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.
@cite_11: Changepoints are abrupt variations in the generative parameters of a data sequence. Online detection of changepoints is useful in modelling and prediction of time series in application areas such as finance, biometrics, and robotics. While frequentist methods have yielded online filtering and prediction techniques, most Bayesian papers have focused on the retrospective segmentation problem. Here we examine the case where the model parameters before and after the changepoint are independent and we derive an online algorithm for exact inference of the most recent changepoint. We compute the probability distribution of the length of the current run,'' or time since the last changepoint, using a simple message-passing algorithm. Our implementation is highly modular so that the algorithm may be applied to a variety of types of data. We illustrate this modularity by demonstrating the algorithm on three different 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 .