The assessment of evolutionary algorithms for analyzing the positional accuracy and uncertainty of...

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Expert Systems with Applications xxx (2014) xxx–xxx

ESWA 9294 No. of Pages 15, Model 5G

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Contents lists available at ScienceDirect

Expert Systems with Applications

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

The assessment of evolutionary algorithms for analyzing the positionalaccuracy and uncertainty of maps

http://dx.doi.org/10.1016/j.eswa.2014.04.0250957-4174/� 2014 Published by Elsevier Ltd.

⇑ Corresponding author. Tel.: +34 950214501.E-mail addresses: [email protected] (F. Manzano-Agugliaro), [email protected]

(F.G. Montoya), [email protected] (C. San-Antonio-Gómez), [email protected] (S. López-Márquez), [email protected] (M.J. Aguilera), [email protected] (C. Gil).

Please cite this article in press as: Manzano-Agugliaro, F., et al. The assessment of evolutionary algorithms for analyzing the positional accuracy andtainty of maps. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa.2014.04.025

Francisco Manzano-Agugliaro a, Francisco G. Montoya a,⇑, Carlos San-Antonio-Gómez d,Sergio López-Márquez b, Maria J. Aguilera c, Consolación Gil b

a Department of Engineering, University of Almería, 04120 Almería, Spainb Department of Informatics, University of Almería, 04120 Almería, Spainc Department of Applied Physics, University of Cordoba, Cordoba, Spaind Department of Cartographic Engineering, Geodesy and Photogrammetry, Polytechnic University of Madrid, 28040 Madrid, Spain

27282930313233

a r t i c l e i n f o

Keywords:Pareto frontEvolutionary algorithmHistorical mapCoordinate transformationMap transformationMulti-objective optimization

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a b s t r a c t

Pre-geodetic maps are an important part of our cultural heritage and a potential source of information forhistorical studies. Historical cartography should be evaluated in terms of precision and uncertainty priorto their use in any application. In the last decade, the majority of papers that address multi-objectiveoptimization employed the concept of Pareto optimality. The goal of Pareto-based multi-objective strat-egies is to generate a front (set) of nondominated solutions as an approximation to the true Pareto-opti-mal front. This article proposes a solution for the problems of multi-objective accuracy and uncertaintyanalysis of pre-geodetic maps using four Pareto-based multi-objective evolutionary algorithms: HVSEA,NSGAII, SPEAII and msPESA. ‘‘The Geographic Atlas of Spain (AGE)’’ by Tomas Lopez in 1804 provides thecartography for this study. The results obtained from the data collected from the kingdoms of Extrema-dura and Aragon, sheets of maps (54-55-56-57) and (70-71-72-73), respectively, demonstrate the advan-tages of these multi-objective approaches compared with classical methods. The results show that theremoval of 8% of the towns it is possible to obtain improvements of approximately 30% for HVSEA,msPESA and NSGAII. The comparison of these algorithms indicates that the majority of nondominatedsolutions obtained by NSGAII dominate the solutions obtained by msPESA and HVSEA; however, msPESAand HVSEA obtain acceptable extreme solutions in some instances. The Pareto fronts based on multi-objective evolutionary algorithms (MOEAs) are a better alternative when the uncertainty of map ana-lyzed is high or unknown. Consequently, Pareto-based multi-objective evolutionary algorithms establishnew perspectives for analyzing the positional accuracy and uncertainty of maps.

� 2014 Published by Elsevier Ltd.

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1. Introduction

Pre-geodetic maps are an important part of our cultural heri-tage (Jenny & Hurni, 2011) and provide a suitable cartographicbase for historical urban and landscape analyses (San-Antonio-Gómez, Velilla, & Manzano-Agugliaro, 2014). These maps have tra-ditionally been examined by historians and geographers ratherthan specialists in modern mapping sciences. Thus, early mapsare considered to be a typical archive that serves as a historical tes-timony of territories and cities. In previous decades, due toadvancements in new computational technologies, the study of

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the metric properties of old maps and the numerical approach tothis issue has improved (Balletti, 2006). Currently, early maps arefrequently incorporated into geographical information systems(GISs) for historical analysis (Audisio, Nigrelli, & Lollino, 2009;Hu, 2010).

Quality is a basic requirement for the users of any product.Since the 1980s, the interest in the quality of spatial data hasincreased due to two developments: the emergence of GIS in the1960s and the beginning of the 1970s and a substantial increasein available spatial data from satellites. As pre-geodetic maps havelimited quality, their quality must be analyzed prior to their use inhistorical studies. In previous decades, a large number of studies,which evaluate the quality of spatial data in early maps, have beenpublished (Boutoura & Livieratos, 2006; Giordano & Nolan, 2007;Leyk, Boesch, & Weibel, 2005; Pearson, 2005; Ravenhill & Gilg,1974).

uncer-

http://dx.doi.org/10.1016/j.eswa.2014.04.025

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2 F. Manzano-AgugliaroQ1 et al. / Expert Systems with Applications xxx (2014) xxx–xxx


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Among the components of spatial data quality, positional accu-racy is the most common in academia. Traditionally, positionalaccuracy assessments in cartography have been based on a compar-ison between the positions of a set of points on a map and the posi-tions of the same points obtained from a more accurate source(Mozas & Ariza, 2011). Positional accuracy is determined by a statis-tical evaluation of random and systematic errors and is specified bythe root-mean-square error (RMSE) or the mean value error (l) andtheir standard deviation (r) (Ariza López & Atkinson Gordo, 2008).The International Organisation for Standardisation (ISO) considerspositional accuracy to be one of the quantitative quality elementsof spatial data (Ariza López & Atkinson Gordo, 2008). InternationalStandard 19113 (ISO2002) proposes a general framework fordescribing and reporting the quality of geographic information,and International Standard 19114 (ISO2003) presents a generalquality evaluation methodology for geographic information.

