Computers, Environment and Urban Systems 35 (2011) 452–463

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Computers, Environment and Urban Systems

journal homepage: www.elsevier .com/locate /compenvurbsys

Positional accuracy, positional uncertainty, and feature change detectionin historical maps: Results of an experiment

Michele Tucci, Alberto Giordano ⇑Department of Geography, Texas State University, San Marcos, TX 78666, USA

a r t i c l e i n f o

Article history:Received 15 February 2011Received in revised form 19 May 2011Accepted 21 May 2011Available online 15 June 2011

Keywords:Positional accuracyPositional uncertaintyFeature change detectionHistorical cartographyHistorical GIS

0198-9715/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compenvurbsys.2011.05.004

⇑ Corresponding author.E-mail address: ag22@txstate.edu (A. Giordano).

a b s t r a c t

The measurement and management of positional accuracy and positional uncertainty is especially prob-lematic in historical cartography and Historical GIS applications, for at least two reasons: first, historicalsources, and especially historical maps, generally carry a higher degree of positional inaccuracy anduncertainty compared to contemporary geographic databases; second, it is always difficult and oftenimpossible to reliably measure the positional accuracy and positional uncertainty of the spatial attributeof historical data. As an added complication, the terms ‘‘inaccuracy’’ and ‘‘uncertainty’’ are often used assynonyms in the literature, with relatively little attention given to issues of uncertainty.

In this article we propose a methodology for detecting the positional inaccuracy and positional uncer-tainty of measurements of urban change using historical maps at a very high spatial resolution (the build-ing). A widely accepted and routinely employed method for detecting urban change, and spatial change ingeneral, consists in overlaying two or more maps created at different dates, but the technique can lead tothe formation of spurious changes—typically, sliver polygons—that are the product of misclassificationerror or map misalignment rather than actual modifications in land cover. In this paper we develop analgorithm to detect such spurious changes. More in general, we extend the discussion to examine theeffects of positional uncertainty and positional inaccuracy in feature change detection analysis. Thecase-study is the city of Milan, Italy.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In the last two decades, the field of Geographic InformationScience (GIScience) has witnessed a rise in the number and varietyof uses and users of geospatial data, as well as an increment in theanalytical and computational powers of GIScience tools and meth-ods (Fisher, Comber, & Wadsworth, 2006; Harding, 2006; Stein &Van Oort, 2006). At least in part as a result of these developments,considerable attention has been paid by the scientific communityto issues of spatial data accuracy and spatial data uncertainty(Burrough & Frank, 1996; Foody & Atkinson, 2002; Guptill &Morrison, 1995; Hunsaker, Goodchild, Friedl, & Case, 2001; Lowell& Jaton, 1999; Shi, Fisher, & Goodchild, 2002; Shi, Goodchild, &Fisher, 2003; Zhang & Goodchild, 2002). Considering that historicalmaps typically carry a higher degree of inaccuracy and uncertaintywhen compared to contemporary cartographic databases, it is notsurprising that these two issues are of particular concern in histor-ical cartography studies and Historical GIS applications (Plewe,2002). By Historical GIS, we mean the use of GIScience methods,tools, and techniques to help understand the history and the his-

ll rights reserved.

torical geography of a place (Bender, Boehmer, Jens, & Schumacher,2005; Dragicevic, Marceau, & Marois, 2001; Gregory, Bennett,Gilham, & Southhall, 2002; Gregory & Ell, 2006, 2007; Kienast,1993; Knowles, 2000; Hillier & Knowles, 2008).

Building on past work (Tucci, Giordano, & Ronza, 2010), in thisarticle we propose a methodology for identifying, classifying andmeasuring how positional accuracy and positional uncertaintyaffect the detection and measurement of urban change. The geo-graphical unit chosen for the analysis is the building footprint. Asa case study, we compare an 1884 historical map of Milan with a2005 cartographic representation of the city; the two maps chosenare at comparable scales, were created for similar purposes, andare well-known and highly regarded cartographic representationof Milan’s urbanscape. In the next section, we frame the discussionin the broader conceptual context of the ontological differencesbetween positional accuracy and positional uncertainty. Such dis-cussion is useful considering that often the two terms are often,and incorrectly, used as synonyms in the literature.

