In order to capture the dependence in the upper tail of a time series, we develop non-negative regularly-varying time series models that are constructed similarly to classical non-extreme ARMA models. Rather than fully characterizing tail dependence of the time series, we define the concept of weak tail stationarity which allows us to describe a regularly-varying time series through the tail pairwise dependence function (TPDF) which is a measure of pairwise extremal dependencies. We state consistency requirements among the finite-dimensional collections of the elements of a regularly-varying time series and show that the TPDF's value does not depend on the dimension being considered. So that our models take nonnegative values, we use transformed-linear operations. We show existence and stationarity of these models, and develop their properties such as the model TPDF's. Additionally, we show the class of transformed-linear MA($\infty$) models forms an inner product space. Motivated by investigating conditions conducive to the spread of wildfires, we fit models to hourly windspeed data and find that the fitted transformed-linear models produce better estimates of upper tail quantities than traditional ARMA models or than classical linear regularly varying models.
Accurate modelling of the joint extremal dependence structure within a stationary time series is a challenging problem that is important in many applications.\ Several previous approaches to this problem are only applicable to certain types of extremal dependence in the time series such as asymptotic dependence, or Markov time series of finite order.\ In this paper, we develop statistical methodology for time series extremes based on recent probabilistic results that allow us to flexibly model the decay of a stationary time series after witnessing an extreme event.\ While Markov sequences of f…
We propose a new framework for modeling high-dimensional matrix-variate time series by a two-way transformation, where the transformed data consist of a matrix-variate factor process, which is dynamically dependent, and three other blocks of white noises. Specifically, for a given $p_1\times p_2$ matrix-variate time series, we seek common nonsingular transformations to project the rows and columns onto another $p_1$ and $p_2$ directions according to the strength of the dynamic dependence of the series on the past values. Consequently, we treat the data as nonsingular linear row and column tran…
OLinear: A Linear Model for Time Series Forecasting in Orthogonally Transformed Domain
State Key Laboratory of General AI Peking University（人工智能通用基础研究国家重点实验室，北京大学）；School of Software, BNRist Tsinghua University（清华大学软件学院，北京人工智能研究院）；Center for Data Science Peking University（北京大学数据科学中心）；Department of Control Science and Engineering Zhejiang University（浙江大学控制科学与工程学院）
This study is concerned with estimating the inequality measures associated with the underlying hypothetical income distribution from the times series grouped data on the Lorenz curve. We adopt the Dirichlet pseudo likelihood approach where the parameters of the Dirichlet likelihood are set to the differences between the Lorenz curve of the hypothetical income distribution for the consecutive income classes and propose a state space model which combines the transformed parameters of the Lorenz curve through a time series structure. Furthermore, the information on the sample size in each survey …
We consider forecasting functional time series of extreme values within a generalised extreme value distribution (GEV). The GEV distribution can be characterised using the three parameters (location, scale and shape). As a result, the forecasts of the GEV density can be accomplished by forecasting these three latent parameters. Depending on the underlying data structure, some of the three parameters can either be modelled as scalars or functions. We provide two forecasting algorithms to model and forecast these parameters. To assess the forecast uncertainty, we apply a sieve bootstrap method t…
University of Southern California（南加州大学）；IBM T.J. Watson Research Center（IBM T.J. Watson研究中心）
In many applications of time series models, such as climate analysis and social media analysis, we are often interested in extreme events, such as heatwave, wind gust, and burst of topics. These time series data usually exhibit a heavy-tailed distribution rather than a Gaussian distribution. This poses great challenges to existing approaches due to the significantly different assumptions on the data distributions and the lack of sufficient past data on extreme events. In this paper, we propose the Sparse-GEV model, a latent state model based on the theory of extreme value modeling to automatic…
This paper considers a general class of parameter-driven models for time series of counts. A comprehensive simulation study is conducted to evaluate the accuracy and efficiency of three estimators: the maximum likelihood estimators of the generalized linear model, 2-state finite mixture model, and 2-state hidden Markov model. Standard errors for these estimators are derived. Our results show that except in extreme cases, the maximum likelihood estimator of the generalized linear model is an efficient, consistent and robust estimator with a well-behaved estimated standard error. The maximum lik…
We introduce a new type of estimator for the spectral tail process of a regularly varying time series. The approach is based on a characterizing invariance property of the spectral tail process, which is incorporated into the new estimator via a projection technique. We show uniform asymptotic normality of this estimator, both in the case of known and of unknown index of regular variation. In a simulation study the new procedure shows a more stable performance than previously proposed estimators.
