We address the problem of data-driven pattern identification and outlier detection in time series. To this end, we use singular value decomposition (SVD) which is a well-known technique to compute a low-rank approximation for an arbitrary matrix. By recasting the time series as a matrix it becomes possible to use SVD to highlight the underlying patterns and periodicities. This is done without the need for specifying user-defined parameters. From a data mining perspective, this opens up new ways of analyzing time series in a data-driven, bottom-up fashion. However, in order to get correct results, it is important to understand how the SVD-spectrum of a time series is influenced by various characteristics of the underlying signal and noise. In this paper, we have extended the work in earlier papers by initiating a more systematic analysis of these effects. We then illustrate our findings on some real-life data.
This paper reviews recent developments of robust estimation in linear time series models, with short and long memory correlation structures, in the presence of additive outliers. Based on the manuscripts Fajardo et al. (2009) and Lévy-Leduc et al. (2011a), the emphasis in this paper is given in the following directions; the influence of additive outliers in the estimation of a time series, the asymptotic properties of a robust autocovariance function and a robust semiparametric estimation method of the fractional parameter d in ARFIMA(p, d, q) models. Some simulations are used to support the u…
Outlying observations are commonly encountered in the analysis of time series. In this paper the problem of detecting additive outliers in integer-valued time series is considered. We show how Gibbs sampling can be used to detect outlying observations in INAR(1) processes. The methodology proposed is illustrated using examples as well as an observed data set.
Parameter change test has been an important issue in time series analysis. The problem has also been actively explored in the field of integer-valued time series, but the testing in the presence of outliers has not yet been extensively investigated. This study considers the problem of testing for parameter change in Poisson autoregressive models particularly when observations are contaminated by outliers. To lessen the impact of outliers on testing procedure, we propose a test based on the density power divergence, which is introduced by Basu et al. (Biometrika, 1998), and derive its limiting …
This paper studies the problem of robustly learning the correlation function for a univariate time series with the presence of noise, outliers and missing entries. The outliers or anomalies considered here are sparse and rare events that deviate from normality which is depicted by a correlation function and an uncertainty condition. This general formulation is applied to univariate time series of event counts (or non-negative time series) where the correlation is a log-linear function with the uncertainty condition following the Poisson distribution. Approximations to the sparsity constraint, …
Dynamic time warping (DTW) is an effective dissimilarity measure in many time series applications. Despite its popularity, it is prone to noises and outliers, which leads to singularity problem and bias in the measurement. The time complexity of DTW is quadratic to the length of time series, making it inapplicable in real-time applications. In this paper, we propose a novel time series dissimilarity measure named RobustDTW to reduce the effects of noises and outliers. Specifically, the RobustDTW estimates the trend and optimizes the time warp in an alternating manner by utilizing our designed …
Ikerlan Technology Research Centre（伊克兰技术研究中心）；University of the Basque Country (UPV/EHU)（巴斯克大学（UPV/EHU））；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心（BCAM））
Recent advances in technology have brought major breakthroughs in data collection, enabling a large amount of data to be gathered over time and thus generating time series. Mining this data has become an important task for researchers and practitioners in the past few years, including the detection of outliers or anomalies that may represent errors or events of interest. This review aims to provide a structured and comprehensive state-of-the-art on outlier detection techniques in the context of time series. To this end, a taxonomy is presented based on the main aspects that characterize an out…
Functional time series whose sample elements are recorded sequentially over time are frequently encountered with increasing technology. Recent studies have shown that analyzing and forecasting of functional time series can be performed easily using functional principal component analysis and existing univariate/multivariate time series models. However, the forecasting performance of such functional time series models may be affected by the presence of outlying observations which are very common in many scientific fields. Outliers may distort the functional time series model structure, and thus…
In this study, we consider a problem of monitoring parameter changes particularly in the presence of outliers. To propose a sequential procedure that is robust against outliers, we use the density power divergence to derive a detector and stopping time that make up our procedure. We first investigate the asymptotic properties of our sequential procedure for i.i.d. sequences, and then extend the proposed procedure to stationary time series models, where we provide a set of sufficient conditions under which the proposed procedure has an asymptotically controlled size and consistency in power. As…
