A new symbolic representation of time series, called ABBA, is introduced. It is based on an adaptive polygonal chain approximation of the time series into a sequence of tuples, followed by a mean-based clustering to obtain the symbolic representation. We show that the reconstruction error of this representation can be modelled as a random walk with pinned start and end points, a so-called Brownian bridge. This insight allows us to make ABBA essentially parameter-free, except for the approximation tolerance which must be chosen. Extensive comparisons with the SAX and 1d-SAX representations are included in the form of performance profiles, showing that ABBA is often able to better preserve the essential shape information of time series compared to other approaches, in particular when time warping measures are used. Advantages and applications of ABBA are discussed, including its in-built differencing property and use for anomaly detection, and Python implementations provided.

The block version of the rational Arnoldi method is a widely used procedure for generating an orthonormal basis of a block rational Krylov space. We study block rational Arnoldi decompositions associated with this method and prove an implicit Q theorem. We relate these decompositions to nonlinear eigenvalue problems. We show how to choose parameters to prevent a premature breakdown of the method and improve its numerical stability. We explain how rational matrix-valued functions are encoded in rational Arnoldi decompositions and how they can be evaluated numerically. Two different types of deflation strategies are discussed. Numerical illustrations using the MATLAB Rational Krylov Toolbox are included.

We derive explicit formulas for converting between rational interpolants in barycentric, rational Krylov (RKFUN), and Newton form. We show applications of these conversions when working with rational approximants produced by the AAA algorithm [Y. Nakatsukasa, O. Sète, L. N. Trefethen, arXiv preprint 1612.00337, 2016] within the Rational Krylov Toolbox and for the solution of nonlinear eigenvalue problems.

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A new symbolic representation of time series, called ABBA, is introduced. It is based on an adaptive polygonal chain approximation of the time series into a sequence of tuples, followed by a mean-based clustering to obtain the symbolic representation. We show that the reconstruction error of this representation can be modelled as a random walk with pinned start and end points, a so-called Brownian bridge. This insight allows us to make ABBA essentially parameter-free, except for the approximation tolerance which must be chosen. Extensive comparisons with the SAX and 1d-SAX representations are included in the form of performance profiles, showing that ABBA is able to better preserve the essential shape information of time series compared to other approaches. Advantages and applications of ABBA are discussed, including its in-built differencing property and use for anomaly detection, and Python implementations provided. Keywords time series • symbolic aggregation • dimension reduction • Brownian bridge 1 IntroductionSymbolic representations of time series are an active area of research, being useful for many data mining tasks including dimension reduction, motif and rule discovery, prediction, and clustering 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 . (b) It starts at a value of about −3, then climbs up to a value of about 0 within 25 time steps, then it stays at about 0 for 100 time steps, after which it goes up to a value of about 3 within 25 time steps, and so on. (c) It starts at a value of about −3, then goes up rapidly by about 3 units, followed by a longer period with almost no change in value, after which it again goes up rapidly by about ...