Maximum likelihood estimation of Poisson and Hawkes processes and extensions to Hawkes process analysis

The purpose of this work is to improve our ability to extract information from data generated by Poisson and Hawkes processes. Our principal focus is to provide improvements to the theory for parameter estimation under these models. To this end, we present novel bounds on the estimation of model parameters for linear Poisson processes. Earlier results applied to Poisson counting processes, but we improve upon these while addressing the broader class of Poisson arrival processes. For Hawkes processes, we present asymptotic parameter estimation bounds. These provide an enhanced understanding of how the structure of a Hawkes process affects our ability to estimate its parameters. We explore the capability of Hawkes processes to model telecommunication networks and to aid in the discovery of connections within such networks. We also discuss the considerations and extensions that should be employed when utilizing this model for this application. Finally, we discuss a way that the Hawkes process can be used to recognize event cascades within a network, rather than mere nodal connectivity. This is particularly relevant to telecommunication applications, as relay nodes often serve as intermediaries between others.

Date

2018-08-22

Resource Type

Text

Resource Subtype

Dissertation

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