Boundary problem (spatial analysis): Difference between revisions

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== Suggested solutions and evaluations on the solutions ==

Several strategies for resolving geographic boundary problems in measurement and analysis have been proposed.<ref>Martin, R. J. (1987) Some comments on correction techniques for boundary effects and missing value techniques. Geographical Analysis 19, 273–282.</ref><ref name=Wong_Fotheringham90>Wong, D. W. S., and Fotheringham, A. S. (1990) Urban systems as examples of bounded chaos: exploring the relationship between fractal dimension, rank-size and rural-to-urban migration. Geografiska Annaler 72, 89–99.</ref> To identify the effectiveness of the strategies, Griffith reviewed traditional techniques that were developed to mitigate the edge effects:<ref name="Griffith83"/> ignoring the effects, undertaking a torus mapping, construction of an empirical butterbuffer zone, construction of an artificial butterbuffer zone, extrapolation into a buffer zone, utilizing a correction factor, etc. The first method (i.e., the ignorance of the edge effects), assumes andan infinite surface in which the edge effects do not occur. In fact, this approach has been used by traditional geographical theories (e.g., [[central place theory]]). Its main shortcoming is that empirical phenomena occur within a finite area, so an infinite and homogeneous surface is unrealistic.<ref name=Griffith_Amrhein83/> The remaining five approaches are similar in that they attempted to produce unbiased parameter estimation, that is, to provide a medium by which the edge effects are removed.<ref name="Griffith83"/> (He called these ''operational solutions'' as opposed to ''statistical solutions'' to be discussed below.) Specifically, the techniques aim at a collection of data beyond the boundary of the study area and fit a larger model, that is, mapping over the area or over-bounding the study area.<ref>Ripley, B. D. (1979) Tests of "randomness" for spatial point patterns. Journal of the Royal Statistical Society, Series B 41, 368–374.</ref><ref name=Wong_Fotheringham90/> Through simulation analysis, however, Griffith and Amrhein identified the inadequacy of such an overbounding technique.<ref name=Griffith_Amrhein83/> Moreover, this technique can bring about issues related to large-area statistics, that is, ecological fallacy. By expanding the boundary of the study area, micro-scale variations within the boundary can be ignored.

As alternatives to operational solutions, Griffith examined three correction techniques (i.e., ''statistical solutions'') in removing boundary-induced bias from inference.<ref name="Griffith83"/> They are (1) based on [[generalized least squares]] theory, (2) using dummy variables and a regression structure (as a way of creating a buffer zone), and (3) regarding the boundary problem as a missing values problem. However, these techniques require rather strict assumptions about the process of interest.<ref>Yoo, E.-H. and Kyriakidis, P. C. (2008) Area-to-point prediction under boundary conditions. Geographical Analysis 40, 355–379.</ref> For example, the solution according to the generalized least squares theory utilizes time-series modeling that needs an arbitrary transformation matrix to fit the multidirectional dependencies and multiple boundary units found in geographical data.<ref name=Griffith80/> Martin also argued that some of the underlying assumptions of the statistical techniques are unrealistic or unreasonably strict.<ref>Martin, R. J. (1989) The role of spatial statistical processes in geographic modeling. In D. A. Griffith (ed) Spatial Statistics: Past, Present, and Future. Institute of Mathematical Geography: Syracuse, NY, pp.&nbsp;107–129.</ref> Moreover, Griffith (1985) himself also identified the inferiority of the techniques through simulation analysis.<ref>Griffith, D. A. (1985) An evaluation of correction techniques for boundary effects in spatial statistical analysis: contemporary methods. Geographical Analysis 17, 81–88.</ref>