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

Abstract

Abstract. 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.

Distance calculation between Level-2 fuzzy regions

“…The above definition was extended to allow for grouping of points with the same membership grade ( [5]). For this purpose, the domain was altered from R 2 to ℘(R 2 ); the powerset of R 2 .…”

“…First all points in a fuzzy region are considered independently, making it impossible for a user to specify that some points belong together. A solution to this was presented in [5], where points carrying the same membership can be grouped. This solution however had its limitations and did not solve the second shortcoming of the original model, which is more subtle and concerns interpretations.…”

Using Level-2 Fuzzy Sets to Combine Uncertainty and Imprecision in Fuzzy Regions

“…When a fuzzy region is defined by means of a limited number of subregions, the concept bears resemblance to the concept of plateau regions [3]. The operations distance and surface have been considered in [12]. The extended concept of fuzzy regions is illustrated on figure 2, we refer to [12] for more details on the surface area.…”

“…The operations distance and surface have been considered in [12]. The extended concept of fuzzy regions is illustrated on figure 2, we refer to [12] for more details on the surface area. Simply put, the fuzzy surface area represents every possible surface area; this implies that when points are grouped in a subregion (e.g.…”

“…While at first sight this makes no real difference to the representation of the region, this change in definition impacts its use. The fuzzy surface area [12] of the lake for instance will now be represented more accurately, possibly having less possible values than before, as points at the same altitude are not counted individually but as a group.…”