Boundary problem (spatial analysis): Difference between revisions

Revision as of 15:08, 15 February 2020

Revision as of 15:43, 15 February 2020

Line 1:

{{About|geographical research|the boundary problem in philosophy of science|Demarcation problem}}

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]]'''.

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.

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.

== 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]].<ref>{{cite book |last1=Burt |first1=James E. |last2=Barber |first2=Gerald M. |title=Elementary statistics for geographers |date=2009 |publisher=Guilford Press |isbn=978-1572304840 |edition=3rd}}</ref> 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,<ref>{{cite book |last1=Henley |first1=S. |title=Nonparametric Geostatistics |date=1981 |publisher=Springer Netherlands |isbn=978-94-009-8117-1}}</ref> 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).<ref>{{cite book |last1=Haining |first1=Robert |title=Spatial Data Analysis in the Social and Environmental Sciences by Robert Haining |date=1990 |publisher=Cambridge University Press |language=en|doi=10.1017/CBO9780511623356 |isbn=9780511623356 }}</ref> 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.

Line 16 ⟶ 15:

== 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'''. 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}}</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 }}</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}}</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 ==

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 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.<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>

[https://web.archive.org/web/20111004153452/http://lh3.ggpht.com/_fHD7C68ZqPM/S-VBehIvDkI/AAAAAAAAARo/QQNdGka6s0E/s512/F4.jpg '''Figure 4.''' A solution to the boundary problem: overbounding]

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, 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.<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, 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.

[https://web.archive.org/web/20111004212336/http://lh5.ggpht.com/_fHD7C68ZqPM/S-VBel89ZlI/AAAAAAAAARs/JoMFID51sxo/s640/F5.jpg '''Figure 5.''' A solution to the boundary problem: sensitivity analysis]

== See also ==

* [[Central place theory]]

* [[Demarcation problem]] (''boundary problem'' in the philosophy of science)

* [[Fuzzy architectural spatial analysis]]

* [[Generalized least squares]]

* [[Geographic information system]]

* [[Level of analysis]]

* [[Modifiable areal unit problem]]

* [[Sensitivity analysis]]

* [[Spatial analysis]]

* [[Spatial autocorrelation]]

== References ==

{{reflist}}

==External links==

*[https://web.archive.org/web/20111004212256/http://lh5.ggpht.com/_fHD7C68ZqPM/S-VBeClLMcI/AAAAAAAAARc/Fg3kytstuUg/s640/BP.jpg '''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.}

*[https://web.archive.org/web/20111004212256/http://lh5.ggpht.com/_fHD7C68ZqPM/S-VBeClLMcI/AAAAAAAAARc/Fg3kytstuUg/s640/BP.jpg '''Figure 1.''' 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)]

*[https://web.archive.org/web/20111004220028/http://lh6.ggpht.com/_fHD7C68ZqPM/S-VBeeRfHdI/AAAAAAAAARg/xiA3mo38_Bk/s640/F2.jpg '''Figure 2.''' Boundary problem: horizontal boundaries]

*[https://web.archive.org/web/20111004153446/http://lh3.ggpht.com/_fHD7C68ZqPM/S-VBedpAdoI/AAAAAAAAARk/L7RzDyBz4SM/s640/F3.jpg '''Figure 3.''' Boundary problem: vertical boundaries]

[[Category:Geography]]