State splitting and merging in probabilistic finite state automata for signal representation and analysis

Probabilistic finite state automata (PFSA) are often constructed from symbol strings that, in turn, are generated by partitioning time series of sensor signals. This paper focuses on a special class of PFSA, which captures finite history of the symbol strings; these PFSA, called D-Markov machines, have a simple algebraic structure and are computationally efficient to construct and implement. The procedure of PFSA construction is based on (i) state splitting that generates symbol blocks of different lengths based on their information contents; and (ii) state merging that assimilates histories by combining two or more symbol blocks without any significant loss of the embedded information. A metric on the probability distribution of symbol blocks is introduced for trade-off between loss of information (e.g., entropy rate) and the number of PFSA states. The underlying algorithms have been validated with three test examples. While the first and second examples elucidate the key concepts and the pertinent numerical steps, the third example presents the results of analysis of time series data, generated from laboratory experimentation, for detection of fatigue crack damage in a polycrystalline alloy.