Density estimation and adaptive bandwidths: a primer for public health practitioners

Abstract

Background: Geographic information systems have advanced the ability to both visualize and analyze point data. While point-based maps can be aggregated to differing areal units and examined at varying resolutions, two problems arise 1) the modifiable areal unit problem and 2) any corresponding data must be available both at the scale of analysis and in the same geographic units. Kernel density estimation (KDE) produces a smooth, continuous surface where each location in the study area is assigned a density value irrespective of arbitrary administrative boundaries. We review KDE, and introduce the technique of utilizing an adaptive bandwidth to address the underlying heterogeneous population distributions common in public health research.

Results: The density of occurrences should not be interpreted without knowledge of the underlying population distribution. When the effect of the background population is successfully accounted for, differences in point patterns in similar population areas are more discernible; it is generally these variations that are of most interest. A static bandwidth KDE does not distinguish the spatial extents of interesting areas, nor does it expose patterns above and beyond those due to geographic variations in the density of the underlying population. An adaptive bandwidth method uses background population data to calculate a kernel of varying size for each individual case. This limits the influence of a single case to a small spatial extent where the population density is high as the bandwidth is small. If the primary concern is distance, a static bandwidth is preferable because it may be better to define the "neighborhood" or exposure risk based on distance. If the primary concern is differences in exposure across the population, a bandwidth adapting to the population is preferred.

Conclusions: Kernel density estimation is a useful way to consider exposure at any point within a spatial frame, irrespective of administrative boundaries. Utilization of an adaptive bandwidth may be particularly useful in comparing two similarly populated areas when studying health disparities or other issues comparing populations in public health.

Figure 1

Point density. Equal values at all locations within the neighborhood (the circle) around the case (star in center).

Figure 2

Kernel Density Estimation. The decay function is illustrated with the highest values located under the case giving way to lower values.

Figure 3

3 D rendering of KDE.

Figure 4

Map of alcohol outlets.

Figure 5

Kernel Density Estimation of alcohol outlets.

Figure 6

Texas at night, as seen from space[23].

Figure 7

Adaptive Bandwidth Kernel Density Estimation of alcohol outlets.

Figure 8

Two overlapping cases and the related density surface. In A, the case in the upper right has a smaller spatial impact, as the expected population was reached earlier than the case in the lower left. B is a rotated 3 dimensional image of the surface shown in A.

Figure 9

Cartographic comparison of density estimation of alcohol outlets near San Antonio, Texas. A-D show San Antonio in the center and Austin in the upper right with more rural areas to the south and west, E-G are zoomed in to show just San Antonio. A and E illustrate KDE using a static bandwidth of ~10 km. B and F illustrate KDE using an adaptive bandwidth with an expected population of 1,000 people and a maximum distance of ~25 km. C shows a LandScan™ dataset where each pixel represents a population count. D shows point data representing alcohol outlets. G is a map of census tracts showing the percentage of families below the poverty level.