Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

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An Introduction To The Theory Of Point Processes

A first course in stochastic processes

(2003). Comparison Methods for Stochastic Models and Risks. Technometrics: Vol. 45, No. 4, pp. 370-371.

Comparison Methods for Stochastic Models and Risks

Dependence modeling with copulas

Rigorously characterizing the statistical properties of cyber attacks is an important problem. In this paper, we propose the first statistical framework for rigorously analyzing honeypot-captured cyber attack data. The framework is built on the novel concept of stochastic cyber attack process, a new kind of mathematical objects for describing cyber attacks. To demonstrate use of the framework, we apply it to analyze a low-interaction honeypot dataset, while noting that the framework can be equally applied to analyze high-interaction honeypot data that contains richer information about the attacks. The case study finds, for the first time, that long-range dependence (LRD) is exhibited by honeypot-captured cyber attacks. The case study confirms that by exploiting the statistical properties (LRD in this case), it is feasible to predict cyber attacks (at least in terms of attack rate) with good accuracy. This kind of prediction capability would provide sufficient early-warning time for defenders to adjust their defense configurations or resource allocations. The idea of “gray-box” (rather than “black-box”) prediction is central to the utility of the statistical framework, and represents a significant step towards ultimately understanding (the degree of) the predictability of cyber attacks.

Characterizing Honeypot-Captured Cyber Attacks: Statistical Framework and Case Study

Analyzing cyber incident data sets is an important method for deepening our understanding of the evolution of the threat situation. This is a relatively new research topic, and many studies remain to be done. In this paper, we report a statistical analysis of a breach incident data set corresponding to 12 years (2005–2017) of cyber hacking activities that include malware attacks. We show that, in contrast to the findings reported in the literature, both hacking breach incident inter-arrival times and breach sizes should be modeled by stochastic processes, rather than by distributions because they exhibit autocorrelations. Then, we propose particular stochastic process models to, respectively, fit the inter-arrival times and the breach sizes. We also show that these models can predict the inter-arrival times and the breach sizes. In order to get deeper insights into the evolution of hacking breach incidents, we conduct both qualitative and quantitative trend analyses on the data set. We draw a set of cybersecurity insights, including that the threat of cyber hacks is indeed getting worse in terms of their frequency, but not in terms of the magnitude of their damage.

Modeling and Predicting Cyber Hacking Breaches

Let X1, …, Xn be independent random variables with Xi having survival function Fλi, i = 1, …, n, and let Y1, …,Yn be a random sample with common population survival distribution Fλ, where λ=Σi=1 nλi/n. Let Xn:n and Yn:n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:n is greater than Yn:n in terms of likelihood ratio order. It is also proved that the sample range Xn:n X1:n is larger than Yn:n Y1:n according to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo [6] and Kochar and Rojo [11], respectively.

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Stochastic comparisons of parallel systems when components have proportional hazard rates

It is important to understand to what extent, and in what perspectives, cyber attacks can be predicted. Despite its evident importance, this problem was not investigated until very recently, when we proposed using the innovative methodology of gray-box prediction. This methodology advocates the use of gray-box models, which accommodate the statistical properties/phenomena exhibited by the data. Specifically, we showed that gray-box models that accommodate the long-range dependence phenomenon can predict the attack rate (i.e., the number of attacks per unit time) 1-h ahead-of-time with an accuracy of 70.2%–82.1%. To the best of our knowledge, this is the first result showing the feasibility of prediction in this domain. We observe that the prediction errors are partly caused by the models’ incapability in predicting the large attack rates, which are called extreme values in statistics. This motivates us to analyze the extreme-value phenomenon, using two complementary approaches: 1) the extreme value theory (EVT) and 2) the time series theory (TST). In this paper, we show that EVT can offer long-term predictions (e.g., 24-h ahead-of-time), while gray-box TST models can predict attack rates 1-h ahead-of-time with an accuracy of 86%–87.9%. We explore connections between the two approaches, and point out future research directions. Although our prediction study is based on specific cyber attack data, our methodology can be equally applied to analyze any cyber attack data of its kind.

Predicting Cyber Attack Rates With Extreme Values

Background
Pharmacy databases are commonly used to assess medication usage, and a number of measures have been developed to measure patients’ adherence to medication. An extensive literature now supports these measures, although few studies have systematically compared the properties of different adherence measures.

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Maochao Xu

Papers

Characterizing Honeypot-Captured Cyber Attacks: Statistical Framework and Case Study

Modeling and Predicting Cyber Hacking Breaches

Stochastic comparisons of parallel systems when components have proportional hazard rates

Predicting Cyber Attack Rates With Extreme Values

Comparison of pharmacy-based measures of medication adherence.