1.1. On positional accuracy and positional uncertainty

Harding (2006) distinguishes the data producer’s perspective(internal) from the data user’s perspective (external) in spatial dataquality, noting that both play an important role in the creation of

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reliable geographic products and analysis. Primarily from the dataproducer’s perspective, models describing the components of geo-graphical data quality have been proposed in standards such as ISO19115 by the International Standard Organization. Among thecomponents of geographic data quality, positional accuracy has re-ceived considerable attention both in the academic world and inthe broader community of GIS users. Positional accuracy is definedas the difference between the recorded location of a feature in aspatial database or in a map and its actual location on the ground,or its location on a source of known higher accuracy. New technol-ogies seem to have facilitated the work of researchers, especially inthe evaluative aspect of measuring positional accuracy (Leyk, Wei-bel, & Boesch, 2005), and generally a dichotomous relationship isestablished when discussing accuracy between what is accurateand what is not—called error—starting from the assumption thatthe objective level of inaccuracy is known. When the inaccuracyof a measurement is not objectively known, which is frequentlythe case with historical maps, the measured feature is defined asuncertain (Hunter & Goodchild, 1993).

The problem of estimating the positional accuracy of an histor-ical map has typically been tackled using spatial–analytical tools(Hessler, 2006; Livieratos, 2006; Oetter, Gregory, Ashkenas, &Minear, 2004; Pontius & Lippitt, 2006), and this is also the ap-proach that we follow in this article. However, by employing meth-ods for the evaluation of positional accuracy based on thecalculation of the Root Mean Squared Error, or RMSE, for the entiremap (Andrews, 1974; Giordano & Nolan, 2007; Hsu, 1978; Murphy,1979; Pearson, 2001, 2004, 2005; Strang, 1998), previous studieshave often neglected to systematically analyze and attempt tomodel how positional accuracy varies across the map.

Another issue that usually receives little attention in the litera-ture—including the articles cited in the preceding paragraph—ispositional uncertainty. Although uncertainty and inaccuracy areontologically distinct concepts, in practice it is often difficult tomeasure the two separately. This is particularly true when dealingwith historical maps: technological advancements, changes in car-

Fig. 1. The Fisher’s un

tographic production techniques, and progress in the field of sur-veying contribute to blur the line between the two, as do thevariety of map purposes, periods of creation, and socio-culturalcontexts in which maps are created.

According to Buttenfield (1993), uncertainty is an ambiguousconcept, which arises from the imperfect understanding and mod-eling of the phenomenon under study, coupled with the use ofimprecise, outdated and incomplete data (Harrower, 2003). As sta-ted by Couclelis (2003), uncertainty occurs simply because part ofthe information is unknown or, as Fisher puts it (1999), it cannotbe known with precision. Although there are many definitions ofuncertainty, there seems to be agreement on a few key concepts.Kardos, Moore, and Benwell (2005) identify three types of uncer-tainty: attribute uncertainty, which applies to differences existingbetween the semantic characteristic of a feature and the corre-sponding data stored; spatial uncertainty, which is relative to differ-ences between the actual physical position of a feature and thecorresponding data stored; and temporal uncertainty, the time dif-ference between data acquisition and data utilization. Leyk et al.(2005) make a distinction between production-oriented uncertainty,the case in which maps contain errors specific to the survey of a cat-egory of information; transformation-oriented uncertainty, which isrelative to data processing and generalization; and application-ori-ented uncertainty, which is caused by semantic incomparabilityand incompatibility between a historical map and a present dayapplication. Probably the best known conceptual model for dealingwith spatial data uncertainty, and one widely cited in the literature(Fisher et al., 2006), was developed by Fischer in 1999. According tothis model (Fig. 1), in well defined objects—those that can be unam-biguously classified—uncertainty is caused by positional and attri-bute errors that can be calculated using a probabilistic approach.This approach is generally less suitable in the case of imprecise, lim-ited or conflicting information, or when the information is qualita-tive (Comber, Wadsworth, and Fisher, 2006). On the other hand,poorly defined objects—those that are difficult to unambiguously as-sign to categories—are affected by uncertainty because of vagueness

certainty model.

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and ambiguity. Vagueness refers to objects with unclear boundariesthat can be modeled with Fuzzy-set theory approaches; ambiguityoccurs when there are differences in the perception of the phenom-enon (for example, the same person can be perceived as being tall orshort by different observers; in this context, ‘‘tall’’ and ‘‘short’’ areexamples of ambiguity).

