24.3 Exponential smoothing
The recursively defined filters of exponential smoothing provides the possibility of inclusion of additional observations in a simple way. By using the minimum t-norm it is possible to determine the arithmetic mean N* of observations (xt*)t∈T recursively. If the mean value N* is given and an additional observation xN+1* is observed, the mean value N+1* can be calculated as
If is replaced by a general weight β with 0 < β < 1 a filter of simple exponential smoothing is obtained.
In exponential smoothing the values yt*, t = 2 (1) N, of the transformed series (yt*)t∈T of fuzzy data (xt*)t∈T as above are calculated recursively from xt* and the smoothed value yt−1* by
with y1* = x1*. The smoothing parameter β is controlling the smoothing behavior. For small β the last observation xt* has more influence, whereas β near 1 produces more smooth transformed series (yt*