ABBA: Adaptive Brownian bridge-based symbolic aggregation of time series

Symbolic time series representations allow for the use of algorithms from text processing and bioinformatics, which often take advantage of the discrete nature of the data. Our focus in this work is to develop a symbolic representation which is dimension reducing whilst preserving the essential shape of the time series. Our definition of shape is different from the one commonly implied in the context of time series: we focus on representing the peaks and troughs of the time series in their correct order of appearance, but we are happy to slightly stretch the time series in both the time and value directions. In other words, our focus is not necessarily on approximating the time series values at the correct time points, but on representing the local up-and-down behavior of the time series and identifying repeated motifs. This is obviously not appropriate in all applications, but we believe it is close to how humans summarize the overall behavior of a time series, and in that our representation might be useful for trend prediction, anomaly detection, and motif discovery. To illustrate, let us consider the time series shown in Figure 1. This series is sampled at equidistant time points with values t 0,t 1,...,t N R, where N 230. There are various ways of describing this time series, for example: (a) It is exactly representable as a high-dimensional vector T [t 0,t 1,...,t N ] R N 1 .