Bernstein Polynomial Processes for Continuous Time Change Detection

TL;DR

This work tackles continuous-time change-point detection by deriving a heterogeneous continuous-time Markov chain with noninformative priors that yield exact posterior inference via the forward-backward algorithm, avoiding MCMC. It introduces the Bernstein polynomial process (BPP) as the transition prior, linking state trajectories to Dirichlet-distributed segment lengths and enabling efficient EM-based Estimation of the latent states and segment parameters. A robust, flexible methodology is developed, including heavy-tailed error modeling, principled priors on the number of segments, and a loss-based Bayes estimator for change-point locations; this is demonstrated through simulations and three satellite-based case studies on deforestation, crop rotation, and drought-driven phenology. The paper further extends to semiparametric phenological modeling with interannual harmonics under continuity constraints and provides extensive appendices with proofs, convergence results, and supplementary results. Overall, the approach delivers scalable, accurate continuous-time change detection in irregularly spaced remote-sensing data and offers a practical framework for incorporating environmental variability and robustness into change inference.

Abstract

There is a lack of methodological results for continuous time change detection due to the challenges of noninformative prior specification and efficient posterior inference in this setting. Most methodologies to date assume data are collected according to uniformly spaced time intervals. This assumption incurs bias in the continuous time setting where, a priori, two consecutive observations measured closely in time are less likely to change than two consecutive observations that are far apart in time. Models proposed in this setting have required MCMC sampling which is not ideal. To address these issues, we derive the heterogeneous continuous time Markov chain that models change point transition probabilities noninformatively. By construction, change points under this model can be inferred efficiently using the forward backward algorithm and do not require MCMC sampling. We then develop a novel loss function for the continuous time setting, derive its Bayes estimator, and demonstrate its performance on synthetic data. A case study using time series of remotely sensed observations is then carried out on three change detection applications. To reduce falsely detected changes in this setting, we develop a semiparametric mean function that captures interannual variability due to weather in addition to trend and seasonal components.

Figures (24)

• Figure 1: (Left) An example of noninformative discrete time hypergeometric state marginals(dotted), $n=10$ and $k=4$ with green as segment 1 and pink as segment 4. Noninformative continuous time state marginal are the corresponding solid lines. (Right) Discrete time noninformative transition probabilities for $n=25$ and $k=10$ for illustration. The colors range from $\pi_{10}(z_{i+1}=1|z_i=1)$(purple) to $\pi_{10}(z_{i+1}=10|z_i=10)$(pink).
• Figure 2: An example of continuous time transitions on 25 uniformly distributed times and 5 segments. (Left) Self-transitions $\pi_5(z_{t_{i+1}}=j|z_{t_{i}}=j)$. Pink is $j=1$, with the remaining in gray, following in increasing order of probability. (Right) Transitions $\pi_5(z_{t_{i+1}}=h|z_{t_{i}}=1)$ for $h=1,\ldots,k-1$. Pink is $h=1$, followed by green, yellow, orange, and brown.
• Figure 3: Two priors on number of segments are compared. Twenty time points are simulated from a uniform distribution on $[0,1]$, and $\pi(k)$ is plotted for each. The prior on $k$ assuming equally likely change point sequences across $k=1,\dots,4$ is in gray reaching maximum probability at $k=4$. The prior on $k$ assuming $k$ is noninformative with respect to the volume of its model is in pink, having maximum probability at $k=1$.
• Figure 4: True positive rate against false positive rate on a synthetic data set with varying sizes of change, time distribution, outlier magnitude, and number of segments. The first row compares BPP,PELT,and BinSeg on all of the data from the factorial study and the second row compares them on a subset when $\nu=3$.
• Figure 5: Breakdown of commission rate, omission rate, and F1 score for the BPP, PELT, and BinSeg models.

Theorems & Definitions (25)

• Proposition 1: Marginal noninformative prior in discrete time
• Proposition 2
• Theorem 1: Noninformative marginal convergence: Hypergeometric to Bernstein
• Theorem 2: Distributional equivalence in continuous time
• Theorem 3
• Theorem 4
• Proposition 3
• proof : Proof of Proposition \ref{['prop:hypergeo']}
• proof : Proof of Proposition \ref{['prop:trans']}
• proof : Proof of Theorem \ref{['thm:hgeobern']}