A Novel Approach to Weighted Fuzzy Rules for Positive Samples

Martina Da

nkov

University of Ostrava, CE IT4Innovations,

30. dubna 22, 701 03 Ostrava 1,Czech Republic

Keywords:

Fuzzy Relation, Relational Model, Fuzzy Approximation, Implicative Model, Fuzzy IF–THEN Rules.

Abstract:

In this contribution, we propose a novel approach to automated fuzzy rule base generation based on underlying

observational data. The core of this method lies in adding information to a particular fuzzy rule in the form of

attached weight given as a value extracted from a relational data model. In particular, we blend two approaches

to receive particular models that overcome their speciﬁc drawbacks.

1 INTRODUCTION

Weighted fuzzy rules have been used intensively in

fuzzy modeling since the early days of the ﬁeld.

Weights were applied to various parts of fuzzy rules:

to input variables, antecedents, consequences, or

whole rules; see, e.g., (Nauck, 2000; Ishibuchi and

Nakashima, 2001; Alcal

a et al., 2003; delaOssa et al.,

2009). Learning techniques for weighted fuzzy rules

usually involve optimization methods such as neu-

ral networks, evolutional computing, or genetic algo-

rithms to ﬁnd the best possible fuzzy rules together

with their associated weights. This typical approach

is repeated also in various recent works, e.g. (Bemani-

N. and Akbarzadeh-T., 2019; Shiny Irene et al., 2020;

Navarro-Almanza et al., 2022).

In this contribution, we focus on weighted fuzzy

rules formalized by the so-called normal forms intro-

duced in (Perﬁljeva, 2004). Normal forms-based for-

malization covers the wide spectrum of the weighted

fuzzy rules mentioned above. Their relationship was

described and studied in (Da

nkov

a, 2007).

Furthermore, we deal with observed data that are

considered as positive ones, i.e. the given data are

the prototypical representatives of a relationship be-

tween input and output spaces. In the standard fuzzy

sets framework, where the scale for the truth values

is [0, 1], we assign F(c, d) = 1 to the observed data

(c, d) and the relationship F.

Having positive sample data at our disposal, we

can freely build fuzzy rules for each data using

the sample-based generation of fuzzy rules provided

https://orcid.org/0000-0001-5806-7898

in (H

ajek, 1998) and called H

ajek’s approach within

this paper.

At this stage, we can face the problem of a huge

number of rules that need to be further reduced for

a ﬁnal computation efﬁciency and a rule-base trans-

parency. We propose to solve this problem by com-

bining the H

ajek approach with normal forms, where

we can ﬁx a number of fuzzy rules in the fuzzy rule-

base to keep a sample-based learned information as

small and compact as needed.

The paper is organized as follows. In Section 2,

we recall a formalization of the basic fuzzy relational

models from (H

ajek, 1998) and provide some selected

properties. The normal conjunctive and disjunctive

norms are presented in Section 3 together with similar

results as in the preceding section. Next, in Section 4,

we introduce special normal forms based on positive

samples that blend the approach of H

ajek and the nor-

mal forms and provide their basic selected theoretical

results. Finally, we summarize the results and propose

some future directions in Section 5.

2 SAMPLE-BASED FUZZY

RULES

As already stated in the Introduction, there is a

vast amount of methods for generating fuzzy rules.

Mostly, they can be characterized as the result of

expert knowledge, generated from observed data, or

their combination. A method based on observed data

formalized in (H

ajek, 1998) automatically generates

a particular fuzzy rule using fuzzy similarity relations

for each input data. In the following, let us recall

nková, M.

A Novel Approach to Weighted Fuzzy Rules for Positive Samples.

DOI: 10.5220/0011548800003332

In Proceedings of the 14th International Joint Conference on Computational Intelligence (IJCCI 2022), pages 209-216

ISBN: 978-989-758-611-8; ISSN: 2184-3236

209

ajek’s approach:

• Consider a complete residuated lattice of the form

L = ⟨L, &, →, ∧, ∨, 0, 1⟩,

with the lattice operations (∧, ∨), the residuated

pair (&, →), the bottom and top elements 0 and 1,

respectively. Additionally, deﬁne

x ↔ y = (x → y) & (y → x).

