The challenge of efficiently identifying anomalies in data sequences is an important statistical problem that now arises in many applications. Whilst there has been substantial work aimed at making statistical analyses robust to outliers, or point anomalies, there has been much less work on detecting anomalous segments, or collective anomalies. By bringing together ideas from changepoint detection and robust statistics, we introduce Collective And Point Anomalies (CAPA), a computationally efficient approach that is suitable when collective anomalies are characterised by either a change in mean, variance, or both, and distinguishes them from point anomalies. Theoretical results establish the consistency of CAPA at detecting collective anomalies and empirical results show that CAPA has close to linear computational cost as well as being more accurate at detecting and locating collective anomalies than other approaches. We demonstrate the utility of CAPA through its ability to detect exoplanets from light curve data from the Kepler telescope.
In point-based sensing systems such as coordinate measuring machines (CMM) and laser ultrasonics where complete sensing is impractical due to the high sensing time and cost, adaptive sensing through a systematic exploration is vital for online inspection and anomaly quantification. Most of the existing sequential sampling methodologies focus on reducing the overall fitting error for the entire sampling space. However, in many anomaly quantification applications, the main goal is to estimate sparse anomalous regions in the pixel-level accurately. In this paper, we develop a novel framework named Adaptive Kernelized Maximum-Minimum Distance AKM$^2$D to speed up the inspection and anomaly detection process through an intelligent sequential sampling scheme integrated with fast estimation and detection. The proposed method balances the sampling efforts between the space-filling sampling (exploration) and focused sampling near the anomalous region (exploitation). The proposed methodology is validated by conducting simulations and a case study of anomaly detection in composite sheets using a guided wave test.
Several important families of computational and statistical results in machine learning and randomized algorithms rely on uniform bounds on quadratic forms of random vectors or matrices. Such results include the Johnson-Lindenstrauss (J-L) Lemma, the Restricted Isometry Property (RIP), randomized sketching algorithms, and approximate linear algebra. The existing results critically depend on statistical independence, e.g., independent entries for random vectors, independent rows for random matrices, etc., which prevent their usage in dependent or adaptive modeling settings. In this paper, we show that such independence is in fact not needed for such results which continue to hold under fairly general dependence structures. In particular, we present uniform bounds on random quadratic forms of stochastic processes which are conditionally independent and sub-Gaussian given another (latent) process. Our setup allows general dependencies of the stochastic process on the history of the latent process and the latent process to be influenced by realizations of the stochastic process. The results are thus applicable to adaptive modeling settings and also allows for sequential design of random vectors and matrices. We also discuss stochastic process based forms of J-L, RIP, and sketching, to illustrate the generality of the results.
In the multiple changepoint setting, various search methods have been proposed which involve optimising either a constrained or penalised cost function over possible numbers and locations of changepoints using dynamic programming. Such methods are typically computationally intensive. Recent work in the penalised optimisation setting has focussed on developing a pruning-based approach which gives an improved computational cost that, under certain conditions, is linear in the number of data points. Such an approach naturally requires the specification of a penalty to avoid under/over-fitting. Work has been undertaken to identify the appropriate penalty choice for data generating processes with known distributional form, but in many applications the model assumed for the data is not correct and these penalty choices are not always appropriate. Consequently it is desirable to have an approach that enables us to compare segmentations for different choices of penalty. To this end we present a method to obtain optimal changepoint segmentations of data sequences for all penalty values across a continuous range. This permits an evaluation of the various segmentations to identify a suitably parsimonious penalty choice. The computational complexity of this approach can be linear in the number of data points and linear in the difference between the number of changepoints in the optimal segmentations for the smallest and largest penalty values. This can be orders of magnitude faster than alternative approaches that find optimal segmentations for a range of the number of changepoints.
We consider the problem of identifying universal low-dimensional features from high-dimensional data for inference tasks in settings involving learning. For such problems, we introduce natural notions of universality and we show a local equivalence among them. Our analysis is naturally expressed via information geometry, and represents a conceptually and computationally useful analysis. The development reveals the complementary roles of the singular value decomposition, Hirschfeld-Gebelein-R\'enyi maximal correlation, the canonical correlation and principle component analyses of Hotelling and Pearson, Tishby's information bottleneck, Wyner's common information, Ky Fan $k$-norms, and Brieman and Friedman's alternating conditional expectations algorithm. We further illustrate how this framework facilitates understanding and optimizing aspects of learning systems, including multinomial logistic (softmax) regression and the associated neural network architecture, matrix factorization methods for collaborative filtering and other applications, rank-constrained multivariate linear regression, and forms of semi-supervised learning.