A NEW CASE-BASED APPROXIMATE REASONING
BASED ON SPMF IN LINGUISTIC APPROXIMATION
Dae-Young Choi
Dept. of MIS, Yuhan University, Koean-Dong, Sosa-Ku, Puchon City, Kyungki-Do, South Korea
Ilkyeun Ra
Dept. of Computer Science & Engineering, Univ. of Colorado Denver, CO 80204 U.S.A.
Keywords: Case-based approximate reasoning (CBAR), Linguistic approximation.
Abstract: A new case-based approximate reasoning (CBAR) based on SPMF in linguistic approximation is proposed.
It provides an efficient mechanism for linguistic approximation within linear time complexity.
1 INTRODUCTION
Case-based reasoning (CBR) is a problem-solving
technique that reuses past experiences to find a
solution. It is quite simple to implement in general,
but it often handles complex and unstructured
decision making problems very effectively.
Moreover, it is maintained in an up-to-date state
because the case-base is revised in real time, which
is a very important feature for the real world
applications. Due to its strength, CBR has been
applied to various problem-solving areas including
manufacturing, finance and marketing, intelligent
product catalogs for Internet shopping malls, conflict
resolution in air traffic control, semiconductors
design, medical diagnosis (Ahn et al., 2007, Chiu,
2002, Chiu et al., 2003). While other major artificial
intelligence techniques depend on generalized
relationships between problem descriptors and
conclusions, CBR utilizes specific knowledge of
previously experienced and concrete problem
situations, so it is effective for complex and
unstructured problems and it is easy to update (Ahn
et al., 2007). In recent years, CBR has received a
great deal of attention and has been used
successfully in diverse application areas. As a
general problem solving methodology intended to
cover a wide range of real-world applications, CBR
must face the challenge to deal with uncertain,
incomplete, and vague information. Building hybrid
approaches by combining CBR with methods of
uncertain and approximate reasoning (Zadeh, 1973,
Mizumoto et al., 1982) plays an important role in
many real-world applications. In this connection, a
new case-based approximate reasoning based on
SPMF in LA is proposed.
2 SPMF
Technology standards help ensure that packages and
application services do not become piecemeal
solutions so that you can leverage them across other
initiatives. Additionally, enterprises with standards
can respond more quickly to changing conditions
than those without standards because creating
information systems from compatible components is
easier and less costly (Tanrikorur, 2001). As
Mamdani (Mamdani, 2001) pointed out in 2001
BISC (Berkeley Initiative in Soft Computing)
workshop on fuzzy logic and the Internet, it is time
to think about ‘standardization on fuzzy sets’.
Let A be a fuzzy set for a linguistic term and be
a subset of the universal set X, then, for x∈X, a
triangular-type membership function can be
represented by using 3 points μ
A
(x
L
, x
M
, x
H
), where
x
L
<x
M
<x
H
, and if the result of this membership
function is normalized to [0, 1] then μ
A
(x
L
, x
M
, x
H
) =
0 for every x∈[-∞, x
L
]∪[x
H
, ∞] and μ
A
(x
L
, x
M
, x
H
) =
1 at x
M
. A trapezoidal-type can be represented by
using 4 points μ
A
(x
L
, x
I1
, x
I2
, x
H
), where
x
L
<x
I1
<x
I2
<x
H
, and if the result of this membership
259
Choi D. and Ra I. (2009).
A NEW CASE-BASED APPROXIMATE REASONING BASED ON SPMF IN LINGUISTIC APPROXIMATION.
In Proceedings of the 11th International Conference on Enterprise Information Systems - Artificial Intelligence and Decision Support Systems, pages
259-262
DOI: 10.5220/0001816002590262
Copyright
c
 SciTePress

function is normalized to [0, 1] then μ
A
(x
L
, x
I
1
, x
I
2
, x
H
)
= 0 for every x∈[-∞, x
L
]∪[x
H
, ∞] and μ
A
(x
L
, x
I
1
, x
I
2
,
x
H
) = 1 at [x
I
1
, x
I
2
]. A more comprehensive study of
standardized parametric membership functions
(SPMF) can be found in (Chang et al., 1991).
3 THE PROPOSED METHOD
Based on their behavioral experiment, they
recommended the five good distance measure (DM)
(i.e., S
4
, q
∞
, q
*
, Δ
∞
, Δ
*
) between fuzzy subset A and
B of a universe of discourse U (Zwick et al, 1987).
We note that the five good DM concentrate their
attention on a single value rather than performing
some sort of averaging or integration. In the case of
S
4
, attention focuses on the particular x-value where
the membership function of A∩B is largest; in q
∞
and Δ
∞
, attention focuses on the α-level set where
the x-distance is largest; in q
*
and Δ
*
, attention
focuses on the x-distance at the highest membership
grade. Considering the result of their behavioral
experiment, we know that the reduction of
complicated membership functions to a single ‘slice’
may be the intuitively natural way for human beings
to combine and process fuzzy concepts. Moreover,
we know that the DM between two fuzzy subsets can
be efficiently represented by a limited number of
features. From this idea, a new case-based
approximate reasoning (CBAR) based on SPMF in
LA is proposed. In the case-based reasoning process,
case indexing and retrieval are the most important
steps because the performance of CBR systems
usually depends on them (Ahn, 2007).
