Detecting change regions on spheres

Abstract

While change point detection in time series data has been extensively studied, little attention has been given to its generalisation to data observed on spheres or other manifolds, where changes may occur within spatially complex regions with irregular boundaries, posing significant challenges. We propose a new class of estimators, namely, Change Region Identification and SeParation (CRISP), to locate changes in the mean function of a signal-plus-noise model defined on $d$-dimensional spheres. The CRISP estimator applies to scenarios with a single change region, and is extended to multiple change regions via a newly developed generic scheme. The convergence rate of the CRISP estimator is shown to depend on the VC dimension of the hypothesis class that characterises the change regions in general. We also carefully study the case where change regions have the geometry of spherical caps. Simulations confirm the promising finite-sample performance of this approach. The CRISP estimator's practical applicability is further demonstrated through two real data sets on global temperature and ozone hole.

Figures (7)

• Figure 1: (a) A scan region $B$ contains multiple true change regions; an inner disc $A$ can aggregate parts of several regions. (b) A small scan region $B$ captures only a fragment of a true change region $R$, yet can still yield a significant contrast.
• Figure 2: Empirical loss of single change region estimation using CRISP, averaged over 100 Monte Carlo repetitions, plotted against the sample size (top panels) and signal strengths (bottom panels) on log-log scale, for different dimensions. Data generating mechanism described in Section \ref{['Sec:EmpPerf']}.
• Figure 3: Empirical results for multiple change-region estimation using CRISP, averaged over 100 Monte Carlo repetitions. The loss is plotted against the sample size (top panels) and signal strengths (middle panels) on log-log scale, and the average number of estimated regions (true value: four) is shown in the bottom panels, for different dimensions. The data-generating mechanism is described in Section \ref{['Sec:EmpPerf']}.
• Figure 4: Empirical results for multiple change region estimation using CRISP and competitors from lidR averaged over $100$ Monte Carlo repetitions when $d=3$. Upper panel: losses plotted against the sample size $n$ on log-log scale. Lower panel: Adjusted Rand Index plotted against $n$ on log-log scale.
• Figure 5: Maps of temperature difference between consecutive decades using the ERA5 and ESA data. Observations at 2000 randomly sampled grid points (blue crosses) are used to compute abnormal regions of temperature difference using the CRISP methodology. The estimated abnormal regions are shown as red discs on the maps.

Theorems & Definitions (40)

• Theorem 1
• Corollary 1
• Theorem 2
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• proof : Proof of Theorem \ref{['thm:consistency']}