Jörg Verstraete

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.

Fuzzy Regions: Theory and Applications

Traditionally, information in geographic information systems (GIS) is represented as crisp information. While for many applications, this is a good enough approximation of reality, some models would benefit from having the inherent imprecision or uncertainty incorporated in the model. In literature, several ideas and concepts to improve on the crisp models have been considered. In the past, we have presented different models to represent and to work with the concept of regions, defined using fuzzy set theory in GIS systems. For such fuzzy regions, a number of approaches already have been described in detail. In this paper, we will elaborate on a fuzzy set approach and practical implementations of the concept. Apart from the concept, two developed techniques (one based on triangulated networks, one based on bitmap models) are presented along with some of the operators. An overview of application fields is provided to illustrate where and how the techniques can be used.

Solving the map overlay problem with a fuzzy approach

The map overlay problem occurs when mismatched gridded data need to be combined, the problem consists of determining which portion of grid cells in one grid relates to the partly overlapping cells of the target grid. This problem contains inherent uncertainty, but it is an important and necessary first step in analysing and combining data; any improvement in achieving a more accurate relation between the grids will positively impact the subsequent analysis and conclusions. Here, a novel approach using techniques from fuzzy control and artificial intelligence is presented to provide a new methodology. The method uses a fuzzy inference system to decide how data represented in one grid can be distributed over another grid using any additionally available knowledge, thus mimicking the higher reasoning that we as humans would use to consider the problem.

Conjunctive Aggregation of Extended Possibilistic Truth Values and Flexible Database Querying

Extended possibilistic truth values are a flexible means to model the linguistic (un)certainty about the truth value of a proposition. With respect to flexible database querying, they are suited to express the extent to which a database instance satisfies a query condition. Since a query can impose several conditions, aggregation is necessary. In this paper, three definitions of conjunctive aggregation operators for extended possibilistic truth values are presented and compared with each other on the basis of their capability to rank the possible alternatives in the result of a (flexible) database query.

Assigning membership degrees to points of fuzzy boundaries

There has been a growing interest in the field of geographic information systems (GIS) to model vague and uncertain regions for which information about a given property is available (e.g. soil structure). Vague and uncertain mean in general that some knowledge is available about the position and the circumference of such a region, but that the information is mostly imprecise or lacks certainty. Either this lack of precise data is inherent to the property (e.g. the degree of soil pollution), or it is due to physical limitations in making accurate measurements (e.g. temperature). There has been some research in modeling such regions using fuzzy regions. Fuzzy regions are modeled by means of broad boundaries. A broad boundary is defined as the area enclosed between two non-intersecting crisp edges including these edges, one completely located within the other. Inside and on the inner edge lay the points for which the definition of the property is completely met; outside and on the outer edge lay the points for which the definition of the property is not at all met. Points of the broad boundary meet the definition of the property up to a certain extent. Up to now, mainly the applicability of the traditional set operators on fuzzy regions (intersection, inclusion and union) has been addressed in the literature. Furthermore, almost no contributions treat the degree to which the points in the fuzzy region come up to the given definition of a property. We present two methods to solve the latter problem, both from a theoretical as well as from a practical point of view.

Similarity between multi-valued thesaurus attributes: theory and application in multimedia systems

In this paper, the theoretical aspects of calculating the similarity between sets, and its generalizations multisets, fuzzy sets and fuzzy multisets, is presented. Afterwards, this theory is applied to enhance the facilities for accessing a multimedia system, namely when searching for correspondence between multi-valued attributes, which are coupled with a thesaurus. Furthermore, to allow flexibility in this search, thesauri with similarities defined between the thesaurus terms are considered. As a possible application, the DEKKMMA project is introduced, a project about an audio archive of African music.

The Spatial Disaggregation Problem: Simulating Reasoning Using a Fuzzy Inference System

The spatial disaggregation problem is an interesting problem when investigating and processing geographically correlated data; it is a special case of the map overlay problem. The map overlay problem occurs when data that are presented in incompatible grids need to be combined or compared; the spatial disaggregation problem is a special case in which the cells of one grid partition the cells of the other grid. The main reason for spatial disaggregation is to increase the spatial resolution of the data. Traditionally, similar methods as for the map overlay problem are used; fairly straightforward assumptions on the geographical distribution of the data are made, but these assumptions do not match the real-world situation. The approach presented in this contribution uses additionally available data ( proxy data) that provides information on the geographic distribution of the input data. As will be illustrated, having this proxy data still does not provide for a straightforward solution and incorporating it inquires some form of reasoning which will be achieved by means of a dynamically constructed fuzzy inference system.

A quantitative approach to topology for fuzzy regions

There has been lots of research in the field of fuzzy spatial data and the topology of fuzzy spatial objects. In this contribution, an extension to the 9-intersection model is presented, to allow for the relative position of overlapping fuzzy regions to be determined. The topology will be determined by means of a new intersection matrix, and a set of numbers, expressing the similarity between the topology of the given regions and a number of predefined cases. The approach is not merely a conceptual idea, but has been built on our representation model and can as such be immediately applied.

Fuzzy Regions: Adding Subregions and the Impact on Surface and Distance Calculation

In the concept of fuzzy regions we introduced before, a region was considered to be a fuzzy set of points, each having its own membership grade. While this allows the modelling of regions in which points only partly belong to the region, it has the downside that all the points are considered independently, which is too loose a restriction for some situations. The model is not able to support the fact that some points may be linked together. In this contribution, we propose an extension to the model, so that points can be made related to one another. It will permit the user to, for instance, specify points or even (sub)regions within the fuzzy region that are linked together: they all belong to the region to the same extent at the same time. By letting the user specify such subregions, the accuracy of the model can be increased: the model can match the real situation better; while at the same time decreasing the fuzziness: if points are known to be related, there is no need to consider them independently. As an example, the use of such a fuzzy region to represent a lake with a variable water level can be considered: as the water level rises, a set of points will become flooded; it is interesting to represent this set of points as a subset of the region, as these points are somewhat related (the same can be done for different water levels). The impact of this extension to the model on both surface area calculation an distance measurement are considered, and new appropriate definitions are introduced.

Compliance for uncertain inventories via probabilistic/fuzzy comparison of alternatives

A direct comparison among highly uncertain inventories of emissions is inadequate and may lead to paradoxes. This issue is of particular importance in the case of greenhouse gases. This paper reviews the methods for the comparison of uncertain inventories in the context of compliance checking. The problem is treated as a comparison of uncertain alternatives. It provides a categorization and ranking of the inventories which can induce compliance checking conditions. Two groups of techniques to compare uncertain estimates are considered in the paper: probabilistic and fuzzy approaches. They show certain similarities which are revealed and stressed throughout the paper. The group of methods most suitable for the compliance purpose is distinguished. They introduce new conditions for fulfilling compliance, depending on inventory uncertainty. These new conditions considerably change the present approach, where only the reported values of inventories are accounted for.