Currently, several standards complete the ISO norm and definethe positional accuracy of cartographic products based on the anal-ysis of a set of points (Ariza López & Atkinson Gordo, 2008):National Map Accuracy Standards (NMAS)-United States Geologi-cal Survey (USGS) 1947 (USA), engineering map accuracy standards(EMAS)-American Society of Civil Engineers (ASCE) 1983, accuracystandards for large-scale maps (ASLSM)-American Society for Pho-togrammetry and Remote Sensing (ASPRS) 1999, National stan-dards for Spatial Data Accuracy (NSSDA)-Federal Geographic DataCommittee (FGDC) 1998, and Standardisation Agreement (STA-NAG) 2215 (Bozic & Radojcic, 2011) by the North Atlantic Treat-ment Organisation (NATO).

A comparison of historical maps with contemporary mapsrequires additional analysis as the spatial elements and their delin-eations that are represented in historical documents contain con-siderable inherent uncertainty (Leyk et al., 2005; Plewe, 2002).The term uncertainty is more complex and extensive than the con-cept of accuracy. Fisher (1999) distinguished three forms of uncer-tainty that develop in the process of deriving a spatial data set fromthe real world: error, vagueness, and ambiguity. Error is defined asthe difference between the value of a property of a measuredobject and the true value of the same property; it can only be mea-sured for well-defined objects. Vagueness is attributed to poor def-inition, poor documentation, or fuzzy objects. Ambiguity stemsfrom disagreement about the definition of objects and fundamen-tal differences in opinion. Thus, positional inaccuracy only refers tothe concept of error. The uncertainty term must be applied whenaccuracy is not feasible (Hunter & Goodchild, 1993).

When studying positional uncertainty in early maps, it isimportant to consider that uncertainty is not limited to the differ-ent stages of map production. The paper or other material used byearly maps are not inert materials and maps can be deformed overthe years, which alters the geometry of a map (Jenny & Hurni,2011). Although positional accuracy and positional uncertaintyare different concepts, it is often difficult or impossible to distin-guish between the two concepts (Tucci & Giordano, 2011). This sit-uation occurs when analyzing the quality of historical maps.

An additional problem in the analysis of historical cartographicquality is caused by the uncertainty of the datum and the projec-tion on which historical maps are based. It is not always possibleto know if the observed displacement between feature locationson a historical map relative to a contemporary map is attributedto inaccuracy, different geodetic reference systems, or differentmethods for transferring features to the map plane (Pearson,2005).

The comparison of early maps with modern cartography allowsthe study of the geometric content and deformation of early maps.If the maps are expressed in different geodetic reference systems,this comparison can be performed after georeferencing the earlymap. Two methods for georeferencing exist: transformation

Please cite this article in press as: Manzano-Agugliaro, F., et al. The assessment otainty of maps. Expert Systems with Applications (2014), http://dx.doi.org/10.10

between two coordinate systems, in which the transformationparameters are known, and transformation using identical groundpoints, in which the transformation parameters are unknown(Podobnikar, 2009). The second method is usually used to refer-ence an early map, as only projection and datum information fromthe late nineteenth century is known with certainty (Pearson,2005). In addition, the influence of the geodetic coordinate systemcan be neglected in the study of old maps as its effects are minorcompared with positional uncertainty (Jenny & Hurni, 2011).

Thus, a comparison between early and contemporary maps canbe performed using best-fitting techniques of different schemes forrelevant transformations of sets of points on an early map to corre-sponding sets of points on a modern reference map (Boutoura &Livieratos, 2006). The transformation can comprise a global trans-formation, which alters the coordinates of the control points afterthe transformation, or a local transformation, which retains thecoordinates of the control points (Balletti, 2006). The global trans-formations are derived from a polynomial system of equations.The commonly employed global transformations consist of confor-mal, affine, and projective linear transformations and polynomial(usually second-order) transformations. Local transformations con-sist of finite element transformations and transformations that aretypically referred to as feature-based warping (Balletti, 2006). Anaffine transformation between geographic coordinates are used inthis study. Using five parameters, this transformation considersrotation, horizontal, and vertical scale errors and latitude and longi-tude displacements.

A crucial aspect of obtaining transformation parameters is theelection of common points in early maps and reference maps(Podobnikar, 2009), which should be carefully chosen as erroneouscontrol points can alter the comparisons of maps. Here, we mustconsider the heterogeneous positional uncertainty of early maps.Some control points may correspond to different spatial positions,possible landscape changes, and, a priori, we do not have criterionto identify them.

Multi-objective evolutionary algorithms are known for theirability to simultaneously optimize several objective functions toobtain a representative set of the Pareto front (Baños, Gil, Reca, &Martìnez, 2009; Márquez et al., 2011; Voorneveld, 2003). The goalof Pareto-based multi-objective strategies is to generate a front(set) of nondominated solutions as an approximation to the truePareto-optimal front (Alcayde et al., 2011). The majority of papersthat have addressed multi-objective optimization employed theconcept of Pareto-optimality to perform a comparison among sev-eral genetic algorithms (Anagnostopoulos & Mamanis, 2011;Baños, Ortega, Gil, Fernández, & de Toro, 2013; Fernández, Gil,Baños, & Montoya, 2013; Gómez-Lorente, Triguero, Gil, & Estrella,2012) or traditional analyses (Dovgan, Javorski, Tušar, Gams, &Filipic, 2013; Sánchez, Montoya, Manzano-Agugliaro, & Gil, 2013).

Previous studies show that the evolutionary algorithm msPESAis a viable alternative for transforming historical map coordinates,particularly where the quality of the position of the set of points tobe used for the transformation cannot be assured (Manzano-Agugliaro, San-Antonio-Gómez, López, Montoya, & Gil, 2013). Theaim of this study is the assessment of different evolutionary algo-rithms to determine the gross error in the control points and theaccuracy of pre-geodetic maps. These algorithms will be evaluatedusing the Geographic Atlas of Spain (AGE) produced by the Spanishcartographer Tomás López (1730–1802). The AGE comprises ananthology of maps of Spanish regions that were drawn in the sec-ond half of the eighteenth century; it is the most ambitious andsuccessfully completed cartographic work (Manzano-Agugliaro,Fernández-Sánchez, & San-Antonio-Gómez, 2013). These pre-geo-detic maps (San-Antonio-Gómez, Velilla, & Manzano-Agugliaro,2011) exhibit substantial heterogeneous positional uncertainty(Chias and Abad).

f evolutionary algorithms for analyzing the positional accuracy and uncer-16/j.eswa.2014.04.025


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Fig. 1. Flow diagram of a typical evolutionary algorithm.