• Let X

, X

̸=

0 and ≈

be a similarity relation on

, j = 1, 2, that is,

(x ≈

x) = 1,

(x ≈

y) ≤ (y ≈

x),

(x ≈

y) & (y ≈

z) ≤ (x ≈

z),

for all x, y, z ∈ X

• Let (c

, d

) ∈ X

× X

, for i ∈ I, where I =

{1, 2, . . . , n}.

• Moreover, let the following properties be valid for

a binary fuzzy relation F on X

× X

– Extensionality of F:

(x ≈

′

) & (y ≈

′

) & F(x, y) ≤ F(x

′

, y

′

), (1)

for all x, x

′

∈ X

, y, y

′

∈ X

– Functionality of F:

(x ≈

′

) & F(x, y)& F(x

′

, y

′

) ≤ (y ≈

′

), (2)

for all x, x

′

∈ X

, y, y

′

∈ X

– Positive samples of F:

i∈I

F(c

, d

) = 1. (3)

• Deﬁne:

Mamd

(x, y) =

i∈I



(x ≈

) & (y ≈

)



, (4)

Rules

(x, y) =

i∈I



(x ≈

) → (y ≈

)



, (5)

for all x ∈ X

, y ∈ X

We call Mamd

and Rules

relational data mod-

els.

Example 2.1. Consider the data given in Figure 1,

where X

= [0, 6] and X

= [0, 50], moreover, deﬁne

(x, y) = 0 ∨ (1 − k|x − y|),

for all x ∈ X

, y ∈ X

, k ∈ R.

Furthermore, let ≈

≡ S

1.7

, ≈

≡ S

0.2

. Then the

relational The models Mamd

in the standard alge-

bras of Łukasiewicz, Product and G

odel are shown in

Figures 2, 3, and 4, respectively. The relational model

Rules

in the standard algebras of Łukasiewicz alge-

bra is in Figure 5. Models Rules

for the remaining

algebras are omitted since their values are 0 nearly

everywhere.

But if we change ≈

as follows: ≈

≡ S

0.02

, we

obtain relational models Rules

in the standard alge-

bras of Łukasiewicz, Product and G

odel as shown in

Figures 6, 7, and 8, respectively.

Figure 1: Input data.

Figure 2: Mamd

for the input data from Figure 1 in

Łkasiewicz standard algebra.

Figure 3: Mamd

for the input data from Figure 1 in the

Product standard algebra.

FCTA 2022 - 14th International Conference on Fuzzy Computation Theory and Applications

210

Figure 4: Mamd

for the input data from Figure 1 in G

odel

standard algebra.

Figure 5: Rules

for the input data from Figure 1

in Łukasiewicz standard algebra.

Figure 6: Rules

for the input data from Figure 1

in Łukasiewicz standard algebra with a ﬁner similarity.

In (H

ajek, 1998), we can ﬁnd many interesting

properties, e.g.,

Mamd

(x, y) ≤ F(x, y), (6)

Figure 7: Rules

for the input data from Figure 1 in the

product standard algebra with a ﬁner similarity.

Figure 8: Rules

for the input data from Figure 1 in G

odel

standard algebra with a ﬁner similarity.

F(x, y) ≤ Rules

(x, y), (7)

i∈I

(x ≈

)

≤ Mamd

(x, y) ↔ F(x, y), (8)

i∈I

(x ≈

)

≤ F(x, y) ↔ Rules

(x, y), (9)

i∈I

(x ≈

)

≤ Mamd

(x, y) ↔ Rules

(x, y). (10)

for all x ∈ X

, y ∈ X

. Hence, Mamd

is below F

and Rules

is above F. The quality of approxima-

tion is expressed here using the equivalence relation

↔, which is dual to pseudometric.

