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MAP COMPARISON KIT

1. MAP COMPARISON KIT User manual research institute for knowledge systems MAP COMPARISON KIT User manual Research Institute for Knowledge Systems BV RIKS P O Box 463 6200 AL Maastricht The Netherlands Tel 31 43 3883322 Fax 31 43 3253155 http www riks nl info riks nl Submitted to National Institute for Public Health and the Environment RIVM Bilthoven The Netherlands July 2003 Introduction 2 2 2 3 3 Getting started __ 1 Open a log file 2 The Analyse application window 1 2 1 The Menu bar 1 2 2 The Toolbar 3 Create your own LOG file Compare Maps _______________ 1 Perform a comparison 2 Exporting results 3 The Map Comparison Methods 2 3 1 Per category 2 3 2 Cell by Cell 2 3 3 Fuzzy Inference System 2 3 4 Fuzzy Set 2 3 5 Numerical comparison 2 3 6 Other operations Customizing the views ____________ 1 The Legend editor 2 The palette editor The Menu System_____________ 1 File menu 2 Edit Menu 3 View menu 4 Options menu 5 Window menu 6 Help menu Files in the Map Comparison Kit ______ 1 The log file 2 Legend files 3 Palette files 4 Map files 5 4 1 ArcASCII format 5 4 2 LLO format 5 4 3 Idrisi file formats 5 4 4 The region map Appendix l Kappa variations Appendix Il Fuzzy Inference System Appendix Ill Fuzzy Set map comparison TABLE OF CONTENTS INTRODUCTION This manual expla
2. 80 C Power et al Fuzzy set theory has also been used in GIS applications most notably in the analysis of uncertainty propagation in GIS operations Veregin 1989 and the devel opment and manipulation of fuzzy relational databases Burrough 1989 Wang et al 1990 Kollias and Voliotis 1991 Sui 1992 Fuzzy sets have also been used in the development of a fuzzy method of accuracy assessment of thematic maps Gopal and Woodcock 1994 Despite the increased use of fuzzy set theory in GIS and remote sensing several authors Gong 1993 Gopal and Woodcock 1994 have expressed the need for research involving fuzzy sets for map comparison Edwards and Lowell 1996 suggest that fuzzy set theory should also be used to develop a single measure of map accuracy such as a fuzzy Kappa statistic Hierarchical fuzzy pattern matching addresses both of these issues 3 Methodology Hierarchical fuzzy pattern matching is designed to emulate human reasoning when comparing multiple maps While performing a visual comparison of maps a person intuitively identifies a hierarchy of similarities between the maps Specifically he would first notice the overall agreement between the maps but would eventually recognize localized patterns of dissimilarities To simulate a visual comparison of maps hierarchical fuzzy pattern matching is similarly performed on both a local and global level 3 1 Local matching The preliminary step in the local matching process is
3. palette is used If you wish to customize the colours used there are two options besides selecting an alternative palette 1 Define the colour of legend items with the Legend item editor ii Modify the palette using the Palette editor see Section 3 2 oka Important Modifying the palette will affect all legends that use this palette Ga Modifying the colours in the legend will only affect the legend belonging to Tip the particular theme or comparison method If in doubt do not use the Palette editor but use the Legend item editor instead When you click in a colour box of a legend a Legend item dialogue opens In this dialogue you can define the names of the labels of the legend classes and set the lower lo and upper hi limits of the class range Also you can select a new colour to represent the cells belonging to the class If you have configured the legend editor to create a linear scale then you can only set a new colour with the legend editor itself Legend item lo hi 0 349 0 449 Color Label Cancel Your configurations are saved as part of a legend file associated with the theme that you are defining the new legend for This file is saved as soon as you press the OK button in the Legend editor dialogue 3 2 The palette editor In the Palette editor dialogue you can edit the colour palettes that the legends of the MAP COMPARISON KIT use It is also possible to create new colour palette files SMP extensi
4. pp 1 1 3 6 NGUYEN H T and WALKER E A 1997 A First Course in Fuzzy Logic Boca Rafon FL CRC Press pp 21 60 ROSENFIELD G H and FITZPATRICK LINS 1986 Coefficient of agreement as a measure of Thematic Classification accuracy Photogrammetric Engineering and Remote Sensing 48 131 137 SIMPSON J J and KELLER R H 1995 An improved fuzzy logic segmentation of sea ice 100 Fuzzy regional comparison of land use maps clouds and ocean in remotely sensed Arctic imagery Remote Sensing of Environment 54 290 315 SINGH A 1989 Digital change detection using remote sensing data International Journal of Remote Sensing 10 989 1003 Sul D 1992 A fuzzy GIS modelling approach for urban land evaluation Computers Environment and Urban Systems 16 101 1185 TAYLOR P 1977 Quantitative Methods in Geography Boston Houghton Mifflin Company pp 177 179 VEREGIN H 1989 Error modelling for the map overlay operation In The Accuracy of Spatial Databases edited by M Goodchild and S Gopal London Taylor amp Francis pp 3 18 WANG F 1990 Improving remote sensing image analysis through fuzzy information repres entation Photogrammetric Engineering and Remote Sensing 56 1163 1169 WANG F HALL G B and SUBARYANO P 1990 Fuzzy information representation and processingin conventional GIS software database design and application International Journal of Geographical Information Syst
5. This fuzziness of location is taken into account by letting the fuzzy representation of a cell be partly defined by the cells found in its proximity The level to which neighbouring cells influence a cell is set with a function Three types of functions are supported in the Parameters dialogue Exponential decay Linear decay and Constant value Each of these functions takes a parameter respectively oe i Fuzzy Kappa Settings Set neighbourhood Settings radius in cell units Radius of neighbourhood Accept or reject latest E changes in settings nehne ceine Apply settings without Linear decay closing the dialogue Slope Expand window with advanced settings C Constant value Value Select distance Set parameter for decay function distance decay function Halving distance Slope and constant Value In the Advanced part of the Parameters dialogue it is possible to apply two different functions for the two maps Using this option requires a thorough understanding of the Fuzzy set map comparison algorithm Fuzziness of categories The definition of categories in maps is often imprecise This is especially true if some or all categories on the map have in fact an ordinal definition such as for instance the categories high medium and low density residential area on a land use map The boundaries between such categories are less clear cut than what seems to be the TA case from the legend Th
6. column number for x values column number for y values column number for z values The body of cell information is organised in colums and must always contain a column with x y and z values Every line in the body refers to a cell The value in the x and y column are used to find the cell in the matrix and the z column gives the particular cell value Each column contains values and the columns are tab or space separated Column and row numbers are calculated as follows col x xllcorner cellsize row y yllcorner cellsize 32 These row and col values are always rounded down to integers Flour algorithm If cells are found more then once in the list then the last value in the list persists Cells that are not found in the list obtain a nodata value LLO files are stored in ASCII format thus they can be edited in ASCII editors such as Notepad The following image gives an example of an LLO file opened in Notepad This example is taken from the Chessboard variations directory of the example files that are optionally installed with the MCK P map1 LLO Notepad 5 4 3 3 4 222500 227500 232500 237500 202500 207500 212500 217500 617500 617500 617500 617500 612500 612500 612500 612500 e a e D he D eA A Idrisi file formats For documentation of the Idrisi file formats we refer to the Idrisi manual One important characteristic of Idrisi files should be mentioned here Idrisi files come in
7. matched Fuzzy regional comparison of land use maps 93 Table 8 Boolean and fuzzy global similarity statistics Fuzzy global Boolean Boolean Dataset matching Kappa CAA Boolean Tau Cincl 0 71 0 64 0 75 0 70 Forest1 0 78 0 76 0 81 0 78 agreement Congalton et al 1983 state that the CAA is an unreliable measurement of map similarity because it overestimates the agreement between maps by not accounting for chance agreement Based on this information a preliminary require ment of an acceptable global similarity procedure is that its output value for a particular map comparison fall between the computed Kappa and the CAA values The Tau coefficients and global matchings both satisfy the above requirement Furthermore the Tau values of 0 70 and 0 78 are very similar to the global matchings values of 0 71 and 0 78 see table 8 Ma and Redmond 1995 describe how the use of Tau over Kappa and CAA is justified for its ability to incorporate probabilities into the calculations which avoids overestimating the random agreement between maps However the authors failed to consider that Tau depends on a pixel by pixel comparison to obtain the observed agreements for the map categories Misregistration of one or both of the maps could decrease the computed agreement value By accounting for locational and attribute uncertainties in the computation of the local matching the fuzzy global matching procedure is an appropriate alternat ive to the Boolean
8. 1 The Legend editor To customize the legend of a theme or a discrepancy map select a window displaying a map of the desired theme or comparison method Use the Legend command from the Edit menu in the menu bar to open the Legend editor dialogue In the figure below the dialogue is shown and the different options are explained Click box to start Select palette and Choose the order of p legend item editor Hiki number of classes legend colours g g Palette settings m M Top of legend is E 0 85 0 95 Palette babyloy smp amp Palette color Palette color N Lancel 0 75 0 85 l L Accept or decline 1 65 0 75 H Classes E Falette editor Start new settings Select type to fit data in map Legend type F 0 35 0 45 C Categoric Numeric Shortcut to palette Cli I l l editor Choose lowest and Numeric legend settings EEEE f Dieplay interval Number format igNest value o ee range Min fo ar f Decimals 2 0 05 U Scale Labels Obtain suggestion for Linear scale legend range Select linear scale A Frequenes dependant scale Pa for equal intervals eee ra custom for user Lo Hil 2 ee a defined intervals Top legend is display Display Max C Custom C Display Min i Lo Hi Choose the order of Choose the appearance __ Apply latest 19 i It is possible to apply the colours from a ready made palette from the palette directory In the illustration above the babylov smp
9. 8 S V4 Va Lii HB 1 Min UA 2 HB 2 Min gt ees UAc HB clmin max 8 In equation 8 S V 1 Vg stands for the similarity between a cell in map A and one at the same location in map B Zadeh 1965 indicates the same expression by the letter M and refers to it as the maximal grade for the intersection ANB This similarity index is chosen because it is functional relatively simple and intuitive Many other fuzzy similarity measures have been researched and proposed however and a better alternative may be found Zwick et al 1987 Shyi Ming 1995 Xuzhu 1995 Tolias Panas and Tsoukalas 2001 Equation 8 calculates the similarities if the Fuzzy Neighbourhood Vectors of the two central cells found in figure 3 The membership settings and notations are those used before in figure 2 SCV nbn a gt V bh B 1 0 5 Mins 0 2 1 Mins oe 0 5 OD iia leas 0 5 9 The value for similarity ranges from O to 1 S V Vg will equal O for two completely dissimilar neighbourhoods and 1 for neighbourhoods with matching central cells The value of 0 5 resulting from the operation is to be interpreted as considerably similar It is noted however that this similarity value is due to the fact that both central cells neighbour a grey cell Thus the calculated similarity is based on the neighbours rather than the cells themselves 2 3 2 Two way comparison By directly comparing the fuzzy representations of two cells a part
10. ADemo map3 img of 12 Result map Fuzzy Set algorithm oe Result map HH Statistical analysis Fuzzy Set algorithm C Program Files Map Comparison Kit Log bestanden Demo map1 img Slope jas Log bestanden Demo map3 ima Result statistics Value fo 5 EEEE ae Fuzzy Kappa 0 485 Average similarity ee EO O e The Map window contains the first map to compare analyse To change the contents of 1 Map window choose another map from the combo box next to the button on the Toolbar If the 7 Map window is not open yet then you can do so by clicking the button e The 2 Map window contains the second map to compare analyse To change the contents of the 2 Map window choose another map from the combo box next to the 2 button on the Toolbar If the 2 Map window is not yet open then you can do so by clicking the 2 button e The Result map window contains the result map This map shows spatial result of the last performed map comparison Depending on the selected method the results are presented in a continuous scale or a nominal scale e The Result statistics window contains the statistical results of the last performed map comparison e The Comparison settings dialogue allows setting and viewing the settings belonging to the active comparison method 1 2 1 The Menu bar The menu of the MAP COMPARISON KIT is situated on the menu bar of the Map Comparison Kit application window The commands are ordered in
11. Journal of Geographical Information Systems 5 209 223 Ma Z and REDMOND R H 1995 Tau coefficients for accuracy assessment of classification of remote sensing data Photogrammetric Engineering and Remote Sensing 61 435 439 MACLEOD R D and CONGALTON R G 1998 A quantitative comparison of change detection algorithms for monitoring eelgrass from remotely sensed data Photogrammetric Engineering and Remote Sensing 64 207 216 MACMILLIAN W D 1978 An introduction to the theory of fuzzy sets in the context of the construction of representational spatial economic theory In Towards the Dynamic Analysis of Spatial Systems edited by R L Martin N J Thrift and R J Bennett London Pion Limited pp 36 52 MAMDANI E H 1976 Advances in the linguistic synthesis of fuzzy logic controllers International Journal of Man Machine Studies 8 669 679 Mas J F 1999 Monitoring land cover changes a comparison of change detection techniques International Journal of Remote Sensing 20 139 152 MASELLI F RUDOLPH A and CONESE C 1996 Fuzzy classification of spatially degraded Thematic Mapper data for the estimation of sub pixel components International Journal of Remote Sensing 17 537 551 MATLAB 1994 Fuzzy Logic Toolbox Software 1994 Mathworks Inc 5 40 MEADES W J and Moorgs L 1989 Forest Site Classification Manual A Field Guide to the Damman Forest Types of Newfoundland Forestry Canada
12. accordance with the Windows conventions thus ensuring quick familiarization with the software The following overview gives a short description of each menu The menus are described in Chapter 4 File manage your files The printing facilities are also located in this menu If you want to exit the program you can do it from here 1 2 2 The Toolbar a Just underneath the Menu bar there is a Toolbar The Toolbar also known as Speed bar gives a fast access to the principal functions of the MAP COMPARISON KIT that are also found in the main menu Opening the map and table windows as well as selecting the maps for the comparison can be done from the Toolbar gt 40 O Landise amp Demo mapt ima _ Demo map3 ima ryt ara 19 Use this button To open a log file from the disk edit the log file start the Legend editor zoom in The size of the map increases 2x with each click zoom out The size of the map decreases 2x with each click select the theme to compare open the 7 Map window Demo mapt img select the map to be shown in the 1 Map window open the 2 Map window Demo mapa img select the map to be shown in the gou Map window select a comparison method open the Comparison settings dialogue perform comparison and open the Result map window perform comparison and open the Result statistics window 10 1 3 Create your own LOG file The MAP COMPARISON KIT alw
13. and disagreements on the fuzzy map With the resolution of the input data being 30m the fuzzy inference system is sensitive to the possibility of random disagreements between the maps while the Boolean model identifies every pixel by pixel disagreement as change 6 Conclusion Historically the comparison of thematic maps has been the basis for many land use change detection procedures Traditional pixel by pixel map comparison tech niques are suspect because of possible map registration and error propagation problems These Boolean similarity operations often can not adequately account for the uncertainty and complexity inherent in spatial information A fuzzy regional polygon by polygon comparison methodology mitigates these difficulties In this paper it has been demonstrated that Hierarchical Fuzzy Pattern Matching can be successfully used to measure both map similarities and land use change between maps while accounting for the uncertainties in the datasets It has been shown that a fuzzy local polygon by polygon land use comparison is less affected by possible map registration problems because the fuzzy inference system indirectly fuzzifies the boundaries of the polygons The local matching results from the fuzzy inference system for the project datasets demonstrate the advantage of the fuzzy approach over the Boolean comparison methods Specifically the fuzzy land use change possibility maps provide a better interpretation of the land use agr
14. for similarity in the central cell will be high because the two cells match both black regardless the circumstance that the neighbourhoods grey and white are dissimilar The calculated similarity of the central cell is 1 Figure 3 Six situations in which the left and right map are compared with consideration of fuzziness of location 4 Results The multi method similarity assessment is applied on a case of validating of model results The particular model is a constrained cellular automata White Engelen and Uljee 1997 applied for the study of the urban development of Dublin as part of the Murbandy project White Engelen Uljee Lavalle and Ehrlich 2000 The objective of the case is to compare model results with observed data the two maps are displayed in Figure 4 1998 observed data 1998 model results C Arable land Permanent crops Pastures E Commercial areas WR Public and private services L Port areas Heterogeneous agricultural areas F Construction sites T Forests D Abandoned area C Shrub and or herbaceous vegetation associations E Road and rail networks and associated land L irport F Mineral extraction sites B Dump sites Sparsely vegetated areas C Wetlands E Residential continuous dense urban Fabric WS Residential continuous medium dense urban Fabric P Artificial non agricultural vegetated areas I Residential discontinuous urban Fabric P water bodies D Residential
15. how well fuzzy land use pattern matching detected forest succession during the six year study period It should be mentioned that forest regeneration and succession are complex and complicated processes that are often difficult to model with traditional Boolean techniques This is partially due to the inability of such techniques to represent intermediate growth patterns Unless a major event such as a forest fire has occurred the patterns of change in forest inventory over six years will tend to be sporadic and fragmented Meades and Moores 1989 The sensitivity of the fuzzy pattern matching model to complex growth patterns was determined by concentrating the change detection analysis on the cleared and non forested categories figure 12 These forest inventory types were considered to be the ones most likely to produce mixed succession and regeneration results The Boolean classification identifies the discrete change and no change classes for each forest inventory type but fails to find areas of mixed change The intermediate change information is lost because the Boolean approach constrains and simplifies the change detection process The transitional range of change on the fuzzy land use possibility map figure 13 shows that the fuzzy model detected intermediate and definitive change patterns Several areas for the non forested category have an intermediate possibility of change and are displayed in a medium grey on the disagreement membership
