Local Weighted Linear Combinationpublish.uwo.ca/~jmalczew/Malczewski_Local Weighted Linear...

Research Article

Local Weighted Linear Combination

Jacek MalczewskiDepartment of GeographyUniversity of Western Ontario

AbstractThe article focuses on one of the most often used GIS-based multicriteria analysismethods: the weighted linear combination (WLC). The WLC model has traditionallybeen used as a global approach based on the implicit assumption that its parametersdo not vary as a function of geographical space. This assumption is often unrealisticin real-world situations. The article proposes a new approach to GIS-based multi-criteria analysis. It develops a local form of the global WLC model. The rangesensitivity principle is used as a central concept for developing the local WLC model.The principle postulates that the greater the range of criterion values is, the greaterthe weight assigned to that criterion should be. Consequently, the local criterionweight can be defined for each neighborhood within a study area as a function of therange of criterion values in a given neighborhood. The range of criterion valuesprovides also the base for defining the local value function. The article presents thetheory behind the local WLC modeling and illustrates an implementation of themodel in a GIS environment.

1 Introduction

GIS-based multicriteria analysis (GIS-MCA) consists of a set of procedures and modelsthat are concerned with spatial patterns of decision alternatives. A spatial decisionalternative consists of at least two elements: action (what to do?) and location (where todo it?) (Malczewski 1999, Chakhar and Mousseau 2008). The spatial component of adecision alternative can be specified explicitly or implicitly. A distinctive future ofGIS-MCA is that the results of the analysis depend not only on the spatial pattern ofdecision alternatives (that is, the locations of decision alternatives and their attributes)but also on the value judgments involved in the evaluation/decision-making process. Atthe most fundamental level, GIS-MCA is a procedure that transforms and combinesgeographic data (input maps) and the decision maker’s (expert’s or agent’s) preferences

Address for correspondence: Jacek Malczewski, Department of Geography, University of WesternOntario, 1151 Richmond Street, London, Ontario N6A 5C2, Canada. E-mail: [email protected]

Transactions in GIS, 2011, 15(4): 439–455

© 2011 Blackwell Publishing Ltddoi: 10.1111/j.1467-9671.2011.01275.x

into a decision (output) map. The procedure involves the use of geographical data, thedecision maker’s preferences and the integration of the data and preferences according toa specified decision (combination) rule (Malczewski 1999, Chakhar and Mousseau 2008,Jankowski et al. 2008).

The last 20 years or so have evidenced remarkable progress in the quantity andquality of research on spatial multicriteria analysis and integrating multicriteria analysismethods into GIS (e.g. Carver 1991; Banai 1993; Pereira and Duckstein 1993; Eastmanet al. 1993; Jankowski 1995; Laaribi et al. 1996; Malczewski 1996, 1999; Thill 1999;Chakhar and Mousseau 2008). A number of multicriteria methods have been imple-mented in the GIS environment including weighted linear combination (WLC) (e.g.Eastman et al. 1993, Malczewski 2000), reference point methods (e.g. Pereira andDuckstein 1993, Tkach and Simonovic 1997), analytic hierarchy process (e.g. Banai1993, Rinner and Taranu 2006), and outranking analysis (e.g. Joerin et al. 2001, Martinet al. 2003, Chakhar and Mousseau 2008). Among these procedures, the WLC methodis considered the most straightforward and most often employed (Eastman et al. 1993;Malczewski 2000, 2006).

The GIS-MCA procedures have mostly been derived from the general decisiontheory and analysis (e.g. Malczewski 1999). They typically involve spatial variabilityonly implicitly by defining evaluation criteria based on the concept of spatial relationssuch as proximity, adjacency, and contiguity (Herwijnen and Rietveld 1999, Ligmann-Zielinska and Jankowski 2008). The conventional approaches use the average or totalimpacts that are deemed appropriate for the whole area under consideration (Tkach andSimonovic 1997). They assume a spatial homogeneity of the decision maker’s preferencesor value judgments within the study area. This implies that the two main components ofMCA (that is, the criterion weights and value functions) are assumed to be spatiallyhomogeneous. For example, the WLC procedure assigns the same criterion weight toevery decision alternative (location) of a given criterion map (Eastman et al. 1993,Malczewski 2000). Also, the procedure uses a single value function (or standardizationprocedure) for the whole study area ignoring the fact that the form of the function maydepend on the local context. Therefore, the conventional WLC procedure is referred toas the global WLC.

