We propose a new approach, the calibrated nonparametric scan statistic (CNSS), for more accurate detection of anomalous patterns in large-scale, real-world graphs. Scan statistics identify connected subgraphs that are interesting or unexpected through maximization of a likelihood ratio statistic; in particular, nonparametric scan statistics (NPSSs) identify subgraphs with a higher than expected proportion of individually significant nodes. However, we show that recently proposed NPSS methods are miscalibrated, failing to account for the maximization of the statistic over the multiplicity of subgraphs. This results in both reduced detection power for subtle signals, and low precision of the detected subgraph even for stronger signals. Thus we develop a new statistical approach to recalibrate NPSSs, correctly adjusting for multiple hypothesis testing and taking the underlying graph structure into account. While the recalibration, based on randomization testing, is computationally expensive, we propose both an efficient (approximate) algorithm and new, closed-form lower bounds (on the expected maximum proportion of significant nodes for subgraphs of a given size, under the null hypothesis of no anomalous patterns). These advances, along with the integration of recent core-tree decomposition methods, enable CNSS to scale to large real-world graphs, with substantial improvement in the accuracy of detected subgraphs. Extensive experiments on both semi-synthetic and real-world datasets are demonstrated to validate the effectiveness of our proposed methods, in comparison with state-of-the-art counterparts.
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Learning Patterns for Detection with Multiscale Scan Statistics
This paper addresses detecting anomalous patterns in images, time-series, and tensor data when the location and scale of the pattern and the pattern itself is unknown a priori. The multiscale scan statistic convolves the proposed pattern with the image at various scales and returns the maximum of the resulting tensor. Scale corrected multiscale scan statistics apply different standardizations at each scale, and the limiting distribution under the null hypothesis---that the data is only noise---is known for smooth patterns. We consider the problem of simultaneously learning and detecting the anomalous pattern from a dictionary of smooth patterns and a database of many tensors. To this end, we show that the multiscale scan statistic is a subexponential random variable, and prove a chaining lemma for standardized suprema, which may be of independent interest. Then by averaging the statistics over the database of tensors we can learn the pattern and obtain Bernstein-type error bounds. We will also provide a construction of an epsilon-net of the location and scale parameters, providing a computationally tractable approximation with similar error bounds.
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- Award ID(s):
- 1712996
- NSF-PAR ID:
- 10061398
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research (31st Annual Conference on Learning Theory))
- Volume:
- 75
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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