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2. A brief overview of evolutionary algorithms

Evolutionary algorithms (EAs) are mathematical techniquesthat search and find quasi-optimal solutions for complex problems(for which an exact mathematical solution does not usually exist)by applying strategies inspired by the evolution of living beings(Dekkers & Aarts, 1991; Goldberg et al., 1989; Horst, Pardalos, &Romeijn, 2002). Therefore, we can define an evolutionary algo-rithm as a probabilistic technique that converges towards a feasi-ble and quasi-optimal solution after several executions oriterations (Fonseca & Fleming, 1993). Evolutionary algorithmsoperate on a population of individuals in each execution. Each indi-vidual represents a potential solution to the problem; which iscoded as a complex data structure. Each solution is evaluated ateach iteration to obtain a measurement of its optimality. A newpopulation is formed by selecting the best individuals accordingto their optimality, which is based on the definition of a fitnessfunction (objective function). To produce new solutions and intro-duce diversity in the search space, some members of the new pop-ulation are subjected to certain transformations in their structure,which is achieved using genetic operators such us selection, cross-over and mutation. This process is repeated until a number of pre-viously defined iterations or generations. The quality of thesolutions improves through iterations (Goldberg & Holland,1988). Fig. 1 shows the flow diagram of a typical evolutionaryalgorithm.

The advantage that EAs offer over traditional search algorithmsis based in its ability to explore vast search spaces with a lower riskof getting trapped in local minimum. This means that one desiredquality of any EA is to keep a population with the higher variety, sothat the search space can be covered more efficiently.

As a single-objective optimization problem may only have asingle optimal solution, Multiobjective Optimization Problemspresent a possibly uncountable set of solutions, which when eval-uated, produce vectors whose components represent a trade-off inthe objective space between the different objectives of the problem(Coello, Lamont, & Van Veldhuisen, 2007; Deb, 2001). The multi-objective optimization attempts to solve problems with more thanone objective without the need to combine them in a single objec-tive (Baños et al., 2011).

A Multiobjective Optimization Problem (MOP) can be defined(Osyczka, 1985) as the problem of finding a vector of decision vari-ables that satisfies constraints and optimizes a vector functionwhose elements represent the objective functions. These functionsform a mathematical description of performance criteria that areusually in conflict with each other. A MOP is defined as

MOP ,min FðxÞ ¼ ðf1ðxÞ; f2ðxÞ; . . . ; fnðxÞÞ; x 2 S ð1Þ

where n is the number of objectives, and n P 2; FðxÞ ¼ ðf1ðxÞ;f2ðxÞ; . . . ; fnðxÞÞ is the vector of objectives to be optimized, and S rep-resents the set of feasible solutions associated with equality andinequality constraints and explicit bounds. Since in real-life MOPsthe criteria are usually in conflict, there is a need to establish otherconcepts to consider optimality. In that sense, a partial orderrelation, known as Pareto dominance relation can be defined(Edgeworth, 1881; Pareto). Fig. 2 shows the graphical descriptionof the Pareto-dominance concept for a function with two objectivesto minimize (Zitzler & Thiele, 1999). The figure shows the existenceof points (shown in red) that are dominated by other points (shownin purple). A set of points that form the Pareto front is also shown;thus, no better solutions than this front exist. A solution s1 domi-nates another solution s2 (s1 � s2) when at least one objective ins1 is better than at least one objective in s2 and not worse thanthe remainder of the objectives. Similarly, two solutions are indif-ferent or incomparable (s1 � s2) if none of the two solutions domi-


nates the remaining solutions. In this manner, it is possible toconstruct a set of nondominated solutions (there may be severalsolutions). These form the optimal Pareto front. As this problem isNP-complete (Garey & Johnson, 1979), we only obtain an approxi-mation to the front; formation of the complete front is notguaranteed.

Fig. 3 shows the solution ordering with respect to an arbitrarysolution G. Regions in which the solutions are worse (G dominatesthese solutions) or better (G is dominated by these solutions) thanG for both objectives are observed. Regions in which the solutionsare incomparable or indifferent (there is some improvement in oneobjective but the other object is worse) are also observed. The aimof the multi-objective technique is to obtain a complete set of non-dominated solutions (the Pareto front). However, in practice and inreal problems, this task can be complex. The search space can beextremely large, and therefore, excessive computing times arerequired.

The best way to approximate the Pareto front is via the utiliza-tion of heuristics, such as the heuristics utilized in this study. We



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Fig. 2. Graphical description of the Pareto-dominance concept for a function withtwo objectives to minimize.

Fig. 3. Ordering of solutions with respect to an arbitrary solution G.



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have selected four evolutionary algorithms: the msPESA, which hasbeen proven suitable for the resolution of this multi-objectiveproblem and has been employed for maps composed of a singlesheet (Manzano-Agugliaro et al., 2013), the NSGAII, the SPEAIIand the HVSEA.

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2.1. NSGAII

The nondominated sorting genetic algorithm II (NSGAII) (Deb,Agrawal, Pratap, & Meyarivan, 2000) is different from other tech-niques as it employs two properties or metrics, which are referredto as irank and idistance. Based on the irank, it is possible to mea-sure the dominance depth of each solution. The idistance enablesthe estimation of the population density in the vicinity of one par-ticular individual. This estimation is dependent on the computa-


tion of the‘‘cuboid’’ (Deb et al., 2000), which is associated withthe mentioned individual. NSGAII proposes that two solutionscan be partially ordered. The order is generated using a comparisonoperator named crowded, which adheres to the following the crite-rion: the ranks of the two solutions are verified. The solution with alower rank is considered to be the most adequate and is selected asthe best rank. If the ranks are equivalent, their densities are veri-fied (based on the cuboid computation) and the rank with thelower value is selected, that is, the rank that exposes a less-popu-lated vicinity. NSGAII employs a tournament selection. Duringalgorithm execution, the population size that forms the fronts isconsidered. If this size is exceeded, then the less-diverse frontsare eliminated. As expected, this pruning technique is conditionedby dominance criteria offered by the comparison operators.

