Time-series decomposition and forecasting

Abstract

Any stationary time-series can be decomposed by means of an optimization operator, called the ζ-optimator, into several components (the time-series){Y _t ⁱ}, i =1,2,…, p, such that the first component {V _t ⁱ} t = 1,2,…,v is a smooth process having a larger autocorrelation in comparison with the original process {Y _t}, i.e. ρ_vi > ρ_y. Usually only a few such components are sufficient for approximating the time-series with good accuracy. The ζ-optimator involves a shape parameter a, so the decomposition is unique provided that a. is fixed. Since the component {V _t ¹} involves much of the useful information it can be used for computing predictors for control purposes. Thus, given the observations Y_v, Y_v-1, Y_v-2,…, a predictor of Y_v+1 is ρ_vi V _v ¹ (q) where, V_v ¹(q) = qY^v + q(1-q)² Y_v-2, …, the weights q(1-q)^r, r=0,1,2,…, decreasing rapidly as q = q(α) ε (0,1) Further, one may choose q rather than choosing α, since q(α) is a one-one mapping. Once q is fixed, the predictor ρ_v1 V _v ¹(q) is obtained in a straightforward way by using the formula above. It is shown that ρ_v1 V _v ¹(q) converges to the best predictor as α → 0. Some examples are worked out, illustrating both the decomposition and the forecasting procedures.