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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]].<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 |
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 |date=1993 |publisher=Wiley |isbn=9781119115151 |
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.
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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>{{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>{{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 |
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/>
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