Estimating the size of temporal memory for symbolic analysis of time-series data

This paper presents an approach for symbolic analysis of time-series data from a dynamical system perspective that uses Markov modeling of symbol sequences. A key aspect of this approach is the selection of relevant histories or depth of the symbol sequence to capture the evolution of the observed dynamical system in its phase-space. An approach based on the spectral properties of the one-step transition matrix is proposed. A key advantage of this approach compared to the state-of-the-art is that it does not require several passes through the time-series data to search for the optimal model given some metric. The proposed approach makes use of the decay-rate of conditional influence of the current symbol to the n-step future of the dynamical system. Using a bound on this decay rate, the optimal depth can be computed in exactly one pass though the time-series data. The effectiveness of the proposed methodology is demonstrated on a known low-dimensional chaotic system and the efficacy of the approach for anomaly detection is demonstrated.