Accurate forecasting of the U.K. gross value added (GVA) is fundamental for measuring the growth of the U.K. economy. A common nonstationarity in GVA data, such as the ABML series, is its increase in variance over time due to inflation. Transformed or inflation-adjusted series can still be challenging for classical stationarity-assuming forecasters. We adopt a different approach that works directly with the GVA series by advancing recent forecasting methods for locally stationary time series. Our approach results in more accurate and reliable forecasts, and continues to work well even when the…
In this paper, we investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns within a cluster. Since these patterns take only the ordinal structure of consecutive data points into account the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators a…
This paper provides extensions of the work on subsampling by Bertail et al. (2004) for strongly mixing case to weakly dependent case by application of the results of Doukhan and Louhichi (1999). We investigate properties of smooth and rough subsampling estimators for distributions of converging and extreme statistics when the underlying time series is η or λ-weakly dependent.
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…
We propose a contemporaneous bilinear transformation for a $p\times q$ matrix time series to alleviate the difficulties in modeling and forecasting matrix time series when $p$ and/or $q$ are large. The resulting transformed matrix assumes a block structure consisting of several small matrices, and those small matrix series are uncorrelated across all times. Hence an overall parsimonious model is achieved by modelling each of those small matrix series separately without the loss of information on the linear dynamics. Such a parsimonious model often has better forecasting performance, even when …
A measure of primal importance for capturing the serial dependence of a stationary time series at extreme levels is provided by the limiting cluster size distribution. New estimators based on a blocks declustering scheme are proposed and analyzed both theoretically and by means of a large-scale simulation study. A sliding blocks version of the estimators is shown to outperform a disjoint blocks version. In contrast to some competitors from the literature, the estimators only depend on one unknown parameter to be chosen by the statistician.
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…
The Value-at-Risk (VaR) is a widely used instrument in financial risk management. The question of estimating the VaR of loss return distributions at extreme levels is an important question in financial applications, both from operational and regulatory perspectives; in particular, the dynamic estimation of extreme VaR given the recent past has received substantial attention. We propose here a two-step bias-reduced estimation methodology called GARCH-UGH (Unbiased Gomes-de Haan), whereby financial returns are first filtered using an AR-GARCH model, and then a bias-reduced estimator of extreme q…
Einmahl, de Haan and Zhou (2016, Journal of the Royal Statistical Society: Series B, 78(1), 31-51) recently introduced a stochastic model that allows for heteroscedasticity of extremes. The model is extended to the situation where the observations are serially dependent, which is crucial for many practical applications. We prove a local limit theorem for a kernel estimator for the scedasis function, and a functional limit theorem for an estimator for the integrated scedasis function. We further prove consistency of a bootstrap scheme that allows to test for the null hypothesis that the extreme…
Identifying risk spillovers in financial markets is of great importance for assessing systemic risk and portfolio management. Granger causality in tail (or in risk) tests whether past extreme events of a time series help predicting future extreme events of another time series. The topology and connectedness of networks built with Granger causality in tail can be used to measure systemic risk and to identify risk transmitters. Here we introduce a novel test of Granger causality in tail which adopts the likelihood ratio statistic and is based on the multivariate generalization of a discrete auto…
Classical moment based change point tests like the cusum test are very powerful in case of Gaussian time series with one change point but behave poorly under heavy tailed distributions and corrupted data. A new class of robust change point tests based on cusum statistics of robustly transformed observations is proposed. This framework is quite flexible, depending on the used transformation one can detect for instance changes in the mean, scale or dependence of a possibly multivariate time series. Simulations indicate that this approach is very powerful in detecting changes in the marginal vari…
Xsensor Technology Corp（Xsensor科技公司）；University of Alberta（阿尔伯塔大学）
Multivariate time series classification is a high value and well-known problem in machine learning community. Feature extraction is a main step in classification tasks. Traditional approaches employ hand-crafted features for classification while convolutional neural networks (CNN) are able to extract features automatically. In this paper, we use dilated convolutional neural network for multivariate time series classification. To deploy dilated CNN, a multivariate time series is transformed into an image-like style and stacks of dilated and strided convolutions are applied to extract in and bet…
Modeling univariate block maxima by the generalized extreme value distribution constitutes one of the most widely applied approaches in extreme value statistics. It has recently been found that, for an underlying stationary time series, respective estimators may be improved by calculating block maxima in an overlapping way. A proof of concept is provided that the latter finding also holds in situations that involve certain piecewise stationarities. A weak convergence result for an empirical process of central interest is provided, and, as a case-in-point, further details are worked out explici…
