of m IR . Finally in Section 4 we suggest a way to classify data which will be useful in the problem of finding existence results for weighted least squares estimator .

(2) where n P is the set of polynomials of degree at most n . The motivation for this short note comes from a mistake in the proof of Theorem 1 in [5] and explained in the Remark 2 below. The goal of this paper is to clarify the dimension of some vector spaces encountered in solving this problem, establish a property useful for proving the existence of a WLSE for exponential models [2] , and suggest a way to classify data using the best polynomial fits. For a standard presentation of the theory related to best (polynomial) least squares fit see [1,3,7,8,9] .
The best polynomial fit problem can be solved by considering an orthogonal projection onto n P or, equivalently, by considering an orthogonal projection onto a subspace of m IR . In Section 2 we briefly review the solution of the problem in n P and specify the dimension of subspaces of polynomials. In the first part of the Section 3 we consider the subspaces of m IR that play a role in solving the problem in m IR . In the second part of this Section 3 we solve the problem using a projection onto a subspace of m IR . Finally in Section 4 we suggest a way to classify data which will be useful in the problem of finding existence results for weighted least squares estimator [2] .

POLYNOMIAL WEIGHTED LEAST SQUARES FITTING IN n P
In the first part of this section we present the underlying subspaces of Lin P j related to the polynomial weighted least squares problem. In the second part we solve the problem using a projection onto a subspace of P .
We consider also the following two other polynomial subspaces for any nonnegative integer , 2 , The next two results specify the dimension of these subspaces.
Theorem 2: Let k be any nonnegative integer and let Polynomial weighted least squares fitting: Under the condition that m n < , we introduce the scalar product on n P defined by for any pair of polynomials p and q in n P . In this case (1) becomes where . is the norm on n P induced by the scalar product. For the i f 's we use the notation It is well known that * In this setting, to simplify the computation of * n p , we can find a sequence of orthogonal polynomials by applying the Gram-Schmidt orthogonalization process to the standard basis { } n t t t , , , , 1 2 of n P . These orthogonal polynomials are given by , Hence the best n -degree least squares polynomial * n p can be written as The next two results will be useful for finding sufficient conditions for the existence of the WLSE for a 3-parametric exponential model [2] .

Proof.
For 0 = n it is obvious because 0   t .
Remark 2: In [5] it is asserted that − 2 V is of dimension m which is clearly false except for . As a consequence the proof given in [5] for the existence of a WLSE for a 3-parametric exponential function is not correct. There are also errors in the proof of the existence of a WLSE in [6] . increasing, and essentially concave, resp. convex. The stationarity and linearity properties are not modified by this transform.

CONCLUSION
We have revisited the polynomial weighted least squares analysis. Doing so we have specified the dimension of three vector subspaces of P (Theorem 1 and Theorem 2) and of m IR (Theorem 5 and Theorem 6) used for solving this problem. We also have established a property (Theorem 3 and Theorem 7) and suggested a classification of data (Definition 1) which will play a role in finding sufficient conditions for the existence of a WLSE for a 3-parametric exponential model [2] .

ACKNOWLEDGMENTS
This work has been supported by a NSERC (Natural Sciences and Engineering Research Council of Canada) individual discovery grant for the first author