In the context of Historical GIS applications, Plewe (2002, 2003)describes the nature of uncertainty and its propagation across his-torical data in the Uncertainty Temporal Entity Model. According tothe model, when transposed in digital format, the World has to bestructured in entities. Entities are assigned identities through aprocess that involves two major steps, conceptual and physical mod-eling, with the former representing the understanding of realityand the latter representing the conversion of this understandingto a form suitable to be processed in a digital environment. Bothare relevant in this study, as the cartographic representation of abuilding involves both conceptual and physical modeling, and bothcan be affected by uncertainty, since entities are mere generaliza-tions of the real referents; consequently the generated geographicdataset, also known as asserted extent, will be uncertain as well.Additionally, by the act of processing spatial information, we pro-duce a new asserted dataset more uncertain than the original. Inother words, uncertainty—like inaccuracy—is propagated. This isan important point, as the overlay operation—which is at the coreof our proposed methodology—is a classical example of error prop-agation, and because there are consequences to the possible misin-terpretation of analytical outcomes based on error-filled generateddatasets (Brown, Duh, & Drzyzga, 2000; Carmel, Dean, & Flather,2001; Langford, Gergel, Dietterich, & Cohen, 2006; Mas, 2005;Shao, Liu, & Zhao, 2001; Wickham, Oneill, Riitters, Wade, & Jones,1997).

In a study on the nature and magnitude of uncertainty in thecartographic context, Lowell (2006) adds to the discussion by not-ing that the definition of uncertainty itself varies depending onwhether we are dealing with geopolitical or with interpretive maps.The two are defined in relation to the concepts of fiat and bona fideboundaries as developed by Smith (2001). In the first case, geopo-litical maps (corresponding to bona fide objects) are characterizedby boundaries which exist by definition (e.g., political bound-aries—Fisher’s well defined objects), and the question to be askedis whether or not they are located in the correct position, withoutinquiring upon their existence. The second type, generally linked toremotely sensed data and corresponding to fiat objects, is exempli-fied by maps in which the existence of the boundaries itself can bein question (existential uncertainty—Fisher’s poorly defined ob-jects), given their subjective nature based on a classification sys-tem (e.g. land cover categories). In the case of urban historicalmaps, because of positional inaccuracy, likely deformation of thepaper medium, and technological limitations, the digital represen-tation of geographical features is intrinsically uncertain (Tucci etal., 2010). Assuming that positional uncertainty increases movingoutwardly from the core of a feature and according to the egg-yolkmodel representation (Cohn & Gotts, 1996), we argue that what isinitially thought of as the cartographic representation of a sharpboundary between built-up and unbuilt areas (Lowell’s geopoliticalmap), is instead more properly represented by a transition zonewith varying width around every feature, and is therefore moretypical of Lowell’s interpretive maps. In such a case, a map thatcould be classified as geopolitical in Lowell’s sense—as is the caseof the 1884 map used for this study—tends to occupy a less clearposition as concerns the differentiation between geopolitical andinterpretive maps. This semantic ambiguity is one of the reasonsit is so difficult to deal separately with positional accuracy andpositional uncertainty in Historical GIS applications.

The above discussion frames our work. A systematic, if brief,exam of the differences between positional accuracy and positional

uncertainty is useful to formulate research questions and hypoth-eses, understand the weaknesses and strengths of the analyticalmethods proposed, derive a set of appropriate assumptions, dis-cuss the results, and develop procedures for the sensitivity analysisstage. In this article, we are dealing with spatial uncertainty as de-fined by Kardos et al. (2005) and in the context of Fisher’s (1999)well defined objects (buildings footprints in a historical map). Suchobjects are affected by production-oriented and transformation-ori-ented uncertainty (Leyk et al., 2005) that can be measured in prob-abilistic terms (Fisher, 1999). Following Plewe’s terminology(2002, 2003), the features examined (buildings) are subject touncertainty as concerns their conceptual and physical modeling,with the latter likely to be more critical in determining the out-comes of the analysis. Furthermore, by employing spatial function-alities in a GIS environment—the overlay of two maps—uncertaintyis propagated. Finally, we will use a contemporary (2005) map assource of higher accuracy to test the reliability of a historical(1884) map; this point is further discussed in the next section.

Although positional accuracy and positional uncertainty areconceptually different phenomena, it is very often difficult orimpossible to differentiate between the two from an operationalpoint of view—that is, to measure one separately from the other.This is especially the case in historical cartography, for the obviousreason that recourse to fieldwork is not an option. Assumptionsthen need to be made about the nature of what is being measuredwhen detecting urban change using historical maps.

