Bivariate kernel estimation produces a smooth estimate of the intensity (i.e., mean number of events per unit area) of a spatial point process across a study region (Bailey and Gatrell, 1995). A spatial point process is one in which underlying forces or systems (e.g., social, economic, etc.) produce point events in geographical space. The approach adopted in this report has been used elsewhere to characterize spatial patterns of point events including crime (Levine, 2006), the location of firms (Maoh and Kanaroglou, 2007), environmental processes, and the incidence of disease (Bailey and Gatrell, 1995). Kernel estimation is used in this study to explore regional variation in the spatial intensity of retail development by format type across the GTA and through time. The approach is exploratory and empirical, and does not produce predictions or forecasts of retail square footage across space.
In certain contexts it is sufficient to use unweighted cases; however, when studying a process such as retail trade, where floorspace correlates with attraction potential, weighting retail location data by some measure of attractiveness can be instructive. In this study, each retail location was assigned a weight equivalent to the location's retail square footage. The weighted bivariate kernel density applied in this research takes the form:
where K{} is a bivariate probability density function referred to as the kernel (e.g., Gaussian, quartic, triangular), which in this example corresponds to the Gaussian, wi is a weight attached to each retail location i (retail square footage), Ii is the intensity of the spatial point process at each observed retail location (i.e., the mean number of retail locations per unit area), x,y are planar x and y coordinates, and s is a scale parameter or bandwidth (in measurement units). Kernel surfaces are estimated separately for each of the three retail formats, and for each year of the study period (1996-2005). Adding the temporal dimension facilitates exploration of the extent to which patterns of development have changed both spatially and temporally.
There are several decisions left to the analyst when applying kernel estimation. First, the kernel function K{} can take one of several possible forms (e.g., Gaussian, quartic, triangular), and second, a decision needs to be taken regarding the value for the scale parameter, s (Bailey and Gatrell, 1995; Levine, 2006). The choice regarding K{} can be guided by the application context. For example, regional-scale analyses are potentially better suited to the application of the Gaussian function, because the function returns estimates of spatial intensity for locations across the study area (Levine, 2006). With respect to the scale parameter, the degree of smoothing is influenced by the specification of s, with larger values providing a smoother estimate of intensity (i.e., points per unit area, or the value of some weight attached to each point - i.e., retail square footage, per unit area).
Because the Gaussian has been chosen for the kernel function, s is equivalent to the standard deviation of a bivariate normal probability distribution. The bivariate case implies there is a requirement for the specification of s in both the x and y axes. Here it is assumed that s is directionally invariant or isotropic (s = sx = sy). While empirical approaches to identify an appropriate bandwidth have been introduced to the literature (Rowlingson and Diggle, 1993; Bailey and Gatrell, 1995; Levine, 2006), selection can also arise from a quasi-empirical approach where the analyst qualitatively evaluates successive interpolations. A decision regarding the scale parameter will be influenced by the degree to which a particular value of s provides an instructive view of the spatial process under examination. This latter approach was adopted for this study.
A demonstrative example of the input and result of the kernel estimation process is shown in Figure A.1. (s = 4,500 m). A bivariate kernel estimate has been created for the spatial pattern of power centre retail locations identified for the year 2000. The estimation was carried out using the weighted Gaussian bivariate kernel function (density.ppp) implemented in the spatstat library developed by Baddley and Turner (2005) for the R language. Each event has been weighted by its retail square footage, with the resulting surface providing an informative cross-section view of the multiple foci of power retailing in the GTA. Units for the density result have been transformed from retail square feet per unit area (i.e., pixel area), to retail square feet only -- a matter of convenience that permits a more intuitive assessment of the figure's contents (summing across all pixels yields an estimate of total retail square footage within the retail system for the estimation year). The kernel estimation approach offers visual clarification of the geographical proximity of retail locations, and the heterogeneous distribution of retail capacity over space. The resulting surface highlights regional trends in power centre retailing, clearly indicating greater intensity of retail development in the outer suburbs of the study area.
Figure A.1: Weighted Gaussian bivariate kernel estimation (units are retail sq. ft.)
