Cepstral identification of autoregressive systems

Abstract

Recent research has provided a better understanding of the power cepstrum, which has led to several applications in time series clustering, classification, and anomaly detection. It has also provided a deeper understanding of the theoretical framework that relates the power cepstrum with some system theoretic properties of the underlying dynamics. In this paper, we pursue the intricate connections between the power cepstrum of a signal and the pole polynomial of the underlying generative model. In this way, we develop a simple and extremely efficient method to identify an autoregressive (AR) system, starting from the power cepstrum of its output signal. This general framework uses Newton’s identities to set up a system of elementary symmetric polynomials over the cepstral coefficients and results in an identification algorithm that is independent of the length of the power cepstrum, with computational complexity only linearly dependent on the order of the model. We provide several numerical examples, first on synthetic time series, then on the classical Yule sunspot numbers modeling problem, and finally on a contemporary application involving structural health monitoring. Subsequently, the novel system identification algorithm is employed to provide insight in the results of weighted cepstral clustering, showing that the model estimated from the center of a cluster provides a good estimator for the dynamics in that cluster.