Over the last decade or so, there has been a growing awareness of the limitations ofthe implicitly spatial GIS-MCA and the value of explicitly spatial GIS-MCA (Tkach andSimonovic 1997; Herwijnen and Rietveld 1999; Feick and Hall 2004; Makropoulos andButler 2006; Rinner and Heppleston 2006; Chakhar and Mousseau 2008; Ligmann-Zielinska and Jankowski 2008, 2011). Herwijnen and Rietveld (1999), and Chakhar andMousseau (2008) have proposed frameworks for incorporating explicitly spatial com-ponents into GIS-MCA. The problem of an uneven spatial distribution of criteriaoutcomes associated with different courses of action is addressed by Tkach and Simo-novic (1997). Their approach determines the course of action for each location within astudy area. Makropoulos and Butler (2006) incorporate spatially variable attitudetowards risk into a GIS-MCA procedure. Rinner and Heppleston (2006) provide anaccount of geographically defined evaluation criteria based on three classes of spatialrelations: location, proximity, and direction. Then they investigate the effect of distance-based adjustment of evaluation scores on the criterion outcomes. They argue that thedistance-based adjustment of evaluation scores allows decision makers to consider alocation’s environment. Ligmann-Zielinska and Jankowski (2011) have advanced Rinnerand Heppleston’s (2006) study by incorporating proximity-adjusted preferences into

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© 2011 Blackwell Publishing LtdTransactions in GIS, 2011, 15(4)

GIS-MCA. They use the concept of proximity-adjusted criterion weights to differentiatepreferences over geographical area (that is, the decision alternatives are given non-uniform criterion weighting). Several GIS-MCA studies have focused on spatial sensitiv-ity analysis as a mechanism for ‘spatializing’ or ‘localizing’ the conventional GIS-MCA(Feick and Hall 2004, Feick 2005, Gómez-Delgado and Tarantola 2006, Ligmann-Zielinska and Jankowski 2008, Chen et al. 2010). Feick and Hall (2004) examine thespatial dimensions of multicriteria weight sensitivity. They test the effect of criterionweighting by decision makers on the ranking of decision alternatives by mapping theweight sensitivity in order to detect localized variations of decision outcomes. Ligmann-Zielinska and Jankowski (2008) provide the most comprehensive account of spatialsensitivity analysis in the GIS-MCA procedures. They examine the spatial sensitivityanalysis in three dimensions: spatial versus aspatial procedures, single versus groupdecision making, and local versus global models. The last dimension is of particularsignificance for this research. Ligmann-Zielinska and Jankowski (2008) suggest that thedistinction between the local and global spatial sensitivity analysis is similar to thelocal-global notion in spatial statistics (e.g. Fotheringham et al. 2000, Lloyd 2010,O’Sullivan and Unwin 2010). Specifically, the spatial sensitivity analysis is concernedwith examining ‘one or more spatial relations within the extent of either the proximal(neighborhood) space or the whole space of the study area’ (Ligmann-Zielinska andJankowski 2008, p. 222). This article suggests the local-global dichotomy in the spatialsensitivity analysis can be extended to the GIS-MCA procedures in general.

Notwithstanding the significant contributions made by the studies to advance GIS-MCA, they have been less successful in extending the theories and principles of MCAinto the explicitly spatial components of GIS-MCA. MCA is based on a series oftheoretical and empirically tested principles (Keeney and Raiffa 1976). One of thefundamental principles of MCA is that the criterion weights should represent the relativeimportance of unit changes in their criterion value functions (Hwang and Yoon 1981,Massam 1988). Thus, there is a relationship between the criterion weight and thecriterion values. The weights assigned to the criterion maps are related to the attributevalues represented on the criterion maps. Specifically, the criterion weight depends on therange of criterion values. This relationship is known as the range sensitivity principle (e.g.von Nitzsch and Weber 1993, Fischer 1995). The principle postulates that the greater therange of criterion values is, the greater the criterion weight should be. This article extendsthe range sensitivity principle into the GIS-MCA research. The principle is used todevelop an explicitly spatial GIS-based WLC (GIS-WLC) model. Central to the proposedmodel are interrelated concepts of local criterion weight and local range of criterionvalues. Therefore, the model is referred to as the local WLC. The remainder of the articleis organized as follows. Section 2 looks at the conventional, global WLC model. Section3 presents the range-sensitivity principle and demonstrates how the principle can be usedfor defining local criterion weights. In Section 4 the local WLC is defined. Section 5provides illustrative examples of the WLC modeling in a GIS environment. Section 6concludes with future directions of the local MCA.

