Scalable Bayesian Multiple Changepoint Detection via Auxiliary Uniformisation

Abstract

Summary In this paper, we perform a sparse filtering recursion for efficient changepoint detection for discrete‐time observations. We attach auxiliary event times to the chronologically ordered observations and formulate multiple changepoint problems of discrete‐time observations into continuous‐time observations. Ideally, both the computational and memory costs of the proposed auxiliary uniformisation forward‐filtering backward‐sampling algorithm can be quadratically scaled down to the number of changepoints instead of the number of observations, which would otherwise be prohibitive for a long sequence of observations. To avoid model bias, a time‐varying changepoint recurrence rate across different segments is assumed to characterise diverse scales of run lengths of the changepoints. We demonstrate the methods through simulation studies and real data analysis.

Bayesian multiple changepoint detection with missing data and its application to the magnitude‐frequency distributions

“…When there exists exactly one jump, $$$ \log p\left(\mathbf{X}\left({t}_{k+1}\right)=i+1,\mathbf{Y}\Big({t}_k,{t}_{k+1}\Big]|\mathbf{X}\left({t}_k\right)=i\right) $$$ is maximized exactly at $$$ {t}_k $$$ or $$$ {t}_{k+1} $$$. Therefore, the MAP estimate of the set of changepoints is equivalently obtainable from the Viterbi algorithm in discrete time, see Proposition 1 in Lu (2022). Nevertheless, both the computational cost and the memory cost are quadratic to $$$ n $$$ in this case, which is prohibitive for a long sequence of observations.…”

“…In this case, both the computational and memory costs can be scaled down quadratically to the number of changepoints, see Lu (2021). The model formulation allows the multiple changepoint detection for both continuous‐time observations and discrete‐time ones in a unified approach (Lu, 2022). Assume $$$ m $$$ changepoints appear at unknown locations $$$ 0\triangleq {\tau}_0<{\tau}_1<\cdots <{\tau}_m<{\tau}_{m+1}\triangleq T $$$, such that $$$ {\mathrm{y}}_{1:n} $$$ is partitioned into $$$ m+1 $$$ segments by $$$ m $$$ changepoints $$$ \boldsymbol{\tau} =\left({\tau}_1,\dots, {\tau}_m\right) $$$.…”

“…For these variable‐dimensional problems, the posterior means are not properly defined. Simulation studies in Lu (2021, 2022) show that the performance of MAP estimates of the number of changepoints and their locations is desirable. We consider a stochastic EM algorithm.…”

“…The two steps iterate until convergence. Often, a continuous‐time version of the Viterbi algorithm (Lu, 2022) is implemented in the M‐step. However, the MAP estimates of the set of changepoints are potentially selected from an infinite number of trajectories of the latent Markov chain, which is inaccessible except for some special cases.…”

Bayesian Change Point Detection via a General Sparse Recursive Filter

“…Thus, potential model bias by assuming all q i are equal is avoided. Because the posterior of the number of change points and their locations is sensitive to the specification of the change point recurrence rate (Lu 2023) and because a subjective selection or a rough estimate of these hyperparameters is suboptimal, we assume that all q i s are unknown and that a conjugate prior Γ(a, b) is used. These hyperparameters are also estimated via an empirical Bayesian approach; see Du et al (2016).…”