The energy distance and energy scores became important tools in multivariate statistics and multivariate probabilistic forecasting in recent years. They are both based on the expected distance of two independent samples. In this paper we study dependence uncertainty bounds for these quantities under the assumption that we know the marginals but do not know the dependence structure. We find some interesting sharp analytic bounds, where one of them is obtained for an unusual spherically symmetric copula. These results should help to better understand the sensitivity of these measures to misspecifications in the copula.
We propose an extension of the univariate Lorenz curve and of the Gini coefficient to the multivariate case, i.e., to simultaneously measure inequality in more than one variable. Our extensions are based on copulas and measure inequality stemming from inequality in every single variable as well as inequality stemming from the dependence structure of the variables. We derive simple nonparametric estimators for both instruments and apply them exemplary to data of individual income and wealth for various countries.
A multivariate score-driven filter is developed to extract signals from noisy vector processes. By assuming that the conditional location vector from a multivariate Student's t distribution changes over time, we construct a robust filter which is able to overcome several issues that naturally arise when modeling heavy-tailed phenomena and, more in general, vectors of dependent non-Gaussian time series. We derive conditions for stationarity and invertibility and estimate the unknown parameters by maximum likelihood (ML). Strong consistency and asymptotic normality of the estimator are proved an…
In this paper, we obtain an upper bound for the Gini mean difference based on mean, variance and correlation for the case when the variables are correlated. We also derive some closed-form expressions for the Gini mean difference when the random variables have an absolutely continuous joint distribution. We then examine some particular examples based on elliptically contoured distributions, and specifically multivariate normal and Student-$t$ distributions.
Multivariate time series prediction has attracted a lot of attention because of its wide applications such as intelligence transportation, AIOps. Generative models have achieved impressive results in time series modeling because they can model data distribution and take noise into consideration. However, many existing works can not be widely used because of the constraints of functional form of generative models or the sensitivity to hyperparameters. In this paper, we propose ScoreGrad, a multivariate probabilistic time series forecasting framework based on continuous energy-based generative m…
In recent years, probabilistic forecasting is an emerging topic, which is why there is a growing need of suitable methods for the evaluation of multivariate predictions. We analyze the sensitivity of the most common scoring rules, especially regarding quality of the forecasted dependency structures. Additionally, we propose scoring rules based on the copula, which uniquely describes the dependency structure for every probability distribution with continuous marginal distributions. Efficient estimation of the considered scoring rules and evaluation methods such as the Diebold-Mariano test are d…
North Carolina State University（北卡罗来纳州立大学）；Intuit Inc.（财捷公司）
Robust anomaly detection is a requirement for monitoring complex modern systems with applications such as cyber-security, fraud prevention, and maintenance. These systems generate multiple correlated time series that are highly seasonal and noisy. This paper presents a novel unsupervised deep learning architecture for multivariate time series anomaly detection, called Robust Seasonal Multivariate Generative Adversarial Network (RSM-GAN). It extends recent advancements in GANs with adoption of convolutional-LSTM layers and an attention mechanism to produce state-of-the-art performance. We condu…
As developing countries continue to face challenges associated with infectious diseases, the need to improve infrastructure to systematically collect data which can be used to understand their outbreak patterns becomes more critical. The World Health Organization (WHO) Integrated Disease Surveillance and Response (IDSR) strategy seeks to drive the systematic collection of surveillance data to strengthen district-level reporting and to translate them into public health actions. Since the analysis of this surveillance data at the central levels of government in many developing nations has tradit…
The current literature in probabilistic forecasting is focused on quantifying the uncertainty of each random variable individually. This leads to the failure in informing about interdependence structure of uncertainty at different locations and/or different lead times. When there is a positive or negative association between a number of random variables, the prediction regions for them should be reflected by multivariate or joint uncertainty sets. The existing literature is very primitive in the area of multivariate uncertainty sets modeling. In this paper, uncertainty regions are generated in…