Expectiles define the only law-invariant, coherent and elicitable risk measure apart from the expectation. The popularity of expectile-based risk measures is steadily growing and their properties have been studied for independent data, but further results are needed to use extreme expectiles with dependent time series such as financial data. In this paper we establish a basis for inference on extreme expectiles and expectile-based marginal expected shortfall in a general $β$-mixing context that encompasses ARMA, ARCH and GARCH models with heavy-tailed innovations. Simulations and applications …
Changes in extreme weather may produce some of the largest societal impacts of anthropogenic climate change. However, it is intrinsically difficult to estimate changes in extreme events from the short observational record. In this work we use millennial runs from the CCSM3 in equilibrated pre-industrial and possible future conditions to examine both how extremes change in this model and how well these changes can be estimated as a function of run length. We estimate changes to distributions of future temperature extremes (annual minima and annual maxima) in the contiguous United States by fitt…
It is increasingly being realised that many real world time series are not stationary and exhibit evolving second-order autocovariance or spectral structure. This article introduces a Bayesian approach for modelling the evolving wavelet spectrum of a locally stationary wavelet time series. Our new method works by combining the advantages of a Haar-Fisz transformed spectrum with a simple, but powerful, Bayesian wavelet shrinkage method. Our new method produces excellent and stable spectral estimates and this is demonstrated via simulated data and on differenced infant ECG data. A major addition…
In this paper, we discuss the application of extreme value theory in the context of stationary $β$-mixing sequences that belong to the Fréchet domain of attraction. In particular, we propose a methodology to construct bias-corrected tail estimators. Our approach is based on the combination of two estimators for the extreme value index to cancel the bias. The resulting estimator is used to estimate an extreme quantile. In a simulation study, we outline the performance of our proposals that we compare to alternative estimators recently introduced in the literature. Also, we compute the asymptoti…
Protecting Federated Learning from Extreme Model Poisoning Attacks via Multidimensional Time Series Anomaly Detection
Department of Computer, Control and Management Engineering, Sapienza University of Rome（计算机、控制与管理工程系，罗马萨皮恩扎大学）；National Research Council in Pisa（比萨国家研究理事会）；Department of Statistics, Ludwig-Maximilians-University of Munich（统计学系，慕尼黑路德维希-马克西米利安大学）；Department of Computer Science, Sapienza University of Rome（计算机科学系，罗马萨皮恩扎大学）
We propose a novel strategy for multivariate extreme value index estimation. In applications such as finance, volatility and risk present in the components of a multivariate time series are often driven by the same underlying factors, such as the subprime crisis in the US. To estimate the latent risk, we apply a two-stage procedure. First, a set of independent latent series is estimated using a method of latent variable analysis. Then, univariate risk measures are estimated individually for the latent series to assess their contribution to the overall risk. As our main theoretical contribution…
Extreme events are occurrences whose magnitude and potential cause extensive damage on people, infrastructure, and the environment. Motivated by the extreme nature of the current global health landscape, which is plagued by the coronavirus pandemic, we seek to better understand and model extreme events. Modeling extreme events is common in practice and plays an important role in time-series prediction applications. Our goal is to (i) compare and investigate the effect of some common extreme events modeling methods to explore which method can be practical in reality and (ii) accelerate the deep…
Single-mission altimetric water level observations of rivers are spatially and temporally limited, and thus they are often unable to quantify the full extent of extreme flood events. Moreover, only missions with a short-repeat orbit, such as Envisat, Jason-2, or SARAL, could provide meaningful time series of water level variations directly. However, long or non-repeat orbit missions such as CryoSat-2 have a very dense spatial resolution under the trade-off of a repeat time insufficient for time series extraction. Combining data from multiple altimeter missions into a multi-mission product allo…
Accurately predicting customer churn using large scale time-series data is a common problem facing many business domains. The creation of model features across various time windows for training and testing can be particularly challenging due to temporal issues common to time-series data. In this paper, we will explore the application of extreme gradient boosting (XGBoost) on a customer dataset with a wide-variety of temporal features in order to create a highly-accurate customer churn model. In particular, we describe an effective method for handling temporally sensitive feature engineering. T…
We consider a transformed Ornstein-Uhlenbeck process model that can be a good candidate for modelling real-life processes characterized by a combination of time-reverting behaviour with heavy distribution tails. We begin with presenting the results of an exploratory statistical analysis of the log prices of a major Australian public company, demonstrating several key features typical of such time series. Motivated by these findings, we suggest a simple transformed Ornstein-Uhlenbeck process model and analyze its properties showing that the model is capable of replicating our empirical findings…