Due to Lemma 7.2.10, the relational data model

Mamd

is extensional and functional. Contrary to

Mamd

, the relational data model Rules

is not func-

tional (see Remark 7.2.11 in (H

ajek, 1998)). How-

ever, it can be proved that Rules

is extensional.

Proposition 2.2. Let F, ≈

, ≈

be as above.

Then the fuzzy relation Rules

is extensional.

A Novel Approach to Weighted Fuzzy Rules for Positive Samples

211

Proof. We have to prove the following inequality

(x ≈

′

) & (y ≈

′

) & Rules

(x, y) ≤ Rules

′

, y

′

for all x, x

′

∈ X

, y, y

′

∈ X

We start from the left-hand side of the above in-

equality:

(x ≈

′

)& (y ≈

′

i∈I



(x ≈

) → (y ≈

)



≤

(x ≈

′

) & (y ≈

′

) & [(x ≈

) → (y ≈

)] =

(x ≈

′

)& [1 → (y ≈

′

)]& [(x ≈

) → (y ≈

)] ≤

(x ≈

′

) & [(x ≈

) → ((y ≈

′

) & (y ≈

))] ≤

(x ≈

′

) & [(x ≈

) → ((y

′

≈

)]

From the transitivity of ≈

, it follows

x ≈

′

≤ (x

′

≈

) → (x ≈

Thus, we obtain

(x ≈

′

) & (y ≈

′

) &

i∈I



(x ≈

) → (y ≈

)



≤ [(x

′

≈

) → (x ≈

)] & [(x ≈

) → ((y

′

≈

)]

≤ (x

′

≈

) → (y

′

≈

for all x, x

′

∈ X

, y, y

′

∈ X

and i ∈ I. From this the

extensionality of Rules

follows directly.

3 WEIGHTED FUZZY RULES

FOR EXTENSIONAL FUZZY

RELATIONS

Unfortunately, H

ajek’s approach without any further

fuzzy rule reduction method is suitable only for a

small number of data samples. One of the possible

solutions is to ﬁx the number of rules and use quanti-

ﬁers of Fuzzy General Unary Hypotheses Automaton

(FGUHA) methods (Hole

na, 1998; Ralbovsk

y, 2009)

(a generalization of the GUHA methods (H

ajek and

Havr

anek, 1978)) to conﬁrm or reject each particular

fuzzy rule designed. Examples of successful solutions

to real-world problems in various ﬁelds of vague na-

ture can be found in (Turunen, 2008).

In the subsequent section, we propose another

novel solution to the problem, where we add to each

ﬁxed fuzzy rule a special weight based on the given

sample data. Since both relational models are exten-

sional, we can use the so-called normal forms intro-

duced in (Perﬁljeva, 2004). First, recall the deﬁnition

of normal forms.

Deﬁnition 3.1. Let G be a fuzzy binary relation on

×X

, X

, ≈

be as above, and p

∈ X

, q

∈

, for j ∈ J, where J = {1, 2. . . . , k}. Then

• The disjunctive normal form for G is deﬁned as

DNF

(x, y) =

j∈J



(x ≈

) & (y ≈

) & G(p

, q

)



, (11)

for all x ∈ X

, y ∈ X

• The conjunctive normal form for G is deﬁned as

CNF

(x, y) =

j∈J



((x ≈

) & (y ≈

)) → G(p

, q

)



, (12)

for all x ∈ X

, y ∈ X

Observe that the relational model Mamd

can be

identiﬁed with DNF

, provided G(x, y) = F(x, y), for

all x ∈ X

, y ∈ X

, I = J, and (c

, d

) = (p

, q

), for

all i ∈ I. An analogous observation cannot be made

for Rules

and CNF

. Because the relational model

Rules

is based on functionality, while CNF

is based

on extensionality.

