Joseph Glaz

Scan statistics and applications

Scan statistics are used in many areas of science and technology to analyze the occurence of observed clusters of events in time and space. The goal is to determine whether an observed cluster of events occurred by chance if it is assumed that the observed events follow a specified probability model. Scan Statistics and Applications is a comprehensive, edited survey that brings together the work of leading authorities to present the most current advances in theory and methodology for this new area of statistical research and application. The chapters contain broad coverage of theory and new analytical and computational methods and techniques in four categories: introductory survey, discrete scan statistics, continuous scan statistics, and applications. Features and Topics: * Comprehensive introductory survey chapter * Discrete scan statistics * Finite Markov chain imbedding * Continuous scan statistics * Spatial scan statistics * Applications in DNA sequence analysis * Monte Carlo approaches to testing order statistics and spacing The book is a valuable resource and state-of-the-art reference for all practitioners, researchers, and professionals in applied probability and statistics who use scan statistics in their work.

Simultaneous Confidence Intervals and Sample Size Determination for Multinomial Proportions

Abstract Simultaneous confidence interval procedures for multinomial proportions are used in many areas of science. In this article two new simultaneous confidence interval procedures are introduced. Numerical results are presented to evaluate these procedures and compare their performance with established methods that have been used in statistical literature. From the results presented in this article, it is evident that the new procedures are more accurate than the established ones, where the accuracy of the procedure is measured by the volume of the confidence region corresponding to the nominal coverage probability and the probability of coverage it achieves. In the sample size determination problem, the new procedures provide a sizable amount of savings as compared to the procedures that have been used in many applications. Because both procedures performed equally well, the procedure that requires the least amount of computing time is recommended.

Approximations and Bounds for the Distribution of the Scan Statistic

Abstract The scan statistic is used in many areas of science to test the null hypothesis of uniformity against a clustering alternative. This article derives approximations and bounds for the distribution of scan statistics. An extensive simulation study is carried out to compare the approximations and the bounds derived in this article with other known approximations and bounds. The new lower and upper bounds that have been derived are very useful in evaluating the approximations and the simulation. A quantity of average number of points in the scanning interval of length d, μd , is introduced to study the approximations for the distribution of the scan statistic. For low values of μd the approximation derived in this article is the most accurate one. The importance of the scan statistics arises from their applications in many areas, including nuclear physics, geology, radio-optics, photography, and epidemiology.

Two-dimensional discrete scan statistics☆

This article investigates the accuracy of approximations for distributions of two-dimensional discrete scan statistics. A product-type approximation, a Bonferroni-type inequality, two Poisson approximations and a compound Poisson approximation are studied. A simulation study is presented to evaluate the accuracy of these approximations. Based on the numerical results it is evident that the standard Poisson approximation can be significantly improved.

Probability Inequalities for Multivariate Distributions with Dependence Structures

Abstract Let X 1, …, Xn be a sequence of random variables with a given positive or negative dependence structure. In this article we exploit the assumed dependence structure to construct a sequence of bounds for the P(Xi e Ci ; i = 1, …, n), where Ci are infinite intervals of the same type. These bounds are superior to the well-known product bounds that are based solely on the marginal probabilities. Moreover, the new bounds can serve as respectable approximations for the P(Xi e Ci ; i = 1, …, n).

Distributed Target Detection in Sensor Networks Using Scan Statistics

We introduce a sequential procedure to detect a target with distributed sensors in a two dimensional region. The detection is carried out in a mobile fusion center which successively counts the number of binary decisions reported by local sensors lying inside its moving field of view. This is a two-dimensional scan statistic-an emerging tool from the statistics field that has been applied to a variety of anomaly detection problems such as of epidemics or computer intrusion, but that seems to be unfamiliar to the signal processing community. We show that an optimal size of the field of view exists. We compare the sequential two-dimensional scan statistic test and two other tests. Results for system level detection are presented.

Multiple clusters on the line

Given N events occurring over time, define an n:t cluster as n consecutive events all contained within an interval of length t. In this paper we derive the expectation, variance and approximate distribution of the number of n:t clusters. The results have applications in epidemiological studies of rare diseases.

Simultaneous confidence intervals for multinomial proportions

In this article approximate parametric bootstrap confidence intervals for functions of multinomial proportions are discussed. The interesting feature of these confidence intervals is that they are obtained via an Edgeworth expansion approximation for the rectangular multinomial probabilities rather than the resampling approach. In the first part of the article simultaneous confidence intervals for multinomial proportions are considered. The parametric bootstrap confidence interval appears to be the most accurate procedure. The use of this parametric bootstrap confidence region in the sample size determination problem is also discussed. In the second part of the article approximate parametric bootstrap equal-tailed confidence intervals for the minimum and maximum multinomial cell probabilities are derived. Numerical results based on a simulation study are presented to evaluate the performance of these confidence intervals. We also indicate several problems for possible future research in this area.

A Martingale Approach to Scan Statistics

Scan statistics are commonly used in biology, medicine, engineering and other fields where interest is in the probability of observing clusters of events in a window at an unknown location. Due to the dependent nature of the number of events in a large number of overlapping window locations, even approximate solutions for the simplest scan statistics may require elaborate calculations. We propose a new martingale method which allows one to approximate the distribution for a wide variety of scan statistics, including some for which analytical results are computationally infeasible.

Multiple Window Discrete Scan Statistics

In this article, multiple scan statistics of variable window sizes are derived for independent and identically distributed 0-1 Bernoulli trials. Both one and two dimensional, as well as, conditional and unconditional cases are treated. The advantage in using multiple scan statistics, as opposed to single fixed window scan statistics, is that they are more sensitive in detecting a change in the underlying distribution of the observed data. We show how to derive simple approximations for the significance level of these testing procedures and present numerical results to evaluate their performance.