2.2. msPESA

Due to the continuous search for faster and more efficientmethods, hybrid techniques have been proposed. In particular,the hybridisation of the Pareto envelope-based selection algorithmPESA (Corne, Knowles, & Oates, 2000) and the NSGAII, which areknown as the mixed spreading between PESA and NSGAII(msPESA) (Gil, Baños, Montoya, Márquez, & Ortega, 2005), has beendesigned to improve the concept of spreading. The msPESA algo-rithm implements a variation in the ‘‘fast non-dominated sortingalgorithm’’ of NSGAII. In this case, only one front and a new storagestrategy of the external population are computed. Specifically, thecandidate solutions are stored in the file of external solutions. Thisfile maintains a list of the best candidates obtained from the searchprocess. If the maximum size is exceeded during the execution,then some solutions are eliminated according to the concept‘‘squeeze factor’’, which is used in PESA. The ‘‘squeeze factor’’ isdefined as the number of elements or chromosomes that exist ina particular cell of the space associated with the objective function.Therefore, the final selection is obtained after searching for themaximum value in a given population and eliminating an indis-tinct (arbitrary) chromosome that contains the ‘‘squeeze factor’’.

2.3. HVSEA

HVSEA (HyperVolume-based Search Evolutionary Algorithm)(Antonio López Márquez, Consolacion Gil Montoya, & Márquez,2011) makes use of a sorted archive (sorted by the values of oneobjective) to store the solutions that become part of the generatedPareto front. For a solution S to enter the archive, it must increasethe hypervolume. In order to do so, HVSEA uses the concept of localhypervolume. The local hypervolume is the value of the S-metricwhen it only takes into account the solutions closest to the inser-tion point of S in the sorted archive (Bader & Zitzler, 2011). If thelocal hypervolume is greater than zero, then the solution Sincreases the local hypervolume, and therefore, the value of theS-metric. Using a local hypervolume means that there is no needto recalculate the whole S-metric each time a new solution is savedto the archive, since only the solutions that improve the localhypervolume are chosen. For faster access, solutions are saved tothe archive in order to finding the closest individuals to S. For gen-erating the solutions that make up the working population, it usesa tournament selection (with tournament size 4) for the crossoverselection of the solutions within the archive. The selection criterionis based on Pareto dominance, so non-dominated solutions will bepreferred in the tournament. In order to keep solution diversity,achieve an even spread of solutions along the Pareto front, and topurge the archive, a (N-1)-dimensional hypergrid (Gil, Márquez,Baños, Montoya, & Gómez, 2007) is implemented, where N is thenumber of dimensions of the objective space.



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2.4. SPEAII

SPEAII (Zitzler et al., 2001) uses a strength indicator in order tomeasure the solution quality of the individuals stored in thearchive. At the end of the procedure, the archive becomes the finalsolution, storing the generated Pareto front. The main operations inthis MOEA consist of generating the fitness of the solutions, calcu-lating the density information for each solution within the solutionset, and then truncating the archive once it becomes full, byremoving the worst quality solutions in the densest areas.

3. Materials and methods

3.1. Material: the Geographic Atlas of Spain of 1804 (AGE)

In 1788, Tomas Lopez began preliminary work on AGE. Thiswork was based on maps that he designed. In a type of dressrehearsal, which was published in 1790, the Private Atlas of thekingdoms of Spain, Portugal and adjacent islands was commis-sioned by Carlos III Manzano-Agugliaro et al. (2013). The carto-graphic method used by Lopez, which is known as deskcartography, was based on a questionnaire that was sent to eachvillage priest (San-Antonio-Gómez et al., 2011). Each priest wasrequired to provide a sketch of his village and two sketches ofthe surrounding leagues. Lopez composed a new sketch of theregion from the sketches provided by the village priests, existing

Fig. 4. AGE image of Extremadu


maps from the 16th, 17th and 18th centuries, and numerous otherdocuments (answers to questionnaires, local histories, geographi-cal descriptions, and cartographic sketches) (San-Antonio-Gómezet al., 2011).

The AGE was the culmination of forty years of work experience ofTomas Lopez Manzano Agugliaro, Martínez García, and San AntonioGomez (2012). His detailed map, which represented mainland Spainas no other map had accomplished, was distributed in 102 sheetswith geographic information about 36 territories. In this study, weexamined the kingdoms of Extremadura (sheets 54-55-56-57) andAragon (sheets 70-71-72-73). Each sheet of AGE has its own refer-ence frame, even when they belong to the same kingdom. This isbecause each sheet was made independently of each other. Thischoice is based on a previous study (Manzano-Agugliaro et al.,2013), in which all errors of AGE were determined to have an aver-age error of 6.5 km. Thus, Extremadura had an average error (RMS)of 10 km between the matched position of 958 towns of early car-tography and current cartography. With regards to Aragon, the pre-vious results showed an RMS error of 7.6 km for 1735 towns. In thiswork, we establish the initial error limit of 20 km to consider a townas correctly matched, this reduced the towns to 501 for Extremadu-ra and to 1399 for Aragon. Distributed as 84, 83, 205 and 129 townsfor sheets 54-55-56-57, respectively; and 137, 552, 363, and 347towns for sheets 70-71-72-73, respectively.

Figs. 4 and 5 display the four maps or sheets that comprise theExtremadura province and the Kingdom of Aragon. The towns that

ra with digitalised towns.



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Fig. 5. AGE image of Aragon with digitalised towns.



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have been digitalised appear in red; they correspond to existingtowns.

3.2. Methods

A software program has been developed that focuses on solvinga multi-objective optimization problem with evolutionary algo-rithms in parallel. The evolutionary algorithms consist of msPESA,NSGAII, HVSEA and SPEAII as described in previous sections. Theresults are obtained by 100 independent executions per map ofeach algorithm.

Two objectives are to be optimized: minimize the mean errorwhen calculating distances between existing urban areas andTomás López’s towns, whilst maximizing the number of locationsincluded in the calculations.

