## Undergraduate Theses and Professional Papers

2015

May

#### Document Type

Professional Paper

Bachelor of Arts

#### School or Department

Mathematical Sciences

#### Major

Mathematics – Statistics

Jon Graham

#### Faculty Mentor Department

Mathematical Sciences

#### Keywords

binary, autologistic, spatial

#### Abstract

This project explores the autologistic model for spatially correlated binary lattice data and uses a one-dimensional spiral to approximate two-dimensional data. An example of this type of data is the presence of disease in plants in a lattice framework. Each plant is labeled “diseased” or “non-diseased,” where the presence of disease in one plant might increase, decrease or not affect the likelihood of disease in a neighboring plant. In order to fit an autologistic model to real data, the method of maximum likelihood would ideally be used to estimate the model parameters for the entire lattice. However, the model form involves an intractable normalizing constant preventing this method from being used directly. Although multiple methods have been developed to estimate the model parameters, most notably Markov Chain Monte Carlo (MCMC) maximum likelihood, these methods either rely on approximations of the normalizing constant or ignore the inherent spatial correlation. To calculate the constant directly, every possible lattice realization must be tabulated. However, for even a small lattice of size 20x20, this would mean 2400 different realizations, which is far too many for even a modern computer to compile. This normalizing constant can be computed in theory using two statistics computed from the data: S, the number of diseased sites, and N, the number of neighboring diseased sites from each realization. This project explored a method of generating all S and N combinations for a linearized subset of the two-dimensional lattice, allowing for calculation of the normalizing constant for the subset. For data on a spatial lattice, a spiral of locations can be extracted and an exact normalizing constant for the spiral calculated. Unfortunately one spiral uses only half of the data so must be combined with results from the remaining locations. Further investigation is being done to compare this method to known approximation methods in order to determine its viability.

Yes

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