2. Materials and methods

2.1. Study assumptions

In feature change detection studies, it is customary to employtechniques borrowed from remote sensing, but when the historicalsources compared predate the development of such technologies,or when remote sensing images for the areas examined are notavailable, maps are often used, for a variety of reasons and in dif-ferent contexts (Boots & Csillag, 2006; Foody, 2007). In this study,we compare two maps to estimate the accuracy and uncertainty ofurban change measurements occurred over the span of more thanone century. The starting point is an earlier work on the urban his-tory of Milan (Tucci et al., 2010) in which we used MapAnalyst(Jenny & Hurni, 2011; Jenny, Weber, & Hurni, 2007) to estimatethe level of planimetric distortion in a series of historical maps ofMilan dating back to the 1700s, including the 1884 map we exam-ine more closely in this study. Based on bidimensional regressiontechniques (Tobler, 1965, 1966, 1994; Symington, Charlton, &Brunsdon, 2002; Friedman & Kohler, 2003), in that earlier workwe calculated the RMSE of the 1884 map to be equal to 7.7 m, witha rotation of 4� counter clockwise and at a scale of approximately1:4000. This RMSE value was obtained by comparing the 1884 mapwith the 2005 map. The calculation of this RMSE value is based onthe assumption that when maps are comparable in scale, purpose,objectives, style, and features represented—as is the case here—andwhen the aim of both maps compared is to accurately representthe situation on the ground—also the case here—the more recentmap is generally the most accurate. Considering that the 2005map (Fig. 2) is an example of Carta Tecnica Regionale—the officialItalian large-scale map subdivision and by design a highly accuratemap, both positionally and thematically, it is safe to expect fea-tures on this map to be more accurately represented than featureson the 1884 Bassani map (Fig. 3).

The act of overlaying two maps introduces error in the results, afact well documented in the GIScience literature (Chrisman, 1989;Edwards & Lowell, 1996; João, 1998; Mas, 2005; Zhang & Good-child, 2002). Langford et al. (2006) have shown that virtually every

Fig. 2. The 2005 CTR map (detail) at the scale 1:5000.

Fig. 3. The 1884 Bassani map (Biblioteca Trivulziana, Milano) at the scale 1:4000.

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published work employing feature change detection techniques isaffected by a considerable amount of error of due to spuriouschanges—typically, sliver polygons. As defined in this context, spu-rious changes are not imputable to actual modifications in landcover but are due solely to classification error and map misalign-ment in the overlay operation (Chrisman, 1989; Goodchild,1978). They are therefore considered the result of uncertaintyrather than inaccuracy. Using the terminology introduced by Leyket al. (2005), we are dealing in this case with transformation-ori-ented uncertainty, which can be measured in probabilistic terms;and—according to Plewe’s terminology (2002, 2003)—the uncer-tainty is specifically introduced during the physical modeling stepwhen converting real world entities to objects suitable to be pro-cessed in a digital environment.

2.2. Methodology

The first step in the procedure consisted in the generation of alayer showing urban change as represented in the two maps. Sinceour focus is methodological, we selected a sample area (Fig. 4)rather than digitizing the entire city; the procedure described here,however, could be expanded to the entire study area. The samplearea was chosen because it includes examples of: (a) changes thathave actually occurred in the period under study, i.e., new build-ings being built or old ones being demolished, in whole or in part;(b) areas that have remained unchanged; (c) uncertain cases, i.e.,

Fig. 4. The area under stu

examples of spurious changes (discussed in Section 3 of thearticle).

Features on the resulting overlaid layer were classified as Chan-ged or Unchanged according to a simple principle: un-built (orbuilt) areas on the 1884 map shown as un-built (or built) also onthe 2005 map were classified as Unchanged, whereas built areason the 1884 map corresponding to un-built areas on the 2005, orvice versa, were classified as Changed. The resulting binary mapwas subsequently filled with a stratified random sample of nearly2600 points, in which relatively more points were placed on chan-ged areas (�1 per 5 square meters) than in unchanged ones (�1per 30 square meters). These points were then assigned a valueof 100 (100% change) if a change had occurred—e.g., a new buildinghad been built—or a value of 0 (representing 0% of change) in caseof no change. Although the boundary between a building and astreet (no building) is in reality represented by a sharp border,we settled on a conceptual model in which such boundary was in-stead formalized as a probability of change surface, taking intoconsideration the likely physical deformation of the historicalmap as well as its inaccuracy and uncertainty levels. The probabil-ity of change ranges from 0% to 100%, with a resolution of 0.5 m onthe ground. The interpolation was performed using ArcGIS GeoSta-tistical Analyst extension. When modeling a system characterizedby a short-range variation of the data—a reasonable assumptionin this case, considering that the width of the transition area be-tween a building and a street in a cartographic representation is

dy in 1884 and 2005.