2 Global WLC

For a given set of criterion (attribute) maps, the global WLC is defined as a mapcombination procedure that associates with the i-th decision alternative (location) a set

Local Weighted Linear Combination 441


of criterion weights, w1, w2, . . . ,wn, (0 � wk � 1, and wkk

n

==

∑ 11

, k = 1,2, . . . , n), and

combines the weights with the criterion (attribute) values, ai1, ai2, . . . , ain, (i = 1,2, . . . ,m) as follows:

V A w v ai k ikk

n

( ) = ( )=

∑1

(1)

where V(Ai) is the overall value of the i-th alternative at location, si, defined by the (xi, yi)coordinates (for the sake of simplicity a single subscript, i, is used to indicate the locationof the i-th alternative); v(aik) is the value of the i-th alternative with respect to the k-thattribute measured by means of the value function. The alternative characterized by thehighest value of V(Ai) is the most preferred one.

The value function, v(aik), is a mathematical representation of human judgment(Beinat 1997). It converts different levels of an attribute into value scores. If aik is the levelof the k-th criterion for the i-th alternative, then value function, v(aik), is the worth ordesirability of that alternative with respect to that criterion. The function relates possibledecision outcomes to a scale which reflects the decision maker’s preferences. There are anumber of techniques for assessing value function (Hwang and Yoon 1981, Massam1988, Malczewski 1999). The approach most often used for assessing a value function inGIS-MCA is the score range procedure. This method transforms the input criterion maplayer into standardized (value) scores, v(aik), as follows:

v a

a a

rk

ik

iki

ik

k( ) =

− { }−

min, for the th criterion to be maximized

mmax,i

ik ik

k

a a

rk

{ } −−

⎧

⎨⎪⎪

⎩⎪ for the th criterion to be minimized⎪⎪

(2)

where mini

ika{ } and maxi

ika{ } are the minimum and maximum criterion values for thek-th criterion, respectively, and r a ak

iik

iik= { } − { }max min is the global range of the k-th

criterion. The standardized score values, v(aik), range from 0 to 1, when 0 is the value ofthe least-desirable outcome and 1 is the most-desirable score.

Since the score range in Equation (2) is defined for the whole study area, the rk valueis referred to as the global range. Consequently, v(aik) is the global value function. Thisimplies that the value function does not take into account spatial heterogeneity of therelationship between the criterion score, aik, and the worth of that score, v(aik). Thepreferences are assumed to be homogeneous irrespectively of the local context andfactors that may affect the level of worth associated with a particular criterion score. Forinstance, if different locations (households) experienced the same amount of propertydamage during a flooding (measured in $), then the global value function would translatethe cost into the same ‘worth’ irrespectively of the characteristic of the locations such asthe household income, property value, etc. Given the contextual characteristics, the valuefunction may vary from one residential neighborhood to another (Tkach and Simonovic1997).

One should emphasize that assigning weights to evaluation criteria accounts for: (1)the changes in the range of variation for each evaluation criterion; and (2) the differentdegrees of importance being attached to these ranges of variation; that is, the differencebetween the maximum and minimum value for a given criterion (Hwang and Yoon 1981,

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Massam 1988). The general rule for weighting procedures is that the weight shouldreflect the range of values for a given criterion in relation to the ranges of values for theother criteria under consideration. Like the global value function, the concept of globalcriterion weight makes an assumption about uniformity of preferences over the wholestudy area. Furthermore, it takes for granted spatial homogeneity of the trade-offbetween evaluation criteria. The criterion weights should represent the rate at which oneis willing to trade off one criterion for another. Specifically, the ratio of two criterionweights is inversely proportional to the rate at which the decision maker is willing totrade them off. The trade-off value can be thought of as a relationship between preferenceassociated evaluation criteria. In the global WLC model (Equation 1) the value isassumed to represent the relationship in every part of the study area irrespective of thespatial heterogeneity that can be present due to social, economic, political, environmentalor other contextual concerns. Consider, for example, a problem of evaluating a site forlocating an airport on the basis of two criteria: noise pollution and spatial accessibility.A community located near the proposed airport site might express preferences (trade-offs), with respect to the two criteria, quite different from those of a community located100 km away from the airport site. The global WLC procedure would assume that thetrade-off is constant over the study area. Consequently, the geographical variation ofpreferences with respect to the evaluation criteria would not be captured.

