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By Silviu Guiasu, M. Malitza

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A simple expression for this computation is given in the next section. 2 Innovations and One-Step-Ahead Forecasts If the value for x0 is known, the innovation ε t is a one-step-ahead prediction error. 1b) to obtain E ( y t | y t − 1 , . . , y 1 , x0 ) = E ( y t | x t − 1 ) = w x t − 1 . Then the prediction of y t , given the initial value x0 and observations y1 , . . , y t−1, is w xt−1 . 3b) xt = F xt−1 + gε t . 3c) This transformation will be called general exponential smoothing. It was first outlined by Box and Jenkins (Box et al.

0 100 100 000 010 001 .. .. . 000 ⎤ ··· 0 0 · · · 0 0⎥ ⎥ · · · 0 1⎥ ⎥ · · · 0 0⎥ ⎥ · · · 0 0⎥ . .. ⎥ . ⎦ ··· 1 0 ⎡ ⎤ α ⎢ β⎥ ⎢ ⎥ ⎢γ ⎥ ⎥ and g = ⎢ ⎢0⎥ . ⎥ ⎣ .. 15) shows that the level and seasonal terms are confounded. 15). To avoid this problem, it is desirable to constrain the seasonal component so that any sequence {st , st+1 , . . , st+m−1 } sums to zero (or at least has mean zero). The seasonal components are said to be normalized when this condition is true. Normalization of seasonal factors involves a subtle modification of the model and will be addressed in Chap.

We may wish to compare the accuracy of methods between many series at a single forecast horizon. That is, we compute a single yˆ n+h|n based on data from times t = 1, . . , n for each of m different series. While these are very different situations, measuring forecast accuracy is the same in each case. The measures defined below are described for one-step-ahead forecasts; the extension to h-steps-ahead is immediate in each case and raises no new questions of principle. 1 Scale-Dependent Errors The one-step-ahead forecast error is simply et = yt − yˆt , regardless of how the forecast was produced.

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