The Fundamental Theorem of Exponential Smoothing

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Abstract
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 Snt(x) = αSn−1t(x) + (1 − α) Snt−1(x), where S0t(x) = xt, 0 < α < 1. If one assumes that the time series of observations {xt} is of the form xt = nt + ∑ı=Nı=0 aıtı where nt 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, …, SN+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.

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