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


The minimization of Q gives rise to a well-known gradient-descent optimization method

where k = 1, 2, ..., n and α stands for a learning rate.

The computations of the gradient are standard yet somewhat tedious. First we determine

The inner term unfolds as follows; at this point we specify the t- and s-norm as the product and probabilistic sum (obviously, we can exploit any other t-norms and co-norms; the ensuing formulas need to be modified accordingly),

Now, let us distinguish two cases:

(i)  if i = k the above derivative reads as

(ii)  if i ≠ k the calculations produce

The derivative of the nonlinear function of possibilities requires more attention. We get

The first factor can be determined once the form of the transforming function (w) has been defined. The second one is written down more explicitly as

The derivative assumes values in the {0, 1} set

This binary nature of the derivative may cause some computational problems lowering an efficiency of the overall learning scheme. To alleviate these shortcomings, one can relax the Boolean constraints and proceed with their multivalued version (such as “included in”) that is represented as

where → is any multivalued implication being used to model the relation of inclusion. The derivative returns 1 if the two implications are set to 1. The higher the degree of containment, the higher the truth value of the implication.

If the obtained result is not specified as an entire fuzzy set but rather comes as a single numerical value that is regarded as a result of decoding (commonly referred to as a defuzzification) of D, F(D) in R, the same optimization scheme as studied above applies here, however it is modified by the decoding block. This makes the learning reinforced. Assuming that F is differentiable with respect to d one gets the performance index

Then

For instance, if F assumes the form of a weighted aggregation operator

for xi being the i-th element in the universe of discourse and

then we obtain

The remaining of the computations of are carried out as already discussed.

4.6.3. Approximate reasoning with defaults

Let us remind that the well known reasoning schemes exploiting rules with fuzzy sets (linguistic labels) assume the form

• if A1 then C1
• if A2 then C2
...
• if Ac then Cc

where fuzzy sets Ai and Ci are used to denote linguistic premises (antecedents) and consequents (conclusions), respectively. The notation will identify them as two fuzzy partitions of the space of antecedents and conclusions

Not getting into details, the functional view of the inference scheme can be illustrated as visualized in Fig. 4.33(i).


Figure 4.33  A generic module of approximate reasoning (i) without default fuzzy set (ii) with default fuzzy set

Moreover A and C describe currently available input and the inferred output, respectively. The fuzzy associative memory model yields the well-known formula

Subsequently, the numeric representative of C reads as

(here we have adopted one of the versions of a center of gravity method); yi* describes a modal value of Ci. Interestingly enough, if A = unknown then (assuming that Ais are normal) we get

The result clearly states that if nothing is known about the input, the output should be viewed as an average of the modal values of the fuzzy sets standing in the conclusion part of the rules.

The basic inference module, Fig. 4.33(i), is now augmented by assuming that the default fuzzy set B is defined in the space of antecedents and becomes involved in a certain preprocessing phase of the input as illustrated in Fig. 4.33(ii). Essentially, what occurs there can be put in the form

This formula underlines the fact that A becomes placed in the context of B before being used for any reasoning. The role of the default reference manifests in a visible way when A is nonnumeric. Again, let A = unknown; then we derive

Then

which could be very distinct from the averaging effect supported by the previous decoding.

4.7. Conclusions

This chapter has concentrated on the ideas of computing with fuzzy sets. The extension principle forms a cornerstone of all operations on fuzzy numbers. Rule-based computing is dominant in modular systems. We introduced a number of algorithms aimed at the realization of rule-based systems. We have started from the knowledge representation schemes conveyed by rules and rules with fuzzy conditions and conclusions. The latter are important in reducing the brittleness of the original rules. Fuzzy controllers are the most visible examples of rule-based computing in control engineering - we have revisited this concept in great detail and outlined the key design principles. On the implementation side, it becomes crucial to identify the main design thrusts and articulate control specifications. We have also discussed reasoning with default values (defaults) exploiting fuzzy rules.

4.8. Problems

4.1. The standard transformation from polar (r, φ) to Cartesian (x, y) coordinates reads as

Consider now a fuzzy “point” defined in the polar coordinates whose values are given as

Find its position in the x-y Cartesian coordinates.

4.2. Let us consider a simple nonlinear relationship associating distance with velocity and time travelled

Assuming now that the velocity is fixed yet given as a triangular fuzzy number V(v; 10, 20, 40) compute the distance covered in 2 hours. What happens to the result if the time of this travel is around 2 hours and can be modelled by the triangular fuzzy number T(t; 1.5, 2, 2.5)?

4.3. Calculate arithmetic mean of two triangular fuzzy numbers T(x; 1, 3, 5) and R(x;2, 4, 6). Repeat the calculations for the expression

where now all arguments are the same T1 = T2 = ... = Tn = T(x; 1, 3, 5). Discuss the obtained results when the number of arguments (n) increases.

4.4. Let A and B be two fuzzy numbers with Gaussian membership functions A(x; 1, 2, 3) and

B(x; 2, 5, 8). Determine X such that

4.5. Discuss how to solve a system of equations involving fuzzy numbers


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.