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The capabilities of modern technology are rapidly increasing, spurred on to a large extent by the tremendous advances in communications and computing. Automated vehicles and global wireless connections are some examples of these advances. In order to take advantage of such enhanced capabilities, our need to model and manipulate our knowledge of the geophysical world, using compatible representations, is also rapidly increasing. In response to this one fundamental issue of great concern in modern geographical research is how to most effectively capture the physical world around us in systems like geographical information systems (GIS). Making this task even more challenging is the fact that uncertainty plays a pervasive role in the representation, analysis and use of geospatial information. The types of uncertainty that appear in geospatial information systems are not the just simple randomness of observation, as in weather data, but are manifested in many other forms including imprecision, incompleteness and granularization. Describing the uncertainty of the boundaries of deserts and mountains clearly require different tools than those provided by probability theory. The multiplicity of modalities of uncertainty appearing in GIS requires a variety of formalisms to model these uncertainties. In light of this it is natural that fuzzy set theory has become a topic of intensive interest in many areas of geographical research and applications This volume, Fuzzy Modeling with Spatial Information for Geographic Problems, provides many stimulating examples of advances in geographical research based on approaches using fuzzy sets and related technologies.



Reasoning About Regions, Relations, and Fields

1.. Fuzzy Reasoning about Geographic Regions

Reasoning about geographic regions, like forests, lakes, cities, etc., often involves uncertainty and imprecision. For example, when we talk about a region like the city of Auckland, we usually do not know exactly the boundaries of that region. Nevertheless, we are able to reason about such a region. Or if we hear on the radio that a cold front is moving in from Antarctica, we can estimate when it will reach New Zealand, although we might not be able to determine with certainty the exact relation between the area covered by the cold front and the one that is referred to as New Zealand.
Recently, the RCC theory has gained a particular interest in the AI research community as formalism to reason about regions. This first-order theory is based on a primitive relation, called connectedness, and uses eight topological relations, defined on the basis of connectedness, to provide a framework to reason about regions. Lehmann and Cohn have introduced an extension to the RCC theory, which deals with imprecision in spatial representations. Our work carries on from there by applying fuzzy sets to the RCC theory and introducing a uniform framework to reason about geographic regions under uncertainty and imprecision.
Hans Werner Guesgen

2.. Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects

The importance of topological and directional relationships between spatial objects has been stressed in different fields, notably in Geographic Information Systems (GIS). In an earlier work, we introduced the notion of the F-histogram, a generic quantitative representation of the relative position between two 2D objects, and showed that it can be of great use in understanding the spatial organization of regions in images. Here, we illustrate that the F-histogram constitutes a valuable tool for extracting directional and topological relationship information. The considered objects are not necessarily convex and their geometry is not approximated through, e.g., Minimum Bounding Rectangles (MBRs). The F-histograms introduced in this chapter are coupled with Allen’s temporal relationships based on fuzzy set theory. Allen’s relationships are commonly extended into the spatial domain for GIS purposes, and fuzzy set theoretic approaches are widely used to handle imprecision and achieve robustness in spatial analysis. For any direction in the plane, the F-histograms define a fuzzy 13-partition of the set of all object pairs, and each class of the partition corresponds to an Allen relation. Lots of directional and topological relationship information as well as different levels of refinements can be easily obtained from this approach, in a computationally tractable way.
Pascal Matsakis, Dennis Nikitenko

3.. Field Based Methods for the Modeling of Fuzzy Spatial Data

In this chapter, two different field based techniques for the modeling of fuzzy information spread over a geographic region, are presented and are compared regarding their applicability. The first one is a vector-mode approach, using triangulated irregular networks (or TINs), the second one is a raster (bitmapmode) approach. Appropriate aggregation operators are defined in both approaches and illustrated by means of examples. The feasibility of the implementation of the operators (by approximation whenever required) is studied. Attention has been paid to the applicability, advantages and disadvantages of both methods in flexible querying.
Jörg Verstraete, Guy De Tré, Rita De Caluwe, Axel Hallez

4.. Modeling Localities with Fuzzy Sets and GIS

Without Abstract
Sungsoon Hwang, Jean-Claude Thill

Fuzzy Classification

5.. Mining Weather Data Using Fuzzy Cluster Analysis

The need to analyze the vast quantities of weather data collected has led to the development of new data mining tools and techniques. Mining this data can produce new insights into weather, climatological and environmental trends that have both scientific and practical significance. This chapter discusses the challenges posed by weather databases and examines the use of fuzzy clustering for analyzing such data. It proposes the extension of the fuzzy K-Means clustering algorithm to account for the spatio-temporal nature of weather data. It introduces an unsupervised fuzzy clustering algorithm, based on the fuzzy KMeans and defines a cluster validity index which is used to determine an optimal number of clusters. These techniques are validated on weather data in the South Central US, and global climate data (sea level pressure). It is seen that the algorithm is able to identify and preserve interesting phenomena in the weather data.
Zhijian Liu, Roy George

