Tail Behavior for Bivariate Distributions Based on Pareto Mixtures

The intensification of global warming under the influence of human activities has directly led to an increase in the magnitude of changes in the climate system, further exacerbating the impact on the global water cycle and making extreme weather events more frequent and intense. In this study, daily temperature data from 1960 to 2020 from nine meteorological stations in the Qingyi River basin were used to analyze the changes of 16 extreme temperature indicators using extreme temperature indicators, the trend analysis method, and the MK analysis method. The results show that in terms of spatial distribution, the colder extreme events in the basin mainly occur in the upstream and that the warmer extreme events mainly occur in the midstream and downstream. According to the temporal trends of the indicators, the indicators of extreme events with anomalous warming are dominated by a significant upward trend, in which the warm night index increases by up to 3.8 days/decade, whereas the indicators of extreme events with anomalous cold are dominated by a significant downward trend, in which the cold night index decreases by up to −3.4 days/decade. In terms of the magnitude of change in the indicators, the cold and night indicators are more variable. According the spatial difference of index changes, the extreme value index mainly shows a decreasing trend in the upstream of the basin and an increasing trend in the midstream and downstream. Combined with the characteristic of a low extreme value index in the upstream and high extreme value index in the midstream and downstream of the basin as a whole, the spatial difference in the extreme value index in the basin is expected to further increase.

Extreme value index and tail probability estimates for mixed distributions

Does bias reduction with external estimator of second order parameter work for endpoint?

Applying extreme value statistics in meteorology and environmental science requires accurate estimators on extreme value indices that can be around zero. Without having prior knowledge on the sign of the extreme value indices, the probability weighted moment (PWM) estimator is a favorable candidate. As most other estimators on the extreme value index, the PWM estimator bears an asymptotic bias. In this paper, we develop a bias correction procedure for the PWM estimator. Moreover, we provide bias-corrected PWM estimators for high quantiles and, when the extreme value index is negative, the endpoint of a distribution. The choice of k, the number of high order statistics used for estimation, is crucial in applications. The asymptotically unbiased PWM estimators allows the choice of higher level k, which results in a lower asymptotic variance. Moreover, since the bias-corrected PWM estimators can be applied for a wider range of k compared to the original PWM estimator, one gets more flexibility in choosing k for finite sample applications. All advantages become apparent in simulations and an environmental application on estimating “once per 10,000 years” still water level at Hoek van Holland, The Netherlands.

Transferring knowledge from one domain to another is challenging due to a number of reasons. Since both conditional and marginal distribution of the training data and test data are non-identical, model trained in one domain, when directly applied to a different domain, is usually low in accuracy. For many applications with large feature sets, such as text document, sequence data, medical data, image data of different resolutions, etc. two domains usually do not contain exactly the same features, thus introducing large numbers of when considered over the union of features from both domains. In other words, its marginal distributions are at most overlapping. In the same time, these problems are usually high dimensional, such as, several thousands of features. Thus, the combination of high dimensionality and missing values make the relationship in conditional probabilities between two domains hard to measure and model. To address these challenges, we propose a framework that first brings the marginal distributions of two domains closer by filling up those missing values of disjoint features. Afterwards, it looks for those comparable sub-structures in the latent-space as mapped from the expanded feature vector, where both marginal and conditional distribution are similar. With these sub-structures in latent space, the proposed approach then find common concepts that are transferable across domains with high probability. During prediction, unlabeled instances are treated as queries, the mostly related labeled instances from out-domain are retrieved, and the classification is made by weighted voting using retrieved out-domain examples. We formally show that importing feature values across domains and latent semantic index can jointly make the distributions of two related domains easier to measure than in original feature space, the nearest neighbor method employed to retrieve related out domain examples is bounded in error when predicting in-domain examples. Software and datasets are available for download.

Copula-based conditional tail indices

Let $(\mathcal{X}_1, \mathcal{Y}_1),\dots,(\mathcal{X}_n, \mathcal{Y}_n)$ be a random sample from a bivariate distribution function $F$ in the domain of max-attraction of a distribution function $G$. This $G$ is characterised by the two extreme value indices and its spectral or angular measure. The extreme value indices determine both the marginals and the spectral measure determines the dependence structure of $G$. One of the main issues in multivariate extreme value theory is the estimation of this spectral measure. We construct a truly nonparametric estimator of the spectral measure, based on the ranks of the above data. Under natural conditions we prove consistency and asymptotic normality for the estimator. In particular,the result is valid for all values of the extreme value indices. The theory of (local) empirical processes is indispensable here. The results are illustrated by an application to real data and a small simulation study.