16. ing the global matching values to a number of standard Boolean similarity measures For this purpose the global matching values are compared to the Coefficient of Areal Agreement CAA Taylor 1977 Kappa coefficient of agreement Rosenfield and Fitzpatrick Lins 1986 Singh 1989 and the Tau coefficient of agreement Ma and Redmond 1995 Firstly note that the global matchings in table 8 fall between the calculated Kappa and CAA numbers For example the global matching value for Cincl is 0 71 which is between the Kappa value of 0 64 and the CAA of 0 75 This is the expected result because of the problems with both Kappa and CAA Foody 1992 found that Kappa consistently overestimates chance agreement and underestimates map Table 6 Local matchings for land use polygons of Forest1 of Evaluation of Land use type polygons Match Mismatch similarity No data 20 2 18 10 0 Water 30 28 2 93 3 Cleared 21 4 17 19 1 Non forested 118 105 13 88 9 bF 60 53 7 88 3 MbF T3 62 13 82 7 MO 56 49 7 87 5 Spruce 10 10 0 100 0 Deciduous 12 8 4 66 7 Total 402 321 81 70 7 Table 7 Evaluation of the matches and mismatches of Forest1 Definite matches Definite mismatches 321 polygons 81 polygons 79 6 of the polygons 20 4 of the polygons 81 4 of total area 18 6 of total area 62 of 402 polygons are 1 pixel 56 are matched 6 are mismatched 0 48 of the total area 39 of 402 polygons are 2 pixels 30 are 9 are mismatched 0 60 of the total area
17. interpretation British Journal of Psychiatry 130 79 83 Metternicht G 1999 Change detection assessment using fuzzy sets and remotely sensed data an application of topographic map revision JSPRS Journal of Photogrammetry and Remote Sensing 54 4 221 233 Monserud R A amp Leemans R 1992 Comparing global vegetation maps with the Kappa statistic Ecological Modelling 62 275 293 Pontius Jr R G 2000 Quantification error versus location error in comparison of categorical maps Photogrammetric Engineering and Remote Sensing 66 8 1011 1016 Pontius Jr R G amp Schneider L C 2001 Land cover change model validation by an ROC method for the Ipswich watershed Massachusetts USA Agriculture Ecosystems amp Environment 85 1 3 239 248 Power C Simms A amp White R 2001 Hierarchical fuzzy pattern matching for the regional comparison of land use maps International Journal of Geographical Information Science 15 1 77 100 White R Engelen G Uljee I Lavalle C and Ehrlich D 2000 Developing an Urban Land use Simulator for European Cities In Proceedings of the 5th EC GIS Workshop held in Stresa Italy 38 30 June 1999 edited by E Fullerton Ispra Italy European Commission Joint Research Centre pp 179 190 Winter S 2000 Location similarity of regions JSPRS Journal of Photogrammetry and Remote Sensing 55 3 189 200 Zadeh L 1965 Fuzzy sets Information and C
18. is Large then Local is Perfect Fuzzy regional comparison of land use maps 87 membership function at the height equal to the fuzzy support of the premise of a rule For example figure 6 is a graphical representation of the ten rules in the database of the local matching fuzzy inference system Note that the point of intersection between the vertical lines and the membership functions determines the membership value for each input variable in the rules Depending on the pixel value the height of the output local matching curve is equal to the lowest value of either the areal intersection or areal complement For example the height of the output curve for rule 8 is equal to the areal intersection value Since the purpose of the fuzzy inference system is to map the input variables to an output subset the consequence of each activated rule needs to be combined into a single output distribution Jager 1995 The local matching fuzzy inference system utilizes the Max Min compositional rule of inference for the aggregation of fuzzy rules More specifically the inference scheme is applied as Nguyen and Walker 1997 M x u Max A x Min B u jJH 1 2 0 4 where Max and Min are the logical OR and AND fuzzy connective operators respectively With Max Min composition as the inference rule the local matching for unique polygon X is expressed as Lm X Max Min Area_Inter X Area_Comp X Pixel Group X 5 3 6 Defuzzifi
19. map These are regions where the matching process has determined that approximately half of a template polygon is contained within the 1991 map Consider the large polygon that is outlined in the Northwest corner of the study area The Boolean model subdivides this region into areas of definite change and no change This suggests that entire sections have undergone a complete land use change while other regions have remained unchanged It is unrealistic that a Boolean boundary could separate where forest succession has taken place In contrast the intermediate change Boolean Change Classes N A No Change Cleared Change Cleared No Change Non Forested Change Non Forested Z FS ee SRA 0 Kilometers 1 Figure 12 Boolean change classes for cleared and non forested 96 C Power et al Change Cleared Change Nonforested LL 1 J 0 Kilometers 1 Figure 13 Fuzzy change possibilities for cleared and non forested possibility classification on the fuzzy map indicates that gradual forest infilling has occurred but there has not been a complete transformation in forest inventory type In this case the fuzzy map has more information about the change characteristics of the study area and gives a more appropriate interpretation of dynamics of forest species succession A further advantage of using fuzzy change possibilities rather than Boolean categories is that there are visually fewer one pixel agreements
20. number of rows number of columns x coordinate of the lower left corner y coordinate of the lower left corner cellsize nodata value The body of cell values is organised in lines and columns and the value found at a line and column number in the file corresponds with a cell value for the 31 same row and columns number in the matrix Lines are divided by a carriage return columns may be separated either by spaces or by tabs ArcASCII files are stored in ASCII format no surprise here thus they can be edited in ASCII editors such as Notepad The following image gives an example of an ArcASCII file opened in Notepad This example is taken from the Nodata Test directory of the example files that is optionally installed with the MCK P test2b asc Notepad E Ioj x File Edit Format wiew Help 13000 yllcorner 357500 cellsize 50 nodata 1 0 0 1 1 1 99 1 1 1 5 4 2 LLO format The LLO llo format is a file format developed in conjunction with the National Institute for Public Health and the Environment RIVM It is a simple map format and can be used by other applications as well as the Map Comparison Kit An llo file itself consists of a header block and a body of cell values The header block is structured after the ArcASCII header and holds the following information e number of rows number of columns x coordinate of the lower left corner y coordinate of the lower left corner cellsize nodata value
21. pairs Idrisi stores the header information and the matrix contents in two separate files The 16 bit version of Idrisi stores the header information in a file with a doc suffix and the matrix values in a img file The doc file is an ASCII file that may be edited from any ASCII editor The 32 bit Idrisi maps consist of a RST file with matrix values and a RDC file with header information If you move an Idrisi map from one location to another you should always make sure to copy both of these files Likewise if you rename an Idrisi map you should make sure to give both files the same name except for the suffix Examples of 32 bit Idrisi maps can be found in the Spot the Differences directory of the example files that are optionally installed with the MCK The 16 bit files can be found in the LOV Netherlands directory 5 4 4 The region map Due to the way in which map files are stored to disc they always represent a rectangular area In reality we most often do not want to compare maps of this shape Instead we want to compare a specific region within the rectangular area For instance if we want to compare two maps of the Netherlands then we may want to exclude all the sea as well as Belgium and Germany from the map This can be accomplished by using a region map The region map contains integer values in which every integer value 33 represents one region By definition the region with the value 0 is excluded from the comparison Thus
22. patterns are often inherently complex and can consist of an intricate intermixture of land use types Boolean maps must frequently simplify or otherwise misrepresent land use patterns so that the results of a post classification comparison may be imprecise The accuracy of a comparison procedure based on a more reliable and robust approach could have a marked improvement in the ability to detect and model real world change A third problem with the traditional approaches is that because they are based on a pixel by pixel comparison they do not necessarily capture the qualitative similarities between two maps that is the similarity of patterns This problem becomes important when map comparisons e g of actual and predicted land use are used to evaluate the output of predictive spatial models such as cellular automata based land use models The predictive models are not expected to be accurate at the pixel scale They are however expected to predict the approximate shapes and locations of land use regions The lack of appropriate comparison techniques spe cifically ones that can handle qualitative comparisons of complex land use maps for the purpose of evaluating model output is currently a major problem in the area of cellular automata based predictive simulation modelling White et al 1997 The purpose of this paper is to present a map comparison procedure based on fuzzy set theory that can more fully capture both the complexity and the p
23. probability that both cells match within the cumulative number of cells of the i th ring P n minus the probability that both cells already match within the previous ring P n _ c E ili gt 1 2 a 1b 1 a X Ya X Xp X P n P ni 1 17 Equation 17 calculates for each combination of categories a and b the probabil ity that their determining ring is the i th 0 stands for the Kronecker delta of a and b which has the value 1 if a and b are equal and 0 if they are not The probability of matching central cells is calculated separately and according to the Kappa statistic Monserud and Leemans 1992 equation 18 Ms E ij 0 r xX 18 a 1 The total statistic for the expected percentage of agreement is the weighted summation of all rings according to equation 19 R P E i x M d 19 i 0O In equation 19 R is the number of the furthest ring M is the fuzzy membership function and d is the radius of the i th ring The derivation of K as presented here does not consider the size of the map The size of the maps is relevant however because the neighbourhoods are different at the edges of maps This should be considered in case small or irregularly shaped maps are compared In these cases K is underestimated because P is over estimated A solution to this problem is to find the cumulative number of cells in each neighbourhood ring for every cell calculate the expected similarity f
24. randomly over the maps The following extreme case illustrates the difference between Kappa and fraction correct We have a model to predict the nesting locations of ducks in a park There are two categories for the maps nest and non nest In reality a 14 nest will be found in only one out of a hundred cells This means that a model that ignores the occurrence of nests and therefore assumes all cells to be non nest obtains an impressive fraction correct of 0 99 regardless the fact that it represents all nest cells incorrectly On the other hand a model that assumes all cells to be nests obtains a meagre fraction correct of 0 01 regardless the fact that this model represents all nests correctly Both models have the same distinguishing quality none at all however one scores better than the other The reason is that the fraction correct rewards models that overestimate prevalent categories The Kappa statistic removes this bias and returns the same similarity for both models the value 0 BH Statistical analysis ioixi The Kappa statistic results from two types Method celbycll of similarity similarity of quantity and Map 1 Crlusers ahageni Test AinalyselLog Fake krt1 img similarity of location Here quantity refers lente Gilusers ahagen Test analyselLogiFskelkrt3 ma tot the total number of cells taken in by each ere e oo m a Category found in the legend in other KLocation 0 401 Histo 0 944 wor
25. takes many aspects into consid eration without deliberately trying Local similarities but also global similarities logical coherence patterns etc are recognized Map comparison methods performed by software usually capture one of these aspects but overlook the others Further more they generally lack the flexibility to switch from one aspect to the other when the data requires it The best example of this rigidity is the cell by cell comparison of two checkerboards the first board has a white field in the upper left corner the second a black field The average observer would immediately recognize the two boards as being highly similar in quality however a cell by cell comparison method would find a black cell where a white one is expected and vice versa Hence total disagreement would be concluded International Journal of Geographical Information Science ISSN 1365 8816 print ISSN 1362 3087 online 2003 Taylor amp Francis Ltd http www tandf co uk journals DOI 10 1080 13658810210157822 236 A Hagen Despite these clear disadvantages there are situations where automated map comparison is preferred above visual comparison One reason is that an automated procedure can save time and human effort More important is that automated procedures are explicitly defined and therefore repeatable Thus the method can be analysed and evaluated and the results can be verified A visual comparison will always be subjective and often intuitive T
26. the uncertainty in the linguistic containment expres sions and allow values to have multiple memberships in the function set Simpson and Keller 1995 Based on an analysis of the data and previous research on land use dynamics White et al 1997 a third set of input membership functions is used to account for the effect of the number of pixels comprising the unique polygons There is a strong possibility that many of the polygons identified by the grouping procedure will consist of one or two pixels The problem that arises is whether or not a single pixel disagreement is actually change or a random artefact in the data The calculation of a global matching value could be adversely affected by assigning the same weight to these small unique polygons as to the larger ones Figure 4 displays the pixel group membership functions both being sigmoidal curves The input data ranges from one to four since the pixel information is divided into four distinct categories 1 one pixel 2 two pixels 3 three pixels and 4 gt three pixels The output from the fuzzy inference system is a set of linguistic expressions that describe the local matchings for the unique polygons The output linguistic statements Membership function plots Small Large 1 0 5 Q 0 1 2 3 4 input variable Pixel_Group Figure 4 Membership functions for pixel groupings Fuzzy regional comparison of land use maps 85 are based on a five point evaluation scale table
27. 1 SCV nbn A gt V crisp B 1 Olin 0 2 Lvin gt 0 5 Ol vin Max 0 2 12 StwowaylA B 0 5 0 2 vein 0 2 13 Figure 4 shows six situations to illustrate the preference for the two way compar ison over the direct comparison of Fuzzy Neighbourhood Vectors For each situation both the similarity according to the direct comparison of the Fuzzy Neighbourhood Vectors and the two way comparison are given It demonstrates that only the two way comparison yields the intended similarity results 2 4 Kyuzzy Statistic for overall map similarity The previous paragraphs specify how for each cell a local measure of similarity can be calculated In addition to this it is for some applications useful to obtain an overall value of similarity An overall value can be obtained by integrating the similarity values over the whole map Division by the total area yields a result between 1 for identical maps and O for total disagreement Since regular grid maps are considered this is equivalent to calculating the average similarity of all cells The average similarity however is not necessarily a good measure for overall similarity because the expected value for similarity would be strongly influenced by the number of categories in the map and also by the numerical distribution of cells over those categories In order to make the results of maps with different numerical distributions more comparable a statistic is introduced that corrects the pe
28. 2 and require a membership function for each linguistic value The output membership set figure 5 consists of two sigmoidal and three Gaussian membership curves As with the input membership functions the local matching output membership functions overlap There is no point in the set where the output local matching value can have single membership in a linguistic value Any derived output value will have multiple membership in the linguistic set which is necessary to account for any uncertainties in the calculated local matchings 3 4 Fuzzification The second stage in the development of the fuzzy inference system is the fuzz ification of the input data Fuzzification of an input variable characterizing a unique polygon involves locating the crisp input value on the x axis of the membership functions and estimating the corresponding memberships from the y axis The resulting fuzzy vector consists of the memberships for each linguistic map agreement expression arranged from left to right F fi f2 fs fa such that Ssa 2 Since fuzzification produces as many vectors as there are input variables in this application three fuzzy vectors are generated for each unique polygon in a map comparison analysis Table 2 Linguistic labels of the output membership functions Linguistic label Output function type Very Poor Sigmoidal Poor Gaussian Good Gaussian Very Good Gaussian Perfect Sigmoidal Membership function plots Poo
29. 4 02 of total area 221 of 370 polygons are 1 pixel 69 are 152 are mismatched 3 45 of the total area matched 41 of 370 polygons are 2 pixels 5 are matched 36 are mismatched 1 28 of the total area Local Matchings 0 2 0 3 0 3 0 4 0 4 0 5 0 5 0 6 0 6 0 7 0 7 0 8 0 8 0 9 0 9 1 N es a ee 0 Kilometers 1 Figure 10 Local template polygon matchings for Forestl1 92 C Power et al representing a high degree of containment of the 1991 map within the 1985 template layer The matching information in table 6 shows that there is a high degree of polygonal land use pattern agreement between the maps All of the land use categories except cleared and no data have agreement percentages greater than 83 0 Unlike the previous datasets the matched template polygons outnumber the mismatched poly gons For example 321 of the 402 template polygons see table 7 matched their counterparts on the 1991 map for an 81 4 overall areal agreement between the maps This difference in matching results may be attributed to the 30 m pixel resolution of the land use maps in Forest1 The smaller scale of the template resulted in most of its unique polygons containing more than two pixels Table 7 shows that 101 of the 401 polygons consisted of one or two pixels of which 15 were mismatched 5 1 Evaluation of Boolean versus fuzzy global similarity The performance of the global matching procedure can be estimated by compar
30. DATA ic MODATAs NODATA uoDarAs lt 0 If you wish to compare a numerical map Other operations i first second i first i first using a categorical map comparison method then the definition of categories of the legend is used If categorical maps are compared using a numerical algorithm the numerical value of a category is its rank number in the legend starting at number 0 The two Other operations are numerical operations as well In the following sections the comparisons and other operations are discussed in the order of appearance in the dialogue 13 2 3 1 Per category Select category E jResult Sin both maps in none of the maps B only in map 1 notin map E only in map 2 not in map 1 This comparison method performs a cell by cell comparison with respect to one user selected category It simultaneously gives the user information about the occurrence of the selected category in both maps The category to consider in the comparison is selected in the Parameters a oeeo of the Per category comparison The maps in the example below are pr compared with respect to the category City al The legend belonging to the Result map of this cell by cell categorical comparison is self explanatory and Apply details to what extent the category is present in one or the other map The Result statistics window offers aggregate results in the form of total number of cel
