BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Paul Malcolm (ANU DST) DTSTART:20220607T060000Z DTEND:20220607T070000Z DTSTAMP:20260423T024721Z UID:anumacs/3 DESCRIPTION:Title: New representations for a semi-Markov chain and related filters\nby Pa ul Malcolm (ANU DST) as part of ANU Mathematics and Computational Sciences Seminar\n\nLecture held in Room 1.33\, Hanna Neumann Building #145.\n\nAb stract\nIt is now usual that the null-hypothesis for a finite-state stocha stic process is conveniently taken to be the standard Markov chain. In the absence of any other system knowledge this is the model that is often use d. Some reasons for this are\; Markov chains are relatively simple\, they have been well studied and much is known about these processes. Added to t his there are now decades of history applying the standard Hidden Markov M odel (HMM) to: defence science\, gene sequencing\, health science\, machin e learning\, artificial intelligence and many other areas. In this seminar we will briefly recall two common application domains of estimation with latent Markov processes\, 1) parts-of-speech tagging (POS) in natural lang uage processing and 2) tracking a maneuvering object with a Jump Markov Sy stem. Semi-Markov models relax an implicit feature of every state in a fir st-order time-homogeneous Markov chains\, that is\, the sojourn random var iables of such states are geometrically distributed and are therefore\, (u niquely) memoryless random variables. In contrast\, semi-Markov chains all ow arbitrary sojourn models. Consequently\, a Hidden semi-Markov Model (Hs MM) offers a richer class of model\, but retains the classical HMM as a sp ecial degenerate case.\n\nThe main task we address in this seminar concern s model calibration\, or parameter estimation of a HsMM. We develop an Exp ectation Maximization (EM) algorithm to compute the best fitting (in the M aximum Likelihood sense) HsMM for a given set of observation data. There a re several parts to this task\, the first is to derive a recursive filter and smoother for a partially observed semi-Markov chain. The second and mo re challenging part of the task is to derive filters and smoothers for var ious processes derived from the latent semi-Markov chain\, for example\, a counting process that counts the number of transitions between two distin ct states labelled "i" and "j"\, up to an including time k. We will see th at estimators for such quantities are non-trivial\, largely because of the sojourn dependence in transition probabilities.\n\nThe estimators we pres ent are all for partially observed joint events\, that is\, the state of t he semi-Markov chain at time "k" and the cumulative time it has remained i n this state. This means we are assured of exponential forgetting of initi al conditions in our estimators. Separate estimators for individual quanti ties such as the semi-Markov state alone are easily computed via marginali zation.\n LOCATION:https://researchseminars.org/talk/anumacs/3/ END:VEVENT END:VCALENDAR