The values G(p

, q

), j ∈ J serve as weights of

particular fuzzy rules. In fact, it works as some kind

of shift (and rotate) operator because for each fuzzy

rule in DNF

, it holds

≈

x) & (q

≈

y) & G(p

, q

) ≤ G(p

, q

), (13)

for all j ∈ J, x ∈ X

, y ∈ X

, and analogously, for each

fuzzy rule in CNF

, it holds

G(p

, q

) ≤ (p

≈

x)& (q

≈

y) → G(p

, q

), (14)

for all j ∈ J, x ∈ X

, y ∈ X

. Roughly speaking, apply-

ing the weights to fuzzy rules as done in normal forms

gradually switches on or off the particular fuzzy rules.

Figures 9–14 demonstrate the behavior described

above. Figure 11 shows the aggregation of fuzzy sets

from Figures 9 and 10 in the Łukasiewicz algebra.

The weight 0.5 lowers the values of DNF below 0.5

as shown in Figure 12. Analogously, the same weight

rotates and shifts the values of DNF above 0.5, see

Figure 14. And Figure 13 demonstrates the duality of

DNF and CNF.

Figure 9: x ≈

FCTA 2022 - 14th International Conference on Fuzzy Computation Theory and Applications

212

Figure 10: y ≈

Figure 11: (X ≈

c) & (y ≈

d) in Łukasiewicz standard

algebra.

Figure 12: (X ≈

c) & (y ≈

d) & 0.5 in Łukasiewicz stan-

dard algebra.

If G is an extensional binary fuzzy relation, then

the following inequalities hold; see (Perﬁljeva, 2004):

DNF

(x, y) ≤ G(x, y), (15)

G(x, y) ≤ CNF

(x, y), (16)

Figure 13: (X ≈

c)&(y ≈

d) → 0 in Łukasiewicz standard

algebra.

Figure 14: (X ≈

c) & (y ≈

d) → 0.5 in Łukasiewicz stan-

dard algebra.

i∈I

[(c

≈

& (y ≈

)

] ≤ DNF

(x, y) ↔ G(x, y),

(17)

i∈I

[(c

≈

& (y ≈

)

] ≤ CNF

(x, y) ↔ G(x, y),

(18)

i∈I

[(c

≈

& (y ≈

)

] ≤

DNF

(x, y) ↔ CNF

(x, y), (19)

for all x ∈ X

, y ∈ X

The above-listed inequalities seem to be analo-

gous to the results of H

ajek’s but they are more gen-

eral. Since the fuzzy relation G is not functional, we

have to lower estimate the above equivalences by the

A Novel Approach to Weighted Fuzzy Rules for Positive Samples

213

respective degrees of similarity in domain X

. How-

ever, in the case of H

ajek’s approach, it is sufﬁcient

to take into consideration only the respective simi-

larities in the domain X

. More properties on ap-

proximate reasoning with normal forms can be found

in (Da

nkov

a, 2007).

4 WEIGHTED FUZZY RULES

FROM THE RELATIONAL

DATA MODELS

Obviously, normal forms cannot be applied directly

for given input data. Naturally, the question arises of

how to obtain weights inside normal forms based on

observational positive data? In this section, we pro-

pose a particular solution to this question.

First, let us summarize the requirements:

• ≈

be a similarity relation on X

, j = 1, 2.

• (c

, d

) ∈ X

× X

, for i ∈ I.

• (p

, q

) ∈ X

× X

, for j ∈ j.

• F be a binary fuzzy relation on X

× X

• D = {(c

, d

)}

i∈I

be a set of positive data, that is,

F(c

, d

) = 1 for all i ∈ I.

Notice that (c

, d

), for i ∈ I are input data for

Mamd and Rules, while (p

, q

), j ∈ J are nodes in

which we construct normal forms.

As shown in Section 2, the relational data models

Mamd and Rules given by (4) and (5), respectively,

are extensional; consequently, all the results valid for

the normal forms pass to the following normal forms:

DNF

Mamd

, (20)

DNF

Rules

, (21)

CNF

Mamd

, (22)

CNF

Rules

, (23)

that will be called weighted fuzzy rules for positive

samples.