31. Kappa statistic Ecological Modelling 62 275 293 PoNTIUS Jr R G 2000 Quantification error versus location error in comparison of categorical maps Photogrammetric Engineering amp Remote Sensing 66 1011 1016 Pontius Jr R G and SCHNEIDER L C 2001 Land cover change model validation by an ROC method for the Ipswich watershed Massachusetts USA Agriculture Ecosystems and Environment 85 239 248 Power C Simms A and WuiTeg R 2001 Hierarchical fuzzy pattern matching for the regional comparison of land use maps International Journal of Geographical Information Science 15 77 100 SHYI MING C 1995 Measures of similarity between vague sets Fuzzy Sets and Systems 74 217 223 Toras Y A PANAS S M and TsouKALAS L H 2001 Generalized fuzzy indices for similarity matching Fuzzy Sets and Systems 120 255 270 WHITE R ENGELEN G and ULJE I 1997 The use of constrained cellular automata for high resolution modelling of urban land use dynamics Environment and Planning B Planning and Design 24 323 343 WHITE R ENGELEN G ULJEE I LAVALLE C and EHRLICH D 2000 Developing an Urban Land use Simulator for European Cities In Proceedings of the 5th EC GIS Workshop held in Stresa Italy 28 30 June 1999 edited by E Fullerton Ispra Italy European Commission Joint Research Centre pp 179 190 WINTER S 2000 Location similarity of regions ISPRS Journal of Photogrammetr
32. Klocation 7 Besides calculating Kappa statistics for all categories combined there is the option to calculate Kappa statistics per category For a categorical Kappa statistic the two maps are transformed to a map consisting of only two categories The first new category is the category for which the individual kappa statistic is derived the second category is the combination of all other categories 2 4 Relative Kappa statistics A typical map comparison problem is the question how well a map generated by a model the Model Map resembles a real map the Reality Map The Kappa statistic can be of use here By itself however it offers insufficient information a Kappa statistic with value 0 7 may be considered very high in one case but can indicate a poor result in another For an indication how well two maps look alike a reference level for similarity is needed This reference level can be obtained from a Reference Map for instance in the form of a historical map The procedure is as follows in first instance the Model Map is compared to the Reality Map this comparison yields several statistics Kappa Khisto and Klocation The same operation is performed on the Reality Map and a Reference Map this comparison also yields values for Kappa Khisto and Klocation Finally the individual comparison results are combined and the similarity between of the Model Map and the Reality Map can be expressed relative to the similarity of the reference m
33. RISON KIT Chapter 1 of this manual gives the basic information required for a quick start with the MAP COMPARISON KIT It describes the layout of the program and tells you what buttons to click in order to start comparing your maps or the example maps given with the MCK Chapter 2 gives a brief introduction to all the map comparison methods that are supported in the MCK In short the main principle of the methods is described as well as the parameters that the user can set and the comparison For in depth information about the comparison methods the user is referred to the appendices Chapter 3 explains how the user can fully define the legends to his or her liking Chapter 4 is meant as a reference chapter It gives a short explanation for all commands found in the menu structure of the MCK Chapter 5 explains about all the different files the MCK use log files map files legend files and palette files Both their structure and function are discussed 1 GETTING STARTED After installation the MAP COMPARISON KIT will be present in the Windows Start menu Press the Start button in the Task bar of Windows 98 NT 2000 XP and place the mouse pointer on the Programs group Walk through the menu until you find the group containing the MCK and open it by clicking the Map Comparison Kit icon If the software was installed correctly the Map Comparison Kit application window will open The Open dialogue will appear asking the user t
34. al et al 1999 Abuelgasim et al 1999 More importantly the membership values in the activation level of the network can approximate the values of the membership curves in the fuzzy inference system During the learning process ARTMAP would change the activity patterns and adjusts the network weights until it reached vigilance thus indicating a match between the input areal values and a land use agreement pattern From a trained network the membership values in the activation node for each agreement category could be used to optimize the corresponding membership functions in the fuzzy inference system Finally future research must address the spatial dependency between the land use maps The spatial autocorrelation values between the template and comparison polygons should be calculated and represented as membership functions in the fuzzy inference system The inclusion of spatial autocorrelation into the fuzzy areal map comparison could expand the similarity analysis beyond the direct comparison of polygons to a comparison of the surroundings of the template polygons This would be similar to a remote sensing analysis of texture or context on multi temporal images By enabling the model to be sensitive to spatial dependencies the map comparison could be performed on highly segmented and fragmented land use patterns that are comprised of a complex intermixture of unique polygons Acknowledgments This work was supported by the Social Sciences an
35. alelevation 5400 rst palette directories E upstream Region map oO x Accept or reject C Program Files SimDelkaBUiSimDeltalregions rst l mone recent changes The region map defines the active area of the maps 11 2 1 Perform w Tip 2 COMPARE MAPS a comparison Comparing maps with the MAP COMPARISON KIT is a four step process 1 Select the maps to be compared 2 Select the desired comparison method 3 Set the parameters for this method if applicable 4 View the result map and or result statistics The MCK remembers the last used maps comparison method and parameter settings It is therefore often possible to skip step 1 2 or 3 The four steps can be Map Comparison Kit demo log taken by using File Edit View Options Window Help commands from the Step 1 Step 2 Step 3 Step 4 J Theme Options menu of the Map 1 Menu bar Show Alternatively all these Comparison method O Demolmapl ing commands are also Parameters w 1 Demo maps ing represented in the Result map Toolbar See also Statistics Section 1 2 2 If the selected comparison method does not require any parameters to be set then the Parameter command is unavailable If the Result map and Statistics command are also unavailable this means that the two selected maps are unequal in size and can therefore not be compared against each other The actual comparison calculation is perf
36. any cells are coloured grey they are mostly dark grey The K of the base map can be used as a reference level Models scoring lower Dublin 1998 validation data Figure 9 Dublin 1998 validation data Comparing categorical maps by fuzzy set theory 247 c Dublin 1998 improved model Figure 10 Three comparison results from validation process than 0 90 do more damage than good while models scoring higher achieve better than minimally required The results from the original model figure 10 b contain a relatively large number of cells that are not identical they are grey and their similarity is relatively low they are mostly dark grey As a result K is smaller than that of the 1988 base data Finally the result map of the improved model still contains a large number of non identical cells however the similarity of these cells is relatively high they are lighter grey The resulting K is higher than that of the base data and therefore yields a positive validation of this model 4 Discussion By applying fuzzy set theory for the comparison of categorical maps it is possible to obtain a spatial and gradual analysis of the similarity of two maps The results 248 A Hagen from the comparison are basically in accordance with those of a visual inspection it distinguishes minor deviations and fluctuations within similar areas from major deviations The comparison method considers uncertainty and va
37. ap 3 Similarity assessment with fuzzy set theory In this section fuzzy set theory as introduced by Zadeh 1965 will be applied to compare categorical maps In order to consider fuzziness in the maps it is necessary to change the way in which cells are represented Instead of one single category or value per cell each cell is characterized by a membership vector Each element in the vector declares with a value between 0 and 1 the degree of membership for one category Two sources of fuzziness are considered the first is fuzziness due to vague distinctions between categories the second is fuzziness due to a gliding scale of severity of spatial error The comparison method is documented more extensively in Hagen to appear 3 1 Considering categorical similarity In many maps there exists vagueness in the definition of categories This is especially true if some or all categories on the map have in fact an ordinal definition such as for instance the categories high medium and low density residential area on a land use map It might often be that boundaries between such categories are less clear cut than what seems to be the case from the legend This fuzziness can be made explicit in the vector describing the cell by giving elements that correspond to similar categories higher membership values Figure 2 gives an example how the fuzziness of the categories can be expressed in the membership vector Category a vector H
38. aps will have a K value close to 1 which stands for completely identical As a bare figure the K statistic is not 246 A Hagen Identical a Low similarity Figure 8 Three levels of agreement by the proposed fuzzy comparison method highly informative It is more informative if there is reference material available as in the practical case presented in 3 2 3 2 Practical case The case presented here applies the two way fuzzy comparison method for validation It compares results generated by a model with real data The particular model is a constrained cellular automaton White et al 1997 applied for the study of the urban development of Dublin as part of the Murbandy project White et al 2000 Three maps are compared with the observed 1998 data figure 9 The first map is the 1988 base map figure 10 a which was the starting situation for the model Next is the 1998 map generated by the original model figure 10 b Finally the 1998 map generated by an improved version of the model figure 10 c is used The land use maps are found in the left column the comparison maps in the right figure 10 Lighter cells in the comparison maps indicate larger similarity The comparison with the base data figure 10 a yields a relatively high K 0 90 even though the modelling effort is zero This means that between 1988 and 1998 a small number of cells change land use however the changes are severe not m
39. ards and Lowell 1996 Second fuzzy set theory provides a method of dealing and comparing maps containing a complex mixture of spatial information A fuzzy map is more appropriate for representing a complex land use type such as vegetation coverage because it enables the pixels or polygons to have multiple memberships in the land use classes Furthermore a fuzzy map comparison model can determine the agreement between fuzzy maps while handling the complexity of the land use classes rather than simply ignoring it Therefore the degrees and types of categorical differences between maps should be determined by a fuzzy post classification comparison 2 1 The fundamentals of fuzzy set theory Zadeh 1965 first introduced fuzzy set theory as a means of describing the imprecision and vagueness of human reasoning in information communications The basis of fuzzy set theory is the notion of imprecise membership functions which provide ways of dealing with the limitations of traditional data classifiers Klir 1988 The rigid spatial models consisting of discrete sharply defined homogeneous classes ignore the geographic variability and complexity within nature and the error inherent in the measurement of it Burrough 1989 Thus a considerable amount of informa tion is lost when sharp edged entities are combined Fuzzy set theory provides more appropriate classifiers because it models cases whose attributes have soft transitional rather than hard boun
40. ataset two figure 8 called Forest 1 are classified Landsat TM images that were acquired on 29 July 1985 and 3 August 1991 respectively The images were georegistered with less than 0 5 pixel RMS to the UTM grid on NTS map sheet 12H 04 producing a pixel resolution of 30m A maximum likelihood algorithm classified the images into forest inventory types based on field information However the forest inventory maps used in this paper are subscenes containing 334 rows by 222 columns that were extracted from the original imagery and are centred on a region to the Northwest of Pasadena Newfoundland Fuzzy regional comparison of land use maps 89 Landuse Types Unclassified Commercial Industrial Residential River Railway Roads N EE E Map 1 template Map 2 0 Kilometers 4 ji Figure 7 Land use maps of Cincl Forest Inventory No Data Water Cleared Non forested bF MbF MO Spruce TT a Deciduous i s N 0 Kilometers T Figure 8 Forest inventory maps of Forest1 Canada For the matching process the 1985 map is the template and the 1991 map is the matching layer 5 Results The first section of the results analyses the local matches and mismatches to estimate the degree and nature of the land use agreement between the maps of the datasets The local matching values from the fuzzy inference system are the membership values of the polygons on map two relative to a template map The local matchin
41. ation mapping Remote Sensing of Environment 70 138 152 CHANG C CHEN K WANG J and ALTHOUSE M 1994 A relative entropy based approach to image thresholding Pattern Recognition 27 1275 1289 CONGALTON R G ODERWALD R and MEAD R A 1983 Assessing Landsat classification accuracy using discrete multivariate analysis statistical techniques Photogrammetric Engineering and Remote Sensing 49 1671 1678 Dat X L and KHoRRAM S 1999 Remotely sensed change detection bBased on artificial neural networks Photogrammetric Engineering and Remote Sensing 65 1187 1194 Fuzzy regional comparison of land use maps 99 EASTMAN R J 1992 Idrisi Technical Reference Manual Clark University Publishing pp 77 80 EDWARDS G and LOWELL K 1996 Modelling uncertainty in photointerpreted boundaries Photogrammetric Engineering and Remote Sensing 62 337 391 Foopy G M 1992 On the compensation for chance agreement in image classification accuracy assessment Photogrammetric Engineering and Remote Sensing 58 1459 1460 Foopy G M 1995 Cross entropy for the evaluation of the accuracy of a fuzzy land cover classification with fuzzy ground data ISPRS Journal of Remote Sensing 17 2 12 GONG P 1993 Change detection using Principal Components Analysis and Fuzzy Sets Theory Canadian Journal of Remote Sensing 19 22 29 GOPAL S WOODCOCK C E and STRAHLER A H 1999 Fuzzy neural network classif
42. atistic is introduced that corrects the percentage of agreement for the expected percentage of agreement The statistic is similar to the Kappa statistic and is therefore called K fuzzy Situation 1 The value for similarity in the central cell will be low because the two cells black and white differ and there are no cells of the same category in the neighbourhood The calculated similarity of the central cell is O Situation 2 The value for similarity in the central cell will be intermediate because the two cells black and grey differ but there are cells of the same categories in the neighbourhood The calculated similarity of the central cell is 0 5 Situation 3 As in Situation 2 the value for similarity in the central cell will be intermediate The similarity will be smaller than in Situation 2 because the matching cells are found within a greater radius The calculated similarity of the central cell is 0 25 Situation 4 The value for similarity of the central cell is equal to the one in Situation 3 because the matching cells are found within the same radius The white cells do not influence the comparison The calculated similarity of the central cell is 0 25 Situation 5 The value for similarity in the central cell will be low because the two cells black and grey differ and there are no cells of the same categories in the neighbourhood The calculated similarity of the central cell is O Situation 6 The value
43. atterned quality of spatial data while also addressing the limitations of traditional pixel by pixel comparisons The basis of the approach is a comparison of land use maps on a polygon to polygon basis using unique polygons mapping A fuzzy relational map comparison model is then developed that produces qualitative and quantitative descriptions of land use agreement on regional scales The comparison model is structured to emulate the human reasoning method of identifying a hierarchy of map similarities This requires that the map comparison be performed on both local and global levels Finally the utility of hierarchical fuzzy pattern matching 1s illus trated by analysing two sets of results 1 a comparison of simulation results from a cellular automata based land use prediction model and 2 a comparison of a temporal sequence of forest inventory land use maps 2 Background traditional pairwise pixel by pixel comparisons The aim of a pairwise post classification comparison is to identify areas of categorical disagreement between two maps by determining the pixels with a differ ence in theme This involves overlaying the maps on a pixel by pixel basis to produce a map and attribute table of site specific differences From the information in the table summary agreement statistics are generated to give a measure of areal disagreement Several authors Singh 1989 Mas 1999 Dai and Khorram 1999 have expressed the need for a better post classi
44. ays works with log files This means that if you have a number of maps on which you want to Fie Edit view Options perform comparisons you will need to make a new log file New Open ctr a To start a new log file click select New from the File menu in the Close Menu bar A dialogue will appear allowing the user to specify the contents of the log file using by using an intuitive point amp click system Log files are discussed in more detail in section 5 1 SAVE AS In a log file maps are grouped according to Themes Maps within a Theme are displayed using the same legend and can be compared against one another Edit Log File E x Aada enE l landscape Landscape Coastline and Add Theme H coastline Boundary are some of the Add a map to a Fi basins complete LOG file i Remove E draindirs The Elevation theme fae Remove the pea contains 12 elevation maps Rename hiahliahted elevation Igniignted map or Jo fimDelbalelevation 6500 rst Up j a SimDeltaelevation 6400 rst H SimDelbalelevation 6300 rst Down bs ao SimDelkaeleyation 6200 rst Rename the oo SimDeltajelevation 6100 rst Legends highlighted theme ao imDeltaeleyation 6000 rst 4SimDeltaelevation 5900 rst Change the al of SimDeltalelevation S800 rst appearance y be a SimDeltatelevation 5700 rst moving themes or 2 ao SimDelbaeleyation S600 rst maps up and down ao SimDelbaleleyation S500 rst Set the legend and ee SimDelt
45. cal matchings The uses of these basic methods are discussed and further refinements and modelling possibilities are outlined 1 Introduction The identification of categorical differences between maps is the basis of much land use dynamics research Specifically a wide variety of remote sensing methods have been developed for detecting land use change in bi temporal categorical and multi spectral imagery Weismiller et al 1977 Wickware and Howarth 1981 Hodgson et al 1988 Abuelgasim et al 1999 However there are numerous examples in the literature of concerns about the limitations of the traditional methods Conventional categorical change detection procedures called post classification com parisons perform a pixel by pixel overlay of two thematic maps to generate a similarity map and associated statistics that indicate regions of disagreement Jensen et al 1987 Hodgson et al 1988 Dai and Khorram 1999 One problem with post classification comparison is that the accuracy and usefulness of the comparison results depend on the accuracy of the categorical classifications and geometric registration of the maps A second more important limitation is that the traditional methods can only compare maps that contain Boolean categories By nature land International Journal of Geographical Information Science ISSN 1365 8816 print ISSN 1362 3087 online 2001 Taylor amp Francis Ltd http www tandf co uk journals 78 C Power et al use
