Extremes: Short and Long-Range Dependence; Modeling and Inference with Applications to Computer Networks and Risk Analysis

这项研究计划解决在计算机网络、保险和金融风险数据的建模和分析中出现的问题。具体地说，它开发了全球网络模型，也处理了极值的聚集。还研究了一些相关的统计问题，如网络范围的预测、感兴趣参数的可辨识性以及Hurst指数、尾部指数和极值指数的有效估计。所提出的全局网络模型基于物理上可解释的自下而上方法，其中首先为每个源-目的网络节点对构建低层模型，然后聚合流量。在一定的限制条件下，当用户数增加时，通过适当的时间调整，得到了基于一类新的高斯过程--泛函分数布朗运动的全网业务量波动的极限近似.限制过程捕获由网络拓扑引起的流量依赖关系。在一个相关的方向上，对极值的聚类进行了研究，并研究了极值指数的关键参数的一些新的估计量。这为研究网络流量的突发性提供了一个新的视角。在此基础上，提出了一种灵活的极值之间时间的非渐近模型，该模型可以更好地预测极值发生的频率。当前的工作是受现代计算机网络中的问题的推动，在现代计算机网络中，人们对表征流量波动和突发性非常感兴趣，以便识别瓶颈链路并以路由故障或恶意活动的形式检测网络故障。所提出的全球网络范围的模型以原则性的方式考虑了网络的拓扑结构和时间相关性，使人们能够实现这些目标。此外，极端聚集现象的新方法的开发将被证明对评估网络流量中突发性的存在和影响是有用的。通过拟议的研究获得的理解和洞察将导致网络流量分析的基本原则的核心。通过将时间依赖性纳入极端金融损失，更好地理解和量化极端聚集现象将对衡量风险产生广泛影响。最后，建议的模型和技术将被集成到开源工具中。

This research program addresses problems arising in the modeling and analysis of computer network, insurance and financial risk data. Specifically, it develops global network models and also deals with clustering of extreme values. A number of associated statistical issues, such as network-wide prediction, identifiability of parameters of interest and efficient estimation of the Hurst, tail and extremal indices are also investigated. The proposed global network models are based on a physically interpretable 'bottom-up' approach, where first a low level model is constructed for each source-destination pair of network nodes and subsequently the trafficis aggregated. Under certain limiting regimes, when the number of users grows and with appropriate rescaling of time, a limit approximation of the fluctuations of the network-wide traffic is obtained that is based on functional fractional Brownian motion, a novel class of Gaussian processes. The limit process captures traffic dependencies induced by the topology of the network. In a related direction, the study of clustering of extreme values is undertaken and a number of new estimators for the key parameter of the extremal index are investigated. This provides a new perspective in the study of burstiness in network traffic. Further, a flexible non-asymptotic model of the times between extremes is proposed, which allows better prediction of the frequency at which extreme values occur.The current work is motivated by problems in modern computer networks, where there is a lot of interest in characterizing traffic fluctuations and burstiness, in order to identify bottleneck links and detect network failures in the form of routing faults or malicious activities. The proposed global network-wide models that take into consideration the network topology together with the temporal dependence in a principled manner allow one to achieve these goals. Further, the development of new methodology for the clustering-of-extremes phenomenon will prove useful in assessing the presence and impact of burstiness in network traffic. The understanding and insight gained as a result of the proposed research will lead to a core of basic principles for network traffic analysis. Understanding better and quantifying the clustering-of-extreme phenomenon will have a broad impact on measuring risk, by incorporating the temporal dependence in extreme financial losses. Finally, the proposed models and techniques will be integrated into open source tools.