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## Definition

As distinct from the **demarcation problem** or boundary problem in the philosophy of science, **boundary problem in spatial analysis** refers to a geographical phenomenon as one of the four major issues that interfere with an accurate estimation of the statistical parameter; the four issues include the boundary problem as well as the scale problem, pattern problem (or spatial autocorrelation), and modifiable areal unit problem (Barber 1988). The boundary problem occurs because of the loss of neighbors in analyses that depend on the values of the neighbors. While geographic phenomena are measured and analyzed within a specific unit, identical spatial data can appear either dispersed or clustered depending on the boundary placed around the data. In analysis with point data, dispersion is evaluated as dependent of the boundary. In analysis with areal data, statistics should be interpreted based upon the boundary.

In geographical research, two types of areas are taken into consideration in relation to the boundary: an area surrounded by fixed natural boundaries (e.g., coastlines or streams), outside of which neighbors do not exist (Henley 1981), or an area included in a larger region defined by arbitrary artificial boundaries (e.g., an air pollution boundary in modeling studies or an urban boundary in population migration) (Haining 1990). In an area isolated by the natural boundaries, the spatial process discontinues at the boundaries. In contrast, if a study area is delineated by the artificial boundaries, the process continues beyond the area.

If a spatial process in an area occurs beyond a study area or has an interaction with neighbors outside artificial boundaries, the most common approach is to neglect the influence of the boundaries and assume that the process occurs at the internal area. However, such an approach leads to a significant model misspecification problem (Upton and Fingleton 1985).

That is, for measurement or administrative purposes, geographic boundaries are drawn, but the boundaries per se can bring about different spatial patterns in geographic phenomena (BESR 2002). It has been reported that the difference in the way of drawing the boundary significantly affects identification of the spatial distribution and estimation of the statistical parameters of the spatial process (Cressie 1992; Fotheringham and Rogerson 1993; Griffith 1983; Martin 1987). The difference is largely based on the fact that spatial processes are generally unbounded or fuzzy-bounded (Leung 1987) but the processes are expressed in data imposed within boundaries for analysis purposes (Miller 1999). Although the boundary problem was discussed in relation to artificial and arbitrary boundaries, the effect of the boundaries also occurs according to natural boundaries as long as it is ignored that properties at sites on the natural boundary such as streams are likely to differ from those at sites within the boundary (Martin 1989).

The boundary problem occurs with regard not only to horizontal boundaries but also to vertically drawn boundaries according to delineations of heights or depths (Pineda 1993). For example, biodiversity such as the density of species of plants and animals is high near the surface, so if the identically divided height or depth is used as a spatial unit, it is more likely to find fewer number of the plant and animal species as the height or depth increases.

**Figure 2.** Boundary problem: horizontal boundaries

**Figure 3.** Boundary problem: vertical boundaries

## Types and examples of the boundary problem

By drawing a boundary around a study area, two types of problems in measurement and analysis takes place (Fotheringham and Rogerson 1993). The first is an **edge effect**. This effect originates from the ignorance of interdependences that occur outside the bounded region. Griffith (1980; 1983) and Griffith and Amrhein (1983) highlighted problems according to the edge effect. A typical example is a cross-boundary influence such as cross-border jobs, services and other resources located in a neighboring municipality (Macquire 1995).

The second is a **shape effect** that results from the artificial shape delineated by the boundary. As an illustration of the effect of the artificial shape, point pattern analysis tends to provide higher levels of clustering for the identical point pattern within a unit that is more elongated (Fotheringham and Rogerson 1993). Similarly, the shape can influence interaction and flow among spatial entities (Arlinghaus and Nystuen 1990; Ferguson and Kanaroglou 1998; Griffith 1982). For example, the shape can affect the measurement of origin-destination flows since these are often recorded when they cross an artificial boundary. Because of the effect set by the boundary, the shape and area information is used to estimate travel distances from surveys (Rogerson 1990) or to locate traffic counters, travel survey stations, or traffic monitoring systems (Kirby 1997). From the same perspective, Theobald (2001; retrieved from BESR 2002) argued that measures of urban sprawl should consider interdependences and interactions with nearby rural areas.

In spatial analysis, the boundary problem has been discussed along with the modifiable areal unit problem (MAUP) inasmuch as MAUP is associated with the arbitrary geographic unit and the unit is defined by the boundary (Rogerson 2006). For administrative purposes, data for policy indicators are usually aggregated within larger units (or enumeration units) such as census tracts, school districts, municipalities and counties. The artificial units serve the purposes of taxation and service provision. For example, municipalities can effectively respond to the need of the public in their jurisdictions. However, in such spatially aggregated units, spatial variations of detailed social variables cannot be identified. The problem is noted when the average degree of a variable and its unequal distribution over space are measured (BESR 2002).

## Suggested solutions and evaluations on the solutions

Several strategies for resolving geographic boundary problems in measurement and analysis have been proposed (Martin 1987; Wong and Fotheringham 1990). To identify the effectiveness of the strategies, Griffith (1983) reviewed traditional techniques that were developed to mitigate the edge effects: ignoring the effects, undertaking a torus mapping, construction of an empirical butter zone, construction of an artificial butter zone, extrapolation into a buffer zone, utilizing a correction factor, etc. The first method (i.e., the ignorance of the edge effects), assumes and 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 (Griffith and Amrhein 1983). 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 (Griffith 1983). (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 (Ripley 1979; Wong and Fotheringham 1990). Through simulation analysis, however, Griffith and Amrhein (1983) identified the inadequacy of such an overbounding technique. 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.

