Date of Completion

4-8-2020

Embargo Period

10-5-2020

Keywords

Forward algorithm, Likelihood estimation, Markov process, Occupation time

Major Advisor

Jun Yan

Co-Major Advisor

Vladimir Pozdnyakov

Associate Advisor

Richard Vitale

Associate Advisor

Zhiyi Chi

Associate Advisor

Haim Bar

Field of Study

Statistics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

Statistical modeling of animal movement is of critical importance in addressing fundamental questions about space use, movement, resource selection, and behavior in animal ecology. The explosion of telemetry data on animal movement from the recent advancements in tracking and observation technologies presents a storm of opportunities and challenges.

A moving-resting-handling (MRH) process is introduced to allow predators to have two different motionless states, resting, and handling. In essence, it is a Brownian motion whose infinitesimal variance changes according to a three-state continuous-time Markov Chain. The Markov Chain can be viewed as a telegraph process with one on state and two off states. We derive the distribution of the occupation times of this Markov Chain and develop a maximum likelihood estimation procedure when the stochastic process at hand is observed at discrete, possibly irregularly spaced time points. The likelihood function is evaluated with forward algorithm in the general framework of hidden Markov models.

The second objective of this thesis is to introduce measurement errors to a recently proposed moving-resting (MR) process. MR process is a Brownian motion governed by a telegraph process, which allows periods of inactivity in one state of the telegraph process. It is promising in modeling the movements of predators with long inactive periods such as mountain lions, but the lack of accommodation of measurement errors seriously prohibits its applications in practice. Here we incorporate measurement errors in the MR model and derive basic properties of the model.

Finally, convolutions of independent gamma variables are encountered in many applications, including animal movement modeling. Accurate and fast evaluations of their density and distribution functions are critical for such applications. We review several numerical evaluations of convolution of independent gamma variables and compare them with respect to their accuracy and speed. We also derive a new computationally efficient formula for the probability mass function of the number of renewals by a given time.

Two R packages smam, coga provide efficient C++ based implementations of the discussed methods and are available in CRAN.

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