Boundary problem (spatial analysis): Difference between revisions

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{{Short description| Geographical problem of calculating properties near edges of areas}}

{{About|geographical research|the boundary problem in philosophy of science|Demarcation problem|the boundary value problem in mathematical modeling|Boundary value problem|the boundary problem in political science|Boundary problem (political science)}}

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. 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.

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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.<ref>{{cite book |last1=Upton |first1=Bernard Fingleton |title=Spatial data analysis by example: Volume 1: Point Pattern and Quantitative Data |date=1985 |publisher=Wiley |___location=Chichester, UK |isbn=978-0471905424}}</ref>

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.<ref name=BESR02>{{cite book |title=Community and quality of life : data needs for informed decision making |date=2002 |publisher=Board on Earth Sciences and Resources, Division on Earth and Life Studies, National Research Council, National Academy Press |___location=Washington, DC |isbn=978-0309082600}}</ref> 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.<ref>{{cite book |last1=Cressie |first1=Noel A. C. |title=Statistics for Spatial Data |url=https://archive.org/details/statisticsforsparevised0000cres_r7f4 |url-access=registration |date=1993 |publisher=Wiley |isbn=9781119115151 |language=en|doi=10.1002/9781119115151 |series=Wiley Series in Probability and Statistics }}</ref><ref name=Fotheringham93>{{cite journal |last1=Stewart Fotheringham |first1=A. |last2=Rogerson |first2=Peter A. |title=GIS and spatial analytical problems |journal=International Journal of Geographical Information Systems |date=January 1993 |volume=7 |issue=1 |pages=3–19 |doi=10.1080/02693799308901936}}</ref><ref name="Griffith83">{{cite journal |last1=Griffith |first1=DA |title=The boundary value problem in spatial statistical analysis. |journal=Journal of Regional Science |date=August 1983 |volume=23 |issue=3 |pages=377–87 |pmid=12279616|doi=10.1111/j.1467-9787.1983.tb00996.x |bibcode=1983JRegS..23..377G }}</ref><ref>{{cite journal |last1=Martin |first1=R. J. |title=Some Comments on Correction Techniques for Boundary Effects and Missing Value Techniques |journal=Geographical Analysis |date=3 September 2010 |volume=19 |issue=3 |pages=273–282 |doi=10.1111/j.1538-4632.1987.tb00130.x|doi-access=free }}</ref> The difference is largely based on the fact that spatial processes are generally unbounded or fuzzy-bounded,<ref>{{cite journal |last1=Leung |first1=Yee |title=On the Imprecision of Boundaries |journal=Geographical Analysis |date=3 September 2010 |volume=19 |issue=2 |pages=125–151 |doi=10.1111/j.1538-4632.1987.tb00120.x|doi-access=free }}</ref> but the processes are expressed in data imposed within boundaries for analysis purposes.<ref>{{cite journal |last1=Miller |first1=Harvey J. |title=Potential Contributions of Spatial Analysis to Geographic Information Systems for Transportation (GIS-T) |journal=Geographical Analysis |date=3 September 2010 |volume=31 |issue=4 |pages=373–399 |doi=10.1111/j.1538-4632.1999.tb00991.x}}</ref> 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.<ref>{{cite book |last1=Martin |first1=R. J. |editor1-last=Griffith |editor1-first=D. A. |title=Spatial Statistics: Past, Present, and Future |date=1989 |publisher=Institute of Mathematical Geography |___location=Syracuse, NY |pages=107–129 |chapter-url=https://www.researchgate.net/publication/30858907 |chapter=The role of spatial statistical processes in geographic modeling}}</ref>