Fig. 6. The reliability of results: original (centered) grid and derived grids (verticalor diagonal shift).

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likely to be small—a Local Polynomial Interpolator is generally em-ployed (Akima, 1970; Johnston, Ver Hoef, & Krivoruchko, 2003).Assuming that each point affects the value of the surface only inits immediate vicinity (Mitas & Mitasova, 1999), the algorithm pro-duces surfaces that represent local variation reasonably well (Tucciet al., 2010, 54–55):

Xn

i¼1

wiðZðxi; yiÞ � l0ðxi; yiÞÞ2

The interpolation is based on a moving window centered atl0(x,y), where n is the number of points within the window, andwi is the weight:

wi ¼ e3di0

a

� �

where di0 is the distance between the point and the center of thewindow, a is the parameter that controls the distance decay, Z isthe datum at the location i, and l0(xi, yi) is the value of the polyno-mial (Johnston et al., 2003).

The result is shown in the background of Fig. 8. The assignmentof probabilities takes into consideration the effects of positionalinaccuracy and positional uncertainty on the results of the analysis,and assumes that the egg-yolk method introduced earlier is appro-priate to model how positional inaccuracy and positional uncer-tainty vary around the perimeter of buildings. By applying thislogic throughout the map, however, we neglect to consider howthe two might vary locally; in a sense, then, we are subject tothe same limitations we noted while discussing studies that onlyuse RMSE as an indication of the accuracy of the historical map.

In the second step of the analysis, we estimated the combinedeffects of positional inaccuracy and positional uncertainty on eachfeature separately from the others. To do so, we developed an algo-rithm, similarly to what Linke et al. (2009) did, centered on a reit-erative segment distance calculation between the 1884 and 2005maps. The polygons representing the 1884 and 2005 building foot-prints (Fig. 5 – 1) were converted into their outlines (Fig. 5 – 2) andthen overlaid on each other (Fig. 5 – 3). A 5-m grid (Fig. 5 – 4) wasthen placed on the resulting layer. For each segment in each cell,we calculated the midpoint (Fig. 5– 5). We then developed a scriptto calculate, for each cell in the 5-m grid, the distance between the1884 and 2005 segment midpoints (Fig. 5 – 6). Note that, although

Fig. 5. A visual representation of

this method does not always necessarily return the minimum dis-tance between two segments, it does insure that a distance value iscalculated for any two segments, even when they intersect eachother—which, by definition, would return zero as minimum dis-tance. Even with this assumption, measurement errors are not rel-evant considering the grid cell size. The script executing thealgorithm is available from the authors upon request.

To test the reliability of the results, we also conducted a seriesof experiments shifting the original grid used by the algorithm indifferent directions (vertically and diagonally), as shown in Fig. 6.This was done because the algorithm assumes that whenever thefootprint of a certain building (as shown on the 1884 and 2005maps) appears on two separate 5-m resolution cells, than thechange must have occurred in reality on the ground. However,there are cases in which the footprints of the same building mightbe very close but just happen to be on different cells. In this case,the algorithm would return the established cut-off 10 m value,whereas the true value should in fact be lower. By shifting the gridhorizontally and vertically we can account for these cases. We also

the methodology workflow.

Fig. 7. A flowchart repres

Table 1The algorithm.

VARIABLES DESCRIPTION:� GRID = The grid layer� FIELDS = List of fields contained in GRID� 1884 = Points (centroids) representing the segments generated by the

intersection of the grid with the 1884 footprint building outline� 2005 = Points (centroids) representing the segments generated by the

intersection of the grid with the 2005 footprint building outline� DIST = Field representing the distance between 1884 and 2005 (the

maximum possible value is set to be 10 meters; see text for anexplanation)

� COUNT = Field representing the number of features (both 1884 and 2005)contained inside each instance (Cell) of the grid layer (GRID).

� minDIST = Minimum distance between 1884 and 2005

PSEUDOCODE:Note: the steps in the pseudocode are numbered to match Fig. 6.(1)

GET GRIDREAD FIELDS in GRIDIF DIST is in FIELDS THEN

SET DIST to zeroELSE

ADD Field DISTSET DIST to zero

ENDIF(2)

GET GRIDFOR Cells IN GRIDREAD COUNTIF COUNT < 2 THEN

SET DIST to 10 meters(3)

