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


6.4.2. Variable processing resolution - fuzzy receptive fields

By defining the linguistic terms (modelling landmarks) and specifying their distribution along the universe of discourse we can orient (focus) the main learning effort of the network. To clarify this idea, let us refer to Figure 6.1.


Figure 6.1  Fuzzy partition completed through a series of linguistic terms

The partition of the variable through assigns a high level of information granularity to some regions (say Ω1 and Ω2) and sensitizes the learning mechanism accordingly. On the other hand, the data falling under Ω3 are regarded internally (viz. at the level they are perceived by the networks) as equivalent by having the same numeric representation in the unit hypercube.

6.5. Uncertainty representation in neural networks

The factor of uncertainty or imprecision can be quantified by exploiting some uncertainty measures as commonly exploited in the theory of fuzzy sets. The underlying rationale is to equip the internal format of information available to the network by some indicators describing measures as commonly exploited in the theory of fuzzy sets. The underlying rationale is to equip the internal format of information available to the network by some indicators describing how uncertain the given datum is. Considering possibility and necessity measures this quantification is straightforward: once Poss( X, Ak) ≠ Nec(X, Ak) then X is regarded uncertain (the notion of uncertainty is also context-sensitive and depends on Ak) (Dubois and Prade, 1988). For numerical data one always arrives at the equality of the two measures that clearly points at the certainty of X. In general, the higher the gap between the possibility and necessity measures, Poss( X, Ak) = Nec(X, Ak) + δ, the higher the uncertainty level associated with X. The uncertainty gap attains its maximum for δ = 1. One can also consider the compatibility measure instead of the two used above - this provides us with more flexibility and discriminatory power yet becomes computationally demanding.

The possibility and necessity measures reveal interesting relationships between uncertainty conveyed by X when processed in terms of A. Denote by λ the possibility computed with respect to A,

Additionally, μ describes the complement of the necessity measure,

Here we are in position to introduce the notions of conflict and ignorance as they emerge in the representation studied in context of A. By looking at the values of λ and μ, we identify three characteristic cases :

λ + μ = 1: no uncertainty associated with X (or, at least, the uncertainty does not manifest in the setting provided by A)
λ + μ > 1: conflict (X becomes a conflicting piece of evidence as it invokes both A and its complement)
λ + μ < 1: ignorance (X indicates a lack of a sufficient support making difficult to make any decision that is either in favour of A or against it)

Let us quantify the above statements through the relationships

or equivalently

where α and β quantified in [0, 1] are used in the evaluation of the level of conflict or ignorance, respectively. A convenient graphical illustration can be formed in terms of the so-called ignorance-conflict plane, Fig. 6.2.

The higher the values of these indices (α or β), the higher the uncertainty (conflict or ignorance) associated with X. If X is a singleton (or it is perceived as such when processed in the specific context of A), the corresponding element in the plane moves along its diagonal.


Figure 6.2  Ignorance - conflict plane

The way of treating the linguistic term makes a real difference between the architecture enhanced by the uncertainty representation layer, Fig. 6.3, and RBF neural networks. The latter ones do not have any provisions to deal with and quantify uncertainty. The forms of the membership function (RBFs) are very much a secondary issue. In general, one can expect that the fuzzy sets used therein can exhibit a variety of forms (triangular, Gaussian, etc.) while RBFs are usually more homogeneous (e.g., all Gaussian). Furthermore, there are no specific restrictions on the number of RBFs used as well as their distribution within the universe of discourse. For fuzzy sets one usually confines this number to a maximum of 9 terms (more exactly, 7 ± 2); additionally we should make sure that the fuzzy sets are kept distinct and thus retain their semantic identity.


Figure 6.3  Possibility - necessity oriented preprocessing layer

In general, when processing data in the fuzzy set environment, we distinguish two main approaches: a parametric and nonparametric data representation. The possibility - necessity mechanism quantifies uncertainty in a nonparametric way. The complementary parametric approach towards uncertainty quantification concerns a direct representation of a fuzzy datum (Hathaway et al., 1996). This representation depends upon the form of the nonnumeric information one has to deal with. A list of several commonly envisioned scenarios includes a number of interesting cases of membership functions; for details refer to Fig. 6.4.


Figure 6.4  Classes of membership functions

  sets. Each set (interval) is fully described by its bounds (a and b).
  triangular fuzzy sets: These are characterized by a modal value (m) and the two bounds (α and β).
  trapezoidal fuzzy sets: Here we need the bounds α and β as well as m and n.
  Gaussian-like fuzzy sets. It is enough to specify their modal values and spreads.

The parametric characterization suitable for one form of the fuzzy sets may not suitable to describe fuzzy sets coming from some other class. To cope with all types of fuzzy sets, one should go for a direct quantization of the input space. This, however, is not feasible. Assume that we have admitted 100 discrete points to complete a uniform quantization of the universe of discourse (discretization points distributed uniformly across the universe of discourse). Then any fuzzy set, no matter what its membership function looks like, becomes represented as an 100-element vector. This gives rise to 100 nodes in the input layer of the network. In case of even a few input variables, this leads to unacceptable large architectures of neural networks.


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.