## Exponential Smoothing (ES)

It is a forecast method used in technical analisys of price time series (without Stagionality and Trend) and derived from Exponential Moving Average. The forecast is obtained as a weighted average of all available observations:

The weights of the linear combination are chosen in order to give greater weight to more recent observations and less weight to the observations in the distant past. The weights are defined by the exponential sequence:

and therefore:

The value of the expected price in T+1 (given the information in T) is obtained from the expected value in T (given the information in T-1), correct the error committed in the prediction of T (also called surprise ) weighted by a factor α (called smoothing parameter).

The exponential smoothing is, therefore, a rule for updating the prediction based on the expected value at the previous time, adjusted by a term proportional to the error of prediction fulfilled. The smoothing parameter give maximum weight to surprise with value equal to one and no weight with a value equal to zero. The new forecast can be thought as the weighted average between the last available observation and the old forecast. The α value determines how much of the current observation influence the future value. As near to zero is α less the current value affects the prediction (so the new forecast will be very similar to the old) and vice versa (for α tending to one the new forecast will be very close to the last value of the series ).

Considering backwards the entire time axis, we can generalize the above formula as:

Note that, for a given α, any prediction with a horizon k greater than one is equal to the value of the prediction in T +1. In fact:

and substituting:

and so on for a generic k horizon. The expression, as a function of k, is a function of prediction constant from T.

Editor: Giuliano DI TOMMASO