6.. Modelling the Fuzzy Spatial Extent of Geographical Entities

In several situations the spatial extent of geographical entities is uncertain or fuzzy. In such cases the entities may be represented using fuzzy sets through the construction of the herein called “fuzzy geographical entities”. Four sources of fuzziness are identified in the process of constructing geographical entities characterized by predefined attributes through the classification of a tessellation. For each, a method to compute the membership grades to the fuzzy geographical entity is proposed, based on the appropriate semantic interpretation of the grades of membership. The interpretations used are the likelihood view of membership grades, the random set view and the similarity view. A practical example is presented for each case.
Cidália Costa Fonte, Weldon A. Lodwick

7.. Multi-Dimensional Interpolations with Fuzzy Sets

Geographic phenomena are continuous and dynamic but are often represented with data that are static, discrete and crisp. Interpolation is a technique that uses discrete sample data to generate a continuous spatial representation of geographic phenomena. Further, fuzzy set theory represents one of the avenues to overcome the problems of static and crisp data representations. This chapter explores the benefits of integrating fuzzy sets theory and spatio-temporal interpolation techniques within geographical information systems (GIS) to address the multidimensionality of geographic phenomena. The fundamental theory of spatial interpolation using geographic sample data, together with an assessment of the inexactness of such data is presented. Moreover, fuzzy interpolation methods that use the concepts of fuzzy data, fuzzy numbers and fuzzy arithmetic to generate fuzzy surfaces are elaborated. Four case studies that use GIS-based fuzzy set reasoning to build multidimensional spatial or spatio-temporal interpolation methods are discussed.
Suzana Dragićević

8.. Talking Space — A Social & Fuzzy Logical GIS Perspective On Modelling Spatial Dynamics

Talking Space is drafted as a GIS-based communication platform to map spatial knowledge, which contains inherent uncertainty. This uncertainty is argued to be due to the semantics of categorization using linguistic symbols as applied in a communication process, which is argued to create and shape space and spatial phenomena. Inherent uncertainty is nothing to be eliminated but is an indispensable part in communicating (spatial or non-spatial) knowledge and therefore needs to be talked about. Space is shaped in a deterministic and objective way — yet, in all probability, this overlooks the perceptions, assessments and interests of many space protagonists. The formation and information of actors in space implies relations among different points of view. Perception and assessment of space is understood inadequately. Conventional planning and GIS do not meet requirements on communicating space. GIS is sometimes even referred to as socially empty space. This emptiness may be filled with our ability to talk about space, to perceive space and talk about perceptions and to visualize what we are talking about. This paper is proposing perspectives on different notions of spatial phenomena and their impact on creating spatial knowledge while limiting ourselves to the logical and techno-logical requirements of GIS. Alternative views on spatial categories and their contribution to communication in space are introduced. Three settings are used to develop our perspectives with respect to a Talking Space. An Introduction (A) focuses on challenges of visualizing relations among individual perspectives in space, which is shaped and constructed by social actors. Social as well as cognitive differences among social actors in geographical space are at the core of a Talking Space Development. In Communicating Spatial Knowledge (B) theoretical foundations are introduced, which are necessary in a Talking Space to draft social perspectives on constructing space. It deems necessary to open up the notion of space to Social Science in enabling all actors in space to comprehend spatial phenomena. Theoretical issues on constructing space are discussed. The third setting describes implications of the perspectives introduced in (A) and (B) to a Talking Space Environment (C). This kind of environment will be discussed as a framework of symbols, models and codes addressing social construction, logical proceedings and visual engagements, respectively. Examples using the notion of a Meeting Point and the modeling of noise are used to support the arguments.
Susanne Kratochwil, Josef Benedikt

9.. A Valuation of the Reliability of a GIS Based on the Fuzzy Logic in a Concrete Case Study

The great difficulty of evaluating the quality of the applications concerning environmental problems, mainly for the heterogeneity of input data and their indeterminateness in the error estimations, is a well known problem. The usage of the Fuzzy Logic can be adequate in the treatment of this kind of information, especially when using approximate linguistic labels to define the input data. We have applied this idea in a previous work and here proposed for the study of a GIS of the PROCIDA island (located near Napoli), realized with technology of the Environmental Systems Research Institute and implemented by means of a software tool called FUZZY-SRA.
Ferdinando Di Martino, Vincenzo Loia, Salvatore Sessa, Michele Giordano