3 The Range-Sensitivity Principle and Local Criterion Weights

The critical aspect of the global WLC is that a criterion weight is intricately associatedwith the corresponding value function. Consequently, a meaningful estimate of a weightrequires that at least the upper and lower limits of the value function (that is, the rangeof criterion values) have been specified (Keeney and Raiffa 1976, Hwang and Yoon 1981,Malczewski 2000). The interrelated concepts of the criterion range (value function) andcriterion weight provide the foundation for developing the local form of WLC. Therelationship is encapsulated in the range-sensitive principle (e.g. Keeney and Raiffa 1976,von Nitzsch and Weber 1993, Fischer 1995). Range sensitivity is a normative propertyconcerned with the dependence of criterion weights on the ranges of criterion values. Theprinciple suggests that, other things being equal, the greater the range of values for thek-th criterion, the greater the weight, wk, that should be assigned to that criterion (e.g.von Nitzsch and Weber 1993, Fischer 1995). Thus, the criterion weights vary as afunction of the range of criterion values, rk. The local form of the range, rk

q, can bedefined as follows:

r a akq

i qikq

i qikq= { } − { }max min

, , (3)

where max,i q

ikqa{ } and min

,i qikqa{ } are the minimum and maximum values of the k-th

criterion in the q-th subset of locations (decision alternatives), respectively.The subset of locations (q = 1, 2, . . . , g) can be defined using one of the two methods

(see e.g. Fotheringham et al. 2000, 2002; Anselin 2005). First, the study area can besubdivided into discrete units (neighborhoods, zones or regions). For example, the subsetcan be specified in terms of economic regions, urban neighborhoods, land use zones,geomorphologic units, watersheds, etc. For the raster data, the subset can be defined in



the context of the zonal overlay functions or the nonoverlapping neighborhoods (blocks).For the vector data model, the neighborhood can be generated using a defined (xj, yj) pairfalling within a given polygon (neighborhood, zone or region) (j = 1, 2, . . . , p).

Second, the subset of locations can be defined using the concept of moving windows(Fotheringham et al. 2000, 2002; Lloyd 2010). In this case, q consists of a focal location(alternative) and locations in its vicinity. The i-th location (xi, yi) is the focal alternativeand the set of neighboring locations defined by the (xj, yj) coordinates. There are manymethods for defining the shape and size of moving windows. For example, one can usethe distance and shared boundary based methods. Using the shared boundary method theq-th neighborhood can be defined as follows: j ∈ q if the i-th and j-th alternatives sharea common boundary, and j ∉q otherwise. This method of defining neighborhood caninvolve Rook′s or Queen′s criteria for identifying a common boundary between twoareas. Although, the first order contiguity is typically used, the second or higher orderneighborhoods can also be generated. This approach is operationalized in the raster GISenvironment in terms of the overlapping neighborhood (or focal) functions (McCoy andJohnston 2001). Alternatively, one can use a distance based method. Given the distance,dij, between two locations, si and sj, and some threshold distance, d, the neighborhood(window), q, is defined as follows: j ∈ q if dij � d, and j ∉ q otherwise. This approach canbe used for raster and vector (polygon centroid) data. Given the distance threshold value,all points (representing polygons or rasters) within the threshold band are included intothe neighborhood. Also, the p-nearest neighbor method can be used to define a set ofoverlapping neighborhoods. It is a distance-based method where p is a given number ofnearest locations of the i-th location. The method uses the distance between the i-thlocation and the number ( p) of nearest neighbor locations. It generates neighborhoodsthat consist of the same number ( p) of locations (polygons, rasters).

Given the definition of the q-th neighborhood, the local criterion weight, wkq, for the

k-th criterion can be defined as a function of the global weight, wk, the global range, rk,and the local range, rk

q. Specifically;

w

w rrw r

r

w wkq

k kq

k

k kq

kk

n kq

kq

k

n

= ≤ ≤ =

=

=∑∑

1

1

0 1 1, and (4)

Since the spatial variability of the local weight, wkq, is a function of the local criterion

range, rkq, the value of a local weight depends on the neighborhood scheme used for

subdividing a study area into neighborhoods (zones or regions). Therefore, this type ofcriterion weighting can also be referred to as the neighborhood-based criterion weights(Feick and Hall 2004).

Notice an important difference between the concepts of the global and local weights.It is related to the methods for obtaining the weights. The global weights are obtained byelucidating the decision maker’s preferences with respect to the relative importance ofevaluation criteria. It is determined empirically using one of the methods for assessingcriterion weights (e.g. Hwang and Yoon 1981, Malczewski 1999). Unlike the globalweights, the local weights are estimated on the basis of a normative theory; that is, therange-sensitivity principle. It is assumed that once the global weight for a given criterionhas been determined, the local weight for that criterion can be estimated as a function ofthe global criterion weight modified by the relationships between the criterion local and

444 J Malczewski


global ranges. The local weighting scheme can be referred to as the spatial weighting(Ligmann-Zielinska and Jankowski 2008). It estimates the relative importance of crite-rion varying over geographical space and assigns a criterion weight to each decisionalternative (location).

