Spatio-Temporal Low Count Processes with Application to Violent Crime Events

There is significant interest in being able to predict where crimes will happen, for example to aid in the efficient tasking of police and other protective measures. We aim to model both the temporal and spatial dependencies often exhibited by violent crimes in order to make such predictions. The temporal variation of crimes typically follows patterns familiar in time series analysis, but the spatial patterns are irregular and do not vary smoothly across the area. Instead we find that spatially disjoint regions exhibit correlated crime patterns. It is this indeterminate inter-region correlation structure along with the low-count, discrete nature of counts of serious crimes that motivates our proposed forecasting tool. In particular, we propose to model the crime counts in each region using an integer-valued first order autoregressive process. We take a Bayesian nonparametric approach to flexibly discover a clustering of these region-specific time series. We then describe how to account for covariates within this framework. Both approaches adjust for seasonality. We demonstrate our approach through an analysis of weekly reported violent crimes in Washington, D.C. between 2001-2008. Our forecasts outperform standard methods while additionally providing useful tools such as prediction intervals

The Power to See: A New Graphical Test of Normality

Many statistical procedures assume the underlying data generating process involves Gaussian errors. Among the well-known procedures are ANOVA, multiple regression, linear discriminant analysis and many more. There are a few popular procedures that are commonly used to test for normality such as the Kolmogorov-Smirnov test and the ShapiroWilk test. Excluding the Kolmogorov-Smirnov testing procedure, these methods do not have a graphical representation. As such these testing methods offer very little insight as to how the observed process deviates from the normality assumption. In this paper we discuss a simple new graphical procedure which provides confidence bands for a normal quantile-quantile plot. These bands define a test of normality and are much narrower in the tails than those related to the Kolmogorov-Smirnov test. Correspondingly the new procedure has much greater power to detect deviations from normality in the tails

Forecasting Evolving Curves

This dissertation describes methodologies for forecasting and testing integer valued time series that consist of Poisson counts. In Chapter 2 we look at univariate time series in which the counts are large and can be approximated using Gaussian models. These models are motivated by data gathered from the call center of a large financial institute. The goal of the models in this application is to predict accurately the one-day-ahead arrival process of incoming calls to the call center. A secondary goal is to provide a Bayesian algorithm to dynamically update these forecasts as more data becomes available. To test the underlying Gaussian assumption we develop in Chapter 1 a simple new graphical procedure which provides confidence bands for a normal quantile-quantile plot. These bands define a test of normality which is both a powerful and visually insightful tool compared to the common used testing procedures. In Chapter 3 we develop a Bayesian nonparametric model for multivariate Poisson time series which have small counts. We demonstrate this model using simulations and by forecasting violent crimes pattern across Washington D.C

Forecasting Evolving Curves

This dissertation describes methodologies for forecasting and testing integer valued time series that consist of Poisson counts. In Chapter 2 we look at univariate time series in which the counts are large and can be approximated using Gaussian models. These models are motivated by data gathered from the call center of a large financial institute. The goal of the models in this application is to predict accurately the one-day-ahead arrival process of incoming calls to the call center. A secondary goal is to provide a Bayesian algorithm to dynamically update these forecasts as more data becomes available. To test the underlying Gaussian assumption we develop in Chapter 1 a simple new graphical procedure which provides confidence bands for a normal quantile-quantile plot. These bands define a test of normality which is both a powerful and visually insightful tool compared to the common used testing procedures. In Chapter 3 we develop a Bayesian nonparametric model for multivariate Poisson time series which have small counts. We demonstrate this model using simulations and by forecasting violent crimes pattern across Washington D.C