Part VII

FUZZY TIME SERIES

Many time series are the results of measurement procedures and therefore the obtained values are more or less fuzzy. Also, many economic time series contain values with remarkable uncertainty. In particular, development data and environmental time series are examples of time series whose values are not precise numbers. If such data are modeled by fuzzy numbers the resulting time series are called fuzzy time series.

Based on the results from Part I of this book it is necessary to consider time series with fuzzy data. In standard time series analysis the values xt of a time series (xt)t∈T, where T is the series of time points, i.e. T = {1, 2, …, N}, are asumed to be real numbers. By the fuzziness of many data, especially all measurement data from continuous quantities, it is necessary to consider time series whose values are fuzzy numbers xt*. Such time series (xt*)t∈T are called fuzy time series.

Mathematically a fuzzy time series is a mapping from the index set T to the set () of fuzzy numbers. The generalization of time series analysis techniques to the situation of fuzzy values should fulfill the condition that it specializes to the classical techniques in the case of real valued time series.

In this part, first the necessary mathematical techniques are ...