One should make a distinction between the local weighting scheme defined for localWLC and the concept of spatial weighting in spatial statistics. The estimation of the localcriterion weights involves value judgments. The concept of weighting scheme in spatialstatistics is not concerned with any aspect of value judgment with respect to evaluationcriteria. It is used as an element of measuring the level of interaction between objects(locations) in geographical space (e.g. Fotheringham et al. 2000, 2002; Getis and Alds-tadt 2004; Lloyd 2010). In spatial statistics the computation of a statistic is localized toa point (xi, yi) by weighting each observation in the data set according to its proximity tothat point. Given the range sensitivity principle, the criterion weights in the WLC modelare localized by identifying the local range in the vicinity of a given decision alternative.

4 Local WLC

Given the local criterion weight, wkq , defined on the basis of the range-sensitive principle,

the local form of WLC can be written as follows:

V A w v aiq

kq

ikq

k

n

( ) = ( )=

∑1

(5)

where V Aiq( ) is the overall value of the i-th alternative estimated locally (in the q-th

neighborhood), v aikq( ) is the value of the k-th criterion measured by means of the local

value function in the q-th neighborhood, and wkq is the local criterion weight. The

decision alternative with the highest value of V Aiq( ) is the most preferred alternative in

the q-th neighborhood.The local value function, v aik

q( ), converts different levels of the k-th attributeassociated with the alternatives located in the q-th neighborhood. Like in the case of theglobal value function, the score range method is used for estimating the local valuefunction. The method transforms the input map layer into standardized scores using thefollowing equation:

v a

a a

rk

ikq

ikq

i qikq

kq

( ) =

− { }−

min,, for the th criterion to be maxiimized

for the th criterion to be minimax

,,i qikq

ikq

kq

a a

rk

{ } −− mmized

⎧

⎨

⎪⎪

⎩

⎪⎪

(6)

where min,i q

ikqa{ } and max

,i qikqa{ } are the minimum and maximum criterion values for the

k-th criterion in the q-th neighborhood, respectively, and rkq is the local range (see

Equation 3). The standardized values, v aikq( ) , range from 0 to 1, with 0 being the value

of the least-desirable outcome and 1 is the value assigned to the most-desirablealternative in the q-th neighborhood.



Notice that the results of both the global and local WLC can be mapped; that is, theV(Ai) and V Ai

q( ) values are mappable. However, unlike the parameters of the globalWLC model, the local parameters vary from place to place. While the global WLCassumes that a single value function can be used to generate a standardized criterion mapand a single weight can be assigned to that map, the local WLC model focuses onestimating the two components (value function and criterion weight) of the model in asubset of locations in the q-th neighborhood. Consequently, these two components canbe mapped and examined within a GIS. Furthermore, the local WLC model can generatea set of mappable outputs such as the local ranges, the parameters of the local valuefunctions, and the trade-offs between criteria. These parameters can provide furtherinsights into the nature of a spatial decision problem.

5 Illustrative Examples

5.1 Example 1

Let us consider a decision situation involving combination of two criterion maps: theterrain slope (%) and proximity-to-water (km). The evaluation criteria are to be mini-mized. Suppose that the study area is represented in a raster format and that each cell(parcel of land) is an alternative decision (location). The study area consists of 16 cells ofan equal size of 1 km2 (Figure 1). Each alternative is described by means of its locationalattributes (the xi, yi coordinates) and attribute data (attribute values, aik). The location ofeach cell is defined by the (xi, yi) coordinates: s1 = (1, 1), s2 = (1, 2), . . . , s16 = (4, 4); thatis, the cell designated by i = 1 is in the top left-hand corner of the grid-cell map and thecells are numbered left to right for each row; the cell i = 16 is located in the bottomright-hand corner of the raster map. The task is to order the decision alternatives andidentify the best one.

The value function (Equation 2) is used to convert the two criterion scores intostandardized values (value function). The functions are based on the global rangevalues: r1 = 12%, and r2 = 4.2 km. The slope and proximity-to-water criterion scoresare converted to the standardized values with the following value functions:

Figure 1 The criterion maps: (a) the terrain slope in %, ai1, and (b) the proximity-to-waterin km, ai2

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v (ai1) = (12.0 - ai1)/(12.0 - 0.0) = 1 - (0.0833 ¥ ai1), and v (ai2) = (4.2 - ai2)/(4.2 -0.0) = 1 - (0.2381 ¥ ai2), respectively. Further, let us assume that one of the criterionweighting methods (see Malczewski 1999, 2000) is used to assign criterion weights, w1

= 0.6 and w2 = 0.4, to the slope and proximity-to-water criteria, respectively. Thus, thedecision maker is willing to trade off a unit of the proximity-to-water for 0.666 unitsof the terrain-slope criterion. Given the standardized criterion maps, Equation (1) isused to combine the maps and corresponding criterion weights according to the globalWLC model. Figure 2a shows the results. The maximum overall value V(Ai) of 0.80 isassociated with location s5 (the first ranking alternative).