The distance standard deviation, which arises in distance correlation analysis of multivariate data, is studied as a measure of spread. The asymptotic distribution of the empirical distance standard deviation is derived under the assumption of finite second moments. Applications are provided to hypothesis testing on a data set from materials science and to multivariate statistical quality control. The distance standard deviation is compared to classical scale measures for inference on the spread of heavy-tailed distributions. Inequalities for the distance variance are derived, proving that the…
National University of Singapore（新加坡国立大学）；University of California, Berkeley（加州大学伯克利分校）；ST Electronics (Info Security) Pte Ltd（新科电子（信息安全）私人有限公司）
The prevalence of networked sensors and actuators in many real-world systems such as smart buildings, factories, power plants, and data centers generate substantial amounts of multivariate time series data for these systems. The rich sensor data can be continuously monitored for intrusion events through anomaly detection. However, conventional threshold-based anomaly detection methods are inadequate due to the dynamic complexities of these systems, while supervised machine learning methods are unable to exploit the large amounts of data due to the lack of labeled data. On the other hand, curre…
Emory University（埃默里大学）；Amazon Research（亚马逊研究院）；Zalando SE
We propose Multivariate Quantile Function Forecaster (MQF$^2$), a global probabilistic forecasting method constructed using a multivariate quantile function and investigate its application to multi-horizon forecasting. Prior approaches are either autoregressive, implicitly capturing the dependency structure across time but exhibiting error accumulation with increasing forecast horizons, or multi-horizon sequence-to-sequence models, which do not exhibit error accumulation, but also do typically not model the dependency structure across time steps. MQF$^2$ combines the benefits of both approache…
Even though the Rao's score tests are classical tests, such as the likelihood ratio tests, their application has been avoided until now in a multivariate framework, in particular high-dimensional setting. We consider they could play an important role for testing high-dimensional data, but currently the classical Rao's score tests for an arbitrary but fixed dimension remain being still not very well-known for tests on correlation matrices of multivariate normal distributions. In this paper, we illustrate how to create Rao's score tests, focussed on testing correlation matrices, showing their as…
There has been a wide interest to extend univariate and multivariate nonparametric procedures to clustered and hierarchical data. Traditionally, parametric mixed models have been used to account for the correlation structures among the dependent observational units. In this work we extend multivariate nonparametric procedures for one-sample and several samples location problems to clustered data settings. The results are given for a general score function, but with an emphasis on spatial sign and rank methods. Mixed models notation involving design matrices for fixed and random effects is used…
High-dimensional tests are applied to find relevant sets of variables and relevant models. If variables are selected by analyzing the sums of products matrices and a corresponding mean-value test is performed, there is the danger that the nominal error of first kind is exceeded. In the paper, well-known multivariate tests receive a new mathematical interpretation such that the error of first kind of the combined testing and selecting procedure can more easily be kept. The null hypotheses on mean values are replaced by hypotheses on distributional sphericity of the individual score responses. T…
Distribution-free tests such as the Wilcoxon rank sum test are popular for testing the equality of two univariate distributions. Among the important reasons for their popularity are the striking results of Hodges-Lehmann (1956) and Chernoff-Savage (1958), where the authors show that the asymptotic (Pitman) relative efficiency of Wilcoxon's test with respect to Student's $t$-test, under location-shift alternatives, never falls below $0.864$ (with the identity score) and $1$ (with the Gaussian score) respectively, despite the former being exactly distribution-free for all sample sizes. Motivated…
Quality of Life (QOL) outcomes are important in the management of chronic illnesses. In studies of efficacies of treatments or intervention modalities, QOL scales multi-dimensional constructs are routinely used as primary endpoints. The standard data analysis strategy computes composite (average) overall and domain scores, and conducts a mixed-model analysis for evaluating efficacy or monitoring medical conditions as if these scores were in continuous metric scale. However, assumptions of parametric models like continuity and homoscedastivity can be violated in many cases. Furthermore, it is e…
