A boundary problem in spatial analysis refers to a phenomenon in which geographical patterns are differentiated by the shape and arrangement of boundaries that are drawn for administrative or measurement purposes. This is distinct from and must not be confused with the boundary problem in the philosophy of science that is used as a synonym for the demarcation problem.
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In spatial analysis, four major problems interfere with an accurate estimation of the statistical parameter: the boundary problem, 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.
Boundary problem: urban sprawl in central Florida (an evaluation by land cover analysis with raster datasets vs. an evaluation by population density bounded in the census tract)
Notes: Land cover datasets were obtained from USGS and population density from FGDL.
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).