ELSEIF 1884 <> 0 AND 2005 <> 0 THEN

CALCULATE minDIST between 1884 and 2005SET DIST = minDIST

(4)ELSE

SET DIST to 10 metersENDIF

ENDIF

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experimented with grids of different dimensions—respectively, of10 and 2.5 m—before settling on the 5-m grid used in the analysis.In the case of the 2.5-m grid, the size of the cell was so small that inthe large majority of cases only one feature from either the 1884 orthe 2005 map fell into the cell, thus leaving a large number of cellswith no values. The opposite case—too many features—occurredwhen we employed a 10-m grid. In the end, a 5-m grid is a goodcompromise. Note that the choice of an appropriate cell size de-pends on factors such as the cartographic properties—scales,mostly—of the maps used in the analysis as well as the urban his-tory of the area under study. Should the algorithm be used to de-tect urban changes that might have occurred in Venice (Italy), forexample, a smaller cell would be advisable, considering that build-ings are frequently very close to each other (sometimes less than2 m apart) in that city. Regardless of context, the algorithm isapplicable to cells of any size.

2.3. The algorithm

Initial data preparation was conducted in ArcGIS 9.3.1 using thesoftware’s Model Builder functionalities, while the Geoprocessor ob-ject and geoprocessing tools were used to create the script. Thescript was written in Python, an interpreted language that isquickly becoming a standard in the scientific community (Ceder,2010; Langtangen, 2009; Lutz, 2009). The pseudocode, a mixtureof code and plain English (Aho, Hopcroft, & Ullman, 1974; Baase,1978; Horowitz & Sahni, 1984) describing the algorithm, is pre-sented in Table 1 and visually described in Fig. 6.

The first part of the algorithm (Fig. 7 – 1 and number 1 in Table1) opens the grid layer and checks if a field called DIST is present. IfDIST is present, the field value is set to zero; if it is not, a field iscreated with a value of zero. After the initial data preparation, aconditional statement counts how many features are containedinto each instance of the grid layer (Cells) by checking the valueof the field named ‘‘COUNT’’. If this value is less than two(Fig. 7– 2 and number 2 in Table 1), the value of DIST is set to an

enting the algorithm.

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arbitrary high distance of 10 m. By using 10 m as a cut-off value,we imply that whenever footprints from the same building appearin different cells, then the building must have actually changed be-tween 1884 and 2005. The robustness of this assumption is testedin the sensitivity analysis section of the paper. Note also that thealgorithm checks if the number of features is less than two becausecases in which this value is equal to one indicate that the buildingfootprint is present in only one map; if the returned value is equalto zero, no building footprints are present in the cell.

The third and the fourth parts of the algorithm (Fig. 7 – 3 and 4and number 3 and 4 in Table 1) check for the remaining cases. If

Fig. 8. The combined effects of positional inaccuracy and positional uncertainty on featurand (3) diagonally shifted.

the count returned in the conditional statement is equal to orgreater than two—and thus meaningful for our study—we needto make sure that the features in the cell come from both the1884 and the 2005 footprints, i.e., we are not dealing with the caseof multiple features from the same map: this is case three in thenumbered list on Table 1 (see also Fig. 7 – 3). We should pointout that even when cell contains one 1884 feature and more thanone feature of the 2005 map, or vice versa, the minimum distancefunction would work on the closest feature. This is insured by thecharacteristics of the tool used (ESRI, 2009). Case number 4 inTable 1 and in Fig. 7 – 4 assigns the arbitrary value (10 m) to the

e change probability for the three grids: (1) original (centered), (2) vertically shifted

Fig. 10. Descriptive statistics, frequency histogram and box plot for the

Fig. 9. Contour map showing the combined levels of positional inaccuracy and posi

Table 2Distance measurement categories: combined levels of inaccuracy and uncertainty.

� Very low: 4 m <= DIST <= Highest evaluated distance (�6 m)� Low: 3 m <= DIST < 4 m� High: 2 m <= DIST < 3 m� Very high: DIST < 2 m

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remaining cases, that is, to multiple features of either the 1884 orthe 2005 building footprints.

Once the DIST field was populated with all the distances be-tween the 1884 and the 2005 segment footprints, the resulting val-ues were classified into four ordinal categories (Table 2), tofacilitate the discussion of results in Section 4 of the article.

original (centered), vertically shifted, and diagonally shifted grids.

tional uncertainty for the 1884 map relative to the 2005 map. See also Table 2.

Table 4Probability of change and combined levels of inaccuracy and uncertainty.

