Fuzzy Spatial Algebra (FUSA): Formal Specification of Fuzzy Spatial Data Types and Operations for Databases and GIS

Spatial database systems and geographical information systems are currently only able to support geographical applications that deal with crisp spatial objects, that is, objects whose extent, shape, and boundary are precisely determined. Examples are land parcels, school districts, and state territories. However, many new, emerging applications are interested in modeling and processing geographic data that are inherently characterized by spatial vagueness or spatial indeterminacy. This requires novel concepts due to the lack of adequate approaches and systems. In this chapter, we focus on an important kind of spatial vagueness called spatial fuzziness. Spatial fuzziness captures the property of many spatial objects in reality that do not have sharp boundaries and interiors or whose boundaries and interiors cannot be precisely determined. We will designate this kind of entities as fuzzy spatial objects. Examples are polluted areas, temperature zones, and lakes. We propose an abstract, formal, and conceptual model of so-called fuzzy spatial data types (that is, a fuzzy spatial algebra) introducing fuzzy points, fuzzy lines, and fuzzy regions in the two-dimensional Euclidean space. This chapter provides a definition of their structure and semantics, which is supposed to serve as a specification of their implementation. Furthermore, we introduce fuzzy spatial set operations like fuzzy union, fuzzy intersection, and fuzzy difference, as well as fuzzy topological predicates as they are useful in fuzzy spatial joins and fuzzy spatial selections. We also sketch implementation strategies for the whole type system and show their integration into databases. An outlook on future research challenges rounds out the chapter.

Spatial Plateau Algebra for implementing fuzzy spatial objects in databases and GIS: Spatial plateau data types and operations

Spatial database systems and Geographical Information Systems (GIS) are currently only able to handle crisp spatial objects, i.e., objects whose extent, shape, and boundary are precisely determined. However, GIS applications are also interested in managing vague or fuzzy spatial objects. Spatial fuzziness captures the inherent property of many spatial objects in reality that do not have sharp boundaries and interiors or whose boundaries and interiors cannot be precisely determined. While topological relationships have been broadly explored for crisp spatial objects, this is not the case for fuzzy spatial objects. In this paper, we propose a novel model to formally define fuzzy topological predicates for simple and complex fuzzy regions. The model encompasses six fuzzy predicates (overlap, disjoint, inside, contains, equal and meet), wherein here we focus on the fuzzy overlap and the fuzzy disjoint predicates only. For their computation we consider two low-level measures, the degree of membership and the degree of coverage, and map them to high-level fuzzy modifiers and linguistic values respectively that are deployed in spatial queries by end-users.

Nowadays Geographical Information Systems (GIS) and spatial database systems are mainly able to handle crisp spatial objects, i.e., objects in space whose locations, extents, shapes, and boundaries are precisely determined. However, geoscientists and advanced GIS applications are increasingly interested in handling spatial objects that are characterized by the feature of spatial fuzziness and do not have sharp but blurred boundaries and/or interiors; hence, we call them fuzzy spatial objects. Examples are air polluted areas and temperature zones. In the same way as fuzzy spatial objects are blurred, the topological relationships (e.g., overlap, inside) between them are blurred too. In this conceptual paper, we propose a novel model to formally define fuzzy topological relationships for simple and complex fuzzy regions. Such a fuzzy relationship computes a membership degree between 0 and 1 that indicates to which extent this relationship holds. It is mapped to a high-level fuzzy modifier and transformed into a Boolean predicate that can be embedded into spatial queries.