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.<ref name=Fotheringham93/> The first is an ''edge effect''.<ref name=Gao1>{{cite journal |last1=Gao |first1=Fei |last2=Kihal |first2=Wahida |last3=Meur |first3=Nolwenn Le |last4=Souris |first4=Marc |last5=Deguen |first5=Séverine |title=Does the edge effect impact on the measure of spatial accessibility to healthcare providers? |journal=International Journal of Health Geographics |date=2017 |volume=16 |issue=1 |page=46 |doi=10.1186/s12942-017-0119-3 |pmid=29228961 |pmc=5725922 |doi-access=free }}</ref> This effect originates from the ignorance of interdependences that occur outside the bounded region.<ref name=Gao1/> Griffith<ref name=Griffith80>{{cite journal |last1=Griffith |first1=Daniel A. |title=Towards a Theory of Spatial Statistics |journal=Geographical Analysis |date=3 September 2010 |volume=12 |issue=4 |pages=325–339 |doi=10.1111/j.1538-4632.1980.tb00040.x}}</ref><ref name="Griffith83"/> and Griffith and Amrhein<ref name=Griffith_Amrhein83>{{cite journal |last1=Griffith |first1=Daniel A. |last2=Amrhein |first2=Carl G. |title=An Evaluation of Correction Techniques for Boundary Effects in Spatial Statistical Analysis: Traditional Methods |journal=Geographical Analysis |date=3 September 2010 |volume=15 |issue=4 |pages=352–360 |doi=10.1111/j.1538-4632.1983.tb00794.x|doi-access=free }}</ref> 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.<ref>{{cite book |last1=Mcguire |first1=James |title=What works : reducing reoffending : guidelines from research and practice |date=1999 |publisher=J. Wiley |isbn=978-0471956860}}</ref>

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.<ref name=Fotheringham93/> Similarly, the shape can influence interaction and flow among spatial entities.<ref>{{cite journal |last1=Arlinghaus |first1=Sandra L. |last2=Nystuen |first2=John D. |title=Geometry of Boundary Exchanges |journal=Geographical Review |date=January 1990 |volume=80 |issue=1 |pages=21 |doi=10.2307/215895|jstor=215895 |bibcode=1990GeoRv..80...21A }}</ref><ref>{{cite journal |last1=Ferguson |first1=Mark R. |last2=Kanaroglou |first2=Pavlos S. |title=Representing the Shape and Orientation of Destinations in Spatial Choice Models |journal=Geographical Analysis |date=3 September 2010 |volume=30 |issue=2 |pages=119–137 |doi=10.1111/j.1538-4632.1998.tb00392.x|doi-access=free }}</ref><ref>{{cite journal |last1=Griffith |first1=Daniel A. |title=Geometry and Spatial Interaction |journal=Annals of the Association of American Geographers |date=1982 |volume=72 |issue=3 |pages=332–346 |issn=0004-5608|jstor=2563023 |doi=10.1111/j.1467-8306.1982.tb01829.x }}</ref> 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,<ref>{{cite journal |last1=Rogerson |first1=Peter A. |title=Buffon's needle and the estimation of migration distances |journal=Mathematical Population Studies |date=July 1990 |volume=2 |issue=3 |pages=229–238 |doi=10.1080/08898489009525308|pmid=12283029 }}</ref> or to locate traffic counters, travel survey stations, or traffic monitoring systems.<ref>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.</ref> From the same perspective, Theobald (2001; retrieved from<ref name=BESR02/>) argued that measures of urban sprawl should consider interdependences and interactions with nearby rural areas.

By drawing a boundary around a study area, two types of problems in measurement and analysis takes place.<ref name=Fotheringham93/> The first is an ''edge effect''. This effect originates from the ignorance of interdependences that occur outside the bounded region. Griffith<ref name=Griffith80>{{cite journal |last1=Griffith |first1=Daniel A. |title=Towards a Theory of Spatial Statistics |journal=Geographical Analysis |date=3 September 2010 |volume=12 |issue=4 |pages=325–339 |doi=10.1111/j.1538-4632.1980.tb00040.x}}</ref><ref name="Griffith83"/> and Griffith and Amrhein<ref name=Griffith_Amrhein83>{{cite journal |last1=Griffith |first1=Daniel A. |last2=Amrhein |first2=Carl G. |title=An Evaluation of Correction Techniques for Boundary Effects in Spatial Statistical Analysis: Traditional Methods |journal=Geographical Analysis |date=3 September 2010 |volume=15 |issue=4 |pages=352–360 |doi=10.1111/j.1538-4632.1983.tb00794.x|doi-access=free }}</ref> 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.<ref>{{cite book |last1=Mcguire |first1=James |title=What works : reducing reoffending : guidelines from research and practice |date=1999 |publisher=J. Wiley |isbn=978-0471956860}}</ref>