3.2.1. Software objectivesMinimize the average distance between actual coordinates and

Tomás López’s coordinates as

D ¼Xn

i¼1

haversindi

r

� �

¼Xn

i¼1

haversin /i2 � /i1ð Þ þ cos /i2 cos /i1 haversin wi2 � wi1ð Þ½ �

ð2Þ


where

� D is the average distance (the mean error in km between thecoordinates of Tomás López’s towns and the equivalent coordi-nates for existing locations)� haversin is the haversine function expressed as

f evolu16/j.esw

haversinðhÞ ¼ sinh2

� �2

¼ 1� cos h2

ð3Þ

� di is the distance between the actual coordinates and ThomasLópez’s coordinates,� r is the radius of the Earth,� /i1;/i2 represents the latitude of point i1 and the latitude of

point i2 and� wi1;wi2 represents the longitude of point i1 and the longitude of

point i2.

Minimize the number of excluded locations N

N ¼ n�Xn

i¼1

ei ð4Þ

where N is the number of excluded locations and n is the number oftotal locations. ei ¼ 1 when the location is excluded and 0otherwise.

tionary algorithms for analyzing the positional accuracy and uncer-a.2014.04.025


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3.2.2. Calculation parameters (genotype)The parameters in the evolutionary algorithm change until the

solution is obtained (which is known as the genotype). Theseparameters are used to perform the three transformations on themap coordinates and to obtain the final coordinates (Lat0; Lon0).

Rotation origin (Lat0; Lon0): The center coordinates on which thepoints are rotated; they are calculated at the beginning of the solu-tion and represent the center of mass of the towns

Lat0 ¼1n

Xn

1Lati Lon0 ¼

1n

Xn

1

Loni ð5Þ

Using the center of mass, we obtain coordinates for the approx-imate center of the nucleus of the towns; thus, the variation isminor. If we use the coordinates of 0� latitude and 0� longitudeas the rotation origin, we would have to increase the latitudinaland longitudinal displacement.

Rotation (a): the rotation applied, in degrees. It is limited to 0.5�as a large rotation skews the results. For this reason, a rotationgreater than 0.1 is typically not used.

Lat0 ¼ Lat � cos a� Lon� sinaLon0 ¼ Lat � sina� Lon� cos a

ð6Þ

Latitudinal displacement in sexagesimal degrees(DLat): sum of theoriginal value and the value calculated by the algorithm.

Lat00 ¼ Lat0 þ DLat ð7Þ

Longitudinal displacement in sexagesimal degrees (DLon):

Lon00 ¼ Lon0 þ DLon ð8Þ

Horizontal scale factor (SLon): the scale based on the coordinatesalong the longitudinal axis. It is limited to 20% to prevent excessiveshifting of the calculations (between 0.8 and 1.2).

Lon000 ¼ Lon00 � SLon ð9Þ

Vertical scale factor (SLat): the scale based on the coordinatesalong the latitudinal axis.

Lat000 ¼ Lat00 � SLat ð10Þ

The evaluation process calculate both objectives by applyingthe changes described in the genotype for each individual to eachof the coordinates on the Tomás López map (TL), for each locationin the individual (solution) for the town. This process yields thetransformed coordinates; we calculate the distance to the equiva-lent town in the existing nucleus for each coordinate. Calculate themean error with the arithmetical mean as

Error ¼ 1n

Xn

i¼1

Di ð11Þ

where Di is the distance between locations (TL map and current)

3.2.3. Genotype operatorsThis section offers a brief explanation on the operators used in

the algorithms. For the initial solution no town was excluded. Afterthis, a mutation is performed for the child solution. The mutationoperator excluded a town with a probability of 10%. Crossoveroperator for excluded towns was:

child[0].SelectorChance[i]

=(1.5*this.SelectorChance[i]

+ 0.5*parent.SelectorChance[i])/2.0;child[1].SelectorChance[i]

=(0.5*this.SelectorChance[i]

+ 1.5*parent.SelectorChance[i])/2.0;


For nonexcluded towns, the selection do a cross with 50% ofprobability for the selection/deselection of a town among its par-ents. E.g., for each town there is a 50% of probability of takingthe state (selected or removed) from parent A, and 50% take thestate of parent B.

The following formula is used to these genotypes:

//Change the value of a variable

if (Rand > 1/ mutating.Genotype.Length)

{double u = Rand;double y = 0;//Changeif (u < 0.5)

y = mutating.Bounds[i][1]

- mutating.Genotype[i];else

y = mutating.Genotype[i]

- mutating.Bounds[i][0];double delta = y * (1.0 -

(mutating.Stats.Evaluations * (0.5 + 0.5 * Rand))/mutating.Config.MaxEvaluations);

//Perform the change

if (u < 0.5)mutating.Genotype[i]

= mutating.Genotype[i] + delta;else

mutating.Genotype[i]

= mutating.Genotype[i] - delta;}

4. Results and discussion

4.1. Kingdom of Extremadura

4.1.1. Sheet 54The results of the analysis of sheet 54 of Extremadura kingdom

are shown in Fig. 6, where the classical solution obtained and aver-age error of 12.41 km with nonexcluded locations. When we runthe program based on algorithm SPEAII, a solution with less errorappears when 7 towns are excluded (11.22 km). Better solutionscan be found but at the price of higher number of towns excluded.The other three algorithms have better performance than the SPE-AII. The NSGA II algorithm presents cero excluded towns and anerror of 8 km, having a very flat Pareto front. To improve the error,7 towns are excluded, reducing the error to only 7.73 km. The algo-rithm HVSEA also get an initial solution (with an error of 8.59 km)with zero excluded towns like NSGAII did; but to achieve the sameerror of 7.73 km it needed to exclude 12 towns. The algorithmmsPESA begins the Pareto front excluding 1 town to achieve anerror of 8 km, however reaching the same error (7.82 km) fasterthan NSGA II with 3 towns excluded. The fronts of NSGAII, HVSEAand msPESA algorithms looks acceptable from 3 towns excludedonwards, representing 3.5% of excluded towns and lowering theerror by 37%.

