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 Sⁿ_t(x) = αSⁿ⁻¹_t(x) + (1 − α) Sⁿ_t−1(x), where S⁰_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=0a_ıt^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, S², …, 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.