Introducing $g$-normal fuzzy relational models

Abstract

In this paper, a harmonious family of fuzzy relational models (FRMs) called $g$-normal FRMs is introduced. By selecting a value for the scalar parameter $g$, a member of this family is determined for which all parts of the model are configured accordingly. In this regard, a specific configuration for FRMs as well as an appropriate derivative-based iterative identification procedure are proposed. While preserving the high modeling capability of FRMs, the proposed model configuration along with the proposed identification algorithm alleviate both the lack of analyzability and the possible existence of hard conflicts between the rules in the rule-base. Furthermore, high relative errors in output computations are prevented using the proposed scheme. The modeling scheme is applied finally to the Box–Jenkins gas furnace benchmark problem as well as a new classification problem. The results are verified by simulation.

Appendix: The algorithm for generation of the pseudo-random data used in Sect. 4.2 for classification

The data used in Sect. 4.2 for classification can be reproduced by six steps given in this section. The algorithm is based on the well-known linear congruential generator.

First note that for a set $A$ and a function $h$, we define $h(A):= \{h(a):a \in A \}$. Also note that all sets are ordered in this section, namely, $A= \{ a(1), \ldots , a(n)\}$. The algorithm is represented by the following six Steps:

• 1. Obtain $W_0,W_1,\ldots ,W_6$ by $W_i= \{ t_i(1), \ldots , t_i(n)\}$ for $i= 0,1,\ldots ,6$, where $t_i(k+1)= f ( t_i(k);a_i,b_i,m_i )$ for $k= 1,2,\ldots ,n-1$, $f(x;a,b,m)= { mod (ax+b , m)}/{m}$, and $a_i,b_i,m_i,n$ are given in Table 1.

• 2. Obtain $X_0,X_1,\ldots ,X_6$: $X_0= W_0, X_1= W_1, X_2= W_2 - 1, X_3= W_3 - 1, X_4= W_4, X_5= W_5 - 0.5, X_6= W_6 - 0.5.$

• 3. Obtain $Z_1,\ldots ,Z_6$ by $Z_i= (\frac{X_0}{a_i'})^2 - (X_i-b_i')^2$ for $i= 1,2,\ldots ,6$, where $a_i',b_i'$ are given in Table 2.

• 4. Obtain

  $$\begin{aligned} \left\{ \begin{array}{l} X_i'= \{ x': (x',z') \in (X',Z')_i \} \\ Z_i'= \{ z': (x',z') \in (X',Z')_i \} \\ \end{array} \right. , \end{aligned}$$

  for $i= 1,\ldots ,6$, where $(X',Z')_i=$

  $$\begin{aligned} \{( x(j),z(j) ) : x(j) \in X_i, z(j) \in Z_i, z(j)\ge 0 \}_{j=1}^n. \end{aligned}$$

• 5. Obtain $Y_1,\ldots ,Y_6$: $Y_1= - \sqrt{Z_1'} + 1, Y_2= - \sqrt{Z_2'} + 1, Y_3= \sqrt{Z_3'} - 1, Y_4= \sqrt{Z_4'} - 1, Y_5= \sqrt{Z_5'}, Y_6= {- \sqrt{Z_6'}}/{1.2}.$

• 6. Construct $(X',Y)_i:= \{ ( x'(j),y(j) ) : x'(j) \in X'_i, y(j) \in Y_i \}_{j=1}^{n'}$.

For $i=1,\ldots ,4$, $(X',Y)_i$ represent the data in the $i$th class while $(X',Y)_5$ and $(X',Y)_6$ together represent the data in the fifth class. Note that $n' \le n$.

Table 1 The parameters used in step 1 (Appendix)

Table 2 The parameters used in step 3 (Appendix)