4.1.2. Sheet 55Fig. 7 shows the results of the analysis of sheet 55 of Extrema-

dura kingdom, where the classical solution obtained an averageerror of 11.67 km with nonexcluded locations. The Pareto front ofSPEAII shows some solutions being better than classical solutionwhen 5 or more towns are excluded (11.19 km). msPESA andNSGAII fronts are quite similar up to 5 excluded towns with anerror of 9.52 km, i.e. remove 6% of towns to reduce the error by



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Fig. 6. Pareto fronts for sheet 54 of Extremadura kingdom.

Fig. 7. Pareto fronts for sheet 55 of Extremadura kingdom.



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18.5%. In this case the msPESA Pareto front has better performanceovercoming all other methods with less average error. In this mapthe HVSEA pareto front is almost parallel to NSGAII front but bring-ing worst points as it excludes more towns.


dura kingdom, where the classical solution obtained an average


error of 14.93 km with nonexcluded locations. The Pareto front ofSPEAII is better than classical solution when 20 towns are excluded(11.5 km), needing to exclude 45 towns for reaching an error of7.31 km. In this case, only the front end solutions improve theother algorithms, as in previous cases. This means that excluding22% of towns reduce the error in 51%. The NSGAII algorithm pre-sents better front till 19 towns excluded with an error of8.57 km, i.e. removing 9.3% of towns reduce the error in 42.6%.



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From here the msPESA algorithm has almost the same front com-pared to NSGAII. In this map the HVSEA pareto front is parallel toNSGAII front but above it, and it has been better to msPSEA onlyfor the first and second solution of its front.

Fig. 8. Pareto fronts for sheet 56

Fig. 9. Pareto fronts for sheet 57



dura kingdom, where the classical solution obtained an averageerror of 17.51 km with nonexcluded locations. The Pareto front of

of Extremadura kingdom.

of Extremadura kingdom.



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SPEAII is better than classical solution when 11 towns are excluded(16.8 km), needing to exclude 18 towns for reaching an error of11.52 km. This means to exclude 14% of towns to reduce the errorin 34.2%. NSGAII as always, has a very flat front, having good initialsolutions, reducing the error to 46.6% by excluding only 7.8% of thetowns, i.e. 10 excluded towns for an error of 9.34 km, almost thesame msPESA solution (9.53 km) with the same number ofexcluded towns. In this case the HVSEA algorithm gets some pointin its Pareto front below the others fronts, e.g., for 7 excludedtowns the error was 7.5 km, this means to exclude 5.4% of townsin order to reduce 45.7% the error.

Regarding the obtained transformation parameter, as an exam-ple of Extremadura kingdom, the four algorithms are analyzed forsheet 57 and the results are shown in Tables 1 and 2. According toTable 1, NSGAII and msPESA exhibit the same trend: DLat exhibitsa positive value near 0.1� while DLon exhibits a negative valuebetween �0.01� and �0.06�, with the exception of one solution

Fig. 10. Pareto fronts for sheet

Table 1Transformation parameters and error (RMS) with NSGAII and msPESA for the Kingdom of

NSGAII

DLat (�) 0.1037 0.1119 0.1080 0.11DLon (�) �0.0422 �0.0685 �0.0350 �0.0a (�) -0.2338 �0.2160 �0.2258 �0.2SLon 0.9760 1.0009 1.0012 1.03SLat 1.0543 1.0765 1.1373 1.08Error (km) 9.96 9.95 9.64 9.34N (excluded) 3 4 5 10

Table 2Transformation parameters and error (RMS) with HVSEA and SPEAII for the Kingdom of E

HVSEA

DLat (�) 0.1353 0.1164 0.0982 0.10DLon (�) �0.0353 �0.0174 �0.0159 �0.0a (�) �0.2460 �0.2393 �0.2421 �0.2SLon 1.0192 1.0151 1.0577 1.02SLat 1.1741 1.2064 1.2135 1.11Error (km) 10.27 9.73 9.67 9.50N (excluded) 3 5 6 7


for msPESA; a always exhibited a negative value near 0.22�, SLon

exhibited a value near 1.009 and SLat exhibited a value near1.088. HVSEA exhibits the same trend as NSGAII and msPESA, asshown in Table 2. But SPEAII uses quite different parameters inorder to reach the solutions, noteworthy that the solutions with14 and 26 excluded towns are practically equal after rounding to4 decimal, and SLon and SLat exhibit values under 1, except for thesolution of 18 excluded towns.

4.2. Kingdom of Aragon

4.2.1. Sheet 70Fig. 10 shows the results of the analysis of sheet 70 of Aragon

kingdom, where the classical solution obtained an average errorof 10.84 km with nonexcluded locations. The Pareto front of SPEAIIis better than classical solution when 25 towns are excluded(5.7 km), this means to exclude 18% of towns in order to reduce

70 of Kingdom of Aragon.

Extremadura (sheet 57).

msPESA

10 0.0993 0.1033 0.0782 0.1017207 �0.0087 �0.0143 0.1909 �0.0409530 �0.2356 �0.2020 �0.2241 �0.240190 0.9927 1.0269 1.0019 1.036854 1.0375 1.1082 1.1470 1.0595

10.21 9.75 9.59 9.533 7 8 10

xtremadura (sheet 57).