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.<ref>{{cite book |last1=Rogerson |first1=Peter A. |title=Statistical methods for geography : a student guide |date=2006 |publisher=SAGE |isbn=978-1412907965 |edition=2nd}}</ref> 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.<ref name=Openshaw1>{{cite book |last1=Openshaw |first1=Stan |title=The Modifiable Areal Unit Problem |date=1983 |isbn=0-86094-134-5 |url=https://alexsingleton.files.wordpress.com/2014/09/38-maup-openshaw.pdf}}</ref><ref name=Chen1>{{cite journal |last1=Chen |first1=Xiang |last2=Ye |first2=Xinyue |last3=Widener |first3=Michael J. |last4=Delmelle |first4=Eric |last5=Kwan |first5=Mei-Po |last6=Shannon |first6=Jerry |last7=Racine |first7=Racine F. |last8=Adams |first8=Aaron |last9=Liang |first9=Lu |last10=Peng |first10=Jia |title=A systematic review of the modifiable areal unit problem (MAUP) in community food environmental research |journal=Urban Informatics |date=27 December 2022 |volume=1 |issue=1 |article-number=22 |doi=10.1007/s44212-022-00021-1 |bibcode=2022UrbIn...1...22C |doi-access=free }}</ref> 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.<ref name=BESR02/>

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.<ref name=Fotheringham93/> Similarly, the shape can influence interaction and flow among spatial entities.<ref>{{cite journal |last1=Arlinghaus |first1=Sandra L. |last2=Nystuen |first2=John D. |title=Geometry of Boundary Exchanges |journal=Geographical Review |date=January 1990 |volume=80 |issue=1 |pages=21 |doi=10.2307/215895|jstor=215895 }}</ref><ref>{{cite journal |last1=Ferguson |first1=Mark R. |last2=Kanaroglou |first2=Pavlos S. |title=Representing the Shape and Orientation of Destinations in Spatial Choice Models |journal=Geographical Analysis |date=3 September 2010 |volume=30 |issue=2 |pages=119–137 |doi=10.1111/j.1538-4632.1998.tb00392.x|doi-access=free }}</ref><ref>{{cite journal |last1=Griffith |first1=Daniel A. |title=Geometry and Spatial Interaction |journal=Annals of the Association of American Geographers |date=1982 |volume=72 |issue=3 |pages=332–346 |issn=0004-5608|jstor=2563023 |doi=10.1111/j.1467-8306.1982.tb01829.x }}</ref> 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,<ref>{{cite journal |last1=Rogerson |first1=Peter A. |title=Buffon's needle and the estimation of migration distances |journal=Mathematical Population Studies |date=July 1990 |volume=2 |issue=3 |pages=229–238 |doi=10.1080/08898489009525308|pmid=12283029 }}</ref> or to locate traffic counters, travel survey stations, or traffic monitoring systems.<ref>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.</ref> From the same perspective, Theobald (2001; retrieved from<ref name=BESR02/>) 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.<ref>{{cite book |last1=Rogerson |first1=Peter A. |title=Statistical methods for geography : a student guide |date=2006 |publisher=SAGE |isbn=978-1412907965 |edition=2nd}}</ref> 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.<ref name=BESR02/>

== Suggested solutions and evaluations on the solutions ==

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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>

As particularly applicable using GIS technologies,<ref>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.</ref><ref>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.</ref> 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.<ref name=Fotheringham93/> As such, thisThis 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, suchSuch a [[sensitivity analysis]] allows the evaluation of the reliability and robustness of place-based measures that are defined within artificial boundaries.<ref>BESR (2002) Community and Quality of Life: Data Needs for Informed Decision Making. Board on Earth Sciences and Resources: Washington, DC.</ref> In the meantime,{{Cclarify|date=July 2025}} 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.

== See also ==

* [[Arbia's law of geography ]]

* [[Concepts and Techniques in Modern Geography]]

* [[distance decay]]

* [[ecological fallacy]]

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* [[Geographic information system]]

* [[Level of analysis]]

* [[modifiableModifiable arealtemporal unit problem ]]

* [[spatial autocorrelation]]

* [[Tobler's first law of geography]]

* [[Tobler's second law of geography]]

* [[Uncertain geographic context problem]]

== References ==

{{reflist}}

{{Authority control}}

[[Category:Geography]]

[[Category:SpatialProblems in spatial analysis]]