Dynamic factor models have a wide range of applications in econometrics and applied economics. The basic motivation resides in their capability of reducing a large set of time series to only few indicators (factors). If the number of time series is large compared to the available number of observations then most information may be conveyed to the factors. This way low dimension models may be estimated for explaining and forecasting one or more time series of interest. It is desirable that outlier free time series be available for estimation. In practice, outlying observations are likely to ari…
Univariate time series often take the form of a collection of curves observed sequentially over time. Examples of these include hourly ground-level ozone concentration curves. These curves can be viewed as a time series of functions observed at equally spaced intervals over a dense grid. Since functional time series may contain various types of outliers, we introduce a robust functional time series forecasting method to down-weigh the influence of outliers in forecasting. Through a robust principal component analysis based on projection pursuit, a time series of functions can be decomposed int…
We present a new framework for robust estimation and inference on second-order stationary time series and random fields. This framework is based on the Generalized Method of Wavelet Moments which uses the wavelet variance to achieve parameter estimation for complex models. Using an M-estimator of the wavelet variance, this method can be made robust therefore allowing to estimate the parameters of a wide range of time series and spatial models when the data suffers from outliers or different forms of contamination. The paper presents a series of simulation studies as well as a range of applicat…
Time series often contain outliers and level shifts or structural changes. These unexpected events are of the utmost importance in fraud detection, as they may pinpoint suspicious transactions. The presence of such unusual events can easily mislead conventional time series analysis and yield erroneous conclusions. In this paper we provide a unified framework for detecting outliers and level shifts in short time series that may have a seasonal pattern. The approach combines ideas from the FastLTS algorithm for robust regression with alternating least squares. The double wedge plot is proposed, …
In this paper, we propose a robust data-driven process model whose hyperparameters are robustly estimated using the Schweppe-type generalized maximum likelihood estimator. The proposed model is trained on recorded time-series data of voltage phasors and power injections to perform a time-series stochastic power flow calculation. Power system data are often corrupted with outliers caused by large errors, fault conditions, power outages, and extreme weather, to name a few. The proposed model downweights vertical outliers and bad leverage points in the measurements of the training dataset. The we…
Zero adjusted regression models are used to fit variables that are discrete at zero and continuous at some interval of the positive real numbers. Diagnostic analysis in these models is usually performed using the randomized quantile residual, which is useful for checking the overall adequacy of a zero adjusted regression model. However, it may fail to identify some outliers. In this work, we introduce a residual for outlier identification in zero adjusted regression models. Monte Carlo simulation studies and an application suggest that the residual introduced here has good properties and detec…
TerraNIS；University of Toulouse（图卢兹大学）；IRIT-INP-ENSEEIHT；TéSA；IRIT-UPS
This paper studies the detection of anomalous crop development at the parcel-level based on an unsupervised outlier detection technique. The experimental validation is conducted on rapeseed and wheat parcels located in Beauce (France). The proposed methodology consists of four sequential steps: 1) preprocessing of synthetic aperture radar (SAR) and multispectral images acquired using Sentinel-1 and Sentinel-2 satellites, 2) extraction of SAR and multispectral pixel-level features, 3) computation of parcel-level features using zonal statistics and 4) outlier detection. The different types of an…
Revenue management strongly relies on accurate forecasts. Thus, when extraordinary events cause outlier demand, revenue management systems need to recognise this and adapt both forecast and controls. Many passenger transport service providers, such as railways and airlines, control the sale of tickets through revenue management. State-of-the-art systems in these industries rely on analyst expertise to identify outlier demand both online (within the booking horizon) and offline (in hindsight). So far, little research focuses on automating and evaluating the detection of outlier demand in this c…
Wigner Research Center for Physics（维格纳物理研究中心）；Semmelweis University（塞梅尔维斯大学）
Recognition of anomalous events is a challenging but critical task in many scientific and industrial fields, especially when the properties of anomalies are unknown. In this paper, we introduce a new anomaly concept called "unicorn" or unique event and present a new, model-free, unsupervised detection algorithm to detect unicorns. The key component of the new algorithm is the Temporal Outlier Factor (TOF) to measure the uniqueness of events in continuous data sets from dynamic systems. The concept of unique events differs significantly from traditional outliers in many aspects: while repetitiv…
In this paper, we use convolutional neural networks to address the problem of model identification for autoregressive moving average time series models. We compare the performance of several neural network architectures, trained on simulated time series, with likelihood based methods, in particular the Akaike and Bayesian information criteria. We find that our neural networks can significantly outperform these likelihood based methods in terms of accuracy and, by orders of magnitude, in terms of speed.