Inaccuracy and Uncertainty

VeryLow

Low High VeryHigh

Total

Probability ofchange (%)

0–25 98.5% 0 0 1.5% 100%

26–50 86.7% 0.5% 1.3% 11.5% 100%51–75 72.9% 2.4% 2.7% 21.9% 100%76–100 78.2% 3.5% 4.1% 14.3% 100%

Total 87.8% 1.1% 1.5% 9.6% 100%

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3. Results

Table 2 orders the distance values assigned by the algorithm toeach cell in four classes, which can then be used to weight theprobability that changes detected between the 1884 and 2005maps—as shown in the probability surface map in Fig. 8—haveactually occurred. The weight is the theoretical effect on the resultsof step one of the analysis of the combined effects of positionaluncertainty and positional inaccuracy as introduced by the overlayfunction.

The contour map in Fig. 9 depicts levels of combined inaccuracyand uncertainty in the representation of features as shown on the1884 map. These values are also shown in Fig. 8. The contour mapwas produced using the distance values assigned by the algorithmto each of the 5-m resolution cells in the grid. As expected, thesevalues are at their highest around the perimeter of buildings, andespecially when buildings run parallel and in close proximity toeach other (cases 1, 2, 3, 4, 5, and 6 in Fig. 9) and when a buildingfaces a wide open space (cases 8 and 9 in Fig. 9). It is difficult tospeculate as to the reasons for these differences, especially consid-ering that exceptions to the two rules are readily found (see cases 7and 10 in Fig. 9). But, regarding the first six cases, during the gen-eralization process cartographers routinely resort to positionalshifts in the representation of features to make the map more read-able. For cases 8 and 9, the accurate location of a building in the ab-sence of nearby features to be used as visual clues is problematic,especially using the relatively limited technologies available to the1884 cartographer—remote sensing and GPS have greatly facili-tated the work of the contemporary GIScientist.

We also tested the sensitivity of the results to grid configura-tions by shifting the location of the grid diagonally and vertically.As a visual inspection of Fig. 8 reveals, the effect on the results ofshifting the grid is negligible. This is confirmed by summary statis-tics for the three grids (the original grid and the two shifted ones)in cases for which the value of DIST is different than 10 m (Fig. 10).We also ran a Kruskal–Wallis one-way analysis of variance by rankstest (Table 3) to determine if there was a statistically significantdifference among the grids; the null hypothesis was that the pop-ulation distribution functions for the three grids were identical,while the alternative hypothesis called for the three populationsto have different medians. With a Kruskal–Wallis value (H) equalto 0.360 and a p-value (0.835) exceeding the established signifi-cance level of 0.05, we fail to reject the null hypothesis, thereforewe can confirm that there is no statistically significant differenceamong the three samples. This establishes the robustness of ourmodel and allows us to continue the analysis employing the ‘‘cen-tered’’ case.

4. Discussion

We are now in a position to discuss how, taken together, posi-tional accuracy and positional uncertainty inform the discussionon the detection of urban changes. This is done in Table 4. In casesin which there is little combined inaccuracy and uncertainty in themeasurement (‘‘Very low’’ category), the likelihood for the changesdetected to be real rather an artifact of the cartographic productionprocess combined with the error inserted by the overlay function-ality, decrease, as should be expected, but not dramatically. Forfeatures most likely to have changed (‘‘Probability of change (%)’’

Table 3Results of the Kruskal–Wallis test.

Kruskal–Wallis (H) 0.360df 2Asymp. Sig. 0.835

between 76% and 100%), the likelihood that these changes are realhas dropped by 22 percentage points, to 78.2%. Similar patterns canbe observed in the 26–50% category (dropped by 13 percentagepoints, to 86.7%), but the interpolator does not appear to performas well in the 51–75% range. The reason for the problem with thisparticular range is not clear, but we can speculate that the algo-rithm performs better in areas that either have changed substan-tially or in areas that have not changed very much; it is theintermediate cases that are the most difficult to model. In the caseof least likely probabilities—between 0 and 25%—values do notseem to have been significantly affected by positional inaccuracyand positional uncertainty: this is probably due to the fact thatthe area under study includes a large square, virtually unchangedbetween 1884 and 2005 (Fig. 4). Note also that the algorithm onlymeasures changes in the location of a building, not in its actualphysical appearance: a building could have been razed and re-placed by another one with the same shape and perimeter, andthe algorithm would record a no change.