In the following, we list the valid properties for the

introduced weighted fuzzy rules, i.e., the particular

normal forms, derived from (15)–(19).

DNF

Mamd

(x, y) ≤ Mamd

(x, y), (24)

Mamd

(x, y) ≤ CNF

Mamd

(x, y), (25)

DNF

Rules

(x, y) ≤ Rules

(x, y), (26)

Rules

(x, y) ≤ CNF

Rules

(x, y), (27)

j∈J

(x ≈

)

& (y ≈

)

≤

DNF

Mamd

(x, y) ↔ CNF

Mamd

(x, y). (28)

j∈J

(x ≈

)

& (y ≈

)

≤

DNF

Rules

(x, y) ↔ CNF

Rules

(x, y). (29)

for all x ∈ X

, y ∈ X

Moreover, if F is extensional and functional, then

we can use H

ajek’s results (6)–(10), and we obtain:

DNF

Mamd

(x, y) ≤ F(x, y), (30)

F(x, y) ≤ CNF

Rules

(x, y), (31)

j∈J

(x ≈

)

& (y ≈

)

i∈I

(x ≈

)

≤

DNF

Mamd

(x, y) ↔ F(x, y), (32)

j∈J

(x ≈

)

& (y ≈

)

i∈I

(x ≈

)

≤

F(x, y) ↔ CNF

Rules

(x, y). (33)

for all x ∈ X

, y ∈ X

Many more interesting results on approximate in-

ference with DNF

Mamd

and CNF

Mamd

are obtained

directly from the results on normal forms published

in (Da

nkov

a, 2007).

Let us provide an illustrative example for the data

given in Figure 15 and the weighted fuzzy rules for

positive samples of the form DNF

Mamd

. We consider

11 nodes

{(p

, q

)}

i=0

= {(5.4 · i, i

)}

i=0

The resulting DNF

Mamd

for the three basic algebras

are shown in Figures 16–18. The respective weights

are as follows:

0.81 0.81 0.81

0.91 0.91 0.93

0.95 0.96 0.96

0.94 0.94 0.96

0.86 0.87 0.93

0.99 0.99 0.99

0.84 0.85 0.92

0.92 0.92 0.95

0.9 0.9 0.94

0.8 0.81 0.87

0.99 0.99 0.99

where L

denotes Łukasiewicz algebra, L

the prod-

uct algebra, and L

the G

odel algebra.

5 CONCLUSION

In this contribution, we presented a novel approach

to the construction of weighted fuzzy rules based

FCTA 2022 - 14th International Conference on Fuzzy Computation Theory and Applications

214

Figure 15: Input data.

Figure 16: DNF

Mamd

in Łukasiewicz algebra.

Figure 17: DNF

Mamd

in Product algebra.

on positive samples. We combine Perﬁlieva’s nor-

mal forms and the relational data models proposed

by H

ajek. It allows us to ﬁx a number of weighted

fuzzy rules and overcome the problem of a huge un-

clear fuzzy rule basis. Additionally, all theoretical re-

sults known in both approaches are applicable for the

special normal forms, too, as was presented some se-

lected results in the preceding section.

In this phase of research, only theoretical results

Figure 18: DNF

Mamd

in G

odel algebra.

are at the disposal. Therefore, it is now the next

step to test the proposed method. Clearly, the num-

ber of weighted fuzzy rules can be optimized as well

as the position of nodes in which the special normal

forms will be constructed. Intuitively, in the case

of DNF

Mamd

, DNF

Rules

, the nodes (p

, q

), j ∈ J

closer to the data points (c

, d

), i ∈ I will propagate

a better approximation in the sense of the degree of

equivalence. In contrast to the previous fact, the more

distant the nodes are from the given data in CNF

Mamd

and CNF

Rules

, the better the approximation (higher

degree of equivalence).

ACKNOWLEDGEMENTS

This research was supported by the Czech Science

Foundation project No. 20–07851S.

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