46. cation To obtain a crisp local matching value it is necessary to transform the output membership function produced by the inference algorithm into a crisp number Although numerous defuzzification methods have been suggested Jager 1995 Nguyen and Walker 1997 the centroid of area defuzzification is used to calculate the local matching numbers because the output fuzzy sets are one dimensional Jager Intersection Complement Pixels Local __ ll a 07 03 3 0 1 0 73 d Figure 6 Rule base and inference structure of the fuzzy inference system 88 C Power et al 1995 The centroid of area calculates the crisp value of the output variable by finding the centre of gravity value of the aggregated output membership function Nguyen and Walker 1997 This is computed as follows Jager 1995 fyuy ny dy i YUup y dy where Z is the centroid of area and u is the membership value in the output distribution B For example see figure 6 The centroid of area defuzzification gives a local matching value of 0 73 for this sample unique polygon The vertical line through the output membership function depicts the location of the centroid of area of the output distribution Z B 6 3 7 Global matching The computation of a fuzzy global similarity number that expresses the overall areal agreement or estimation of change between two land use maps involves the ageregation of each of the local matchings for the unique polygons The l
47. d Humanities Research Council of Canada under grant 410 95 1409 and by the Land Water Environment Information Technology Programme LWI ICES of the Dutch government Also the helpful suggestions of two anonymous reviewers are greatly acknowledged References ABUELGASIM A A Ross W D GOPAL S and Woopcock C E 1999 Change detection using fuzzy neural networks environmental damage assessment after the Gulf War Remote Sensing of Environment 70 208 223 BLONDA P PASQUARELLO G Losito S Mori A Posa F and RAGNo D 1991 An Experiment for the integration of multitemporal remotely sensed images based on a fuzzy logic approach International Journal of Remote Sensing 12 463 476 BONHAM CARTER G 1994 Geographic Information Systems for Geoscientists modelling with GIS Oxford Pergamon Press pp 235 238 BURROUGH P 1989 Fuzzy mathematical methods for soil survey and land evaluation Journal of Soil Science 40 477 492 CANNON R L JITENDA V D BEZDEK J C and TRIVEDI M M 1986 Segmentation of a Thematic Mapper image using the fuzzy c means clustering algorithm EEE Transactions on Geoscience and Remote Sensing GE24 400 408 CARPENTER G and GROSSBERG S 1997 Fuzzy art In Fuzzy Engineering edited by B Kosko Carmel Prentice Hall pp 467 497 CARPENTER G GOPAL S MACOMBER S MARTENS S and WOODCOCK C 1999 A neural network method for mixture estimation for veget
48. daries Mathematically a fuzzy set A in x is described by a membership function as a set of pairs A X u x x EX 1 where u x is the membership grade of x in A and x x means that x is found in the universe of discourse X The membership value u x ranges from zero to one with a gradual transition from full membership at 1 to no membership at 0 In standard set theory a membership function has only two values 0 or 1 The selection of the appropriate membership function for a fuzzy set is generally based on the subjective opinion of the researcher Zimmerman 1985 However the structure of the membership function will determine the extent to which the memberships change away from the optimal value MacMillian 1978 Fuzzy set theory is gaining increasing support from spatial researchers A number of studies Cannon et al 1986 Wang 1990 Maselli et al 1996 utilizing fuzzy c means clustering for remote sensing image classification have shown that fuzzy set theory can deal with images containing a complex mixture of spatial and spectral informa tion Unlike the traditional classifiers the fuzzy c means clustering algorithm assigns multiple memberships to a pixel to represent land use class mixtures and intermediate conditions Similarly fuzzy rule based systems have used fuzzy membership functions to represent and model the qualitative estimations of interpretation experts during the image classification process Blonda et al 1991
49. discontinuous sparse urban Fabric Gl industrial C Outside metropolitan area ndustrial areas Figure 4 Observed and simulated maps of the metropolitan area of Dublin in 1998 Figure 5 gives the results of the Fuzzy two way comparison in the form of a comparison map indicating per cell the level of agreement The membership function that was applied is one of exponential decay with a halving distance of two cells The comparison map can be an aid to find the cause of the disagreement For instance a large area of strong disagreement is found in the north of the city where the model situates Commercial areas where Airport is expected The comparison map also clearly points out the Road and which represents a motorway that exists in reality but was not foreseen by the model the curved linear shape starting just south of the airport Identical Distinct Figure 5 Spatial assessment of similarity by the fuzzy set approach For validation a reference level was sought It was found in the map of observed data of 1988 Figure 7 this map was also used as the initial situation of the simulation If the model map is more similar to the observed data than the reference level is to the observed data then the validation is positive Taking into account that land use changes only mildly in a period of ten years this is a considerably strict validation The comparison is performed conform the method presented in Sect
50. ds the histogram and location refers to a the spatial distribution of the different Kappa 0 380 0 556 0 332 categories over the map In order to __ eS ea recognise to which extent similarity of eee location and quantity are represented in the Contigency Table Kappa statistic it is split up into two statistics Kappa Histo or KHisto and Kappa Location or KLoc Where Kappa KHisto KLoc KHisto only depends on the total number of cells taken in by each category and KLoc strictly depends on the spatial distribution of the categories over the map Kappa as well as KLoc and KHisto are calculated on the basis of the contingency table which details the cross distribution of categories over the two maps The table is expressed in number of cells The Kappa and related statistics are calculated both for the whole map and for the individual categories found in the legend Appendix I offers detailed information on these statistics 2 3 3 Fuzzy Inference System The evaluation of the spatial similarities between two raster maps is traditionally based on cell by cell comparison techniques However a cell by cell comparison can register a small displacement in cells as land use disagreement even though the land use patterns may be essentially the same The Fuzzy Inference System comparison method offers an alternative approach Rather than cells polygons that are found in both maps are compared on their characteristics The calculation
51. e PALETTE EDITOR See also Section 3 2 of this manual Important Modifying a palette will affect all legends that use this palette Modifying the colours in the legend will only affect the legend belonging to the particular theme or comparison method If in doubt do not use the Palette editor but use the Legend item editor instead 24 4 3 View menu View SEE Use the View menu to change the manner in which the maps are presented Zoom In in the active map window Zoom Out w Show Regions Grid Font w Toolbar w Statusbar Go to Command TT xi Use the Go to command to move the cursor to a specific cell on i the map Selecting this command from the View menu opens the x fs y E Go to dialogue requesting to enter the co ordinates of the desired cell When you have entered the co ordinates and clicked OK the Canca pointer will move to the desired cell in the active map e Zoom in Command Use the Zoom in command to increase the size of the map in the active map window by a factor 2 This command is identical to pressing the Zoom in button from the Toolbar Zoom out Command Use the Zoom out command to increase the size of the map in the active map window by a factor 2 This command is identical to pressing the Zoom out button from the Toolbar Show Regions Command Use the Show regions command to draw or remove the boundaries of the regions on top of the map in the active
52. e comparison and open the Result Map window Statistics Use the Statistics command to perform the comparison and open the Result Statistics window 4 5 Window menu Use the Window menu to arrange the contents of the screen and to activate Window one of the opened windows Cascade Tile Horizontally Tile Vertically Arrange Icons wel Demomapi img Cascade Command Use the Cascade command to arrange multiple opened windows in an overlapped fashion so that the Caption bar of each window is visible Tile Horizontally Command Use the Tile Horizontally command to arrange multiple opened windows one above another in a non overlapped fashion so that all windows are visible 2l Tip Tile vertically Command Use the Tile Vertically command to arrange multiple opened windows side by side in a non overlapped fashion so that all windows are visible Arrange Icons Command Use the Arrange Icons command to arrange the icons of minimized windows at the bottom of the application window Important If the map windows are arranged at the bottom of the application window they may hide some or all of the icons List of Windows 1 2 3 4 At the bottom of the Window menu a list of open windows is presented A check mark marks the name of the active window Choose a window from this list to make it active 4 6 Help menu Help Index Abouk Tip Use the Help menu to select the ty
53. e means series of RGB coordinates Palette files contain up to 256 The files are located in the palette directory which is set in the log file If the palette directory is not set in the log file or if it is a non existent directory then the default directory is used This is the directory Palettes in the same directory where also the Map Comparison Kit executable is located Palette files have an smp suffix and are compatible with the palette files that are used in the Idrisi GIS software 5 4 Map files 5 4 1 The map files used in the Map Comparison Kit are all of the Raster map type This means that they are structured like a matrix containing cells which are ordered in rows and columns Each cell is assigned a value that can either be categorical or numerical It is very important that the map files used in the comparison are of the same size which means that they have to have the same number of rows and columns A separate legend file is used to let the Map Comparison Kit interpret and display the values found in the correctly The supported file formats for the Map Comparison Kit are e Idrisi 16 bit Raster format img e Idrisi 32 bit Raster format rst e ArcAscii raster format asc e Laboratorium Lucht Onderzoek format llo ArcASCll format ArcASCII is a popular GIS format for raster files An ArcASCII file consists of a header block followed by a body of cell values The header block holds the following information
54. eement characteristics of a dataset than do Boolean maps The transitional change categories on a fuzzy map contain more change information and better represent the complex and intermediate change conditions In addition fuzzy maps give a better visual representation of where change has occurred spatially by retaining the form of the Fuzzy regional comparison of land use maps 97 template layers The global matching results for the datasets analyzed outperform a number of commonly used overall similarity statistics The work presented in this paper is a first attempt at developing a fuzzy map comparison model that is a viable alternative to the Boolean map comparison procedures Future research should be directed at several issues in order to expand the applicability of the model First of all the local matching process can be extended beyond the areal comparison of maps The fuzzy inference system can be restructured to include membership functions for the matching of complex polygonal properties such as shape and fractal dimension These additional variables could aid in the explanation and description of the differences between maps For example an increase in fractal dimension from one year to the next may be the result of an increase in the complexity of the land use pattern due to urban expansion Secondly the reliability of the fuzzy map similarity results and the performance of the fuzzy pattern matching model should be field tested agains
55. efore is fundamentally different from its crisp counterpart the Cell by Cell map comparison which considers pairs of cells either to be either equal or unequal The Fuzzy Set approach expresses similarity of each cell in a value between 0 distinct and 1 identical as the following figure illustrates In order to distinguish minor differences from major differences the Fuzzy Set approach takes two types of fuzziness into account fuzziness of categories and fuzziness of location Besides the result map also two global similarity indices are calculated The Average Similarity calculates the average similarity of all cells in the map This similarity index is flawed in the same way as the Fraction Correct A better similarity index is the Fuzzy Kappa which is the fuzzy equivalent of the Kappa statistic See section 2 3 2 for a discussion of Fraction Correct and Kappa The following two sections give information about the parameter settings for the Fuzzy Set map comparison More detailed information about the method can be found in Appendix III Fuzziness of Location In a categorical map most commonly a land use map each cell is taken in by a certain category In reality this does seldom mean that the area of the cell is solely taken in by that particular category In many cases it means that this 16 category is known or expected to be present in that neighbourhood and that the cell is mostly in accordance with that category
56. ems 4 261 283 WICKWARE G M and HowarrH P J 1981 Procedures for change detection using Landsat digital data International Journal of Remote Sensing 2 277 291 WEISMILLER R A KRISTOF S J SCHOLZ D K ANUTA P E and Momin S A 1977 Change detection in Coastal Zone environments Photogrammetric Engineering and Remote Sensing 43 1533 1539 Wun R ENGELEN G and INJEE I 1997 The use of constrained cellular automata for high resolution modelling of urban land use dynamics Environment and Planning B 24 323 343 ZADEH L 1965 Fuzzy sets Information and Control 8 338 353 ZHANG J and Foopy G M 1998 A fuzzy classification of sub urban lland cover from remotely sensed imagery International Journal of Remote Sensing 19 2721 2738 ZIMMERMAN H 1985 Fuzzy Set Theory and its Applications Boston Kluwer Nijhaft Publishing pp 1 150 APPENDIX III FUZZY SET MAP COMPARISON Taylor amp Francis Taylor amp Francis Group INT J GEOGRAPHICAL INFORMATION SCIENCE 2003 VOL 17 NO 3 235 249 Research Article Fuzzy set approach to assessing similarity of categorical maps ALEX HAGEN Research Institute for Knowledge Systems P O Box 463 6200 AL Maastricht The Netherlands e mail ahagen riks nl Received 18 October 2001 accepted 13 May 2002 Abstract For the evaluation of results from remote sensing and high resolution spatial models it 1s often necessary to assess t
57. f the membership function deserves further research as well These settings determine the tolerance of the comparison It is expected that the appropriate tolerance is related to the uncertainty contained in the map There are many sources of uncertainty for instance data quality model complexity spatial scale and definition of map categories Once more is known about the relationship between uncertainty and fuzzy representation of maps it will be worthwhile to further explore the possibilities of differentiation of fuzzy representation the two maps that are compared can be subject to different membership functions the neighbourhood radius may vary per category for model results that look further in the future a larger tolerance may be used and many other refinements can be considered The comparison methods can be of practical use in calibration procedures The overall figure for similarity can be used directly to qualify model results It is potentially more effective to incorporate the spatial results in the procedure and focus the model improvements on those areas or categories with the most severe disagreement The results of remote sensing and high resolution spatial models can be assessed in more detail than before Based upon the spatial comparison results it is possible to specify the discrepancies between observed data and model results Furthermore the comparison map can be used to find correlations between similarity and other spatial
58. fication change detection or map similarity procedure because of the limitations of a pixel by pixel comparison First the procedure is sensitive to image misregistration and the existence of mixed pixels A pixel by pixel comparison of multi temporal maps will interpret any misalignment of one or both of the maps as change Furthermore any misclassification of a pixel on either one or both of the maps will be interpreted as a difference in theme although the disagreement is a result of the inherent errors in the dataset Jensen 1981 Second the comparison techniques will often produce results that are significantly different from the actual land use This is due to their inability to account for the inaccuracies in the maps throughout the comparison operation Macleod and Congalton 1998 Fuzzy regional comparison of land use maps 79 In contrast the flexibility of a fuzzy representation of spatial data offers the potential for avoiding the problems of traditional comparison procedures First of all misregistration and locational inaccuracies can be accounted for by fuzzifying the boundaries of the pixels or polygons of the input maps Generally the width of the fuzzy boundaries will correspond to the level of uncertainty in each of the land use maps Using a fuzzy implication algorithm fuzzy polygons can be compared to determine the sections that are different due to error and those that are different because of actual land use disagreement Edw
59. for the UK amp Ireland example a region map consisting of zeroes for all the sea and ones for all the land will be sufficient However the region map also has a visualisation purpose if the region map is selected in the view menu the outlines of all regions are depicted over the active map In the following example a region map dividing the Netherlands in 40 administrative units COROPS is applied PP gph The Netherlands without using the The Netherlands using the region region map map The above example is taken from the LOV Netherlands directory of the example files that are optionally installed with the MCK 34 APPENDIX KAPPA VARIATIONS Multi method assessment of map similarity Alex Hagen Research Institute for Knowledge Systems P O Box 463 6200 AL Maastricht The Netherlands ahagen riks nl Abstract This paper describes a multi method approach to assessment of similarity of categorical maps The assessment incorporates several newly developed comparison methods Some are related to the Kappa statistic others are applications of Fuzzy Set theory By combining the methods a broad assessment of similarity will be obtained which makes it possible to find the magnitude nature and spatial distribution of similarity between two maps 1 Introduction The growth of high resolution spatial modelling geographical information systems and remote sensing offers many possibilities but also challenges A major i
60. g values for Cincl are illustrated in figure 9 which visually indicates a relatively high degree of agreement between the two maps for most areas The low matching values generally consist of smaller polygons that are dispersed throughout the study area Along the lines of a traditional comparison matrix the incidences of land use agreement are measured by a table of frequency of matches and mismatches for each land use category Similar to the procedure presented by 90 C Power et al Local Matchings 0 2 0 3 03 04 gt 0 4 0 5 0 5 0 6 0 6 0 7 0 7 0 8 0 8 0 9 0 Kilometers 4 gt Z Figure 9 Local template polygon matchings for Cincl Gopal and Woodcock 1994 a fuzzy cut of 0 70 is used to measure the frequency of local matches Formally 1 if u 0 7 Local Match X s 8 0 otherwise A land use polygon on a second map is similar to a template polygon if its local matching membership grade is gt 0 70 Table 4 displays the results for Cincl using the fuzzy threshold agreement value The first column shows the land use type and the second column displays the total number of polygons for each map category The matches and mismatches are given as numbers of polygons in columns three and four while the last column shows the percentage of land use agreement for each land use class The similarity percentages for the rivers and transportation systems are in perfect Table 4 Local matchings fo