We present an approach to clustering time series data using a model-based generalization of the K-Means algorithm which we call K-Models. We prove the convergence of this general algorithm and relate it to the hard-EM algorithm for mixture modeling. We then apply our method first with an AR($p$) clustering example and show how the clustering algorithm can be made robust to outliers using a least-absolute deviations criteria. We then build our clustering algorithm up for ARMA($p,q$) models and extend this to ARIMA($p,d,q$) models. We develop a goodness of fit statistic for the models fitted to …
This article introduces a general class of heavy-tailed autoregressions for modeling integer-valued time series with outliers. The proposed specification is based on a heavy-tailed mixture of negative binomial distributions that features an observation-driven dynamic equation for the conditional expectation. The existence of a unique stationary and ergodic solution for the class of autoregressive processes is shown under a general contraction condition. The estimation of the model can be easily performed by Maximum Likelihood given the closed form of the likelihood function. The strong consist…
Recent paper "TVOR: Finding Discrete Total Variation Outliers Among Histograms" [arXiv:2012.11574] introduces the Total Variation Outlier Recognizer (TVOR) method for identification of outliers among a given set of histograms. After providing a theoretical discussion of the method and verifying its success on synthetic and population census data, it applies the TVOR model to histograms of ages of Holocaust victims produced using United States Holocaust Memorial Museum data. It purports to identify the list of victims of the Jasenovac concentration camp as potentially suspicious. In this commen…
Data cleaning is a crucial part of every data analysis exercise. Yet, the currently available R packages do not provide fast and robust methods for cleaning and preparation of time series data. The open source package tsrobprep introduces efficient methods for handling missing values and outliers using model based approaches. For data imputation a probabilistic replacement model is proposed, which may consist of autoregressive components and external inputs. For outlier detection a clustering algorithm based on finite mixture modelling is introduced, which considers time series properties in t…
Over the past years Robust PCA has been established as a standard tool for reliable low-rank approximation of matrices in the presence of outliers. Recently, the Robust PCA approach via nuclear norm minimization has been extended to matrices with linear structures which appear in applications such as system identification and data series analysis. At the same time it has been shown how to control the rank of a structured approximation via matrix factorization approaches. The drawbacks of these methods either lie in the lack of robustness against outliers or in their static nature of repeated b…
University of A Coruña（拉科鲁尼亚大学）；Sapienza University of Rome（罗马第一大学）；Technological Institute for Industrial Mathematics (ITMATI)（工业数学技术研究所）
Three robust methods for clustering multivariate time series from the point of view of generating processes are proposed. The procedures are robust versions of a fuzzy C-means model based on: (i) estimates of the quantile cross-spectral density and (ii) the classical principal component analysis. Robustness to the presence of outliers is achieved by using the so-called metric, noise and trimmed approaches. The metric approach incorporates in the objective function a distance measure aimed at neutralizing the effect of the outliers, the noise approach builds an artificial cluster expected to co…
Periodicity detection is an important task in time series analysis, but still a challenging problem due to the diverse characteristics of time series data like abrupt trend change, outlier, noise, and especially block missing data. In this paper, we propose a robust and effective periodicity detection algorithm for time series with block missing data. We first design a robust trend filter to remove the interference of complicated trend patterns under missing data. Then, we propose a robust autocorrelation function (ACF) that can handle missing values and outliers effectively. We rigorously pro…
Time series analysis is used to understand and predict dynamic processes, including evolving demands in business, weather, markets, and biological rhythms. Exponential smoothing is used in all these domains to obtain simple interpretable models of time series and to forecast future values. Despite its popularity, exponential smoothing fails dramatically in the presence of outliers, large amounts of noise, or when the underlying time series changes. We propose a flexible model for time series analysis, using exponential smoothing cells for overlapping time windows. The approach can detect and r…
Extracting the underlying trend signal is a crucial step to facilitate time series analysis like forecasting and anomaly detection. Besides noise signal, time series can contain not only outliers but also abrupt trend changes in real-world scenarios. To deal with these challenges, we propose a robust trend filtering algorithm based on robust statistics and sparse learning. Specifically, we adopt the Huber loss to suppress outliers, and utilize a combination of the first order and second order difference on the trend component as regularization to capture both slow and abrupt trend changes. Fur…