SPEAII

06 0.5335 �0.0175 0.1587 �0.0174238 �0.0138 0.5000 �0.0070 0.5000522 �0.0088 �0.1655 �0.2129 �0.165563 0.8021 0.9967 0.8417 0.996721 0.9478 0.8000 1.1943 0.8000

16.83 12.53 11.52 11.0911 14 18 26



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the error by 47%. But the main drawbach is that this front is quitefar (i.e. covered) from the others. msPESA has a front starting with5 towns excluded and getting an error of 9.75 km, i.e. 3.6% ofexcluded towns to reduce the error by 10%. In this sheet the algo-rithm HVSEA has a front that covers msPESA until 15 excludedtowns from which, msPESA is always below. The NSGAII algorithmhas the best initial solutions until 10 excluded towns, i.e. excluding7.3% of towns reduce the error by 46%. From 10 towns onwards,there are algorithms that deliver better solutions such as HVSEA.


kingdom, where the classical solution obtained an average errorof 6.2 km with nonexcluded locations, the lowest error analyzedtill this moment. Thus, for SPEAII, a better solution than classicalone is obtained excluding a high number of towns, i.e. 25% ofexcluded towns to reduce the error by 14%. With regard to NSGAII,excluding 5.6% of towns is able to reduce the error only by 2.4%, for


Table 3Transformation parameters and error (RMS) with NSGAII and msPESA for the Kingdom of

NSGAII

DLat (�) 0.1061 0.1027 0.1004 0.09DLon (�) �0.1387 �0.1128 �0.1360 �0.1a (�) �0.0128 �0.0021 �0.0617 �0.0SLon 1.0154 1.0135 1.0555 1.01SLat 0.9918 0.9553 0.9181 0.92Error (km) 6.46 5.97 5.94 5.59N (excluded) 25 31 32 43

Table 4Transformation parameters and error (RMS) with HVSEA and SPEAII for the Kingdom of A

HVSEA

DLat (�) 0.0852 0.0841 0.0679 0DLon (�) �0.1361 �0.1210 �0.1134 �a (�) �0.0073 �0.0519 0.0138 �SLon 1.0018 1.0213 0.9913 1SLat 0.9241 0.9836 0.9390 0Error (km) 5.77 5.66 5.65 5N (excluded) 99 103 112 1


msPESA excluding 18.5% of towns is able to reduce the error onlyby 2.6%, almost the same of HVSEA that excluding 18% is able toreduce the error only by 5.7%. For these map, the 4 Pareto frontsneed to exclude too much towns to improve the classical solution,this is because the error is low and therefore it is difficult toimprove.

For the obtained transformation parameter, as an example ofAragon kingdom, the four algorithms are analyzed for sheet 71.Both algorithms (NSGAII and msPESA) in this case do not exhibitthe same trend for all parameters, as shown in Table 3. Thus,DLat exhibited a positive value of approximately 0.1� for NSGAIIand 0.08 for msPESA, DLon was approximately �0.13�, SLon wasapproximately 1.03 for both algorithms, and SLat was approxi-mately 0.95 for NSGAII and 0.87 for msPESA. However, the highestdifference was obtained for a, with a value of approximately�0.03�, a positive value for the first solutions of msPESA and a neg-ative value (�0.008�) in the last solution as analyzed by the Paretofront. The algorithm HVSEA, Table 4, has similar behavior that


Aragon.

msPESA

16 0.0881 0.0875 0.0881 0.0648210 �0.1312 �0.1312 �0.1312 �0.1340483 0.0308 0.0180 0.0180 �0.009387 1.0650 1.0045 1.0255 1.054806 0.8000 0.8422 0.9121 0.9400

7.711 6.526 5.960 5.84097 98 102 105

ragon.

SPEAII

.0867 0.5000 0.1227 0.5018 0.50000.1076 0.4159 0.4999 0.4961 0.50000.0392 0.0000 0.0001 0.0000 0.0000.0297 0.8000 0.8225 0.7998 1.0171.9905 0.8000 0.8039 0.9647 0.8000.38 8.49 8.25 7.19 6.8714 132 135 136 137



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NSGAII, but SPEAII has the worst parameters, where the rotation ais zero or close to in all solutions and SLon and SLat exhibited valuesunder 1, except for the solution of 105 excluded towns.


kingdom, where the classical solution obtained an average errorof 10.67 km with nonexcluded locations. The Pareto front of SPEAIIalgorithm begins to improve classical solution when a large num-ber of towns are excluded, thus excluding 23% of towns is only ableto reduce the error by 6%. Something similar happens to msPESAalgorithm, that needs to exclude 18.2% of the towns to reducethe error by 21.6%; and also for HVSEA that needs to exclude17.4% of towns to reduce the error by 23.5%, but excluding 18.7%of towns is able to reduce error by 37%. As happened in the previ-ous maps, the algorithm NSGAII has the best pareto front for low




non-excluded solutions. It begins to improve the classical solutionby excluding on 18 towns, i.e. excluding 5% of towns is able toreduce the error by 19.8%.


kingdom, where the classical solution obtained an average errorof 7.26 km with nonexcluded locations. SPEAII begins to improveclassical solution when a large number of towns are excluded;excluding 17.6% towns only get to reduce the error to 1.5%, beingthis pareto front far from the other 3 fronts. The front of msPESAbegins excluding 15 towns, i.e. excluding 4.3% of towns is able toreduce the error by 11.8%, but evolving steeply, reaching withthe elimination of 5.8% of the towns reduce errors by 39.8%. Forthis map the front of HVSEA begins better than msPESA front, i.e.4.3% of excluded towns to reduce the error by 24.5%. The NSGAII





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Table 5Performance comparison for HVSEA, msPESA, NSGAII and SPEAII.

HVSEA msPESA NSGAII SPEAII

Sheet N (%) Error (%) N (%) Error (%) N (%) Error (%) N (%) Error (%)

54 3.6 35.8 3.6 37.0 3.6 37.0 22.6 12.155 6.0 13.2 6.0 18.4 6.0 18.5 6.0 4.156 4.9 32.5 4.9 35.3 4.9 38.7 22.0 51.057 5.4 45.7 5.4 44.3 3.9 44.9 10.9 28.470 8.0 48.9 8.0 45.7 7.3 46.1 18.2 47.071 17.9 5.7 18.5 2.6 14.3 10.3 25.7 14.272 18.7 37.0 19.0 24.4 13.5 31.9 23.1 6.373 4.3 24.5 4.3 11.8 4.0 35.1 18.7 5.8

Average 8.6 30.4 8.7 27.5 7.2 32.8 18.4 21.1



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front behaves very well for initial solutions, i.e. 0.3% of excludedtowns to reduce the error by 21.3%, and for 4% of excluded townsto reduce the error by 35.1%. However to further reduce this error,other algorithms works better, especially HVSEA and msPESA.

