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


that form a suitable measure quantifying the approximation effect. This function, in turn, can be used to construct a membership function of the term A describing an acceptable linearization error. The membership function of A at x characterizes an extent to which the linearization of the function is acceptable. Assuming that the expression

is finite, the membership function of A is defined as

In general, as a more flexible model of A we can admit any monotonically increasing function

such that φ(0) =0 and φ(1) = 1, and

3.5.5. Membership estimation as a problem of parametric optimization

In a nutshell, this method is not aimed at constructing a certain membership function from scratch. The intent here is to carry out a standard curve fitting to some experimental data consisting of ordered pairs (element, membership value), denoted by (xk, M(xk)). The form of the parameterized membership function is given in advance. It should reflect the nature of the concept to be described. Several examples of commonly used membership functions have been already discussed. For example, the notion of a low temperature can be described by a Γ membership function The parametric membership estimation procedure is as follows.

Let us assume a parameterized membership function A(x;p) where x ∈ X, and p is a vector of its parameters in the appropriate parameter space P. For a given pair of data (xk,M(xk)), k = 1,…, N, the vector of parameters p of the assumed membership function A(x;p) needs to be determined. A commonly used procedure exploits the Mean Squared Errors (MSE) as the estimation criterion. Therefore the problem reads

which actually is a nonlinear optimization task.

In the estimation of the membership functions, one can look at some additional constraints or intuitive hints. To maintain a smooth transition between membership values, we can assume that the rate of change of a membership value is proportional to the membership value of A(x) as well as its complement, 1 - A(x). This produces a differential equation of the first order

where “k” is a positive proportion factor.

In what follows, we illustrate the problem of membership estimation alluding to some experimental data described by McCloskey and Glucksberg (1978). The experiment deals with a number of objects which are classified as belonging to a category of fruits. The membership grades are determined for each object individually. The results are normalized to the unit interval. The results listed below are also plotted in Fig. 3.3 - note a smooth transition between elements fully belonging to the class and those excluded from it.

1 apple 1.000000
2 banana 0.965726
3 pineapple 0.907258
4 strawberry 0.903226
5 cantaloupe 0.831653
6 mango 0.819556
7 watermelon 0.794355
8 papaya 0.770161
9 fig 0.747984
10 cranberry 0.743952
11 raisin 0.730847
12 pomegranate 0.710685
13 coconut 0.688508
14 avocado 0.562500
15 orange juice 0.562500
16 pumpkin 0.546371
17 tomato 0.521169
18 olive 0.407258
19 acorn 0.382056
20 cucumber 0.344758
21 eggplant 0.340726
22 squash 0.327621
23 sweet potato 0.327621
24 beet 0.319556
25 sunflower seed 0.302419
26 peanut 0.294355
27 carrot 0.290323
28 onion 0.290323
29 corn 0.277218
30 chicken 0.104839


Figure 3.2  Membership values of fuzzy set of fruits

3.6. Fuzzy relations

In comparison to fuzzy sets that are defined in a single universe of discourse, fuzzy relations are defined in Cartesian product of some universes of discourse. For instance, a fuzzy relation R in X × Y is defined via a membership function

The difference between fuzzy relations and relations is illustrated in Fig. 3.3.


Figure 3.3  Two-dimensional relation (i) and fuzzy relation (ii)

While formally equivalent with multidimensional fuzzy sets (when X is replaced by the corresponding Cartesian product and one is willing to accept this notation), fuzzy relations stand on their own, forming a separate area of studies.

Likewise, fuzzy sets, fuzzy relations are very common and we can easily enumerate a number of examples:

  We talk about similar elements (giving rise to similarity relations). Then R(x, y) denotes a degree of similarity between x and y. Obviously, R(x, x) =1.
  All two dimensional images are examples of fuzzy relations. The membership value R(x, y) in [0, 1] denotes a level of brightness of (x, y) pixel. In this setting the previously studied operations of contrast intensification, fuzzification, etc. assume a highly convincing interpretation.
  Fuzzy figures are examples of fuzzy relations. The disk equation

induces a {0, 1} relation

The fuzzy disk is equivalent to the fuzzy relation

Each fuzzy relation in X ×Y can be expressed via its two projections

  projection on X

  projection on Y

There are two interesting interpretations of the projection operation, Fig. 3.4.


Figure 3.4  Two interpretations of a projection operation

  the projection ProjxR is a shadow of R. A light source is situated behind the fuzzy relation that, itself, is treated as a semi-transparent object that produces a shadow of this object
  one can consider the fuzzy relation as a sort of a hilly terrain. Then the projection expresses a level of difficulty one is faced with to traverse a certain path.

Finally, we would like to emphasize that a reconstruction of R out of its projections in the form of the Cartesian product of these does not result in the original fuzzy relation but rather its upper estimate,


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.