  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.
