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Markov Modeling via Spectral Analysis: Application to Detecting Combustion Instabilities

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Abstract

Effective representation 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 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 is used to infer an approximate generative Markov model for the data. The time-series data is first symbolized by partitioning the continuous domain to obtain a discrete-valued signal. 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.

Keywords

  • Bayesian inferencing
  • Combustion processes
  • Detection and estimation
  • Dynamic data-driven modeling
  • Hidden Markov modeling
  • Markov processes
  • Model order reduction
  • Novelty detection
  • Thermo-acoustic instabilities
  • Time series analysis

This work has been supported in part by the U.S. Air Force Office of Scientific Research under Grant No. FA9550-15-1-0400. This work was conducted when all authors were at The Pennsylvania State University.

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Acknowledgements

The authors would like to thank Professor Domenic Santavicca and Mr. Jihang Li of Center for Propulsion, Penn State for kindly providing the experimental data for combustion used in this work. Benefits of technical discussion with Dr. Shashi Phoha at Penn State are also thankfully acknowledged.

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Correspondence to Asok Ray .

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Jha, D.K., Virani, N., Ray, A. (2022). Markov Modeling via Spectral Analysis: Application to Detecting Combustion Instabilities. In: Blasch, E.P., Darema, F., Ravela, S., Aved, A.J. (eds) Handbook of Dynamic Data Driven Applications Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-74568-4_6

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  • DOI: https://doi.org/10.1007/978-3-030-74568-4_6

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