61. gueness in the specification of the location of categories fuzziness of location as well as in the definition of the categories fuzziness of category The values for similarity will range from 0 to 1 The average of all cells can be used as a measure of overall similarity of the two maps and also lies between 0 and 1 The comparison method yields results that are more gradual than those from other methods kappa statistic or cell by cell comparison hence it is more likely to give an adequate indication of small differences The introduction of the K statistic makes it possible to compare individual comparison results and therefore makes it possible to rank a collection of maps according to similarity to a reference map In the calculation of Kp zzy the observed level of similarity is corrected for the statistically expected level of similarity The derivation of expected similarity presented in this paper is valid for comparisons considering only fuzziness of location Furthermore the derivation assumes infinitely large maps For small or irregularly shaped maps and for comparisons that also involve fuzziness of category K has not been derived yet Instead of formally deriving the expected level of similarity it is also an option to apply Monte Carlo analysis of randomly generated maps A general expression or procedure for calculation of K will be subject of further research The selection of the appropriate shape and size o
62. he outcome of a visual comparison will therefore depend on the person performing the comparison The comparison method presented here was primarily developed to be of use in the calibration and validation process of cellular models for land use dynamics The method is based on fuzzy set theory Bandemer and Gottwald 1995 Zadeh 1965 Several authors addressed the potential of fuzzy set theory for geographical applica tions Cheng et al 2001 Fisher 2000 and fuzzy set theory has been used before to assess the accuracy of map representations and for map comparisons Metternicht 1999 Lewis and Brown 2001 Power et al 2001 The subject of map comparison is closely related to accuracy assessment of maps in the sense that accuracy assessment is one of its applications Foody 2002 presents an overview of the status of land cover classification accuracy assessment Several issues that are brought to attention in that overview are at least partly addressed in this paper Foody 2002 asks Why cannot some level of positional tolerance be more generally incorporated into thematic map accuracy assessment Also it is stressed that spatial variability of error can be a major concern Finally Foody 2002 states that there is scope for considerable research on the topic of fuzzy classifications in accuracy assessment The objective is to find a method that to some extent mimics the human compar ison and gives a detailed assessment of si
63. he similarity of sets of maps This paper describes a method to compare raster maps of categorical data The method applies fuzzy set theory and involves both fuzziness of location and fuzziness of category The fuzzy comparison yields a map which specifies for each cell the degree of similarity on a scale of 0 to 1 Besides this spatial assessment of similarity also an overall value for similarity is derived This statistic corrects the cell average similarity value for the expected similarity It can be considered the fuzzy equivalent of the Kappa statistic and is therefore called K A hypothetical case demonstrates how the comparison method distinguishes minor changes and fluctuations within patterns from major changes Finally a practical case illustrates how the method can be useful in a validation process 1 Introduction With the growth of high resolution spatial modelling geographical information systems and remote sensing the need for map comparison methods increases Good comparison methods are needed to perform calibration and validation of spatial results in a structured and controllable manner The importance of map comparison methods is recognized and has growing interest among researchers Monserud and Leemans 1992 Metternicht 1999 Winter 2000 Pontius 2000 Pontius and Schneider 2001 Power et al 2001 For most purposes visual human comparison still outperforms automated pro cedures When comparing maps the human observer
64. ication of global land cover from a 1 AVHRR Data Set Remote Sensing of Environment 67 230 243 GoPAL S and Woopcock C E 1994 Theory and methods for accuracy assessment of Thematic Maps using fuzzy sets Photogrammetric Engineering and Remote Sensing 60 181 188 Hopason M E JENSEN J R HALKARD E M and COULTER M 1988 Monitoring wood stork foraging habitat using remote sensing and geographic information systems Photogrammetric Engineering and Remote Sensing 54 1601 1607 JAGER R 1995 Fuzzy Logic in Control Delft The Netherlands Delft University of Technology Publishing pp 44 147 JANG J S R SUN C T and MIZuUTANI E 1997 Neuro Fuzzy and Soft Computing An Computational Approach to Learning and Machine Intelligence Upper Saddle River New Jersey Prentice Hall pp 73 93 JENSEN J R 1981 Urban change detection mapping using Landsat digital data The American Cartographer 8 127 147 JENSEN J R RAMSEY E W HALKARD E M CHRISTENSEN E J and SHARITZ R R 1987 Inland wetland change detection using aircraft MSS data Photogrammetric Engineering and Remote Sensing 53 521 529 KLIR G 1988 Fuzzy Sets Uncertainty and Information New Jersey Prentice Hall pp 2 50 KOoLLiAs V J and VouioTis A 1991 Fuzzy reasoning in the development of geographical information systems FRSIS a prototype soil information system with fuzzy retrieval capabilities International
65. ies in the legend that are sub categories of the same main category is often less distinct than between categories that do not belong to a common group of categories This can also be expressed in the Fuzzy Category Vector as is illustrated by an example in table 3 In the example of table 3 the sub categories citrus sugarcane and banana agriculture are considered more similar to each other than to the other categories residential industry and water It should be kept in mind that the fuzzy representation is in reality an interpreta tion of the original crisp data There are no straightforward rules for assigning membership values The definition of the appropriate set depends for instance on the nature of the map the aim of the comparison and the number of categories present 2 2 Representation of fuzziness of location Besides fuzziness of category also fuzziness of location is considered The calcula tion of fuzziness of location 1s based upon the notion that the fuzzy representation of a cell depends on the cell itself and to a lesser extent also the cells in its neighbourhood The extent to which the neighbouring cells influence the fuzzy representation is expressed by a distance decay function For instance a cone defined by radius an exponential decay defined by halving distance or a 3 D Gausse curve defined by variance see figure 1 Bandemer and Gottwald 1995 Which function is most a
66. igh density residential Ear 2 4 02 0 0 Medium density residential 2 4 0 0 Low density residential Ea l 3 4 0 0 0 1 Agriculture rato 0 010 Industry Ct Figure 2 An example of fuzzy representation of ordinal data 3 2 Considering proximity of similar cells Proximity of similar cells can also be expressed in the membership vector Cells within a certain distance the neighbourhood of a central cell influence the fuzzy representation of that cell To achieve this the proximity of categories is considered to contribute to the degree of membership of those categories The different membership contributions of the neighbouring cells are combined by calculating the union according to fuzzy set theory This is expressed in Equation 8 for a map with N categories and considering a neighbourhood consisting of C cells m stands for the value of the membership function at the i th cell in the neighbourhood and is calculated according to a distance decay funtion F Max p m Aa m ree u o m E Max u m H m Es m m veel 8 Pare o Max p Mi Hyo m Uy c m 3 3 Comparison of fuzzy cells The maps of fuzzy membership vectors obtained by considering proximity and categorical similarity are compared The comparison algorithm is designed to evaluate similarity in accordance with human intuitive criteria This can be achieved by performing a two way comparison proceeding as follo
67. ill be between 0 and 1 according to level of similarity as expressed in equation 4 Hecat 1 Vou 3 eae Original category i gt Heati 1 OS hear lt 1 GAS 4 Table 2 demonstrates for example how the fuzziness of the categories can be expressed in the Fuzzy Category Vector The meaning of this particular fuzzy representation of categories 1s that for instance low density residential 1s considered more similar to high density residential than industry On the other hand low density residential is less similar to high density residential than medium density residential Table 1 Crisp Vector representation of four categories Original Category representation Crisp Vector Urban area 1 1 0 0 0 Undeveloped 2 0 1 0 0 Agriculture 3 0 0 1 0 Water 4 0 0 0 1 238 A Hagen Table 2 Fuzzy representation of ordinal data Category Number Fuzzy Category Vector High density residential 1 1 04 02 0 0 0 Medium density residential 2 0 4 1 04 0 0 QO Low density residential 3 0 2 04 1 0 0 0 Agriculture 4 0 0 0 1 0 0 Industry 5 0 0 0 0 1 0 Water 6 0 0 0 0 0 1 In the previous example it is clear that high medium and low density residen tial are sub categories of residential Maps will more often contain a mixture of categories and sub categories The sub categories are not always ordinal they can also be nominal The difference between categor
68. ing cells are found within the same radius The white cells do not influence the comparison situation 5 The value for similarity in the central cell must be low because the two cells black and grey differ and there are no cells of the same categories in the neighbourhood 5s 0 5 STwolvay 0 Situation 6 The value for similarity in the central cell will be high because the two cells match both black regardless the circumstance that the neighbourhoods grey and white are dissimilar S 1 STwoway 1 Figure 4 Six situations in which the middle cells of the left and right map are compared with consideration of fuzziness of location Weights according exponential decay function with halving distance of 2 fuzziness of categories is not considered The concept of neighbourhood ring needs to be introduced In a raster map cells that are at the same distance from a central cell are said to form a neighbourhood ring In figure 5 the first nine rings are Comparing categorical maps by fuzzy set theory 243 Pt ttt tt Pt tt fet te fetzfetztst felstetstststat ERROR _tefefatijoliistefo Aei J fetsteistetsts P ft fetztet7tet Pt ttt te PE bse a iiliji Figure 5 Numbered rings within a four cell radius numbered 1 to 9 The central cell is numbered 0 In table4 their relevant characteristics are presented The calculation of K as described below applies for fuzziness of location w
69. ins how you can analyse and compare maps with the use of the MAP COMPARISON KIT MCK Besides a number of comparison algorithms the MCK also offers advanced options for visualizing organizing and exporting raster maps The first version of the MAP COMPARISON KIT dates back to 1992 when it was still called the ANALYSE TOOL The software was initially intended for the analysis of series of maps that are generated as output by simulation software of the Research Institute for Knowledge Systems RIKS From 1992 onwards the tool has steadily been further developed as part of RIKS projects for RIVM RIKZ RWS EC JRC and others The most current extension is developed by order and for the account of RIVM within the framework of project S 50002 01 TO Measuring and Modelling New additions are the extended Kappa analysis the Fuzzy Inference System map comparison and Fuzzy Set map comparison All these map comparison methods are the result of research performed by RIKS Another novelty is that the software is not only suited to work in conjunction with other RIKS products but may be used to compare any raster maps in some of the most popular file formats In particular these are ArcASCIHI Idrisi Raster and the LLO format which is used at the Netherlands Institute for Public Health and the Environment RIVM This stretches the use of the tool beyond the analysis of RIKS simulation results and inspired the name change from ANALYSE TOOL to MAP COMPA
70. ion 2 4 The results are presented schematically in Figure 6 The conclusion is a positive validation of the model Kfuzzy 0 91 Khisto 0 99 Klocation 0 97 P Quality report 4908 real Kfuzzy 1 Khisto 2 Klocation 2 Kfuzzy 0 90 1988 real ee 0 97 Klocation 0 99 1998 model Figure 6 Relative comparison results Figure 7 The observed map of 1988 which functioned both as the Reference Map and as the initial situation of the simulation With the Kappa related statistics it is also possible to recognize the contribution per individual category and also to distinct between similarity due to quantity and similarity due to location The result of that analysis can be found in Figure 8 Overal 960 97 oo Arable land 0 95 Pastures 0 94 Forests 1 00 Shrub and or herbaceous vegetation associations 1 00 Sparsely vegetated areas 1 00 Wetlands 1 00 Residential continuous dense urban fabric 0 78 Residential continuous medium dense urban fabric 0 95 Residential discontinuous urban fabric 1 00 Residential discontinuous sparse urban fabric 0 91 Industrial areas 0 96 Commercial areas 0 86 Public and private services 0 95 Port areas 0 85 Construction sites 0 00 Road and rail networks and associated land 0 43 Airport 0 88 Mineral extraction sites 0 97 Dump sites 0 99 Artificial non agricultural vegetated areas 0 93 Water bodies 1 00 Outside metropolitan area 1 00 Figure 8 Detailed Kappa results
71. is is called Save Category Similarity ha atris Land City River 1 O 0 Park Map 11 Map 2 fuzziness of categories In order to take fuzziness of categories into account when comparing maps it 1s necessary to fill out the Category Similarity Matrix This matrix is found in the Advanced part of the Parameters dialogue In the matrix the similarity between each pair of categories from the legend can be specified with a number between 0 crisply distinct and 1 completely identical By default the categories are set to be crisply defined which means that the category matrix is set to unity Clicking the Unity button will restore this setting A Category Similarity matrix can be saved to disk by clicking the Save button A previously saved matrix can be opened via the Load button O l 0 O j 1 j 0 17 2 3 5 Numerical comparison Six different numerical cell by cell comparisons are supported They are listed in the following table abs second first absolute difference second first max abs second first abs second first max abs second first The box NODATA allows the user to specify how to perceive a cell containing the no data value when it is compared to a cell that does contain a normal value The choice is either to give a no data value as the result or to treat the no data cell as if it has the value 0 Compare numerical values i second fi
72. isp the Fuzzy Category Vector V a and the Fuzzy Neighbourhood Vector Van Lhe Crisp Vector does not involve fuzziness at all The Fuzzy Category Vector represents a cell when only fuzziness of category is considered Finally the Comparing categorical maps by fuzzy set theory 231 Fuzzy Neighbourhood Vector represents a cell considering fuzziness of both category and location Equation 1 gives the general form of the Crisp Vector its membership values are set according to equation 2 It signifies that in the Crisp Vector representation of a cell has a degree of membership of 1 for its original category and 0 for all other categories Table 1 gives examples of crispvectors at four different locations each in different categories Herisp 1 wad o 1 hase Original category i gt Merispi 1 Merisp j Q EFS 2 2 1 Representation of fuzziness of categories Vagueness may exist in the definition of categories This is especially true if some or all categories on the map have in fact an ordinal definition such as for instance the categories high medium and low density residential area on a land use map Similarity between categories is expressed in the Fuzzy Category Vector equa tion 3 by assigning a higher degree of membership for categories that are more similar to the original category That means that for the original category it will have a full membership degree of 1 For the other categories the membership w
73. istics can be derived The following three are applied in this paper 1 P A stands for Fraction of Agreement and is calculated according to Equation 1 2 P E stands for Expected Fraction of Agreement subject to the observed distribution and is calculated according to Equation 2 3 P max stands for Maximum Fraction of Agreement subject to the observed distribution and is calculated according to Equations 3 c P A gt P 1 tal c P 2 Pan Pr 2 t c P max gt min P P n 3 1 1 2 2 Kappa statistics In many situations it is preferential to express the level of agreement in a single number When the comparison consists of a number of pair wise comparisons the Kappa statistic can be a suitable approach Carletta 1996 The essence of the Kappa statistic is that the fraction of agreement P A is corrected for the fraction of agreement statistically expected from random relocating of all cells in the map Thus this expected agreement is based on random location subject to the observed distribution it is referred to as P E The Kappa statistic is defined according to Equation 4 lt _ P A PE 1 P E 4 2 3 Kappa dissected into Khisto and Klocation Pontius 2000 clarifies that the Kappa statistic confounds similarity in quantity with similarity of location In this sense quantity means the total presence as a fraction of all cells of a category over the whole map With location i
74. ith a distance decay membership function The membership values depend on the membership function In this case equation 15 it is an exponential decay function with a halving distance of two cells Mia e C 27 42 15 Consider the generic contingency table comparing maps X and Y table 5 where pi fraction of cells which are of category i in map X and category j in map Y and X total fraction of category iin map X In case the two central cells category a in map Y and category b in map X do not match then the probability that both the central cells have their counterpart on Table 4 Ring characteristics Ring 0 1 Z 3 4 5 6 7 8 9 Number of cells 1 4 4 4 8 4 4 8 8 4 Cumulative number of 0 4 8 12 20 24 28 36 44 48 cells excluding central T N o o Distance cells 01 2 2 ao a 3 10 4 13 4 Membership value 1 0 71 O61 0 5 0 46 0 38 0 35 0 33 0 30 0 25 Table 5 Generic contingency table Map Y categories Map X categories 2 c Total 1 Pid Pia tee Pic x 2 Pai P22 ja P2c X gt C Pci Pe e Pce Xc Total Y L a 1 244 A Hagen a cell within a certain distance is calculated as P n equation 16 There n is the number of cells present within that distance P n 1 1 X x 1 C 16 The smallest distance within which the central cells of both cells are matched on the other map determines the similarity in a two way fuzzy comparison The probabil ity that this is the i th neighbourhood ring is the
75. ity into account The Kfuzzy statistic can do both By calculating relative measures for Kfuzzy Khisto and Klocation with the aid of a reference map it 1s possible to give a founded validation of the similarity between a model map and an observed map References Carletta J 1996 Assessing agreement on classification tasks the kappa statistic Computational linguistics 22 2 Cheng T Molenaar M amp Lin H 2001 Formalizing fuzzy objects from uncertain classification results International Journal of Geographical Information Science 15 1 27 42 Fielding A H amp Bell J F 1996 A review of methods for the assessment of prediction errors in conservation presence absence models Environmental Conservation 24 1 38 49 Fisher P 2000 Sorites paradox and vague geographies Fuzzy Sets and Systems 113 1 7 18 Foody G M 2002 Status of land cover classification accuracy assessment Remote Sensing of Environment 80 1 185 201 Hagen A to appear Fuzzy set approach to assessing similarity of categorical maps Lantz C A amp Nebenzahl E 1996 Behavior and interpretation of the K statistic Resolution of two paradoxes Journal of clinical epidemiology 49 4 431 434 Maclure M amp Willet W C 1987 Misinterpretation and misuse of the kappa statistic American Journal of Epidemiology 126 2 161 169 Maxwell W E 1977 Coefficients of agreement between observers and their
