| Summary: | Semi-Markov processes have become increasingly important in probability and statistical modeling, which have found applications in traffic analysis, reliability and maintenance, survival analysis, performance evaluation, biology, DNA analysis, risk processes, insurance and finance, earthquake modeling, etc. In the first part of this thesis, our focus is on applying semi-Markov processes to modeling the on-off duty cycles of different nodes in wireless networks. More specifically, we are interested in restoration of statistics of individual occupancy patterns of specific users based on wireless RF observation traces. In particular, we present a novel approach to finding the statistics of several operations, namely down-sampling, superposition and mislabelling, of a discrete time semi-Markov process in terms of the sojourn time distributions and states transition matrix of the resulting process. The resulting process, after those operations, is also a semi-Markov processes or a Markov renewal process. We show that the statistics of the original sequence before the superposition operation of two semi Markov processes can be generally recovered. However the statistics of the original sequence cannot be recovered under the down-sampling operation, namely the probability transition matrix and the sojourn time distribution properties are distorted after the down-sampling. Simulation and numerical results further demonstrate the validity of our theoretical findings. Our results thus provide a more profound understanding on the limitation of applying semi-Markov models in characterizing and learning the dynamics of nodes' activities in wireless networks.
In the second portion of the thesis a review is provided about several graphical models that have been widely used in literature recently to characterize the relationships between different users in social networks, the influence of the neighboring nodes in the networks or the semantic similarity in different contexts.
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