Characterization of the Weights of Least Squares Adaptive Polynomials
Robert Kent Goodrich, Ranjit M. Passi · SIAM Journal on Applied Mathematics · 1988
Passi and Morel have provided a concept of adaptive least squares polynomials for time series data using exponential weighting $w_k = a^{ - k} w_0 ,0 < a < 1$. Given the least squares polynomial $p(t)$ obtained from data $Y_k ,k\leqq 0$, they derived a computationally efficient recursion algorithm to update the polynomial coefficients when a new data value is received. They showed that the exponential weighting possesses the “update property,” i.e., the polynomial-update is invariant to new data value $Y_1 $ if $Y_1 = p(1)$. It was this update property that made the recursive algorithm possible. In this paper we explore the weights ${\bf w}$ which possess this update property for a more general class of functions that include the polynomials as a special case. In this general case we give a necessary and sufficient condition for a weight function to satisfy the update property. In the case of polynomials it is shown that the exponential weights are the only weight functions of practical significance which satisfy the update property. An estimate of the rate of convergence of the algorithm of Passi and Morel is provided.