Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 56

x centred moving average , a modification of a simple moving average in
which the average is placed in the middle of an interval of n periods, i.e. at
the n /2 point, which holds for odd numbers n
x weighted moving average , an averaging algorithm that discriminates the
participation of individual observations according to their “age”, as shown
in the following equation:

x ( ) t wxt () wxt ( )... wxt n ( . )

w 1 2 n

From the equation it is evident that more recent observations could be given higher
weights by greater values of weights w than the older ones. However, the sum of
all weights used should be equal to one.
The moving average is easy to understand and simple to use, but it gives equal
weight to all past data, of which a large number have to be stored and used for
forecasts. This also holds for the weighted moving average, for which it is difficult
to select the optimal values for individual weights.
Therefore, a more advanced version of the weighted moving average is an
alternative like exponential smoothing , a version with exponentially decreasing
weights as the observation data become older.

2.9.5.2 Forecasting Using Exponential Smoothing
The exponential smoothing approach is particularly convenient for short-time
forecasting. Although it also employs weighting factors for past values, the
weighting factors here decay exponentially with distance of the past values of the
time series from the present time. This enables a compact formulation of the
forecasting algorithm in which only a few most recent data are required and less
calculations are needed, which is highly relevant to on-line applications in
industrial automation, where programmable controllers and signal processors are
used.
Smoothing of observation data is basically required when the data are to a
certain degree erroneous due to the superposition of some error component H( t ) and
the exact value x ( t ), i.e . when the measured signal is expressed as

x

() t

x

() t . xt () H () t

In exponential smoothing, the concept of a weighted moving average is used. In
using exponentially decaying coefficients not all past values are used for
prediction; rather, a reduced number of measured and calculated data are used,
represented by the iterative exponential smoothing algorithm

xt () D xt () (1 D ) ( xt 1),

e e

with the forecast

ĭ ( t + k ) =

x

( tk . )