#### Title

Using a Spiral to Estimate Spatial Lattice Model Parameters

#### Presentation Type

Poster

#### Faculty Mentor’s Full Name

Jon Graham

#### Faculty Mentor’s Department

Mathematics

#### 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 and labeled “diseased” or “non-diseased.” The presence of disease in one plant would either 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, such as 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 structure must be tabulated. However, for even a small lattice of size 20x20, this would mean 2400 different configurations, which is far too many for even a modern computer to compile. This normalizing constant can be computed directly using S, the number of diseased sites, and N, the number of neighboring diseased sites from each configuration. This project explored a method of generating all S and N combinations for a linear subset of the lattice, allowing for calculation of the normalizing constant for the subset. For data on a spatial lattice, a spiral of locations can be “pulled out,” and the spiral’s exact normalizing constant calculated. Unfortunately one spiral uses only half of the data so it 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.

#### Category

Physical Sciences

Using a Spiral to Estimate Spatial Lattice Model Parameters

South UC Ballroom

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 and labeled “diseased” or “non-diseased.” The presence of disease in one plant would either 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, such as 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 structure must be tabulated. However, for even a small lattice of size 20x20, this would mean 2400 different configurations, which is far too many for even a modern computer to compile. This normalizing constant can be computed directly using S, the number of diseased sites, and N, the number of neighboring diseased sites from each configuration. This project explored a method of generating all S and N combinations for a linear subset of the lattice, allowing for calculation of the normalizing constant for the subset. For data on a spatial lattice, a spiral of locations can be “pulled out,” and the spiral’s exact normalizing constant calculated. Unfortunately one spiral uses only half of the data so it 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.