The Fundamental Theorem of Exponential Smoothing

Exponential smoothing is a formalization of the familiar learning process, which is a practical basis for statistical forecasting. Higher orders of smoothing are defined by the operator S n t (x) = αS n−1 t (x) + (1 − α) S n t−1 (x), where S 0 t (x) = x t , 0 < α < 1. If one assumes that the time series of observations {x t } is of the form x t = n t + ∑ ı=N ı=0 a ı t ı where n t is a sample from some error population, then least squares estimates of the coefficients a, can be obtained from linear combinations of the operators S, S 2 , …, S N+1 . Explicit forms of the forecasting equations are given for N = 0, 1, and 2. This result makes it practical to use higher order polynomials as forecasting models, since the smoothing computations are very simple, and only a minimum of historical statistics need be retained in the file from one forecast to the next.