76. ive local matching scales are subjectively devised Figure 3 identifies the shape and parameters of the membership functions for the five linguistic scaling expressions for the areal intersection input data The same membership functions also apply to the areal complements since they are computed from the intersection values Two distinct types of membership functions are evident 1 the sigmoidal curve very low and very high and 2 the generalized bell curve low medium and high Simpson and Keller 1995 describe a sigmoidal membership function as a left or right open curve asymmetrical with respect to its crossover point At the crossover point the values of the membership function are rising toward or falling from a plateau of complete membership The asymmetric open structure of a sigmoidal membership function makes it appropriate for representing concepts such as very low or very high because values above or below a specific point are assigned complete membership or non membership In terms of localized map comparisons the sigmoidal curves depict instances where the land use agreement between maps is known with a high degree of certainty When an input value falls within the Table 1 Input linguistic local matching interpretations Scaling value Description Very low Definite land use differences Boolean areal intersection is very low Low Land use differences very likely areal intersection is low Medium Possible land
77. land use data can be confined to crisp borders Realistically a more appropriate measurement of the local matching between the maps would involve the computation of fuzzy areal intersections and complements 3 2 Development of the fuzzy inference system for local matching The purpose of the fuzzy inference system is to describe the regional similarities between land use maps with linguistic membership functions Formally a linguistic membership function is a mathematical curve that represents a person s intuitive 82 C Power et al perception of the degree of matching between sections of the input maps By converting the linguistic agreement expressions into membership functions the fuzzy pattern matching model quantitatively emulates human reasoning to produce an output agreement value The fuzzy inference system for this project was developed with the Fuzzy Logic Toolbox from Matlab 1994 and is based on Mamdani inferencing Mamdani 1976 Many of the fuzzy inference systems in previous research are based on either Mamdani or Takagi Sugeno TSK inferencing Simpson and Keller 1995 Jang et al 1997 For this project the advantages of a Mamdani system lie in the differences of the consequents of the fuzzy rules and the aggregation and defuzzification proced ures of each system Mamdani fuzzy inference systems are rule based decision models that produce mathematical control statements as output membership functions to handle the interacti
78. ls of each type of comparison result The cell by cell comparison which is discussed in the following section generates additional statistics for each category Per category City C users tahageniTesti nalyseiLogiFakeikrti img C userstahageniTestiAnalyseiLogiFakeikrt3 img Statistics in none of the maps 2277000 only in map 1 not in 73 000 in both maps 92 000 only in map 2 not in 55 000 2 3 2 Cell by Cell Result map Cell by zell The Cell by Cell comparison method is the most straightforward method for comparing raster maps The method simply considers for each pair of cells on the two maps whether they are equal or not This results in a comparison map displaying the spatial distribution of agreement This comparison method does not take any parameters As straightforward as the Cell by Cell comparison is its derived statistic the Fraction Correct This statistic is calculated as the number of equal cells divided by the total number of cells The fraction correct is considered flawed as an overall measure for similarity The reason is that when the fraction correct is used as a similarity index the agreement of the more common categories is weighted too heavily For a better balanced measure of similarity the Kappa statistic is often used It is the fraction correct that has been rescaled to adjust for the fraction correct that would be expected if the given total numbers of categories were distributed
79. methods for the analysis of map similarity Unlike the Boolean approach chance agreement and misregistration problems are handled by the overlap of the output membership functions for the local matchings 5 2 Fuzzy versus Boolean land use comparison results The advantages of fuzzy pattern matching over the Boolean approach are difficult to quantify because both procedures have different purposes As a result a visual interpretation of the differences between the fuzzy and Boolean agreement maps is the basis of the discussion of the advantages of fuzzy pattern matching as a map comparison technique Figure 11 contains the fuzzy and Boolean land use similarity maps for Cincl Map B is a Boolean agreement map containing discrete agreement and disagreement categories The fuzzy land use agreement layer Map A displays the land use differences between the input maps as a continuous range of possibilities of member ship in a land use disagreement class The visualization of the disagreement possibil ities on the fuzzy map is based on a gradation in the intensity and hue of the colour for the disagreement class with the possibility of disagreement between the maps being highest for the darkest polygons and decreasing as the colour lightens The primary advantage of a fuzzy agreement map is that it contains more information and gives a more realistic interpretation of the land use characteristics of a dataset The fuzzy agreement information allows the u
80. milarity The method is aimed at comparing categorical raster maps The assessment results are spatial and gradual additionally an overall figure for similarity is aggregated from the detailed spatial results 2 Methods For the comparison of maps two sources of fuzziness are considered fuzziness of location and fuzziness of category A similar distinction is found in Cheng et al 2000 where thematic and geometric aspects of uncertainty are treated separately In this paper fuzziness means a level of uncertainty and vagueness of a map This fuzziness is not inherently present in the map but follows from an observer s inter pretation Fuzziness of category means the observation that some categories in the legend of a map are more similar to each other than others With fuzziness of location is meant that the spatial specification found in a categorical map is not always as precise as appears a category that in the map is positioned at a specific location may be interpreted as being present somewhere in the proximity of that location In the original map every cell is represented by a single category In the fuzzy representation a cell will partially belong to multiple categories To allow cells to belong to multiple categories simultaneously they are assigned a membership vector The elements of the vector give the degree of belonging to each category In this paper three types of membership vectors will be distinguished the Crisp Vector V
81. n after you have typed the name of the file 22 This command 1s identical to pressing the Open button from the Toolbar Close Command Use the Close command to close the log file you are currently working on If the log file is new or has been changed then you will be asked to save it Save as Command Use the Save as command to save the log file that you are currently working on Export Command Use the Export command to save the map in the active map window on the disk Maps from all the map windows can be saved in this manner 4 When you select the Export command the Save As dialogue window appears It contains all the files in your work directory with the right extensions If you select in the Save as type box the type that you want to save the map you can save maps in Idrisi format IMG Filename My New Map img SOSOS S S S extension or Arc Info ASCII grid format ASC extension Save as type Idrisi Image Files rst img Cancel maps img fae region img Page Setup Command Use the Page Setup command to decide on the size and scale at which you want the MAP COMPARISON KIT to print the active map as ee eae X Unit of measurement fem E cells per unit of measurement Cancel Print grid M major grid As soon as this command is active the Page Setup dialogue window appears enabling you to specify how many cells you want to print pe
82. nference system The rule base only need include the rules for which the areal intersection and complement ratios are opposites Under Mamdani inference the critical step in the implication process is finding the consequence of each rule by combining its strength and output membership function Jager 1995 The consequence of a rule is computed by clipping an output Table 3 Rules for the local matching fuzzy inference system Rule Rule structure 1 If Area_Inter is Very_Low and Area_Comp is Very_High and Pixel_Group is Small then Local is Poor 2 If Area_Inter is Very_Low and Area_Comp is Very_High and Pixel_Group is Large then Local is Very_Poor 3 If Area_Inter is Low and Area_Comp is High and Pixel_Group is Small then Local is Good 4 If Area_Inter is Low and Area_Comp is High and Pixel_Group is Large then Local is Poor 5 If Area_Inter is Medium and Area_Comp is Medium and Pixel_Group is Small then Local is Good 6 If Area_Inter is Medium and Area_Comp is Medium and Pixel_Group is Large then Local is Good 7 If Area_Inter is High and Area_Comp is Low and Pixel_Group is Small then Local is Good 8 If Area_Inter is High and Area_Comp is Low and Pixel_Group is Large then Local is Very_Good 9 If Area_Inter is Very_High and Area_Comp is Very_Low and Pixel_Group is Small then Local is Perfect 10 If Area_Inter is Very_High and Area_Comp is Very_Low and Pixel_Group
83. o open an existing log file A log file is a small file that points the MCK to those maps that may be used in the comparisons Log files can be opened edited and saved It is also possible to build a new log file from scratch 1 1 Open a log file Find the correct file with log extension in the Open dialogue open 2ix The MAP COMPARISON KIT is build lek demo RE according to the Windows standards ia Hence it is possible to find the file by My Recent com_ereciolog browsing thought your own computer or E another computer in your network If you Desktop have found the correct file select it and click the Open button or double click on My Documents the icon of the file My Computer During the installation of the MCK the z option is given to include a number of Dy Fiene femos o example log files These are placed in the S o Fic oo _o same directory as your MCK application 1 2 The Analyse application window The Map Comparison Kit application window consists of the Caption bar the Menu bar the Work pane and the Toolbar You can simultaneously open four windows maximum three map windows and one statistics window Furthermore it is possible to keep the Comparison Settings dialogue open May Comparison Kit demo_no_re jio log E ife x Fil Edit View C Linear decay Constant value Options Window ao 5 LandUss t Demo map1 img Help 7 E Demo map img EET
84. occurrences e g certain categories distances from landmarks geographical and political boundaries etc The applicability of the method is not restricted to geographical problems other fields of potential use are image analysis pattern recognition and video image analysis References BANDEMER H and GOTTWALD S 1995 Fuzzy sets fuzzy logic fuzzy methods with applications Chichester New York J Wiley Comparing categorical maps by fuzzy set theory 249 CARLETTA J 1996 Assessing agreement on classification tasks the kappa statistic Computational Linguistics 22 249 254 CHENG T MOLENAAR M and LIN H 2001 Formalizing fuzzy objects from uncertain classification results International Journal of Geographical Information Science 15 27 42 FISHER P 2000 Sorites paradox and vague geographies Fuzzy Sets and Systems 113 7 18 Foopy G M 2002 Status of land cover classification accuracy assessment Remote Sensing of Environment 80 185 201 Lewis H G and Brown M 2001 A generalized confusion matrix for assessing area estimates from remotely sensed data International Journal of Remote Sensing 22 3223 3235 METTERNICHT G 1999 Change detection assessment using fuzzy sets and remotely sensed data an application of topographic map revision ISPRS Journal of Photogrammetry amp Remote Sensing 54 221 233 MONSERUD R A and LEEMANS R 1992 Comparing global vegetation maps with the
85. of the compar ison result depends on the comparison of the two neighbourhoods excluding the central cell The consequence is that even if two cells at the same location in two maps belong to different categories and these two categories are not similar to any Central cell A Central cell B Crisp vector 1 0 0 0 1 0 Fuzzy Nbh Vector 14 0 2 0 5 0 5 1 0 5 Figure 3 Two neighbourhoods and their central cells Comparing categorical maps by fuzzy set theory 241 of the categories in the neighbourhood there is a possibility that the cells are considered similar because their neighbourhoods are similar This is not intended for the map comparison To avoid an overpowering influence of the similarities between the neighbour hoods the so called two way comparison is introduced It proceeds as follows in first instance the Fuzzy Neighbourhood Vector of cell A is compared to the Crisp Vector of cell B Next the Crisp Vector of cell A is compared to the Fuzzy Neighbourhood Vector of cell B Finally the lower of the two comparison results establishes the similarity at that location equation 10 STwoway A B SCV nbn A gt V crisp B SCV crisp A gt V nbn B Min 10 The calculation of the two way similarity value of the central cells in figure 3 is calculated according to equations 11 13 A lower similarity of 0 2 is found S V nbn A gt V crisp B 10 5 Latin 1 Ol vin 0 5 Ol vin ax 0 5 1
86. of the similarity is based upon a Fuzzy Inference System evaluation of these characteristics The characteristics that are taken into account in this evaluation are area of intersection area of disagreement and size of the polygon The Fuzzy Inference System approach is in essence a symmetrical which means that the comparison of two maps is different depending on which map is considered to be the reference or real map and which is the comparison or model map In many cases it is not possible or preferred to make this 15 Fuzzy Inference System setting Eo eee x Bi distinction For these cases it 1s made possible to f Standard first raster is the reference map combine the two possible a symmetrical Standard second raster is the reference map comparison results into one symmetrical result S Si anise Rules eg tno ate The options are to calculate a cell by cell ey average product minimum or maximum of the two comparison results This option can be set in the Parameters dialogue of the Fuzzy Inference System comparison method Symmetric by taking the product Apply Symmetric by taking the minimum Symmetric by taking the maximum Appendix II offers detailed information on the Fuzzy Inference System comparison method 2 3 4 Fuzzy Set The main purpose of the Fuzzy Set map comparison is to take into account that there are grades of similarity between pairs of cell in two maps The Fuzzy Set approach ther
87. ogic behind the aggregation procedure is that a local matching value is a measurement of areal agreement between two land use polygons By multiplying a local matching number by the area of the unique polygon an agreement area is calculated Then the ageregation of the local matching areas relative to the total area of the unique polygons map produces the global similarity value This is computed as ore Area x E is Pa y s Total Area S 7 where n is the number of unique polygons in the template layer 4 Description of the datasets The data sources for this paper consist of a set of atemporal urban land use maps and a set of multi temporal forest inventory maps The fuzzy inference system compares atemporal maps for a map similarity analysis and multi temporal maps for land use change detection Dataset one figure 7 which will be referred to as Cincl is comprised of two simulated land use maps of Cincinnati Ohio A cellular automata based model of urban dynamics developed by White et al 1997 produced these maps From a set of quasi deterministic transition rules the simulated map was generated by ten iterations of the cellular model with an antecedent land use map as the initial configuration Both maps are 80 rows by 80 columns rasters at a pixel resolution of 250m The problem is to determine how similar the two simulations are Map 1 is the template or reference layer in the matching process The two land use maps in d
88. on When the Palette command is selected from the View menu the Palette editor dialogue window opens In the figure below the window is shown and the relevant settings are explained The changes that you make to a palette can be saved by clicking the Save or Save as button The changes are then saved in a palette file SMP 20 The Palette editor is closely associated with the Legend editor The latter enables to define the way in which data are presented on a map Palette editor file handling SMP Color Index New files a Oper Save 4 buttons for palette Accept or de cline the last changes Cancel ii Save as Set the number of colours in the palette The maximum is 256 Default colour is black Click in a colour box to access the colour editor for that box Palette size 20 App Apply current settings Blend colours that smoothly evolve from the one Indicated with the lower index till the one Rexerze palette indicated with the upper index Reverse the order of the Falette blend function Blend colours in the palette Lowe index jo Upper index fis 21 4 THE MENU SYSTEM This paragraph explains the different functions that are available from the menus of the MAP COMPARISON KIT The menus are treated as they appear in the Menu bar from left to right and per menu from the top to the bottom 4 1 File menu File New Open Cl
89. on of cross entropy see Foody 1995 and Chang et al 1994 As a single index value cross entropy can be readily interpreted to evaluate how well the fuzzy agreement and disagreement patterns represent change on the ground Thirdly research is required into the implementation of optimization techniques to obtain the best structure for the fuzzy inference system It is possible that the local matching results are inaccurate because the shape of the membership curves and the amount of overlap between the functions are less than optimal Preliminary research suggests that the solution may be to replace the fuzzy inference system with an Adaptive Neural Fuzzy Inference System ANFIS An ANFIS is functionally equivalent to a fuzzy inference system except that it uses a backpropagation neural network algorithm to fine tune the internal structure of the system Using fuzzy agreement training data the connective updating capabilities of the ANFIS would continually shape the membership functions of the matching system until a learning error threshold is reached Jang et al 1997 It is important to recognize that other fuzzy neural network systems particularly fuzzy ARTMAP are also applicable for the optimization process The viability of fuzzy ARTMAP should be investigated because it avoids the problems of overfitting and learning forgetfulness associated 98 C Power et al with backpropagation Carpenter and Grossberg 1997 Carpenter et al 1999 Gop
90. ons of the inputs to the system Jang et al 1997 The design of this system requires the developer to create both input and output membership functions from linguistic interpretations of a subject Through the compositional rule of inference and a defuzzification algorithm Mamdani systems produce an overall output value from the output membership functions Jang et al 1997 The advantage of Mamdani fuzzy inference systems is that the fuzzy input and output membership functions are better suited to handle fuzziness and data uncertainty and work better with human input A disadvantage is that the defuzzification process is computa tionally intensive and not easily subjected to rigorous quantitative analysis Unlike Mamdani systems TSK fuzzy inference systems only contain fuzzy input membership functions since the consequences of the rules are crisp polynomial functions Thus the reasoning mechanism of a TSK system can not follow the compositional rule of inference and produces a final output value from the weighted average of the rule consequences By avoiding the mathematical complexities of the defuzzification procedure TSK systems are better suited for mathematical analysis A significant disadvantage for this project is that the crisp rule outputs make a TSK model counterintuitive due to the inability to propagate fuzziness from the input to outputs in a appropriate manner Jang et al 1997 Also the simplification of the consequents with crisp