The local WLC model is demonstrated for two methods of defining neighborhoods:the block and moving window methods (see Section 3). The simplest way of introducingthe local WLC model is to divide the study area into nonoverlapping neighborhoods(blocks) and estimate the overall values, V Ai

q( ), for each block. For example, the studyarea (16 cells) is subdivided into four blocks as follows: block a a a a1 1

121

51

61= { }, , , ,

block a a a a2 32

42

72

82= { }, , , , block a a a a3 9

3103

133

143= { }, , , , and block a a a a4 11

4124

154

164= { }, , , . Alter-

natively, the neighborhoods are generated using the concept of moving windows. Thethreshold distance dij � 2 km is used to define the neighborhood of the i-th location. Forexample, for q = 1 the neighborhood includes the following alternatives: a1

1, a21, a3

1, a51,

a61, a9

1; that is, the neighborhood consists of all alternatives located within 2 km of thefocal location, s1.

Figures 2b and 2c display the overall values, V Aiq( ), for the local WLC models and

Tables 1 and 2 show a selection of localized parameter estimates. As expected the resultsare different for the two models due to the way the neighborhoods have been defined. Inthe case of nonoverlapping neighborhoods, the maximum overall value of 1.00 isassociated with locations s3 and s9 (the first ranking alternatives), while the movingwindows model resulted in the location, a12, as the best alternative (see Figures 2b and c).Notice that none of these alternatives has been identified as the most preferred by theglobal WLC model. The best alternative for the global model (s5) ranks third and secondaccording to the nonoverlapping and overlapping neighborhood models, respectively.More importantly, however, the local models provide a wealth of information about thespatial distribution of preferences with respect to the two evaluation criteria (see Tables 1and 2). Specifically, the value functions and criterion weights vary spatially. The globalweights, w1 = 0.6 and w2 = 0.4, are adjusted locally based on the changes of the localranges, rq

1 and rq2 , relative to the global ranges, r1 and r2. Notice that the relationship

between the local weights follows the relationship between the global weights; that is, w1

Figure 2 The overall values: (a) the global WLC model, (b) the local WLC model fornonoverlapping neighborhoods (blocks), and (c) the local WLC model for overlappingneighborhoods (moving windows defined by dij � 2 km)



Table 1 Local WLC for nonoverlapping neighborhoods (blocks)

i, q rq1 rq

2 v aiq1( ) v ai

q2( ) wq

1 wq2 V Ai

q( )

1,1 8 1.4 1.25–0.13 a111 1.00–0.71 a12

1 0.75 0.25 0.342,1 8 1.4 1.25–0.13 a21

1 1.00–0.71 a221 0.75 0.25 0.07

3,2 2 1.2 1.50–0.50 a312 2.67–0.83 a32

2 0.47 0.53 1.004,2 2 1.2 1.50–0.50 a41

2 2.67–0.83 a422 0.47 0.53 0.56

5,1 8 1.4 1.25–0.13 a511 1.00–0.71 a52

1 0.75 0.25 0.826,1 8 1.4 1.25–0.13 a61

1 1.00–0.71 a621 0.75 0.25 0.56

7,2 2 1.2 1.50–0.50 a712 2.67–0.83 a72

2 0.47 0.53 0.448,2 2 1.2 1.50–0.50 a81

2 2.67–0.83 a822 0.47 0.53 0.23

9,3 7 1.2 1.29–0.14 a913 2.67–0.83 a92

3 0.82 0.18 1.0010,3 7 1.2 1.29–0.14 a10 1

3, 2.67–0.83 a10 2

3, 0.82 0.18 0.64

11,4 12 1.4 1.00–0.08 a1114

, 3.00–0.71 a11 24

, 0.75 0.25 0.7312,4 12 1.4 1.00–0.08 a12 1

4, 3.00–0.71 a12 2

4, 0.75 0.25 0.73

13,3 7 1.2 1.29–0.14 a13 13

, 2.67–0.83 a13 23

, 0.82 0.18 0.9014,3 7 1.2 1.29–0.14 a14 1

3, 2.67–0.83 a14 2

3, 0.82 0.18 0.54

15,4 12 1.4 1.00–0.08 a15 14

, 3.00–0.71 a15 24

, 0.75 0.25 0.0816,4 12 1.4 1.00–0.08 a16 1

4, 3.00–0.71 a16 2

4, 0.75 0.25 0.14

Table 2 Local WLC for overlapping neighborhoods (moving windows defined bydij � 2 km)

i = q rq1 rq

2 v aiq1( ) v ai

q2( ) wq

1 wq2 V Ai

q( )

