Learning under Commission and Omission Event Outliers

TL;DR

This work tackles learning from event streams corrupted by omission and commission outliers by embedding a robust weighting scheme into a $K$-mixture temporal point process framework. A carefully designed influence-based weight function $\phi'_{\rho_1,\rho_2}$ assigns interval-wise weights $W_i(S;\boldsymbol B)$, enabling simultaneous robustness in both estimation and clustering while preserving asymptotic consistency when outliers are absent. The authors establish identifiability, gradient unbiasedness, and local convergence (Theorems Mgrad, Mout, cor.grad, and local), along with a practical algorithm that alternates updating latent labels and model parameters. Empirical results across simulated non-homogeneous Poisson and Hawkes processes, plus a real IPTV dataset, demonstrate substantial improvements in clustering purity and accurate outlier detection, including practical detection guarantees. This approach offers a scalable, theory-backed tool for robust event-stream analysis with broad applicability to next-event prediction, change-point detection, and multilabel clustering in dynamic systems.

Abstract

Event stream is an important data format in real life. The events are usually expected to follow some regular patterns over time. However, the patterns could be contaminated by unexpected absences or occurrences of events. In this paper, we adopt the temporal point process framework for learning event stream and we provide a simple-but-effective method to deal with both commission and omission event outliers.In particular, we introduce a novel weight function to dynamically adjust the importance of each observed event so that the final estimator could offer multiple statistical merits. We compare the proposed method with the vanilla one in the classification problems, where event streams can be clustered into different groups. Both theoretical and numerical results confirm the effectiveness of our new approach. To our knowledge, our method is the first one to provably handle both commission and omission outliers simultaneously.

Figures (7)

• Figure 1: The visualization of the two types of event outliers. The left plot is for the omission case, where the red box indicates a possible missing event. The right plot is for the commission case, where the green box indicates a potential unexpected event.
• Figure 2: Left plot: the setting of zhang2024robust where the first two event sequences are inliers while the third one is assumed to be the outlier. Right plot: our setting where all three event sequences are inliers but each of them may contain several event outliers.
• Figure 3: The visualization of event contamination process. The upper plot: the original event sequences generated from the true underlying intensity function. The middle plot: the event contamination on the event sequences. (Green area indicates the Type-ii time interval and Orange areas indicate Type-i time intervals.) The bottom plot: the observed event sequence after event contamination.
• Figure 4: A toy example of the non-identifiability case. The upper plot gives an observed event sequence, which is equally likely to be generated from the event sequence with type-i contamination (bottom left) or type-ii contamination (bottom right). In other words, we may have at least 50 % chance to give the wrong classification result when the restriction \ref{['eq:adjust:rho']} is removed.
• Figure 5: The left household ID is 54254406, and the right household ID is 54350089. The former has larger weight values during the spring festival and the latter has larger weight values during national holiday.

Theorems & Definitions (28)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• Corollary 1
• Remark 4
• Remark 5
• Theorem 1
• Remark 6
• Remark 7