Fuzzy Representations of Landscape Features

10.. Fuzziness and Ambiguity in Multi-Scale Analysis of Landscape Morphometry

Recent research on the identification of landscape morphometric units has recognised that those units have a vague spatial extent which may be modelled by fuzzy sets. To date most such have looked at the landscape at a single resolution although scale dependence is one of the reasons the concepts are vague. The fact is that the allocation of landscape elements to morphometric classes is ambiguous, and in this chapter we exploit the ambiguity of multi-resolution classification as the basis of the morphometric classes as fuzzy sets. We explore this idea with respect to both the mountains around Ben Nevis in Scotland and the dynamic environment of a coastal dunefield. The results in the first example show that the landscape elements identified correspond to landmarks named in a placename database of the area, although many more peaks are found than are named in the available database. In the second case multi-temporal data on a dynamic coastal dunefield is used to show results for fuzzy set and fuzzy logic analysis to identify patterns of change which contrast with more traditional change analysis. Both examples provide new insights over the types of analysis which are currently available in Geographical Information Systems, and the manipulation of scale to parameterise membership of the fuzzy set is a uniquely geographical method in fuzzy set theory.
Peter Fisher, Jo Wood, Tao Cheng

11.. Fuzzy Representation of Special Terrain Features Using a Similarity-based Approach

Fuzzy representation of terrain positions can be useful in environmental modeling process, especially in soil-landscape studies. Existing methods for deriving this representation from a digital elevation model (DEM) are often neither effective nor efficient, especially when dealing with some special terrain positions that have only regional or local meanings. This paper presents a similarity-based method for deriving fuzzy representation of special terrain features. This method has two general steps. The first is to find the typical locations (cases) of a specified terrain position and assign full fuzzy membership to these typical locations. The typical locations can be identified in two ways: they can be located by using a set of simple rules based on the geomorphologic definition of the terrain position; or they can be pinpointed or delineated directly by experts using a GIS visualization tool. With the typical locations identified, the next step is to compute the similarities between these typical locations and other landscape locations, and the derived similarity values are then used to approximate the fuzzy memberships of those locations for being the terrain position. This process is applied to some special terrain features in two study areas: one in Wisconsin and the other in Tennessee. In the Wisconsin study area, this method is used to derive the fuzzy representations of broad ridge, narrow ridge, and headwater. In the Tennessee study area, this method is used to derive the fuzzy membership of being a “knob”. The resultant fuzzy representations are realistic and meaningful and the whole process is computationally efficient, which indicates that this similaritybased (cased-based) method can be an effective and flexible approach to deriving fuzzy representations of terrain features.
Xun Shi, A-Xing Zhu, Rongxun Wang

Decision Making with GIS and Fuzzy Sets

12.. Spatial Decision-Making Using Fuzzy Decision Tables: Theory, Application and Limitations

In this paper the basic principles of decision-making using fuzzy decision tables (FDTs) are explained and illustrated. The main emphasis is on introducing standard notations and definitions. The point of departure is the crisp decision table formalism and its inability to deal with imprecision and vagueness. As a potential solution, elements of the theory of fuzzy sets are used to develop a new modelling technique, known as FDTs. The properties of FDTs are formally described and illustrated.
Frank Witlox, Ben Derudder

13.. Spatial Decision Making Using Fuzzy GIS

Geographic Information Systems (GIS) and spatial databases are inherently suited for fuzziness, because of the uncertainty inherent in the assimilation, storage, and representation of spatial data. One of the most fertile GIS development areas is integrating multiple criteria decision models into GIS querying mechanisms. The classic approach for this integration has been to use Boolean techniques of decision making with crisp representations of spatial objects to produce static maps as query answers. This paper examines a prototype system, FOOSBALL, which integrates both multiple attribute querying and a fuzzy object-oriented GIS. FOOSBALL addresses many of the inherent weaknesses of current systems by implementing: 1) fuzzy set membership as a method for representing the performance of decision alternatives on evaluation criteria, 2) fuzzy methods for both criteria weighting and capturing geographic preferences, and 3) a fuzzy object oriented spatial database for feature storage. This makes it possible to both store and represent query results more precisely. The end result of all of these enhancements is to provide spatial decision makers with more information so that their decisions will be more informed, and thus, more correct.
Ashley Morris, Piotr Jankowski

14.. Spatially Explicit Individual-Based Ecological Modeling with Mobile Fuzzy Agents

Previous theoretical work illustrated how fuzzy spatial relations can be used to control the movement of mobile agents in spatially explicit individualbased ecological models (Robinson 2002). We present a computational framework and methodology for modeling small mammals as mobile agents making decisions during the dispersal process. It is shown how this object-oriented framework can accommodate the uncertainty of geographic information as well as the inherent fuzziness of the decision process. A fuzzy decision making model is presented along with its corresponding crisp equivalent. Using a realistic landscape, simulations are used to explore model behavior relative to fuzzy compensatory and noncompensatory aggregation operators. Simulations are used to compare fuzzy versus crisp model behaviors. Results are used to evaluate relative strengths and weaknesses of each. It is shown that this approach can be used for developing individual-based models to address spatially explicit ecological problems that are dependent on being based in a geographic information systems environment.
Vincent B. Robinson, Phil A. Graniero


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