Data Modeling With Polynomial Representations and Autoregressive Time-Series Representations, and Their Connections

Two of the data modelling techniques - polynomial representation and time-series representation - are explored in this paper to establish their connections and differences. All theoretical studies are based on uniformly sampled data in the absence of noise. This paper proves that all data from an underlying polynomial model of finite degree q can be represented perfectly by an autoregressive time-series model of order q and a constant term μ as in equation (2). Furthermore, all polynomials of degree q are shown to give rise to the same set of time-series coefficients of specific forms with the only possible difference being in the constant term μ. It is also demonstrated that time-series with either non-integer coefficients or integer coefficients not of the aforementioned specific forms represent polynomials of infinite degree. Six numerical explorations, with both generated data and real data, including the UK data and US data on the current Covid-19 incidence, are presented to support the theoretical findings. It is shown that all polynomials of degree q can be represented by an all-pole filter with q repeated roots (or poles) at z = +1. Theoretically, all noise-free data representable by a finite order all-pole filter, whether they come from finite degree or infinite degree polynomials, can be described exactly by a finite order AR time-series; if the values of polynomial coefficients are not of special interest in any data modelling, one may use time-series representations for data modelling.