Change-Point Detection Using Directional Derivatives

Allan B. Clark, A. Lawson
University of South Carolina School of Public Health, Columbia, SC

Corresponding author: Andrew B. Lawson, Department of Epidemiology and Biostatistics, Norman J. Arnold School of Public Health, 800 Sumter St., Columbia, SC 29208. Telephone: 803-777-6647; Fax: 803-777-2524; E-mail: alawson@gwm.sc.edu.

Abstract

Introduction: Individual-level disease maps, which estimate the risk for disease across a geographic region, usually are based on observing a set of spatial locations of cases and controls. This study examined the extension of where cases and controls form a space-time point process (locations and dates) and focused on assessing whether the intensity of cases changed across time.

Objectives: The objectives of the study were to develop a method for detecting changes in individual-level disease maps. The method was applied to a data set of birth abnormalities in the United Kingdom, which included locations and times of all the live (singleton) births and abnormalities during a 5-year period.

Methods: The change in intensity across time was measured through the directional derivative, with respect to time, of the geotemporal surface. This can be estimated nonparametrically and is used to check for both sudden and gradual changes over time.

Results: The results do not demonstrate the descriptive ability of an approach that relies on a map of changes in risk. The directional derivative was computed at 10 update points, corresponding to when data were available, and summary statistics were produced (Table). Isolated departures from constant risk were indicated, but the measure aggregated over a map did not demonstrate any change.

Conclusions: A directional derivative approach might yield optimal answers and is worthy of further research. The example data (Table) demonstrate that isolated changes are occurring, but data aggregated over a map did not indicate any change. Therefore, the geography of the problem should be considered, but an analysis that is aggregated over geography should not be performed.

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