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 BOCPD @cite_11 generative model assumes that the segments composing a sequence of observations @math are non-overlapping product partitions. This means that every partition @math (i.e., segments for HSMMs) is generated i.i.d. given a fixed observation model @math . The parameters @math are also drawn i.i.d. across partitions from some fixed distribution parameterized by @math . Formally, the UPM represents the distribution @math which generates observations based on the current run length @math and observed values @math . The original formulation of BOCPD did not provide a learning procedure since the model hyperparameters @math were assumed known.
@cite_11
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We investigate the clustering performances of the relaxed @math means in the setting of sub-Gaussian Mixture Model (sGMM) and Stochastic Block Model (SBM). After identifying the appropriate signal-to-noise ratio (SNR), we prove that the misclassification error decay exponentially fast with respect to this SNR. These partial recovery bounds for the relaxed @math means improve upon results currently known in the sGMM setting. In the SBM setting, applying the relaxed @math means SDP allows to handle general connection probabilities whereas other SDPs investigated in the literature are restricted to the assortative case (where within group probabilities are larger than between group probabilities). Again, this partial recovery bound complements the state-of-the-art results. All together, these results put forward the versatility of the relaxed @math means.