To conclude our discussion, we briefly turn to alternative ap-proaches to the measurement of urban change. Previous researchhas shown that fuzzy set theory (Bandemer & Gottwald, 1995; Za-deh, 1965) holds great potential for measuring map accuracy levels(Lewis & Brown, 2001; Metternicht, 1999; Power, Simms, & White,2001) and can be fruitfully employed for comparing the accuracyof different maps of the same area. In this context, Hagen (2003)and Hagen-Zanker, Straatman, and Uljee (2005) make a conceptualdistinction between fuzziness of category and fuzziness of location.The former refers to thematic accuracy and uncertainty, and de-scribes the frequent occurrence in which, in a map legend, somecategories are more similar to each other than is the case for othercategories. The latter is its positional counterpart and refers to an-other common occurrence in which the location of a feature on amap, while precisely indicated, in reality refers to an approximatelocation: the boundary between vegetation zones on a map is a fit-ting example of fuzziness of location. The likelihood that a certainfeature is correctly classified, or is placed at its real location, is indi-cated by three membership vectors—the Crisp, Fuzzy Category, andFuzzy Neighborhood vectors—which together comprise a functionrepresenting the fuzziness of categorical data. Hagen-Zanker, Stra-atman, and Uljee’s work differs from ours in two ways: first, it isbased on the raster data model, while we work with vector data;second, in our case it is not possible to separate inaccuracy fromuncertainty in the measurement, because we cannot ground truththe dataset. On the other hand, the two methods share similarities.Most notably, in both cases a decay function is used to estimatepositional accuracy, consistent with the egg-yolk principle: thelikelihood for a polygon representing a building to be correctly lo-cated decrease as one moves away from the geometrical center ofthe building, and is minimum at or around its perimeter. Differ-ences and similarities notwithstanding, it would be useful in thefuture to compare our results with those obtained by applying fuz-zy set theories and concepts to map comparison in the historicalcontext.

462 M. Tucci, A. Giordano / Computers, Environment and Urban Systems 35 (2011) 452–463

5. Conclusions

In this paper we have proposed a methodology for detecting theeffects of positional inaccuracy and positional uncertainty on theresults of urban change detection analysis at a high spatial resolu-tion (the building). Results indicate that certain spatial configura-tions of features—such as buildings running in parallel—might bemore sensitive to the combined effects of positional accuracy andpositional uncertainty than other configurations, but it is left to fu-ture studies to establish the extent to which these results can begeneralized to other historical maps. We believe that trying tocome up with a taxonomy of spatial patterns of this type is a wor-thy research objective.

Besides contributing to the Historical GIS and historical cartog-raphy literature, the methodology and algorithm presented in thispaper could also be used for testing the reliability of any featurechange detection analysis. As Figs. 8 and 10 and Table 4 show, posi-tional accuracy and positional uncertainty are not uniformly dis-tributed; on the contrary, they affect various features in variouslocations on the map in different ways. Studies in which the posi-tional accuracy of historical maps is based on the calculation ofRMSE values fail to examine systematically how inaccuracies anduncertainties vary across the map, except for selected locationswhere additional information might be available.

Future work will concentrate on the application of our model todifferent and larger datasets in order to test its efficacy and even-tually to refine the conceptual approach for a less specific and moregeneral extension to other applications. Also of future possibleinterest, we believe it would be useful to investigate neighborhoodeffects—especially measures of spatial autocorrelation—to the re-sults presented in this paper to look for spatial regularities.

Throughout this paper, we have also argued for the incorpora-tion of a careful evaluation of the effects of positional accuracyand positional uncertainty in urban change detection studies, spe-cifically as concerns the sensitivity of its analytical results to localvariations. Whenever we try to reduce the continuum of the Worldto a sum of discrete elements with sharp boundaries, as we dowhen modeling this continuum with a raster or vector data model,vagueness and uncertainty are the unavoidable price the act ofmodeling has to pay (Haack, 1996; Williamson, 1994). It is for thisreason that, as discussed in Section 1.1, although accuracy anduncertainty are semantically well-defined and distinct concepts,in practice it is nearly impossible to measure them as separateentities, especially when using historical geographical and histori-cal cartographical sources in the context of Historical GIS applica-tions. In fact, not only is uncertainty an intrinsic characteristic ofevery system, but, for lack of comparable ground truth, in historicalcartography the act of separating it from map accuracy is a nearlyintractable problem, one that nonetheless needs to be addressed, atleast on ontological grounds.

The method we propose in this article achieves the objective ofevaluating the positional inaccuracy and positional uncertainty ofmeasurements of urban change using historical maps, but we arewell aware of the fact that ‘‘whether one uses fuzzy sets, roughsets, a multivalent logic, or some other device to tame the vague-ness beast, no amount of additional or better data can resolvethe uncertainty inherent in looking for boundaries (spatial as wellas conceptual) where there are none’’ (Couclelis, 2003, 169). This isespecially true with historical data.

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