**Figure 4.** A solution to the boundary problem: overbounding

As alternatives to operational solutions, Griffith (1983) examined three correction techniques (i.e., **statistical solutions**) in removing boundary-induced bias from inference. 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 (Yoo and Kyriakidis 2008). 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 (Griffith 1980). Martin (1987) also argued that some of the underlying assumptions of the statistical techniques are unrealistic or unreasonably strict. Moreover, Griffith (1985) himself also identified the inferiority of the techniques through simulation analysis.

As particularly applicable using GIS technologies (Haslett et al. 1990; Openshaw et al. 1987), a possible solution for addressing both edge and shape effects is to an re-estimation of the spatial or process under repeated random realizations of the boundary. This solution provides an experimental distribution that can be subjected to statistical tests (Fotheringham and Rogerson 1993). As such, this strategy examines the sensitivity in the estimation result according to changes in the boundary assumptions. With GIS tools, boundaries can be systematically manipulated. The tools then conduct the measurement and analysis of the spatial process in such differentiated boundaries. Accordingly, such a sensitivity analysis allows the evaluation of the reliability and robustness of place-based measures that defined within artificial boundaries (BESR 2002). In the meantime, the changes in the boundary assumptions refer not only to altering or tilting the angles of the boundary, but also differentiating between the boundary and interior areas in examination and considering a possibility that isolated data collection points close to the boundary may show large variances.

**Figure 5.** A solution to the boundary problem: sensitivity analysis

## See also

- Central place theory
- Demarcation problem ("boundary problem" in the philosophy of science)
- Generalized least squares
- Geographic information system
- Modifiable areal unit problem
- Sensitivity analysis
- Spatial analysis
- Spatial autocorrelation

## References

- Arlinghaus, S. L. and Nystuen, J. D. (1990) Geometry of boundary exchanges. Geographical Review 80, 21–31.
- Barber, G. M. (1988) Elementary Statistics for Geographers. Guilford Press: New York, NY.
- BESR (2002) Community and Quality of Life: Data Needs for Informed Decision Making. Board on Earth Sciences and Resources: Washington, DC.
- Cressie, N. (1992) Statistics for Spatial Data. John Wiley and Sons: New York, NY.
- Ferguson, M. R. and Kanaroglou, P. S. (1998) Representing the shape and orientation of destinations in spatial choice models. Geographical Analysis 30, 119–137.
- Fotheringham, A. S. and Rogerson, P. A. (1993) GIS and spatial analytical problems. International Journal of Geographical Information Systems 7, 3–19.
- Griffith, D. (1980) Towards a theory of spatial statistics. Geographical Analysis 12, 325–339.
- Griffith, D. (1983) The boundary value problem in spatial statistics. Journal of Regional Science 23, 377–387.
- Griffith, D. A. (1982) Geometry and spatial interaction. Annals of the Association of American Geographers 72, 332–346.
- Griffith, D. A. (1985) An evaluation of correction techniques for boundary effects in spatial statistical analysis: contemporary methods. Geographical Analysis 17, 81–88.
- Griffith, D. A. and Amrhein, C. G. (1983) An evaluation of correction techniques for boundary effects in spatial statistical analysis: traditional methods. Geographical Analysis 15, 352–360.
- Haining, R. (1990) Spatial Data Analysis in Social and Environmental Sciences. Cambridge University Press: New York, NY.
- Haslett, J., Wills, G., and Unwin, A. (1990) SPIDER: an interactive statistical tool for the analysis of spatially distributed data. International Journal of Geographical Information Systems 3, 285–296.
- Henley, S. (1981). Nonparametric Geostatistics. Applied Science Publishers: London, UK.
- Kirby, H. R. (1997) Buffon’s needle and the probability of intercepting short-distance trips by multiple screen-line surveys. Geographical Analysis, 29 64–71.
- Leung, Y. (1987) On the imprecision of boundaries. Geographical Analysis 19, 125–151.
- 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. 107–129.
- Martin, R. J. (1987) Some comments on correction techniques for boundary effects and missing value techniques. Geographical Analysis 19, 273–282.
- Miller, H. J. (1999) Potential contributions of spatial analysis to geographic information systems for transportation. Geographical Analysis 31, 373–399.
- Openshaw, S., Charlton, M., and Wymer, C. (1987) A mark I geographical analysis machine for the automated analysis of point pattern data. International Journal of Geographical Information Systems 1, 335–350.
- Ripley, B. D. (1979) Tests of "randomness" for spatial point patterns. Journal of the Royal Statistical Society B 41, 368–374.
- Rogerson, P. A. (1990) Buffon’s needle and the estimation of migration distances. Mathematical Population Studies 2, 229–238.
- Rogerson, P. A. (2006) Statistical Methods for Geography: A Student Guide. Sage: London, UK.
- Upton, J. G. G. and Fingleton, B. (1985) Spatial Data Analysis by Example. Volume 1: Point Pattern and Quantitative Data. Wiley: Chichester, UK.
- 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.
- Yoo, E.-H. and Kyriakidis, P. C. (2008) Area-to-point prediction under boundary conditions. Geographical Analysis 40, 355–379.

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