When transport in networks follows the shortest paths, the union of all shortest path trees G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⋃spt</sub> can be regarded as the "transport overlay network". Overlay networks such as peer-to-peer networks or virtual private networks can be considered as a subgraph of G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⋃spt</sub> . We construct two types of G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⋃spt</sub> : (a) G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⋃spt(α)</sub> where α is the extreme value index of polynomial link weights and (b) G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⋃spt(ρ)</sub> where ρ is the correlation coefficient of the 2-dimensional correlated uniformly distributed link weights in QoS routing. By tuning the extreme value index α of polynomial link weights, a phase transition occurs around a critical extreme value index α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> of the link weight distribution. If α > α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> , transport in the network traverses many links whereas for α < α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> , all transport flows over a critical backbone: the minimum spanning tree (MST). In QoS routing with 2-dimensional link weights, as we decrease the correlation coefficient ρ from 1 to -1, the overlay G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⋃spt</sub> becomes denser, and is equal to the substrate when ρ = -1. With the Erdos-Renyi random graph as the underlying topology, we show that the overlay G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⋃spt(ρ)</sub> is also close to an Erdos-Renyi random graph G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (N), an observation with potential for mobile and wireless ad-hoc networks. The existence of such a controllable transition in the overlay structure may allow network operators to steer and balance flows in their network.

In extreme value analysis, the extreme value index plays a vital role as it determines the tail heaviness of the underlying distribution and is the primary parameter required for the estimation of other extreme events. In this paper, we review the estimation of the extreme value index when observations are subject to right random censoring and the presence of covariate information. In addition, we propose some estimators of the extreme value index, including a maximum likelihood estimator from a perturbed Pareto distribution. The existing estimators and the proposed ones are compared through a simulation study under identical conditions. The results show that the performance of the estimators depend on the percentage of censoring, the underlying distribution, the size of extreme value index and the number of top order statistics. Overall, we found the proposed estimator from the perturbed Pareto distribution to be robust to censoring, size of the extreme value index and the number of top order statistics.

The objective of this paper is to demonstrate the practical construction of a multivariate probability distribution function using an actual soil database containing su(CIUC), OCR, and four piezocone parameters. Five hundred and thirty-five multivariate data points were compiled from 40 clay sites around the world (Brazil, Canada, Hong Kong, Italy, Malaysia, Norway, Singapore, Sweden, UK, USA, and Venezuela). It was found that a multivariate probability distribution can be constructed by transforming each component of a multivariate normal distribution to a Johnson distribution. Existing bivariate regression equations focus on strong correlations. Weak correlations are typically discarded. Site investigation is a costly exercise and ideally, one should exploit all measured geotechnical data for design. The multivariate distribution is a concise model to summarize all available information. Conditional distributions can be easily derived to update the marginal distribution of any one parameter or the multivariate distribution of any group of parameters given information from other parameters. One of the objectives of site investigation is to perform cost-effective field tests and to evaluate design parameters based on these field measurements. Clearly, conditioning involving updating one or more design parameters using one or more field measurements is a natural probabilistic generalization of the current soil property evaluation methodology. INTRODUCTION When multivariate geotechnical data exist in sufficient amount, it is of significant practical usefulness to construct a multivariate probability distribution function. The applications include: (a) deriving the mean and coefficient of variation (COV) of any parameter given the information contained in a subset with possibly more than one parameter, and (b) evaluating if new strong pairwise correlations can be found either among the original components or some derived components. For the former, it is likely for the COV of a design parameter, say the undrained shear strength (su), to decrease when other parameters, say normalized cone tip resistance [(qtv)/ v] and overconsolidation (OCR), have been measured. This aspect is significant for reliability-based design. In fact, COV reduction can be viewed as a measure of the value of information and may eventually provide a sensible method for deciding if it is worthwhile to measure an additional parameter. For the latter, the ability to predict the existence of new correlations not included as part of model calibration provides a stronger scientific underpinning to correlation studies in geotechnical engineering. The reason is that these predictions can be falsified by taking new observations, which is the cornerstone of the scientific method. In other words, it is a lot harder to develop multivariate models, but if they do stand the test of time, they are usually more robust than bivariate models. The objective of this paper is to demonstrate the practical construction of a multivariate probability distribution function using an actual soil database containing su(CIUC), OCR, and

