Markov Modeling of Time Series via Spectral Analysis for Detection of Combustion Instabilities

Abstract

Modeling of temporal patterns to infer generative models from measurement data is critical for dynamic data-driven application systems (DDDAS). Markov models are often used to capture temporal patterns in sequential data for statistical learning applications. This chapter presents a methodology for reduced-order Markov modeling of time-series data based has been used on spectral properties of stochastic matrix and clustering of directed graphs. Instead of the common Hidden Markov model (HMM)-inspired techniques, a symbolic dynamics-based approach to infer an approximate generative Markov model for the data. The time-series data is first symbolized by partitioning of the discrete-valued signal in continuous domain. The size of temporal memory of the discretized symbol sequence is then estimated using spectral properties of the stochastic matrix created from the symbol sequence for a first-order Markov model of the symbol sequence. Then, a graphical method is used to cluster the states of the corresponding high-order Markov model to infer a reduced-size Markov model with a non-deterministic algebraic structure. A Bayesian inference rule captures the parameters of the reduced-size Markov model from the original model. The proposed idea is illustrated by creating Markov models for pressure time-series data from a swirl stabilized combustor where some controlled protocols are used to induce instability. Results demonstrate complexity modeling of the underlying Markov model as the system operating condition changes from stable to unstable which is useful in combustion applications such as detection and control of thermo-acoustic instabilities.