Many geographical applications have to deal with spatial objects that reveal an intrinsically vague or fuzzy nature. A spatial object is fuzzy if locations exist that cannot be assigned completely to the object or to its complement. Spatial database systems and Geographical Information Systems are currently unable to cope with this kind of data. Based on an available abstract data model of fuzzy spatial data types for fuzzy points, fuzzy lines, and fuzzy regions, this paper proposes the Spatial Plateau Algebra that provides spatial plateau data types as an executable type system for fuzzy spatial data types. Each spatial plateau object consists of a finite number of crisp counterparts that are all adjacent or disjoint to each other, are associated with different membership values, and hence form different plateaus. The formal framework and the implementation are based on well known, exact models and available implementations of crisp spatial data types. Spatial plateau set and metric operations as well as spatial plateau topological predicates on spatial plateau objects are expressed as a combination of geometric methods on the underlying crisp spatial objects. This paper offers a conceptually clean specification for implementing a database extension for fuzzy spatial objects and their operations and predicates. Further, we demonstrate the embedding of these new data types as attribute data types in a database schema as well as the incorporation of fuzzy spatial operations and predicates into a database query language.

Spatial libraries are core components in many geographic information systems, spatial database systems, and spatial data science projects. These libraries provide the implementation of spatial type systems that include spatial data types and a large diversity of geometric operations. Their focus relies on handling crisp spatial objects, which are characterized by an exact location and a precisely defined extent, shape, and boundary in space. However, there is an increasing interest in analyzing spatial phenomena characterized by fuzzy spatial objects, which have inexact locations, vague boundaries, and/or blurred interiors. Unfortunately, available spatial libraries do not provide support for fuzzy spatial objects. In this article, we describe the R package named fsr, which is based on the Spatial Plateau Algebra and is publicly available at https://cran.r‐project.org/package=fsr. Our tool provides methods for building fuzzy spatial objects as spatial plateau objects and conducting exploratory spatial data analysis by using fuzzy spatial operations.

Decision support based on spatial (and not only alphanumerical) data has received increasing interest in geographical applications, such as geoscience, agriculture, and economics applications, and has led to Spatial Decision Support Systems (SDSS). SDSS use spatial database systems and Geographical Information Systems as their data management and analysis components in order to get and handle the needed spatial data and perform recommendations, estimations, or predictions. For instance, farmers want to know what the best areas of their farmland are to grow a specific crop. In most cases, the extent and the properties of the spatial phenomena of interest are vague and imprecise. They can be adequately represented by fuzzy spatial objects (e.g., fuzzy points, fuzzy lines, fuzzy regions). In this paper, we formally propose a model named Fuzzy Inference on Fuzzy Spatial Objects (FIFUS), which infers recommendations, estimations, and predictions based on fuzzy rules and knowledge of domain specialists. It incorporates fuzzy spatial objects into the components of the existing fuzzy inference methods in order to take into account the spatial imprecision found in the real world. As a main advantage, FIFUS is a general-purpose model and can thus be applied in many geoscience applications.

Topological relations are one of the most important aspects in GIS modeling. The topological relations between crisp spatial objects have been well identified. However, the topological relations between fuzzy spatial objects need more investigation. This paper discusses different definitions in fuzzy boundary and their differences and relations. One of the definitions is then selected as the fuzzy boundary for GIS. In order to identify the topological relations between fuzzy spatial objects, three intersection matrices, namely 3*3-intersection matrix, 4*4-intersection matrix and 5*5-intersection matrix, are proposed and formalized in a fuzzy topological space. These matrices are all applicable for identification of topological relations between fuzzy spatial objects. 152 topological relations between two simple fuzzy regions are identified based on the 4×4-intersection approach in R 2 .

Spatial database systems and geographical information systems are currently only able to support geographical applications that deal with only crisp spatial objects, that is, objects whose extent, shape, and boundary are precisely determined. Examples are land parcels, school districts, and state territories. However, many new, emerging applications are interested in modeling and processing geographic data that are inherently characterized by spatial vagueness or spatial indeterminacy. Examples are air polluted areas, temperature zones, and lakes. These applications require novel concepts due to the lack of adequate approaches and systems. In this chapter, the authors show how soft computing techniques can provide a solution to this problem. They give an overview of two type systems or algebras that can be integrated into database systems and utilized for the modeling and handling of spatial vagueness. The first type system, called Vague Spatial Algebra (VASA), is based on well known, general, and exact models of crisp spatial data types and introduces vague points, vague lines, and vague regions. This enables an exact definition of the vague spatial data model since we can build it upon an already existing theory of spatial data types. The second type system, called Fuzzy Spatial Algebra (FUSA), leverages fuzzy set theory and fuzzy topology and introduces novel fuzzy spatial data types for fuzzy points, fuzzy lines, and fuzzy regions. This enables an even more fine-grained modeling of spatial objects that do not have sharp boundaries and interiors or whose boundaries and interiors cannot be precisely determined. This chapter provides a formal definition of the structure and semantics of both type systems. Further, the authors introduce spatial set operations for both algebras and obtain vague and fuzzy versions of geometric intersection, union, and difference. Finally, they describe how these data types can be embedded into extensible databases and show some example queries.

