Fuzzy spatial pattern processing using linguistic hidden Markov models

In this work, we propose a hidden Markov model (HMM), called the linguistic HMM, suitable for processing sequences of fuzzy vectors. A fuzzy vector B is an n-tuple of fuzzy numbers. Since fuzzy numbers are often associated with linguistic terms, such as "small," "medium," etc., a fuzzy vector can also be called a linguistic vector. Similarly, an HMM that processes linguistic vectors can be called a "linguistic HMM." The derivation of the linguistic HMM (LHMM) from the continuous HMM is performed using the extension principle and the decomposition theorem. We prove that a LHMM behaves in the same fashion as the CHMM in the degenerate linguistic case when the fuzzy numbers are singletons (real numbers). We also provide an example where an LHMM was used for the recognition of a play (pick-and-shoot) during a basketball game. The positions of the players were described using spatial fuzzy relations. For the recognition experiment, we generated two sets of 100 sequences containing pick-and-shoot and non pick-and-shoot sequences, respectively. The LHMM results obtained for the fuzzy sequences were compared to the CHMM results obtained on a crisp version of the same sequences. The results obtained showed that the fuzzy spatial relations together with the LHMM provide a better description of the movement than the CHMM.