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.
This work has been supported in part by the U.S. Air Force Office of Scientific Research under Grant No. FA9550-15-1-0400.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
P.B. Graben, Estimating and improving the signal-to-noise ratio of time series by symbolic dynamics. Phys. Rev. E 64(5), 051104 (2001)
C.S. Daw, C.E.A. Finney, E.R. Tracy, A review of symbolic analysis of experimental data. Rev. Sci. Instrum. 74(2), 915–930 (2003)
F. Darema, Dynamic data driven applications systems: new capabilities for application simulations and measurements, in 5th International Conference on Computational Science – ICCS 2005, Atlanta, 2005
S. Sarkar, S. Chakravarthy, V. Ramanan, A. Ray, Dynamic data-driven prediction of instability in a swirl-stabilized combustor. Int. J. Spray Combustion 8(4), 235–253 (2016)
A. Ray, Symbolic dynamic analysis of complex systems for anomaly detection. Signal Proc. 84(7), 1115–1130 (2004)
K. Mukherjee, A. Ray, State splitting and merging in probabilistic finite state automata for signal representation and analysis. Signal Proc. 104, 105–119 (2014)
I. Chattopadhyay, H. Lipson, Abductive learning of quantized stochastic processes with probabilistic finite automata. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 371(1984), 20110543 (2013)
C.M. Bishop, Pattern Recognition and Machine Learning (Springer, New York, 2006)
C.R. Shalizi, K.L. Shalizi, Blind construction of optimal nonlinear recursive predictors for discrete sequences, in Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence, (UAI ’04), Banff, 2004, pp. 504–511
Y. Seto, N. Takahashi, D.K. Jha, N. Virani, A. Ray, Data-driven robot gait modeling via symbolic time series analysis, in American Control Conference (ACC ’16), Boston (IEEE, 2016) pp. 3904–3909
K. Deng, Y. Sun, P.G. Mehta, S.P. Meyn, An information-theoretic framework to aggregate a Markov chain, in American Control Conference, (ACC ’09), St. Louis (IEEE, 2009) pp. 731–736
B.C. Geiger, T. Petrov, G. Kubin, H. Koeppl, Optimal Kullback–Leibler aggregation via information bottleneck. IEEE Trans. Autom. Control 60(4), 1010–1022 (2015)
M. Vidyasagar, A metric between probability distributions on finite sets of different cardinalities and applications to order reduction. IEEE Trans. Autom. Control 57(10), 2464–2477 (2012)
Y. Xu, S.M. Salapaka, C.L. Beck, Aggregation of graph models and Markov chains by deterministic annealing. IEEE Trans. Autom. Control 59(10), 2807–2812 (2014)
A. Srivastav, Estimating the size of temporal memory for symbolic analysis of time-series data, in American Control Conference, Portland, 2014, pp. 1126–1131
D.K. Jha, A. Srivastav, K. Mukherjee, A. Ray, Depth estimation in Markov models of time-series data via spectral analysis, in American Control Conference (ACC ’15), Chicago (IEEE, 2015) pp. 5812–5817
D.K. Jha, Learning and decision optimization in data-driven autonomous systems, Ph.D. dissertation, The Pennsylvania State University, 2016
D.K. Jha, A. Srivastav, A. Ray, Temporal learning in video data using deep learning and Gaussian processes. Int. J. Prognostics Health Monit. 7(22), 11 (2016)
S. Sarkar, D.K. Jha, A. Ray, Y. Li, Dynamic data-driven symbolic causal modeling for battery performance & health monitoring, in 2015 18th International Conference on Information Fusion (Fusion) (IEEE, 2015), pp. 1395–1402
J. Lin, E. Keogh, L. Wei, S. Lonardi, Experiencing SAX: a novel symbolic representation of time series. Data Min. Knowl. Disc. 15(2), 107–144 (2007)
D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge, 1995)
S. Garcia, J. Luengo, J.A. Saez, V. Lopez, F. Herrera, A survey of discretization techniques: taxonomy and empirical analysis in supervised learning. IEEE Trans. Knowl. Data Eng. 22(99), (2012)
V. Rajagopalan, A. Ray, Symbolic time series analysis via wavelet-based partitioning. Signal Process. 86(11), 3309–3320 (2006)
R. Xu, D. Wunsch, Survey of clustering algorithms. IEEE Trans. Neural Netw. 16(3), 645–678 (2005)
V.N. Vapnik, Statistical Learning Theory (Wiley, New York, 1998)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Jha, D.K., Virani, N., Ray, A. (2018). Markov Modeling of Time Series via Spectral Analysis for Detection of Combustion Instabilities. In: Blasch, E., Ravela, S., Aved, A. (eds) Handbook of Dynamic Data Driven Applications Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-95504-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-95504-9_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-95503-2
Online ISBN: 978-3-319-95504-9
eBook Packages: Computer ScienceComputer Science (R0)