Symbolic Analysis-based Reduced Order Markov Modeling of Time Series Data

This paper presents a technique for reduced-order Markov modeling for compact representation of time-series data. In this work, symbolic dynamics-based tools have been used to infer an approximate generative Markov model. The time-series data are first symbolized by partitioning the continuous measurement space of the signal and then, the discrete sequential data are modeled using symbolic dynamics. In the proposed approach, the size of temporal memory of the symbol sequence is estimated from spectral properties of the resulting stochastic matrix corresponding to a first-order Markov model of the symbol sequence. Then, hierarchical clustering is used to represent the states of the corresponding full-state Markov model to construct a reduced-order or size Markov model with a non-deterministic algebraic structure. Subsequently, the parameters of the reduced-order Markov model are identified from the original model by making use of a Bayesian inference rule. The final model is selected using information-theoretic criteria. The proposed concept is elucidated and validated on two different data sets as examples. The first example analyzes a set of pressure data from a swirl-stabilized combustor, where controlled protocols are used to induce flame instabilities. Variations in the complexity of the derived Markov model represent how the system operating condition changes from a stable to an unstable combustion regime. In the second example, the data set is taken from NASA's data repository for prognostics of bearings on rotating shafts. We show that, even with a very small state-space, the reduced-order models are able to achieve comparable performance and that the proposed approach provides flexibility in the selection of a final model for representation and learning.Comment: 21 pages, 12 figure

Markov Modeling of Time-Series Data using Symbolic Analysis

Markov models are often used to capture the temporal patterns of sequential data for statistical learning applications. While the Hidden Markov modeling-based learning mechanisms are well studied in literature, we analyze a symbolic-dynamics inspired approach. Under this umbrella, Markov modeling of time-series data consists of two major steps -- discretization of continuous attributes followed by estimating the size of temporal memory of the discretized sequence. These two steps are critical for the accurate and concise representation of time-series data in the discrete space. Discretization governs the information content of the resultant discretized sequence. On the other hand, memory estimation of the symbolic sequence helps to extract the predictive patterns in the discretized data. Clearly, the effectiveness of signal representation as a discrete Markov process depends on both these steps. In this paper, we will review the different techniques for discretization and memory estimation for discrete stochastic processes. In particular, we will focus on the individual problems of discretization and order estimation for discrete stochastic process. We will present some results from literature on partitioning from dynamical systems theory and order estimation using concepts of information theory and statistical learning. The paper also presents some related problem formulations which will be useful for machine learning and statistical learning application using the symbolic framework of data analysis. We present some results of statistical analysis of a complex thermoacoustic instability phenomenon during lean-premixed combustion in jet-turbine engines using the proposed Markov modeling method