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4.3. Algorithms assessment

We have confirmed considerable improvement in the quality ofthe results with respect to the classical techniques used in theanalysis. This improvement is due to the inherent nature of theevolutionary algorithms. These algorithms efficiently explore thezones of promising solutions. By eliminating few solutions, wecan achieve an average error that is significantly lower than theerrors detected in current techniques.

Table 5 summarizes a general comparison of all four algorithmsimplemented, HVSEA, msPESA, NSGAII, and SPEAII. It is shown thepercentage of excluded cities per sheet, and the reduction of rela-tive percentage error compared to classical solution. The criterionschosen for this comparison was: first, choose a percentage ofexcluded towns between 5 and 10% when possible and always asclose to this lower limit; second, the percentage of excluded townsas similar between the 4 algorithms analyzed. When it has notbeen possible to choose these, choose some reference value forthe percent of excluded towns as similar as possible between them.

As shown in Table 5, the worst algorithm to address this prob-lem has been the SPEAII, which needed to exclude the same or sim-ilar percentage of towns to reduce that percentage error. The other3 algorithms (HVSEA, msPESA, and NSGAII,) in general can be con-sidered a quite similar behavior, and good compared to the classi-cal solution. For these three algorithms, by utilizing the algorithmsand eliminating 8% of the cities, it is possible to obtain improve-ments in the range of 30% for HVSEA, msPESA, and NSAG-II, respectto classical solution. The best behavior had especially in the initialsolutions was the NSGAII, and then HVSEA.

We highlight the utility of the different transforming factors ofcoordinates that were been selected. For solutions in the Paretofronts, the displacement (D) is the most important factor. This fac-tor corrects most of the existing errors, which may be related to theoriginal source of the historical maps. Previously and with avail-able means, it was not easy to establish the coordinate frameworkdue to the lack of support for precise cartographic methods. Theseparameters enable a systematic displacement in the historicalmaps with respect to current cartography. For both algorithmsand for all solutions, with the exception of one solution, the param-eters always act in the same direction: SLat > 0 and SLon < 0.

The rotation (a) does not present a distinct trend comparedwith the displacement. Note that this factor shows a negativedirection for NSGAII and HVSEA and a positive direction formsPESA and SPEAII in the map of the Kingdom of Aragon. Con-versely, this map shows a negative trend for all solutions in thefour algorithms for the map of the Kingdom of Extremadura. These


results indicate that this trend corresponds to particular errors inthese maps.

With regards to the scaling (S), it presents similar trends foreach map. For the map of Kingdom of Extremadura, we alwaysobserve SLat > 1 except for SPEAII, and after the second solutionin the Pareto fronts, SLon > 1 except for SPEAII. For the Kingdomof Aragon, both Pareto fronts always present SLon > 1 and SLat < 1except for SPEAII that was always SLat and SLon < 1. Then, takinginto account the three algorithms (NSGAII, msPESA and HVSEA)this could be because of a systematic error in the historical map.Given that the three algorithms (NSGAII, msPESA and HVSEA) workgood, obtaining similar transforming parameters in their best solu-tions, they converge towards ‘‘ideal’’ parameters. The algorithmthat discards fewer towns (NSGAII) ‘‘learns’’ at a faster rate. There-fore, we consider that improves the resolution of this problem.

5. Conclusions

The obtained Pareto fronts, Figs. 6–13 show that SPEAII algo-rithm performs worst for the left side of the front, i.e. findinglow error solutions with low non-excluded towns, in contrast toNSGAII that usually finds better solutions. The NSGAII algorithmpresents Pareto fronts that cover the fronts of the msPESA andHVSEA. This finding indicates that for the same number ofexcluded cities, it obtains better results, at least for the initial solu-tions. However, the msPESA and HVSEA algorithms reduce errorscompared with the NSGAII, at the cost of excluding more cities.That is, they perform better for solutions in the right side extremeof the front. The majority of non-dominated solutions obtained byNSGAII dominate the solutions obtained by msPESA and HVSEA,although msPESA obtains acceptable extreme solutions in someinstances. However it has been observed that NSGAII has a hori-zontal slope needing to exclude too many towns for reducing thenerror from a certain boundary. Therefore, it can be a good optioncombining Pareto fronts of several algorithms, using the envelopefronts obtained.

It has been observed that when the error is small, the uncer-tainty is also small because it’s more difficult for our algorithmsto outperform the classical solution, needing to exclude too manytowns, e.g., sheet 71 (Aragon kingdom whith the lowest error). SeePareto fronts of Fig. 11 and the results of Table 5, where on averagefor the four algorithms, excluding 19% of towns only reduce theerror by 8%. Then we can say that the algorithms studied, espe-cially HVSEA, msPESA and NSGAII, are best alternative when theuncertainty of map analyzed is greater.

In summary, comparisons of four multi-objective evolutionaryalgorithms (HVSEA, msPESA, NSGAII, and SPEAII) for analyzingthe positional accuracy and uncertainty problem of eight earlymaps have been presented. The experiments were performed ontwo historical maps composed of four sheets with high uncer-tainty, and with its own reference frame. The results show that



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by eliminating 8% of the cities it is possible to obtain improve-ments of approximately 30% for HVSEA, msPESA and NSGAII. Basedon the comparison between HVSEA, NSGAII and msPESA, we canconclude that, for this particular problem, the majority of non-dominated solutions obtained by NSGAII dominate the solutionsobtained by msPESA and HVSEA. However, acceptable extremesolutions can be obtained with HVSEA, msPESA and SPEAII in someinstances. It can be a good option combining Pareto fronts of sev-eral algorithms, using the envelope fronts obtained. The Paretofronts based on multi-objective evolutionary algorithms (MOEAs)are a better alternative when the uncertainty of a map to analyzeis high, making them very suitable for solving the problem posed.Consequently this creates new perspectives for analyzing the posi-tional accuracy and uncertainty of historical maps.

Acknowledgements

This study was funded by project HAR2009–12937 (GIS System-atic Analysis for the Planimetric Accuracy of the Geographic Atlasof Spain of Tomas Lopez, 1804) of the Spanish Ministry of Scienceand Innovation. The authors are grateful to Dr. Branko Bozic, a pro-fessor at the University of Belgrade (Serbia), for providing informa-tion for this manuscript.

References

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