We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a p-dimensional time series where the dimension p = p_n converges to infinity when the sample size n increases. We give a short overview of the literature on the topic both in the light- and heavy-tailed cases when the data have finite (infinite) fourth moment, respectively. Our main focus is on the heavytailed case. In this case, one has a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the iid case but also when rows and columns of the data are linearly…
This work proposes a novel method to robustly and accurately model time series with heavy-tailed noise, in non-stationary scenarios. In many practical application time series have heavy-tailed noise that significantly impacts the performance of classical forecasting models; in particular, accurately modeling a distribution over extreme events is crucial to performing accurate time series anomaly detection. We propose a Spliced Binned-Pareto distribution which is both robust to extreme observations and allows accurate modeling of the full distribution. Our method allows the capture of time depe…
German Research Center for Artificial Intelligence（德国人工智能研究中心）
The Industrial Internet of Things drastically increases connectivity of devices in industrial applications. In addition to the benefits in efficiency, scalability and ease of use, this creates novel attack surfaces. Historically, industrial networks and protocols do not contain means of security, such as authentication and encryption, that are made necessary by this development. Thus, industrial IT-security is needed. In this work, emulated industrial network data is transformed into a time series and analysed with three different algorithms. The data contains labeled attacks, so the performan…
Electronic records contain sequences of events, some of which take place all at once in a single visit, and others that are dispersed over multiple visits, each with a different timestamp. We postulate that fine temporal detail, e.g., whether a series of blood tests are completed at once or in rapid succession should not alter predictions based on this data. Motivated by this intuition, we propose models for analyzing sequences of multivariate clinical time series data that are invariant to this temporal clustering. We propose an efficient data augmentation technique that exploits the postulat…
Flexible spatial models that allow transitions between tail dependence classes have recently appeared in the literature. However, inference for these models is computationally prohibitive, even in moderate dimensions, due to the necessity of repeatedly evaluating the multivariate Gaussian distribution function. In this work, we attempt to achieve truly high-dimensional inference for extremes of spatial processes, while retaining the desirable flexibility in the tail dependence structure, by modifying an established class of models based on scale mixtures Gaussian processes. We show that the de…
Employing the framework of regular variation, we propose two decompositions which help to summarize and describel high-dimensional tail dependence. Via transformation, we define a vector space on the positive orthant, yielding the notion of basis. With a suitably-chosen transformation, we show that transformed-linear operations applied to regularly varying random vectors preserve regular variation. Rather than model regular-variation's angular measure, we summarize tail dependence via a matrix of pairwise tail dependence metrics. This matrix is positive semidefinite, and eigendecomposition all…
Both marginal and dependence features must be described when modelling the extremes of a stationary time series. There are standard approaches to marginal modelling, but long- and short-range dependence of extremes may both appear. In applications, an assumption of long-range independence often seems reasonable, but short-range dependence, i.e., the clustering of extremes, needs attention. The extremal index $0<θ\le 1$ is a natural limiting measure of clustering, but for wide classes of dependent processes, including all stationary Gaussian processes, it cannot distinguish dependent processes …
We present a new method for estimating the edge of a two-dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on Haar series and extreme values of the point process. We give conditions for various kind of convergence and we obtain remarkably different possible limit distributions. We propose a method of reducing the negative bias, illustrated by a simulation.
Block maxima methods constitute a fundamental part of the statistical toolbox in extreme value analysis. However, most of the corresponding theory is derived under the simplifying assumption that block maxima are independent observations from a genuine extreme value distribution. In practice however, block sizes are finite and observations from different blocks are dependent. Theory respecting the latter complications is not well developed, and, in the multivariate case, has only recently been established for disjoint blocks of a single block size. We show that using overlapping blocks instead…
At the end of April 20, 2020, there were only a few new COVID-19 cases remaining in China, whereas the rest of the world had shown increases in the number of new cases. It is of extreme importance to develop an efficient statistical model of COVID-19 spread, which could help in the global fight against the virus. We propose a clustering-segmented autoregressive sigmoid (CSAS) model to explore the space-time pattern of the log-transformed infectious count. Four key characteristics are included in this CSAS model, including unknown clusters, change points, stretched S-curves, and autoregressive …