1 9 2.0 1.11–0.11 a111 1.00–0.50 a12

1 0.70 0.30 0.382 9 3.0 1.11–0.11 a21

2 1.00–0.33 a222 0.61 0.39 0.26

3 9 3.2 1.11–0.11 a313 1.00–0.31 a32

3 0.60 0.40 0.754 10 2.6 1.00–0.10 a41

4 1.38–0.38 a424 0.67 0.33 0.68

5 9 3.0 1.11–0.11 a515 1.00–0.33 a52

5 0.61 0.39 0.746 9 3.2 1.11–0.11 a61

6 1.00–0.31 a626 0.60 0.40 0.62

7 12 2.6 1.00–0.08 a717 1.38–0.38 a72

7 0.71 0.29 0.698 10 2.8 1.00–0.10 a81

8 1.50–0.36 a828 0.65 0.35 0.65

9 7 3.2 1.29–0.14 a919 1.00–0.31 a92

9 0.53 0.47 0.7110 12 2.6 1.00–0.08 a10 1

10, 1.38–0.38 a10 2

10, 0.71 0.29 0.57

11 12 2.8 1.00–0.08 a11111

, 1.50–0.36 a11 211

, 0.69 0.31 0.6212 12 2.0 1.00–0.08 a12 1

12, 2.10–0.50 a12 2

12, 0.76 0.24 0.83

13 10 2.6 1.20–0.10 a13 113

, 1.38–0.38 a13 213

, 0.67 0.33 0.2814 10 2.8 1.20–0.10 a14 1

14, 1.50–0.36 a14 2

14, 0.65 0.35 0.65

15 12 2.0 1.00–0.08 a15 115

, 2.10–0.50 a15 215

, 0.76 0.24 0.0716 12 1.4 1.00–0.08 a16 1

16, 3.00–0.71 a16 2

16, 0.82 0.18 0.14

448 J Malczewski


> w2 and w wq q1 2> . However, in some situations the local weights can be very different

than the global weights. For example, for the neighborhood q = 2 in the local WLCnonoverlapping neighborhood model, w w1

2220 47 0 53= < =. . (see Table 1), while for the

global WLC model w1 = 0.6 > w2 = 0.4. This indicates that the local structure of the twocriteria is substantially different from the global structure.

5.2 Example 2

This example is based on datasets derived from the National Topographic Database ofCanada (Canadian Council on Geomatics 2010a) and the National Road Networkdata (Canadian Council on Geomatics 2010b). The study area is located in the BlueMountains region of Southwestern Ontario, Canada (along the Lake Huron/GeorgianBay shoreline). The decision situation involves identifying land suitability pattern bycombining two evaluation criteria: the terrain slope (in %) and the proximity-to-roads(in meters) (see Figure 3). The evaluation criteria are to be minimized. The globalweights (w1 = 0.6 and w2 = 0.4) are assigned to the slope and proximity-to-roadscriteria, respectively. The criteria are represented by 20 m resolution raster maps. Eachmap consists of 895 rows and 550 columns. The study area covers 196.9 km2. Theneighborhoods for the local WLC modelling are defined by a moving window of 1 km¥ 1 km.

Figure 3 The Blue Mountains region: the terrain slope (in %) and the proximity-to-roads(in meters)



The global and local WLC models were implemented in the ArcGISTM SpatialAnalysis environment. Figure 4 shows the spatial patterns of overall scores generated bythe two models. As expected the models suggest that the most suitable lands are locatedalong the roads and the coastal area situated between the Georgian Bay shoreline andBlue Mountains and the least suitable lands are found in the mountainous areas (seeFigures 3 and 4). There is however an essential difference between the results of theglobal and local models. The results of the local WLC modelling are more localized. Thepattern of the land suitability is a reflection of the local conditions captured by the localranges of criterion values (local value functions) and local criterion weights. To this endit is instructive to examine the spatial patterns of those parameters.

Figures 5 and 6 show the spatial distribution of the standardized criterion scoresgenerated according to Equations (2) and (6) for the global and local WLC models,respectively. For the global model, the slope and proximity-to-roads criterion scores areconverted into the standardized values with the following value functions: v(ai1) = (97.47- ai1)/(97.47 - 0.00) = 1.0000 - (0.0103 ¥ ai1), and v(ai2) = (1958.88 - ai2)/(1958.88 -20.00) = 1.0103 - (0.0005 ¥ ai2), respectively (Figure 5). The patterns of standardizedvalues according to the global WLC are substantially different from the pattern of thelocal standardized values (Figure 6). The local patterns point to the high and low valuesof the standardized scores for the slope and proximity-to-roads with considerably higherprecision than the global patterns.

