This paper proposes a new multi-linear projection method for denoising and estimation of high-dimensional matrix-variate factor time series. It assumes that ap1×p2matrix-variate time series consists of a dynamically dependent, lower-dimensional matrix-variate factor process and ap1×p2matrix idiosyncratic series. In addition, the latter series assumes a matrix-variate factor structure such that its row and column covariances may have diverging/spiked eigenvalues to accommodate the case of low signal-to-noise ratio often encountered in applications. We use an iterative projection procedure to reduce the dimensions and noise effects in estimating front and back loading matrices and to obtain faster convergence rates than those of the traditional methods available in the literature. We further introduce a two-way projected Principal Component Analysis to mitigate the diverging noise effects, and implement a high-dimensional white-noise testing procedure to estimate the dimension of the matrix factor process. Asymptotic properties of the proposed method are established if the dimensions and sample size go to infinity. We also use simulations and real examples to assess the performance of the proposed method in finite samples and to compare its forecasting ability with some existing ones in the literature. The proposed method fares well in out-of-sample forecasting. In the appendix, we demonstrate the efficacy of the proposed approach even when the idiosyncratic terms exhibit serial correlations with or without a diverging white noise effect.

Feature-distributed data, referred to data partitioned by features and stored across multiple computing nodes, are increasingly common in applications with a large number of features. This paper proposes a two-stage relaxed greedy algorithm (TSRGA) for applying multivariate linear regression to such data. The main advantage of TSRGA is that its communication complexity does not depend on the feature dimension, making it highly scalable to very large data sets. In addition, for multivariate response variables, TSRGA can be used to yield low-rank coefficient estimates. The fast convergence of TSRGA is validated by simulation experiments. Finally, we apply the proposed TSRGA in a financial application that leverages unstructured data from the 10-K reports, demonstrating its usefulness in applications with many dense large-dimensional matrices.

The detection of anomalies in time series has gained ample academic and industrial attention, yet, no comprehensive benchmark exists to evaluate time-series anomaly detection methods. Therefore, there is no final verdict on which method performs the best (and under what conditions). Consequently, we often observe methods performing exceptionally well on one dataset but surprisingly poorly on another, creating an illusion of progress. To address these issues, we thoroughly studied over one hundred papers, and summarized our effort in TSB-UAD, a new benchmark to evaluate univariate time series anomaly detection methods. In this paper, we describe Theseus, a modular and extensible web application that helps users navigate through the benchmark, and reason about the merits and drawbacks of both anomaly detection methods and accuracy measures, under different conditions. Overall, our system enables users to compare 12 anomaly detection methods on 1980 time series, using 13 accuracy measures, and decide on the most suitable method and measure for some application.

Anomaly detection (AD) is a fundamental task for time-series analytics with important implications for the downstream performance of many applications. In contrast to other domains where AD mainly focuses on point-based anomalies (i.e., outliers in standalone observations), AD for time series is also concerned with range-based anomalies (i.e., outliers spanning multiple observations). Nevertheless, it is common to use traditional point-based information retrieval measures, such as Precision, Recall, and F-score, to assess the quality of methods by thresholding the anomaly score to mark each point as an anomaly or not. However, mapping discrete labels into continuous data introduces unavoidable shortcomings, complicating the evaluation of range-based anomalies. Notably, the choice of evaluation measure may significantly bias the experimental outcome. Despite over six decades of attention, there has never been a large-scale systematic quantitative and qualitative analysis of time-series AD evaluation measures. This paper extensively evaluates quality measures for time-series AD to assess their robustness under noise, misalignments, and different anomaly cardinality ratios. Our results indicate that measures producing quality values independently of a threshold (i.e., AUC-ROC and AUC-PR) are more suitable for time-series AD. Motivated by this observation, we first extend the AUC-based measures to account for range-based anomalies. Then, we introduce a new family of parameter-free and threshold-independent measures, VUS (Volume Under the Surface), to evaluate methods while varying parameters. Our findings demonstrate that our four measures are significantly more robust in assessing the quality of time-series AD methods.

The detection of anomalies in time series has gained ample academic and industrial attention. However, no comprehensive benchmark exists to evaluate time-series anomaly detection methods. It is common to use (i) proprietary or synthetic data, often biased to support particular claims; or (ii) a limited collection of publicly available datasets. Consequently, we often observe methods performing exceptionally well in one dataset but surprisingly poorly in another, creating an illusion of progress. To address the issues above, we thoroughly studied over one hundred papers to identify, collect, process, and systematically format datasets proposed in the past decades. We summarize our effort in TSB-UAD, a new benchmark to ease the evaluation of univariate time-series anomaly detection methods. Overall, TSB-UAD contains 13766 time series with labeled anomalies spanning different domains with high variability of anomaly types, ratios, and sizes. TSB-UAD includes 18 previously proposed datasets containing 1980 time series and we contribute two collections of datasets. Specifically, we generate 958 time series using a principled methodology for transforming 126 time-series classification datasets into time series with labeled anomalies. In addition, we present data transformations with which we introduce new anomalies, resulting in 10828 time series with varying complexity for anomaly detection. Finally, we evaluate 12 representative methods demonstrating that TSB-UAD is a robust resource for assessing anomaly detection methods. We make our data and code available at www.timeseries.org/TSB-UAD. TSB-UAD provides a valuable, reproducible, and frequently updated resource to establish a leaderboard of univariate time-series anomaly detection methods.