Least squares adaptive polynomials

Abstract

A concept of adaptive least squares polynomials is introduced for modelling time series data. A recursion algorithm for updating coefficients of the adaptive polynomial (of a fixed degree) is derived. This concept assumes that the weights w are such that i) the importance of the data values, in terms of their weights, relative to each other stays fixed, and that ii) they satisfy the update property, i.e., the polynomial does not change if a new data value is a polynomial extrapolate. Closed form results are provided for exponential weights as a special case as they are shown to possess the update property when used with polynomials. The concept of adaptive polynomials is similar to the linear adaptive prediction provided by the Kalman filter or the Least Mean Square algorithm of Widrow and Hoff. They can be useful in interpolating, tracking and analyzing nonstationary data.