Theoretical and experimental results on time-series representations of polynomials

Abstract

This study considers time-series representations of polynomials. Often in data modelling and many other applications, accurate estimations of the degree of a polynomial, of the noise standard deviation, and of the coefficient of the highest degree of a polynomial are useful in detection, estimation, and prediction. The major contributions of this paper can be found in the original research offering novel theoretical and experimental results. The theoretical results include an alternative proof of the qth order AR time-series representation, with a constant, of a polynomial of degree q, an alternative proof of the (q + 1)-th order AR time-series representation, without a constant, of a polynomial of degree q, as well as generalized equations (valid for a polynomial of an arbitrary degree) for reduced variance estimation of the polynomial coefficient corresponding to the highest degree. The experimental investigations are the most comprehensive so far, in that they use well over 35 times more realisations than before, use a greater variety of noisy data (Gaussian, Uniform, and Exponential noise), and use a larger range of polynomial degrees as well as of noise standard deviations than before. Experimental results on estimations of the degree of a polynomial, of the noise standard deviation, and of the polynomial coefficient corresponding to the highest degree using seven methods (AIC, AICc, GIC, BIC, Chi-square, F-distribution, and PTS2) are presented. Results indicate clearly that PTS2 performs the best.