Adversarial domain adaptation is a powerful approach to transfer the knowledge of the label-rich source domain to the label-scarce target domain by mitigating domain shifts across distributions. Existing domain adaptation methods align either the marginal distribution with a single-domain discriminator or conditional distributions with multiple-domain discriminators. However, aligning both marginal (global) and conditional (local) distributions should be considered for domain adaptation. This paper proposes a novel adversarial distribution adaptation network (ADAN) to jointly reduce both the global and local distribution discrepancies between different domains for learning domain-invariant representations. ADAN utilizes a single-domain discriminator to adapt the global distribution between two domains, and source decision boundaries to align the local distributions between sub-domains. Furthermore, we extend our ADAN as improved ADAN (iADAN), in which we utilize a feature norm term to regularize the task-specific features to improve model generalization. Extensive experimental results show that our method outperforms other state-of-the-art domain adaptation methods on Office-Home and ImageCLEF-DA datasets.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2012 SIAM International Conference on Data Mining (SDM)Dual Transfer LearningMingsheng Long, Jianmin Wang, Guiguang Ding, Wei Cheng, Xiang Zhang, and Wei WangMingsheng Long, Jianmin Wang, Guiguang Ding, Wei Cheng, Xiang Zhang, and Wei Wangpp.540 - 551Chapter DOI:https://doi.org/10.1137/1.9781611972825.47PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Transfer learning aims to leverage the knowledge in the source domain to facilitate the learning tasks in the target domain. It has attracted extensive research interests recently due to its effectiveness in a wide range of applications. The general idea of the existing methods is to utilize the common latent structure shared across domains as the bridge for knowledge transfer. These methods usually model the common latent structure by using either the marginal distribution or the conditional distribution. However, without exploring the duality between these two distributions, these single bridge methods may not achieve optimal capability of knowledge transfer. In this paper, we propose a novel approach, Dual Transfer Learning (DTL), which simultaneously learns the marginal and conditional distributions, and exploits the duality between them in a principled way. The key idea behind DTL is that learning one distribution can help to learn the other. This duality property leads to mutual reinforcement when adapting both distributions across domains to transfer knowledge. The proposed method is formulated as an optimization problem based on joint nonnegative matrix trifactorizations (NMTF). The two distributions are learned from the decomposed latent factors that exhibit the duality property. An efficient alternating minimization algorithm is developed to solve the optimization problem with convergence guarantee. Extensive experimental results demonstrate that DTL is more effective than alternative transfer learning methods. Previous chapter Next chapter RelatedDetails Published:2012ISBN:978-1-61197-232-0eISBN:978-1-61197-282-5 https://doi.org/10.1137/1.9781611972825Book Series Name:ProceedingsBook Code:PRDT12Book Pages:1-1150Key words:Keywords Transfer Learning, Cross-Domain Classification, Dual Transfer Learning, Nonnegative Matrix Tri-Factorization

A multi-constrained domain adaptation network for remaining useful life prediction of bearings

Analysis of extreme rainfall events can be performed using two main approaches; fitting Generalized Extreme Value distribution to the yearly peaks of events in the observation period or the annual maximum series, and fitting Generalized Pareto distribution to the peaks of events that exceed a given threshold or the partial duration series. Even though partial duration series are able to reduce sampling uncertainty and are useful for analyzing extreme values and tail asymmetries, the series require an optimal threshold. The objective of this study is to compare and determine the best method for selecting the optimal threshold of partial duration series using hourly, 12-hour and 24-hour data of rainfall time series in Peninsular Malaysia. Nine semi-parametric second order reduced-bias estimators are applied to estimate extreme value index and six estimators are used for the external estimation of the second order parameter. A semi-parametric bootstrap is used to estimate mean square error of the estimator at each threshold and the optimal threshold is then selected based on the smallest mean square error. Based on the plots of extreme value index and mean square error, several second order reduced-bias estimators behave reasonably well compared to Hill estimator, as indicated by their stable sample paths and flatter mean square errors.

Due to the fact that for heavy tails the classical Hill estimator of a positive extreme value index is asymptotically biased, new and interesting alternative estimators have appeared in the literature. In this work we compare several classical estimators of the extreme value index based on moments of the upper order statistics. Since several alternative estimators have eventually a null asymptotic bias, for some heavy tailed models, the comparison is performed not only with the Hill and recent generalized means estimators but also with an asymptotically unbiased Hill estimator. The comparison study is performed asymptotically, under a third-order framework, and for finite samples, through a Monte Carlo simulation study.