A popular data-driven method for choosing the bandwidth in standard kernel regression is cross-validation. Even when there are outliers in the data, robust kernel regression can be used to estimate the unknown regression curve [Robust and Nonlinear Time Series Analysis. Lecture Notes in Statist. (1984) 26 163--184]. However, under these circumstances standard cross-validation is no longer a satisfactory bandwidth selector because it is unduly influenced by extreme prediction errors caused by the existence of these outliers. A more robust method proposed here is a cross-validation method that d…
Identification of unexpectedly high values in a time series is useful for epidemiologists, economists, and other social scientists interested in the effect of an exposure spike on an outcome variable. However, the best method to identify spikes in time series is not known. This paper aims to fill this gap by testing the performance of several spike detection methods in a simulation setting. We created simulations parameterized by monthly violence rates in nine California cities that represented different series features, and randomly inserted spikes into the series. We then compared the abilit…
In this article there is no intention to repeat basic concepts about risk management, but we will try to define why often is usefull the time series analysis during the assessment of risks, and how is possible to compute a significative analysis using regression and autoregression. After some basic concepts about trend analysis, will be introduced some methods to identify peaks. This is often usefull when there is no need to use the full time series, because sometimes is more practical to focus only on the extremes. With a correct time series without not-anomalous data, the extremes time serie…
We propose a simple multiple outlier identification method for parametric location-scale and shape-scale models when the number of possible outliers is not specified. The method is based on a result giving asymptotic properties of extreme z-scores. Robust estimators of model parameters are used defining z-scores. An extensive simulation study was done for comparing of the proposed method with existing methods. For the normal family, the method is compared with the well known Davies-Gather, Rosner's, Hawking's and Bolshev's multiple outlier identification methods. The choice of an upper limit f…
Covariate Informed Identification of Heterogeneity and Outliers in Longitudinal Data
University of Louisville（路易斯维尔大学）；Case Western Reserve University（凯斯西储大学）；Mass General Brigham, Harvard Medical School（麻省总医院布里格姆医院，哈佛医学院）；University of Missouri（密苏里大学）
A new technique for the detection of outliers in contingency tables is introduced. Outliers thereby are unexpected cell counts with respect to classical loglinear Poisson models. Subsets of cell counts called minimal patterns are defined, corresponding to non-singular design matrices and leading to potentially uncontaminated maximum-likelihood estimates of the model parameters and thereby the expected cell counts. A criterion to easily produce minimal patterns in the two-way case under independence is derived, based on the analysis of the positions of the chosen cells. A simulation study and a…
We introduce a trimmed version of the Hill estimator for the index of a heavy-tailed distribution, which is robust to perturbations in the extreme order statistics. In the ideal Pareto setting, the estimator is essentially finite-sample efficient among all unbiased estimators with a given strict upper break-down point. For general heavy-tailed models, we establish the asymptotic normality of the estimator under second order regular variation conditions and also show it is minimax rate-optimal in the Hall class of distributions. We also develop an automatic, data-driven method for the choice of…
Near real time change detection is important for a variety of Earth monitoring applications and remains a high priority for remote sensing science. Data sparsity, subtle changes, seasonal trends, and the presence of outliers make detecting actual landscape changes challenging. Adams and MacKay (2007) introduced Bayesian Online Changepoint Detection (BOCPD), a computationally efficient, exact Bayesian method for change detection. Incorporation of prior information allows for relaxed dependence on dense data and an extensive stable period, making this method applicable to relatively short time s…
Smart metering infrastructures collect data almost continuously in the form of fine-grained long time series. These massive data series often have common daily patterns that are repeated between similar days or seasons and shared among grouped meters. Within this context, we propose an unsupervised method to highlight individuals with abnormal daily dependency patterns, which we term evolution outliers. To this end, we approach the problem from the standpoint of Functional Data Analysis (FDA) and we use the concept of functional depth to exploit the dynamic group structure and isolate individu…
In real-world application scenarios, the identification of groups poses a significant challenge due to possibly occurring outliers and existing noise variables. Therefore, there is a need for a clustering method which is capable of revealing the group structure in data containing both outliers and noise variables without any pre-knowledge. In this paper, we propose a $k$-means-based algorithm incorporating a weighting function which leads to an automatic weight assignment for each observation. In order to cope with noise variables, a lasso-type penalty is used in an objective function adjusted…
