Online detection of instantaneous changes in the generative process of a data sequence generally focuses on retrospective inference of such change points without considering their future occurrences. We extend the Bayesian Online Change Point Detection algorithm to also infer the number of time steps until the next change point (i.e., the residual time). This enables us to handle observation models which depend on the total segment duration, which is useful to model data sequences with temporal scaling. In addition, we extend the model by removing the i.i.d. assumption on the observation model parameters. The resulting inference algorithm for segment detection can be deployed in an online fashion, and we illustrate applications to synthetic and to two medical real-world data sets.
The joint probability in is not only insightful for detecting change points, but can also be used to perform predictions about future observations. The predictive distribution for the next observation (shown in ) is derived by marginalizing over the current run length posterior, which is obtained from . Note that an HSMM (Section ) with a single hidden state @math leads to almost the same generative model of BOCPD. One of the key differences is that the HSMMs are traditionally parameterized in terms of the total segment duration, whereas the BOCPD parameterization is based on the run length through the hazard function instead of the duration model. As hinted earlier, the equivalence between the hazard function and the duration model is well-understood---see @cite_11 for details. However, the chosen parameterization has relevant implications for the UPMs, as discussed in Section .
@cite_11
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Dynamic Probabilistic Networks (DPNs) are exploited for modeling the temporal relationships among a set of different object temporal events in the scene for a coherent and robust scene-level behaviour interpretation. In particular, we develop a Dynamically Multi-Linked Hidden Markov Model (DML-HMM) to interpret group activities involving multiple objects captured in an outdoor scene. The model is based on the discovery of salient dynamic interlinks among multiple temporal events using DPNs. Object temporal events are detected and labeled using Gaussian Mixture Models with automatic model order selection. A DML-HMM is built using Schwarz's Bayesian Information Criterion based factorisation resulting in its topology being intrinsically determined by the underlying causality and temporal order among different object events. Our experiments demonstrate that its performance on modelling group activities in a noisy outdoor scene is superior compared to that of a Multi-Observation Hidden Markov Model (MOHMM), a Parallel Hidden Markov Model (PaHMM) and a Coupled Hidden Markov Model (CHMM).