Ph.D Thesis

     My research deals with resampling marked point processes. A one-dimensional marked point process can be thought of as a generalized time series. That is, we have the arrival of points along the real line and at each observation, we have an associated value (or mark). In the case of a time series, the points are evenly spaced.

     In my work, I assume that the points are generated from an inhomogeneous Poisson process. That is, they follow a Poisson distribution with an intensity, λ(t), that changes over time. In higher dimensions, we can think of the points as occuring in space, such as the location of weather stations or oil deposits.

     My work is an extension of work done by my advisor, Dimitris Politis, back in 1999 titled "Resampling Marked Point Processes". In that paper, resampling methods for a homogeneous Poisson process are developed. (In a homogeneous Poisson process, the intensity is a constant, λ, for all t.)

     Karr (1986) developed the real-world theory for the homogeneous Poisson process in "Inference for Stationary Random Fields Given Poisson Samples". I extend his results to the inhomogeneous setting.

     We consider two methods for resampling an inhomogeneous marked (Poisson) process. In one-dimension, we can transform our inhomogeneous Poisson process to a homogeneous Poisson process and then use existing methods to resample the data. Alternatively, we can use what is known as a local block bootstrap to resample our data.

     A local block bootstrap is similar to a block bootstrap (where data is resampled in blocks), but we impose an additional restriction that the resampled block is "close" to the original block. This helps us to better capture trends in our data, such as places where the intensity changes. In higher dimensions, the local block bootstrap is the only method available to resample such data.

     We carry out a variety of simulations for our methods and we see that the local block bootstrap performs quite well, outperforming the transformation methods in most instances and having coverage probabilities close to the true confidence level.

Here is a link to my thesis: Resampling Inhomogeneous Marked Point Processes.

Here are the slides used as my Ph.D defense: Ph.D Defense Slides.



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