91. ontrol 8 338 353 APPENDIX II FUZZY INFERENCE SYSTEM Q INT J GEOGRAPHICAL INFORMATION SCIENCE 2001 voL 15 No 1 77 100 TAY y ed D f apt ended si Research Article Hierarchical fuzzy pattern matching for the regional comparison of land use maps CONRAD POWER MATRIKS Maastricht Technological Research Institute for Knowledge and Systems Maastricht University Maastricht The Netherlands ALVIN SIMMS and ROGER WHITE Department of Geography Memorial University of Newfoundland St John s Canada Received and accepted 20 March 2000 Abstract The evaluation of the spatial similarities and land use change between two raster maps is traditionally based on pixel by pixel comparison techniques However a pixel by pixel comparison can register a small displacement in pixels as land use disagreement even though the land use patterns may be essentially the same The techniques of unique polygons mapping and hierarchical fuzzy pattern matching where the maps are compared on both a local and global level are combined to provide a more robust alternative approach Local matchings determine the degree of containment of each unique polygon in the template map in terms of fuzzy areal intersections Formally the local agreement values are based on polygon property containments and are calculated from a fuzzy logical Max Min compositional algorithm A global agreement value is derived by the fuzzy summation of the lo
92. or each cell and derive the average per cell An alternative for the analytical calculation of P 1s to find an estimate by Monte Carlo analysis 3 Results 3 1 Hypothetical case The two maps in figure 6 were created in order to demonstrate the features of the map comparison method Several types of differences occur minor shifts major shifts growth decline introduction removal and differences of cell categories within clusters of similar content The method is symmetrical this means that there is no difference between comparing map 1 with map 2 or vice versa Therefore growth is equivalent to decline as is introduction to removal A large part of the map is coloured white this does not indicate a so called no data value but rather the white cells represent a category just like the coloured cells Figure 7 gives the results of the direct cell by cell method a and the proposed fuzzy cell by cell method b The fuzzy membership function is that of exponential Comparing categorical maps by fuzzy set theory 245 Major decrease nS and shift landscape alament ao j Most of river L minor shifts E al Ta a Minor decrease I Many differences at the micro level within similar clusters ai the macro level Mapi Map2 Figure 6 The two maps to compare a Cell by cell comparison b Fuzzy comparison of maps Figure 7 Comparison results a Cell by cell comparison b Fuzz
93. ormed when one of the result windows needs to be updated This means that the calculation is only performed after a change in the 7 Map 2 Map Comparison method or its Parameters AND a result window is being opened This means that it is possible to select and view maps in the 7 Map and 2 Map window without immediately performing the comparison You can then choose to perform the comparison once both intended maps have been selected This is especially important to realize when a calculation intensive comparison method has been selected 2 2 Exporting results The MAP COMPARISON KIT features Clipboard support for easy report writing This functionality is commonly known as Copy amp Paste and allows the user to copy information directly from the MCK and paste it into another 12 Tip Windows program Maps and legends are copied to the Windows Clipboard as bitmaps i e images whereas the result statistics are copied as tab delimited ASCII tables i e plain text If you right click on a legend in a map window a Copy menu item will appear clicking this item will send an image of the legend to the Windows Clipboard Likewise you can copy the map that is displayed in a map window or the statistics from a Result Statistics window Instead of right clicking you can also type lt Ctrl C gt to copy the contents from the active window In most Windows programs clicking Paste or typing lt Ctrl V gt will paste the con
94. ose Save as Export map Page Setup Print Print Preview Printer setup Use the File menu to open import or export a file to print maps and close the MAP COMPARISON KIT Chri o Ctrl P 1 england selection log 2 hym mils log 3 CHuserst ILLO Casella log 4 Chlusers spot_Fifteen log Exit New Command Use the New command to create a new log file This implies that the log file that you are currently working on is closed If that file has not been saved to disk yet you will be asked to do so When you create a new log file you can specify the contents using the log file editor as explained in Section 0 Open Command Use the Open command to open a log file stored on disk You cannot open more than one log file at the time However you can combine log files by using the Import function in the log editor see Section 0 When you select the Open command the Open dialogue appears If the name of the file of your choice is not visible in the list box use the scroll bars to move through the list of filenames in the directory or disk you are working in If the file you want to open is not in the current directory or on the current disk use the scroll list or browse symbols in the section named Look in to change directories disks or network sites Double click the name of the file you want to open You can also type the name and path of the file in the Filename box Press the Open butto
95. overall and per individual category 0 96 0 96 1 00 1 00 1 00 1 00 0 78 0 95 1 00 0 91 0 96 0 86 0 95 0 85 0 00 0 82 1 00 1 00 0 99 0 93 1 00 1 00 0 98 0 98 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 1 00 0 08 0 53 0 88 0 97 1 00 1 00 1 00 1 00 The results presented in Figure 8 suggest that although little improvement can still be made most of it can be expected from improving the spatial allocation The categories with relatively weak spatial allocation are Residential continuous dense urban fabric and Construction sites The relatively low scores for Road and Airport are in accordance with the observations made on the comparison map 5 Conclusion The multi method approach to map comparison as presented in this paper offers a refined assessment of similarity Due to the introduction of Khisto it is possible to express Kappa as a combination of similarity in quantity and location By applying the Kappa related statistics per category it becomes clear how the different categories contributed Negative aspects of the kappa statistics are compensated by the fuzzy set method Firstly a spatial assessment of similarity 1s given the comparison map is highly informative and clarifies not only the location of disagreement but also the severity Other negative aspects of the Kappa statistics are that it cannot consider similarity between categories and does not take proxim
96. pe of help that you want the MAP COMPARISON KIT to display on the screen The different commands in this menu will permit to look up information about the MCK its commands options and tools The Index Command Use the Jndex command to get the opening screen of the Help file of the MCK From the opening screen you can jump to step by step instructions for using the MCK Double click the topic that you want help on A help screen will appear Once you open help you can click the Contents button whenever you want to return to the opening screen Important In this version of the MCK the on line help is not operational About Command Use the About command to get the copyright notice and version number of the MAP COMPARISON KIT that you are using The latter is important if you need assistance with the software from the developers or when you request an update of the software 28 5 FILES IN THE MAP COMPARISON KIT The MAP COMPARISON KIT makes use of different types of files To work with the MCK it is not necessary to know about these files however a basic understanding will be most beneficial for the regular user Four types of files are important in the MCK Log files Map files Legend files and Palette files A special Map file is the Region file which designates the area of the map that is being compared These file types are discussed in the following sections 5 1 The log file The log file is the MCK s gateway
97. polynomial functions can lead to loss of membership linguistic meanings Figure 2 is a flowchart of the four basic elements of the Mamdani fuzzy inference system for the matching of the unique polygons The crisp input values are the calculated areal intersection and complement ratios from the unique polygons mapping The output local matching values depend on the fuzzy relational and compositional algorithms that comprise and link the sections of the fuzzy inference network 3 3 Creation of the input and output membership functions The creation of the input membership functions depends on the development of a linguistic scaling of the local matchings for the unique polygons from the Boolean Create Crisp Membership Functions Crisp Rule Based Defuzzification Inference Fuzzification Value Value Figure 2 Four stages of designing a Mamdani fuzzy inference system Fuzzy regional comparison of land use maps 83 areal intersection and complement ratios Formally the semantic expressions are needed as answers to the question What is the possibility that the land use is similar for a specific localized comparison of unique polygons A five point scale is generated ranging from very low to very high The linguistic values and their descriptions are in table 1 To transform the crisp intersection and complement numbers into linguistic values membership functions for each of the qualitat
98. ppropriate and also the size of the neighbourhood depends on the nature of the uncertainty vagueness of the data and the observer s tolerance for spatial error From a theoretical point of view there is not a best alternative hence it is worthwhile to experiment with size and form of the function Table 3 Fuzzy representation of hierarchical data Category Number Fuzzy Category Vector Residential 1 1 0 0 0 0 0 Citrus agriculture 2 0 1 Os 03 0 0 Sugarcane agriculture 3 0 0 3 1 0 3 0 0 Banana agriculture 4 0 0 3 03 1 0 0 Industry 5 0 0 0 0 1 0 Water 6 0 0 0 0 0 1 Comparing categorical maps by fuzzy set theory 239 Exponential Gaussian Figure 1 Some 3D memberships The different membership contributions of the neighbouring cells are combined by calculating the fuzzy union of all neighbouring cells multiplied by their respective distance based membership The vector that results from this operation is the Fuzzy Neighbourhood Vector This is expressed in equations 5 and 6 for a map with C categories and N cells in the neighbourhood Equation 6 shows how cells in the neighbourhood contribute to the fuzzy representation of the central cell With increas ing distance from the central cell the contribution decreases as expressed by the distance based membership m The highest contribution of each category sets the membership value of that category Hnbh 1 V abn rm2 5 Hnbh C Hama Penta is Megan May e
99. r 0 0 1 Os OS Ds ys erect 08 0 9 1 output variable Local Figure 5 Local matching output membership functions 86 C Power et al 3 5 Rule based inference The essential part of a fuzzy inference system is a set of fuzzy rules that are related by means of a fuzzy implication function and a compositional rule of inference Jang et al 1997 Fuzzy rules are a collection of linguistic If Then statements that describe how a fuzzy inference system makes a decision about categorizing an input or controlling an output Simpson and Keller 1995 With fuzzy rule based reasoning the fuzzy rules are represented by a fuzzy implication function The implication process defines the associations between the input membership functions and determines the consequence of a rule Furthermore the fuzzy implication of a rule depends on its If Then connective operator which expresses how a fuzzy rule is delineated by a fuzzy relation Jang et al 1997 The premise variables of the rules in the local matching rule base are connected with a conjunctive T norm which satisfies the condition t a b Min a b 3 where Min sets the upper boundary of the function as the intersection of a and b Formally a T norm refers to a logical AND connective so that fuzzy rules are written as If A and B then C To ensure that the rule base exhibited both consistency and completeness ten rules table 3 are created for the rule base of the local matching fuzzy i
100. r land use polygons of Cincl Evaluation of Land use type of Polygons Match Mismatch similarity Unclassified 80 19 61 23 75 Commercial 79 32 47 40 50 Industrial 106 35 71 33 01 Residential 95 34 61 35 80 River 2 2 0 100 Railway 6 6 0 100 Roads 2 2 0 100 Total 370 130 240 61 87 Fuzzy regional comparison of land use maps 91 100 agreement but the results for the other categories suggest that substantial land use disagreement is evident With 130 matches and 240 mismatches the land use maps should be considerably different but this discrepancy is due to the pixel resolution of the data With a resolution of 250m most of the polygons in the grouping template layer for Cinc 1 consist of one or two pixels These small template polygons account for most of the mismatches between the maps even though they represent a small portion of the study area The matchings evaluations in table 5 show that the smaller number of matched polygons accounted for 85 98 of the template area In addition 188 of the 240 mismatches were for one or two pixel polygons that combined occupy only 4 73 of the template map The local matchings for Forest1 indicate that little land use change has occurred from 1985 to 1991 figure 10 A majority of the matchings range from 0 70 to 1 Table 5 Evaluation of matches and mismatches for Cincl Definite matches Definite mismatches 130 240 32 43 of polygons 67 57 of polygons 85 98 of total area 1
101. r measurement 0 unit cm or inch Furthermore it is also possible to pict 0 Bottom 0 indicate if you want to print the grid on your map and you can also set the margins of the pages to be printed Print Command Use the Print command to print the map displayed in the active map window e Print Preview Command Use the Print Preview command to get a preview of the printed document on the screen Print setup Command Use the Print setup command to change the settings on the printer enabling correct printing 2 List of Recent Files 1 2 3 4 The MAP COMPARISON KIT keeps track of the four most recently opened log files It will display those in the Fi e menu If you select one of the four files it will be opened The Exit Command Use the Exit command to quit the MAP COMPARISON KIT if you are working with a new or modified log file you will be asked to save your changes 4 2 Edit Menu Edit 51 E gt Tip Log file Legend Palette The edit menu offers access to the editors for log files legend files and palette files Log Command Use the Log command to open the LOG FILE EDITOR Legend Command Use the Legend command to open the LEGEND EDITOR The LEGEND EDITOR enables you to adjust the legends of all the maps in the MAP COMPARISON KIT and to create new legends See also Section 3 1 of this manual Palette Command Use the Palette command to open th
102. rcentage of agreement for the expected percentage of agreement based upon the number of cells taken in by each category on each map i e based upon the histograms of the two maps The statistic is similar to the Kappa statistic and is therefore called K The formula for K equation 14 is identical in form to that of the Kappa statistic Carletta 1996 Monserud and Leemans 1992 The difference lies in the calculation of the expected similarity K P P Fuzzy p P 14 where P observed percentage of agreement i e average similarity P expected similarity based upon given histograms In the following paragraphs P is derived for two way comparisons in which 242 A Hagen Situation 1 The value for similarity in the central cell must be low because the two cells black and white differ and there are no cells of the same category in the neighbourhood S 0 5 STwoWay 0 Situation 2 The value for similarity in the central cell will be intermediate because the two cells black and grey differ but there are cells of the same categories in the neighbourhood S 0 5 Stwowey 0 5 Situation 3 As in Situation 2 the value for similarity in the central cell must be intermediate The similarity must be smaller than in Situation 2 because the matching cells are found within a greater radius Situation 4 The value for similarity of the central cell is equal to the one in Situation 3 because the match
103. rithms Theme d Map 1 H Map 2 b Comparison method Parameters Result map Statistics Theme Sub menu Use the Theme Sub menu to select the theme from the numbered items in the Sub menu These are all themes present in the log file you are working on Map1 Map 2 Sub menu Use the Map 1 sub menu to select the 7 Map from the numbered elements Or use the Show command to open the 1 Map window 26 Likewise use the Map 2 sub menu to select the 2 Map from the numbered elements Or use the Show command to open the 2 Map window Map Comparison Kit spot _fifteen log File Edit view Options Window Help Theme d Map 1 d Show SE ne wO Spot the DifferencesispotiS_a rst Comparison method 1 Spot the DifferencesispotiS_bi rst Parameters 2 a pot the Differencesispoti5S_bz rst Result map Statistics The submenus list all maps of the selected theme in the log file that you are working on Comparison method Command Use the Comparison method command to select which comparison algorithm to use In this dialogue window you can select the method of your choice See Chapter 2 of this manual for the individual comparison methods e Parameters Use the Parameters command to open the parameter dialogue belonging to the selected comparison method This dialogue is only available if the selected comparison makes use of parameters Result Map Use the Result Map command to perform th
104. rst abs second first second first masl abal second first J abs second first 7 max abel second first J second 7 first C abs second first NODATA C ANODATA s NODATA f NODATAs 0 x 2 3 6 Other operations The other operations are not map comparison methods They offer some often used GIS functionality and thus help to avoid some tiresome switching from one program to the other The options are to perform a cell by cell addition of the 1 Map and the 2 Map to add a constant value to all cells in the 7 Map or to multiply all cells in the 7 Map with a constant value The result of the operation can be found in the Result map Other operations E first second first 1 f first 18 3 CUSTOMIZING THE VIEWS All maps of one theme are displayed according to the same legend Likewise all discrepancy maps resulting from the same comparison method are displayed according to the same legend These legends are completely customisable The legends may contain the colour information for the different legend items or they may apply colours from a palette file Therefore this chapter contains a section about the legend editor and a section about the palette editor For most users it suffices to only use the legend editor Chapter 5 of this manual discusses the palette and legend files and their relation with other files in the MAP COMPARISON KIT 3
105. s meant the spatial allocation of the quantity over the map Pontius introduces two statistics to separately consider similarity of location and similarity of quantity The statistic for similarity of quantity 1s called Kquantity but the application of this statistic leads to many practical problems The statistic for similarity of location on the other hand is very informative because it gives the similarity scaled to the maximum similarity that can be reached with the given quantities Klocation is calculated according to Equation 5 Klocation BiG laa 5 P max P E An alternative expression for the similarity of the quantitative model results is the maximal similarity that can be found based upon the total number of cells taken in by each category This is called P max P max can be put in the context of Kappa and Klocation by scaling it to P E The resulting statistic 1s newly introduced here and is called Khisto because it is a statistic that can be calculated directly from the histograms of two maps Khisto is defined by Equation 6 Rio AARE 6 1 P E The definition of Khisto has the powerful property that Kappa is now defined as the product of two factors Equation 7 The first factor is Klocation which is a measure for the similarity of spatial allocation of categories of the two compared maps The second factor is Khisto which is a measure for the quantitative similarity of the two compared maps K Khisto
