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Modeling cyber risks has been an important but challenging task in the domain of cyber security. It is mainly because of the high dimensionality and heavy tails of risk patterns. Those obstacles have hindered the development of statistical modeling of the multivariate cyber risks. In this work, we propose a novel approach for modeling the multivariate cyber risks which relies on the deep learning and extreme value theory. The proposed model not only enjoys the high accurate point predictions via deep learning but also can provide the satisfactory high quantile prediction via extreme value theory. The simulation study shows that the proposed model can model the multivariate cyber risks very well and provide satisfactory prediction performances. The empirical evidence based on real honeypot attack data also shows that the proposed model has very satisfactory prediction performances., 25 pages

Causality is an intriguing concept that once tamed, can have many applications. While having been widely investigated in other domains, its relevance and usefulness in the cybersecurity domain has received little attention. In this paper, we present a systematic investigation of a particular approach to causality, known as Granger causality (G-causality), in cybersecurity. We propose a framework, dubbed Cybersecurity Granger Causality (CGC), for characterizing the presence of G-causality in cyber attack rate time series and for leveraging G-causality to predict (i.e., forecast) cyber attack rates. The framework offers a range of research questions, which can be adopted or adapted to study G-causality in other kinds of cybersecurity time series data. In order to demonstrate the usefulness of CGC, we present a case study by applying it to a particular cyber attack dataset collected at a honeypot. From this case study, we draw a number of insights into the usefulness and limitations of G-causality in the cybersecurity domain.

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 {\em 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 (LRD) phenomenon can predict the attack rate (i.e., the number of attacks per unit time) 1-hour 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 {\em extreme values} in statistics. This motivates us to analyze the {\em extreme-value phenomenon}, by using two complementary approaches: the Extreme Value Theory (EVT) and the Time Series Theory (TST). In this paper, we show that EVT can offer long-term predictions (e.g., 24-hour ahead-of-time), while gray-box TST models can predict attack rates 1-hour ahead-of-time with an accuracy of 86.0-87.9\%. 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.