Geographical Information Systems and spatial database systems are well able to handle crisp spatial objects, i.e., objects in space whose location, extent, shape, and boundary are precisely known. However, this does not hold for fuzzy spatial objects characterized by vague boundaries and/or interiors. In the same way as fuzzy spatial objects are vague, the topological relationships (e.g., overlap, inside) between them are vague too. In this conceptual paper, we propose a novel model to formally define fuzzy topological relationships for fuzzy regions. For their definition we consider the numeric measure of coverage degree and map it to linguistic terms that can be embedded into spatial queries.

Quantitative fuzzy topological relations of spatial objects by induced fuzzy topology

For a long time topological predicates between spatial objects have been a main area of research on spatiald ata handling, reasoning, and query languages. But these predicates still suffer from two main restrictions: first, they are only applicable to simplified abstractions of spatial objects like single points, continuous lines, and simple regions, as they occur in systems like current geographical information systems and spatial database systems. Since these abstractions are usually not suficient to cope with the complexity of geographic reality, their generalization is needed which especially has influence on the nature and definition of their topological relationships. This paper gives a formal definition of complex crisp regions, which may consist of several components and which may have holes, and it especially shows how topologicalpre dicates can be defined on them. Second, topological predicates so far only operate on crisp but not on fuzzy spatial objects which occur frequently in geographical reality. Based on complex crisp regions, this paper gives a definition of their fuzzy counterparts and shows how topological predicates can be defined on them.

Topological relations between fuzzy regions in a fuzzy topological space

Fuzzy set theory has found increasing interest in the geosciences, geographic information systems, and spatial database systems to represent objects in the two-dimensional space that are afflicted with spatial fuzziness. From a system perspective, fuzzy spatial data types for fuzzy points, fuzzy lines, and fuzzy regions have been introduced, e.g., by the authors’ formal Fuzzy Spatial Algebra (FUSA). The authors’ Spatial Plateau Algebra (SPA) provides an implementation of FUSA by means of spatial plateau data types for plateau point, plateau line, and plateau region objects. It is based on well known non-fuzzy, crisp spatial data types. In this paper, we deal with the issue of constructing fuzzy region objects as plateau region objects from real point sets by leveraging domain expert knowledge and by assuming that each point is assigned a numerical value of a given application context. For this, we propose a general two-stage data extraction method. The first stage deploys a fuzzification policy to assign membership degrees to the points in the set. Examples of such policies are the execution of fuzzy clustering algorithms or the usage of fuzzy sets. The second stage uses a construction policy to build plateau regions by using the membership degrees generated in the first stage as input. Examples of such construction policies are the computation of fuzzy Voronoi diagrams or fuzzy Delaunay triangulations, or the calculation of fuzzy convex hulls.

In many geographical applications there is a need to model spatial phenomena not simply by sharply bounded objects but rather through vague concepts due to indeterminate boundaries. Spatial database systems and geographical information systems are currently not able to deal with this kind of data. In order to support these applications, for an important kind of vagueness called fuzziness, we propose an abstract, conceptual model of so-called fuzzy spatial data types (i.e., a fuzzy spatial algebra) introducing fuzzy points, fuzzy lines, and fuzzy regions. This paper focuses on defining their structure and semantics. The formal framework is based on fuzzy set theory and fuzzy topology.