106. ser to concentrate on specific characteristics of the results such as whether a specific land use type accounts for most of the darker disagreement areas Since a cellular automata land use prediction model produced dataset one an analyst can use the information about the higher disagreement possibility areas to recalibrate the model to produce better prediction results This may be difficult or impossible with Boolean results because 94 C Power et al A Fuzzy Agreement Possibilities Land Use Disagreement B Boolean Agreement Classes 0 Kilometers 4 Figure 11 Fuzzy A versus Boolean B agreement for Cincl the Boolean approaches often lose agreement information when producing dichotomous similarity categories A second advantage of the fuzzy agreement map is that it retains the form of the template layer This gives a better visual impression of where land use differences are situated spatially For example the areas of lowest disagreement on figure 11 represent the river and transportation system of the study area However the Boolean map consists of a patternless mixture of disagreement and agreement areas that make it difficult to relate the result to the original land use maps It is apparent that Fuzzy regional comparison of land use maps 95 the discrete classification from the Boolean model has simplified the land use similarity results The comparison of the forest inventory maps in Forest1 demonstrates
107. slar Mn latex 6 where F the degree of membership for category i Uapn i j membership of category i for neighbouring cell j in Vasho Heat i Membership of category i for neighbouring cell j in Vea m distance based membership of neighbouring cell j Figure 2 and equation 7 illustrate this for a cell in a neighbourhood with a radius of 2 cells Figure 2 describes the situation Equation 7 applies equations 5 and 6 for the central cell of the particular situation 1x02 1x05 0x02 Hah OxO5 1x1 0x0 5 0x0 2 1x05 0x0 2 yax 0x0 2 0x0 5 1x0 2 1 V abh HUnbh 2 0 x 0 5 0 x 1 0 x 0 5 0 2 7 0x0 2 0x05 1x0 2 yrax 0 5 0x0 2 0x0 5 0x0 2 Unpn3 1x0 5 0x1 1x05 1x0 2 0x05 0x0 2 yax In the example of figure 2 the Fuzzy Category Vector is equal to the Crisp Vector indicating that similarity between categories has not been considered The procedure is identical if the Fuzzy Category Vector does express similarity between categories 240 A Hagen Neighbourhood Membership definition Fuzzy Category Vector M ee Grey 001 Figure 2 Neighbourhood legend and membership definition 2 3 The comparison 2 3 1 Comparison of two fuzzy cells The similarity of two maps can be assessed by cell by cell comparison of the fuzzy vectors assigned to all cells The expression for similarity at each location is based upon the fuzzy set intersection of the two fuzzy vectors and is given in equation
108. ssue in the development of analytical techniques for spatial data is the comparison of maps The need for map comparison methods is recognized and has growing interest among researchers Metternicht 1999 Monserud and Leemans 1992 Pontius 2000 Pontius 2001 Power Simms and White 2001 Winter 2000 In this paper map comparison is approached from two directions The first angle is over the confusion matrix which is presently the core of accuracy assessment Foody 2002 The confusion matrix is mostly used to derive the Kappa statistic Additional statistics are introduced in order to come to an advanced use of Kappa statistics in Section 2 The second angle of approach is fuzzy set theory is found in Section 3 Fuzzy set theory is applied to deal with several map comparison issues that were also recognized by Foody 2002 One issue is to allow some level of positional tolerance in the map comparison Another issue is to find the spatial distribution of error The third issue is to differentiate in error magnitude which means that some errors are more significant than others 2 Advanced use of Kappa statistics The Kappa statistic is much used to assess the similarity between observed and predicted results It is not only applied for geographical problems e g Pontius 2000 Monserud amp Leemans 1992 but in many other fields such as medical and social sciences As a result much has been published about the kappa statistic and its func
109. t a ground truthing dataset Boolean comparison procedures generally assess the accuracy of change detection results with an error matrix and Kappa analysis However Foody 1995 states that a standard error matrix is inappropriate for computing the accuracy of a fuzzy change detection analysis because of its inability to accommodate the fuzziness in both the land use maps and the ground data Ground data can rarely be assumed to be error free and often contain attribute and locational uncertainty Therefore a fuzzy accuracy assessment should handle the uncertainty in the agreement maps and ground data during the similarity analysis For this project the accuracy assessment will be a soft estimation of the closeness of the qualitative fuzzy labels assigned to the change maps and field test sites Note that fuzzy agreement labels will have to be qualitatively assigned to the test sites for the closeness measurement to be possible Since the agreement maps and ground data will be fuzzy the entropy of each data source can be calculated and used to determine an index of accuracy based on cross entropy Zhang and Foody 1998 Cross entropy will use the entropy values to measure the distance or closeness of the probability distribution of the agreement map to the probability distribution of the ground data Formally the closer the agreement map to the ground data the lower the cross entropy and the higher the map similarity accuracy For a detailed discussi
110. ted below In general it is more convenient to use the log file editor as presented in section 0 ff demo log Notepad The legend directory is called Legends File Edit Format view Help The palette directory is Palettes LegendsDir Legends PalettesDir Palettes RegionsMap Regions 1img 3 z Landuse lu_89 ing The regions map is called Regions img Landuse lu_33 img There are three themes in this LOG file Landuse Ecosystem and Population There are four maps of each theme The Landuse maps are lu_ 89 lu_ 93 lu 96 and lu 1970 1u_1970 ing Ecosystem eco_1970 img Population pop 1970 imd The log file consists of lines containing a keyword and a file or directory name The keywords LegendsDir PalettesDir and RegionsMap are recognized and are used to point the MCK to the respective directories or map These lines are optional All other keywords are taken to be names of themes and should be followed by the filename of a map 5 2 Legend files Maps of one theme in the log file are displayed according to the same legend This legend is found in the legend directory and has the name of the theme followed by the txt suffix It is not necessary to place legend files in the legend directory for all themes If the MCK displays a theme for which there is no legend file present then it will generate a legend with default settings Legends contain information about
111. tents of the Clipboard into the document that you are working on A special tip for Microsoft Office users The tab delimited table can be directly pasted into MS Word or MS Excel In MS Word the tab delimited table can be converted to a regular table by applying the Convert Text to Table command from the Table menu in the Menu bar Maps can also be saved in some often used GIS formats you can save the map of the active map window as an Idrisi Raster Map with RST or IMG suffix or an ArcASCII Raster file with ASC suffix Use the Export command from the File menu in the Menu bar to open a Save as dialogue for the map in the active window 2 3 The Map Comparison Methods Comparisons and other operations a x The Comparison Method dialogue offers Compare categories i Per category Cell by cell f Fuzzy Set algorithm Compare numerical values i second first i abel second first Fuzzy Inference System algorithm y three types of operations Compare T categories Compare numerical values and Other operations In principle categorical maps should be compared with categorical map comparison methods and numerical maps with numerical second first 4 max abs second first J map comparison method The MAP i abel second first 4 mas abel second first J COMPARISON KIT is not dogmatic and second first i abel second lt first allows you to ignore these principles NO
112. the categorical definition of maps They decide whether the values found in the map are categorical or numerical For categorical maps the names associated with the rank numbers found in the map are given For numerical maps the display intervals are given as well as the formatting on the display names The legend file also contains information on the colours in which the categories are displayed Here there are two options 1 The legend file contains the colour coordinates for each category display interval 2 The legend file contains a reference to a palette file and the categories display intervals are coloured according to the colours found in the palette file The rule of thumb is that colour sets that are typical for one particular theme are defined in the legend itself whereas colour sets that have a generic value in the sense that they may be applicable for more themes are found in the palette file Therefore the default location for the legend directory is in the 30 same directory as the log file and the default location for the palette directory is in the directory where the Map Comparison Kit executable 1s located The legend files are ASCII files meaning that you can edit them with editors such as Notepad However to be certain that the legend file format is adhered to it is advised to only use the legend editor of the Map Comparison Kit 5 3 Palette files Palette files contain a collection of colours which in practic
113. tionality has been extensively discussed Carletta 1996 Fielding amp Bell 1996 Lantz amp Nebenzahl 1987 Maxwell 1977 In this section the Kappa statistic and the contingency table that forms its basis will be shortly discussed followed by the introduction of some derived statistics and suggestions for practical use of Kappa and its related statistics 2 1 Contingency Table The calculation of Kappa is based upon the so called contingency table sometimes also referred to as confusion matrix Figure 1 gives the generic form of a contingency table The table details how the distribution of categories in map A relates to that of map B The cells contain a value which is the fraction of the cells in the map which is taken in map A by the category specified in the matrix row and in map B by the category specified in the matrix column For example a value of 0 25 for p 2 would indicate that 25 percent of the mapped area is of category 1 in map A and category 2 in map B The last row and column give the column and row totals Each row total represents the total fraction of cells of the related category in map A Similarly each column total represents the total fraction of cells of the related category in map B All fractions together makes up the whole map therefore the total sum equals 1 D co e O Figure 1 The contingency table in its generic form Monserud amp Leemans 1992 On the basis of the contingency table many stat
114. to convert the input raster land use maps into grouped polygon layers using unique polygon mapping By performing the local matching on a polygon by polygon basis the problems of a pixel by pixel comparison are avoided The creation of the unique polygons maps first involves the use of a grouping algorithm to determine the contiguous groupings of identically valued pixels in a raster map and assign them unique integer identifiers The derived groups or polygons are comprised of pixels that have the same attribute value and contact each other in any of the eight possible directions N S E W NE NW SE or SW Eastman 1992 In unique polygons mapping the first grouping map is overlaid with the second to create an overlay image and a relational attribute table The overlay process generates a series of relational polygons from the intersection of both grouping maps Bonham Carter 1994 A unique polygons map is illustrated in figure 1 which shows the overlay of map one and map two producing the unique polygons map and table Each polygon on the map is assigned a unique identifier so that the table has the same number of rows as there are polygons from the overlay process A unique polygons table is ideally suited to model land use change or map similarities because each unique polygon in the table represents the degree of containment and intersec tion of the polygons on map one in the polygons on map two The degree of areal containment for each pol
115. to the maps to compare The log file itself does not contain any spatial data instead it points the MCK to the maps It also contains references to the legends directory and the palette directory Those are the directories where the MCK will look for legend files and palette files and will place them when they are generated A log file organises maps according to themes Maps within a theme are displayed according to the same legend and may be compared with each other Besides the maps belonging to the different themes there is one map with a special task this is the region map This map is used to designate which cells inside the maps lie inside comparison area If no region map is referred to in he log file then all cells in the maps are inside the comparison area All maps in a log file including those belonging to different themes must be of the same size contain the same number of rows and columns The MAP COMPARISON KIT will display maps of different sizes and allow you to adjust the legends and palettes but no comparisons will run when the sizes of the 1 Map the 2 Map and the region map do not coincide The log file contains the following information e The legend directory The palette directory The region map The name of each theme The maps contained in each theme 29 You can still manually edit a log file using an ASCII editor such as Windows Notepad but will need to use the exact structure of a log file as depic
116. use differences areal intersections and complements are similar High Land use differences very unlikely areal agreement is high Very high Land uses are identical areal agreement close to perfect Figure 3 Membership functions for areal interesection linguistic values 84 C Power et al plateau range of either sigmoidal curve a person intuitively believes that the Boolean area measurement represents the actual degree of agreement between the maps A generalized bell membership function is a symmetrical closed curve consisting of two transitional membership slopes connected by a total membership plateau At the two crossover points membership grades rise monotonically towards one plateau while they fall from another The generalized bell functions low medium and high in figure 3 represent instances where the user believes that the Boolean areal informa tion does not accurately describe the local matching between two maps A generalized bell function is appropriate for these situations because its two transitional slopes enable it to determine if a Boolean areal ratio value underestimates or overestimates the actual local agreement By shifting the emphasis of gradual membership to the boundaries of the curves a Boolean ratio value is fuzzified if it falls beyond the lower or upper boundary of the total membership plateau Note that the membership functions in figure 3 overlap The degree of overlap is subjectively estimated to handle
117. window The boundaries drawn are those defined in the Region map See also section 5 4 4 While the function is selected the menu option is preceded with a tick mark 25 Grid Command Grid line options Distance between lines 30 cells Line width i piselg Row offset cells Column offset cells Use the Grid command to draw a major grid on top of the maps When Grid is selected the Grid options dialogue will open and you are requested to switch on or off the Show grid lines check box Next you have to enter the size of the grid expressed in number of cells Finally you can offset the origin of the grid by a certain amount of cells in order to coincide with another reference system This grid is also called the major grid to distinguish it from the minor grid which is the set by the resolution of the map Font Command Use the Font command to change the font font style and size of the character set used to print the legends of maps The Toolbar Command Use the Toolbar command to view or hide the Toolbar in the application window While the function is selected the menu option is preceded with a mark The Statusbar Command Use the Statusbar command to view or hide the Status bar in the application window While the function is selected the menu option is preceded with a tick mark 4 4 Options menu Options Use the Options menu to operate the map comparison algo
118. ws in first instance the fuzzy vector of cell A is compared to the category vector of cell B according to fuzzy set theory Next the category vector of cell A 1s compared to the fuzzy vector of cell B Finally the lower of the two comparison results establishes the similarity By applying the comparison cell by cell for the whole area a similarity map is generated In this similarity map each cell has a value between 0 for total disagreement and 1 for identical cells Figure 3 shows six situations that clarify this point it should be noted that the exact value for the intermediate similarities between total disagreement and identical depend on the membership function that is applied The similarity values in Figure3 are based upon a membership function of exponential decay with a halving distance of V2 cells 3 4 Aggregate map results to obtain overall similarity measure Kfuzzy It is possible to aggregate the similarity map that results from the Fuzzy two way comparison to an overall value of map similarity For instance by integrating the similarity values over the whole map Subsequent division by the total area yields a result between 1 for identical maps and 0 for total disagreement The outcome of the fuzzy comparison depends partly on the number of categories present and also on the numerical distribution of cells over those categories In order to make the results of maps with different numerical distribution better comparable a St
119. y amp Remote Sensing 55 189 200 XUZHU W DE BaAETs B and KERRE E 1995 A comparative study of similarity measures Fuzzy Sets and Systems 73 259 268 ZADEH L A 1965 Fuzzy sets Information and Control 8 338 353 ZWICK R CARLSTEIN E and Bupescu D V 1987 Measures of similarity among fuzzy concepts a comparative analysis International Journal of Approximate Reasoning 1 221 242
120. y comparison of maps decay with a halving distance of two cells and a neighbourhood with a four cell radius The direct cell by cell method consists of the pair wise comparison of the categories in each cell of the two maps cells where the maps are identical in both maps are in white cells where the categories differ are in black In the fuzzy comparison map lighter cells are more similar than darker cells The comparison map that results from the procedure contains values between 0 and 1 This can be more detailed than required Based on the objective of the map comparison it can be worthwhile to include a classifying step For instance it is possible to distinguish between total agreement medium similarity and low similarity Figure 8 gives the map resulting from classification with the use of a threshold level at 0 65 The areas containing new introductions e g the added linear element in the upper left corner or major shifts e g the shifts of two larger oval shapes are distinguished from the areas of minor shifts e g the other linear elements and fluctuations within patterns e g the pattern of coloured cells at the lower right side of the map Kryzzy 18 calculated to be 0 49 This means that the maps are significantly more similar than would be expected solely from the number of cells of each category because that level of similarity has the K value of 0 The maps are however also clearly distinct because highly similar m
121. ygon in the attribute table is used to measure the local matching between polygons on the land use maps The calculation of the areal polygon containment values depends on map one being a template or reference map of the land use characteristics of a study area and map two a predicted land use layer or an actual land use map at a later date Note that containment applies to both land use agreement and disagreement In the local Fuzzy regional comparison of land use maps 81 M ap 1 M ap 2 Unique Polygons Attribute Table Unique ID Figure 1 Structure of a unique polygon map and attribute table matching scheme the calculated areal intersection ratio will be the local agreements between polygons while the areal complement ratio will represent land use disagreements The areal intersection ratio is computed by identifying the rows in the unique polygons table with identical land uses for a specific template polygon summing the unique areas for these rows and dividing the summed agreement area by the total area for the polygon on the template An areal complement value is computed as one minus the areal intersection ratio The areal intersection and complement ratios are only computed for the unique polygons on map one since it is the template for the matching process The calculated intersections and complements ratios are Boolean values that are computed on the assumption that the unique polygon maps are error free and that real world

MAP COMPARISON KIT

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