The analysis of the spatial patterns of local weights provides further insights into thenature of local spatial structure. Figure 6 shows the spatial distribution of the localweights generated by Equation (4) for the slope and proximity-to-roads criteria. The

Figure 4 The Blue Mountains region: the overall scores of land suitability according tothe global WLC and local WLC models

450 J Malczewski


Figure 5 The Blue Mountains region: (a) standardized values, v (aik), for the slope andproximity-to-road criteria according to the global WLC model, and (b) standardizedvalues, v aik

q( ) , for the slope and proximity-to-road criteria according to the local WLCmodel



spatial distribution of local weights should be examined in the context of the globalweights. Given the global weights w1 = 0.6 and w2 = 0.4, the slope criterion is 1.5 timesmore important than the proximity-to-roads. This implies that one would be indifferentbetween a change in v(ai1) of 1 unit and a change in v(ai2) of 1.5 units. The global weightsare homogeneously applied to the whole study area; that is, each location (raster) isassigned the same criterion weights. The local weights reflect heterogeneity of theevaluation criteria. They can be examined using the local trade-offs w wq q

1 2 , which arerelated to the magnitude of criterion values variation locally.

The w wq q1 2 value ranges from 0.0 to 2.68 (see Figure 6). For a flat terrain the local

range rq1 0= and consequently wq

1 0= , wq2 1= , and w wq q

1 2 0= . This implies that theproximity-to-roads is the only criterion considered locally. For neighborhoods with0 11 2< <w wq q , the proximity-to-roads criterion is relatively more important that theslope criterion. These neighborhoods tend to be located in the relatively flat areasnorth-east and south-west of the mountains (Figure 6). For example, if the q-th neigh-borhood located in a flat area (with a low value of the local range rq

1 ) has the localweights for the slope and proximity-to-roads criterion of 0.25 and 0.75, respectively,then 3 1 2w wq q= ; and consequently, one would be indifferent between a change in the slopevalue of 3 units and a change in the proximity-to-roads of one unit. In neighborhoodsthat are characterized by w wq q

1 2 1> the slope criterion is relatively more important thanthe proximity-to-roads criterion. These neighborhoods are located in the mountainousareas. For example, if the q-th neighborhood located in the mountainous terrain (char-acterized by large values of local ranges for the slope criterion) and local weights for theslope and proximity-to-road criteria of 0.75 and 0.25, respectively, then w wq q

1 23= ;

Figure 6 The Blue Mountains region, Ontario: local criterion weights, wkq , for the slope

and proximity-to-road criteria; and the trade-off ( w wq q1 2 ) between the two criteria for

the local WLC model

452 J Malczewski


accordingly, one would be indifferent between a change in the slope value of 1 unit anda change in the proximity-to-roads of 3 units. In general, there is a high trade off( w wq q

1 2 ) between the two criteria in the mountainous areas and low degree of trade-offin relatively flat regions.

Conclusions

This article has advanced spatial multicriteria analysis by developing a local form of theglobal WLC. This has been achieved by considering the range-sensitivity principle as thepivotal element of the transition from the global to local form of the WLC model. Theprinciple has been used for developing the concept of local range (value function) andlocal criterion weight. It has provided a mechanism for making the relationships betweenthe criterion weights and value function explicitly spatial.

The local WLC model opens up new opportunities for visualizing and analysingspatial decision problems in a GIS environment. The results of global WLC are unmap-pable with the exception of the criterion values and overall scores of decision alterna-tives. In contrast, the parameters influencing the local WLC results can be mapped andfurther examined with GIS. Specifically, an analysis of the spatial distribution of localcriterion weights and local value functions can provide new insights into the nature ofspatial decision problem. The spatially distributed parameters of the local WLC modelcan be used for examining the local preferences with respect to relative importance ofevaluation criteria as well as the local trade-offs between criteria.

The future research about local multicriteria modeling should focus on advancingboth theoretical and practical aspects of integrating MCA with GIS. Applied studies areneeded to test the concept of local MCA in real-world situations. The range-sensitivityprinciple can be used to advance research on integrating GIS and MCA by developinglocal forms of such global approaches to GIS-MCA as the analytic hierarchy process andreference point methods. As one can expect the results of local MCA are sensitive to thescale of the spatial units. Consequently, the problem of the most appropriate size andshape of neighborhood (zone or region) should be an important part of local MCA.

Acknowledgments

This research was supported by the GEOIDE Network (Project HSS-DSS-17) of theNetworks of Centres of Excellence and the Social Science and Humanities ResearchCouncil (Grant 410-2009-1831). The author would like to thank anonymous reviewersfor their constructive comments on an earlier version of this manuscript.

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