Reveiling Complexity-Related Time-Series Features with the Monotonic Aggregation Transform

Numerous attempts have been made to assess the complexity and predictability of a time series. In AI applications, the latter may be used to determine or classify the origin of an unknown signal. This paper presents the theoretical background to an empirical time series analysis methodology based on the monotonic aggregation transform. For any given time series, its extrema are assumed to contain more information than intermediate data, which is supported empirically for long-memory financial and technological data. Based on this assumption, properties of k-th order minima and maxima are studied as well as their mutual relations. The latter have allowed us to construct a binary decomposition tree and an extremal hull of a time series observation set. It will be proven that the natural characteristic of decomposition trees can be interpreted as an entropy function of the corresponding observation set. Furthermore, the maximum height as well as the sum of all node orders of a decomposition tree is a measure of its information contents. When considered as a function of a sliding time window of constant length in a stationary time series, the above characteristics give us clues as regards the predictability of the original time series, its differences or integrands. We will show the practical implications of the above method in analyzing various kinds of temporal data.