EarthWeb   
HomeAccount InfoLoginSearchMy ITKnowledgeFAQSitemapContact Us
     

   
  All ITKnowledge
  Source Code

  Search Tips
  Advanced Search
   
  

  

[an error occurred while processing this directive]
Previous Table of Contents Next


3.15. Shadowed sets

It is worth underlining that a purely numeric format of representing fuzzy sets has raised some concerns in the past and, in sequel, has triggered a search for some other models that are less numeric and precise. There have been some concerns about purely numeric processing within fuzzy sets that does not leave any room for symbolic computations. As already mentioned, this trend of questioning excessive precision of fuzzy sets has been motivated by the conceptual shortcoming associated with precise numeric values of membership used to describe vague concepts. The justification of this concern is also on an experimental side. Along this line let us recall several enhancements such as fuzzy sets of type-2 (Mizumoto, 1976), interval valued fuzzy sets (Sambuc, 1975), and probabilistic sets (Hirota, 1982), in particular. The underlying premise is that the grades of membership themselves are rather fuzzy sets, sets, or truncated random variables all being defined in the unit interval. In studies on probabilistic sets (even though they originate on a somewhat different conceptual and computational ground), it has been found that most of uncertainty in the determination of the membership values is associated with those grades situated around 0.5. This finding is quite appealing. In contrast, we are usually far more confident about assigning values close to 1 (thus counting the elements in) or 0 (therefore making the corresponding element excluded from the concept). On the other hand, the membership values (such as those around 0.5) always spark some hesitation and are always more difficult to place on a simple numeric scale. This observation forms a cornerstone of the construct to be developed. Consider a fuzzy set A. We elevate some membership values (usually those that are high enough) and reduce those that are viewed as substantially low. The elevation and reduction mechanisms are quite a radical as we go for 1 and 0, respectively. In other words, we can say that by doing that we eliminate (disambiguate) the original concept described by the fuzzy set. As this reduces vagueness we make some extra provisions for maintaining the overall level of vagueness constant by allowing some other regions of the universe of discourse associated with intermediate membership values that are defined in a more relaxed way or, simply, to let them be totally undefined. To do that, rather than attaching there a single specific membership value, we assign a unit interval that could be regarded as a nonnumeric model of membership grade. Fig. 3.10 summarizes the proposed construct - the shadowed set (Pedrycz, 1997) is induced from the fuzzy set by accepting a specific threshold level.


Figure 3.10  Fuzzy set and induced shadowed set

Observe that this development produces an effect of vagueness allocation as the regions of vagueness (unit intervals) are assigned to some regions of X rather than to the entire space as encountered in fuzzy sets. In essence, we have transformed the fuzzy set to a set with some clearly marked vagueness zones or, put it more descriptively, shadows. Note that the result is a structure that maps X to 0, 1, and [0, 1]. We will call this concept a shadowed set

The elements of X for which attains 1 constitute its core while the elements where (x) = [0, 1] form a shadow of this construct. One can envision some particular cases such as a shadowed set without any core (only shadow available) and shadowed sets with nonexistent shadows.

To proceed with more computational issues, we address an issue of balancing the overall level of vagueness residing within the concept. As some of the regions come with elevated or reduced membership values (1 and 0), this process should happen at an expense of increased uncertainty in the intermediate membership values, namely an introduction of unit intervals to be distributed across some ranges of X. As shown in Fig. 3.11, we study the areas below the membership function and these need to be balanced by selecting a suitable threshold α meaning that the following relationship holds

i.e.,

In other words, the threshold α ∈ [0, 1/2) should lead to V(α) = 0.


Figure 3.11  Determination of a shadowed set

Shadowed sets exhibit some interesting conceptual links with the existing concepts, especially interval valued sets and rough sets. With respect to the first class, shadowed sets are somewhat subsumed by them. The important operational difference lies in the way in which these concepts have been formed. Interval-valued fuzzy sets are developed independently from fuzzy sets and, by no means are implied by them, whereas shadowed sets, as shown above, can be directly implied (induced) by fuzzy sets. Conceptually, shadowed sets are close to rough sets even though the mathematical foundations of these latter are very different. In rough sets we distinguish between three regions:

  the regions whose elements are fully accepted (membership value equals 1) and belonging to the concept under discussion;
  the regions whose elements definitely do not belong to the concept;
  the regions where membership grade is doubtful - these come in the form of the shadows of the introduced shadowed sets.

These regions originate from the predefined indiscernibility relation. In this sense shadowed sets narrow down a conceptual and an algorithmic gap between fuzzy sets and rough sets highlighting how these could be directly related.

Two points are worth underlining in the setting established so far:

  the proposed concept attempts to capture vagueness in a nonnumeric fashion - we do not commit ourselves to any specific (and precise) membership values over the specific regions of the universe of discourse.
  the factor of vagueness becomes localized in the form of shadows as opposed to the situation existing with fuzzy sets where it is spread across the entire universe of discourse.

For discrete universes of discourse when we are dealing with a collection of membership values rather than continuous functions, the previous index (V) involves several sums and comes in the form

One can think of a certain algorithmic generalization of the shadowed set by admitting two separate thresholds that is α and β, α < β. This modifies V making it a function of two arguments

This extension, however, does not change the generic underlying idea and will not be pursued here.


Previous Table of Contents Next

Copyright © CRC Press LLC

HomeAccount InfoSubscribeLoginSearchMy ITKnowledgeFAQSitemapContact Us
Products |  Contact Us |  About Us |  Privacy  |  Ad Info  |  Home

Use of this site is subject to certain Terms & Conditions, Copyright © 1996-2000 EarthWeb Inc. All rights reserved. Reproduction in whole or in part in any form or medium without express written permission of EarthWeb is prohibited. Read EarthWeb's privacy statement.