# Boundary problem (spatial analysis)

A boundary problem in analysis is 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 also called the demarcation problem.

## Definition

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.[1] The boundary problem occurs because of the loss of neighbours in analyses that depend on the values of the neighbours. 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 area 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 neighbours do not exist,[2] 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).[3] 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 neighbours 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.[4]

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.[5] 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.[6][7][8][9] The difference is largely based on the fact that spatial processes are generally unbounded or fuzzy-bounded,[10] but the processes are expressed in data imposed within boundaries for analysis purposes.[11] 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.[12]

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.

## Types and examples

By drawing a boundary around a study area, two types of problems in measurement and analysis takes place.[7] The first is an edge effect. This effect originates from the ignorance of interdependences that occur outside the bounded region. Griffith[13][8] and Griffith and Amrhein[14] 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 neighbouring municipality.[15]

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.[7] Similarly, the shape can influence interaction and flow among spatial entities.[16][17][18] 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,[19] or to locate traffic counters, travel survey stations, or traffic monitoring systems (Kirby 1997). From the same perspective, Theobald (2001; retrieved from[5]) 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.[20] 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.[5]

## References

1. Burt, James E.; Barber, Gerald M. (2009). Elementary statistics for geographers (3rd ed.). Guilford Press. ISBN 978-1572304840.
2. Henley, S. (1981). Nonparametric Geostatistics. Springer Netherlands. ISBN 978-94-009-8117-1.
3. Haining, Robert (1990). Spatial Data Analysis in the Social and Environmental Sciences by Robert Haining. Cambridge University Press. doi:10.1017/CBO9780511623356. ISBN 9780511623356.
4. Upton, Bernard Fingleton (1985). Spatial data analysis by example: Volume 1: Point Pattern and Quantitative Data. Chichester, UK: Wiley. ISBN 978-0471905424.
5. Community and quality of life : data needs for informed decision making. Washington, DC: Board on Earth Sciences and Resources, Division on Earth and Life Studies, National Research Council, National Academy Press. 2002. ISBN 978-0309082600.
6. Cressie, Noel A. C. (1993). Statistics for Spatial Data. Wiley Series in Probability and Statistics. Wiley. doi:10.1002/9781119115151. ISBN 9781119115151.
7. Stewart Fotheringham, A.; Rogerson, Peter A. (January 1993). "GIS and spatial analytical problems". International Journal of Geographical Information Systems. 7 (1): 3–19. doi:10.1080/02693799308901936.
8. Griffith, DA (August 1983). "The boundary value problem in spatial statistical analysis". Journal of Regional Science. 23 (3): 377–87. doi:10.1111/j.1467-9787.1983.tb00996.x. PMID 12279616.
9. Martin, R. J. (3 September 2010). "Some Comments on Correction Techniques for Boundary Effects and Missing Value Techniques". Geographical Analysis. 19 (3): 273–282. doi:10.1111/j.1538-4632.1987.tb00130.x.
10. Leung, Yee (3 September 2010). "On the Imprecision of Boundaries". Geographical Analysis. 19 (2): 125–151. doi:10.1111/j.1538-4632.1987.tb00120.x.
11. Miller, Harvey J. (3 September 2010). "Potential Contributions of Spatial Analysis to Geographic Information Systems for Transportation (GIS-T)". Geographical Analysis. 31 (4): 373–399. doi:10.1111/j.1538-4632.1999.tb00991.x.
12. Martin, R. J. (1989). "The role of spatial statistical processes in geographic modeling". In Griffith, D. A. (ed.). Spatial Statistics: Past, Present, and Future. Syracuse, NY: Institute of Mathematical Geography. pp. 107–129.
13. Griffith, Daniel A. (3 September 2010). "Towards a Theory of Spatial Statistics". Geographical Analysis. 12 (4): 325–339. doi:10.1111/j.1538-4632.1980.tb00040.x.
14. Griffith, Daniel A.; Amrhein, Carl G. (3 September 2010). "An Evaluation of Correction Techniques for Boundary Effects in Spatial Statistical Analysis: Traditional Methods". Geographical Analysis. 15 (4): 352–360. doi:10.1111/j.1538-4632.1983.tb00794.x.
15. Mcguire, James (1999). What works : reducing reoffending : guidelines from research and practice. J. Wiley. ISBN 978-0471956860.
16. Arlinghaus, Sandra L.; Nystuen, John D. (January 1990). "Geometry of Boundary Exchanges". Geographical Review. 80 (1): 21. doi:10.2307/215895. JSTOR 215895.
17. Ferguson, Mark R.; Kanaroglou, Pavlos S. (3 September 2010). "Representing the Shape and Orientation of Destinations in Spatial Choice Models". Geographical Analysis. 30 (2): 119–137. doi:10.1111/j.1538-4632.1998.tb00392.x.
18. Griffith, Daniel A. (1982). "Geometry and Spatial Interaction". Annals of the Association of American Geographers. 72 (3): 332–346. doi:10.1111/j.1467-8306.1982.tb01829.x. ISSN 0004-5608. JSTOR 2563023.
19. Rogerson, Peter A. (July 1990). "Buffon's needle and the estimation of migration distances". Mathematical Population Studies. 2 (3): 229–238. doi:10.1080/08898489009525308. PMID 12283029.
20. Rogerson, Peter A. (2006). Statistical methods for geography : a student guide (2nd ed.). SAGE. ISBN 978-1412907965.
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• 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.
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• 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.