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

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

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.