In this paper, we suggest linguistic case
indexing and retrieval based on SPMF. We try to
find efficiently the fuzzy subset of the linguistic
value that is the most similar to the fuzzy subset
resulting from observation (A′) related to a linguistic
variable in a rule-base. We assume that there is the
pre-defined set of linguistic variables (PSLV) sorted
by alphabetically. Each linguistic variable in the
PSLV has the pointer to indicate its own table with
the relevant linguistic values. For example, the table
regarding the linguistic variable ‘age’ may be
consisted of linguistic values such as ‘young’, ‘very
young’ represented by fuzzy subsets. We assume that
fuzzy subsets for linguistic values are defined by the
SPMF.
The performance of CBR systems usually
depends on case indexing and retrieval (Ahn, 2007).
In the proposed linguistic case indexing and
retrieval, we utilize the partitioning concept that
disjoints the linguistic variables used in the process
of CBAR. It can be used to avoid exploring the
irrelevant linguistic values in the process of CBAR.
Since the resulting fuzzy subset obtained from
observation (A′) is related to its linguistic variable,
we can find the related linguistic variable easily by
referencing the linguistic variable matched in a rule-
base. After the related linguistic variable (LV) is
determined in the PSLV, its corresponding table
(TBL) is obtained by using the pointer (P
i
, i = 1, 2,
…, n) associated with the linguistic variable. Based
on the relevant fuzzy subsets represented by the
SPMF in the table, the distances between all relevant
fuzzy subsets in the table and an observation (A′)
represented in the SPMF are computed by using
Euclidean distance. Thus, the rule with the most
similar linguistic value (i.e., linguistic value with the
minimum distance) could be retrieved for the
CBAR. An approximate transformation method
(ATM) based on the SPMF was proposed in (Choi,
2006). The ATM transforms the non-parametric
membership functions into the SPMF. The detail
algorithms and their properties for the ATM were
described in (Choi, 2006). We note that each table
(TBL
i
, i = 1, 2, …, n) is consisted of the relevant
linguistic values represented by the SPMF. For
example, the triangular-type or the trapezoidal-type
will be defined as (x
L
, x
M
, x
H
) or (x
L
, x
I1
, x
I2
, x
H
),
respectively in Section 2.
Example 1. We consider the linguistic variable ‘age’.
For simplicity, we assume that the table regarding
the linguistic variable ‘age’ is consisted of 2
linguistic values such as ‘young’ and ‘very young’.
They are defined as (x
L
, x
I1
, x
I2
, x
H
)
1
= (15, 20, 30,
35)
1
= A
1
(‘young’) and (x
L
, x
I1
, x
I2
, x
H
)
2
= (17, 20, 27,
30)
2
= A
2
(‘very young’) respectively, by using the
trapezoidal-type. We assume that an observation
(A′) is parameterized to (x
L
′, x
I1
′, x
I2
′, x
H
′) = (16, 20,
25, 29) = A′ as in Figure 1. The detail algorithms for
transforming the non-parametric membership
functions into the SPMF were described in (Choi,
2006). The distances among fuzzy subsets are
achieved by using Euclidean distance as follows :
d
1
(A
1
,A′)
=
2222
)2935()2530()2020()1615( −+−+−+−
=
62 .
d
2
(A
2
,A′)
=
2222
)2930()2527()2020()1617( −+−+−− +
=
6 .
ICEIS 2009 - International Conference on Enterprise Information Systems
260

μ
1
15 16 17 20 25 27 29 30 35 X
Figure 1: Fuzzy subsets on the linguistic variable ‘age’.
In this case, for an observation A′, the linguistic
value ‘very young’ is retrieved because d
1
> d
2
. In a
similar way, it may be extended to i, (i =1, 2, …, m),
relevant linguistic values in the table. Thus, the
linguistic value with min (d
1
, d
2
, …, d
m
) is retrieved
for CBAR in LA.
In this paper, we explain the proposed method
by using trapezoidal-type membership function. It
can be similarly extended to other SPMF such as
Π-type, S-type, etc. The proposed method is
achieved by using only the small number of
parameters in SPMF. In addition, the proposed
linguistic case indexing and retrieval utilize the
partitioning concept that disjoints the linguistic
variables (Zadeh, 1987) used in the process of
CBAR. It can be used to avoid exploring the
irrelevant linguistic values in the process of
CBAR in LA. One of the crucial problems in real
world applications is the computational speed of
the applied method. Acceptable speed is generally
achieved only if the time complexity is at most
polynomial. In this respect, the proposed method
is valuable because its time complexity is linear as
shown in Example 1. It provides an efficient
mechanism for LA within linear time complexity.
4 COMPARISONS
Wenstop (1976) suggested the linguistic
computation that were almost entirely problem
dependent. He may specify only two primary subsets
in a universe of discourse composed of perhaps 25
elements. Obviously, this would not be very
conducive if a close match was required in such a
sparse space. Thus, some care should be taken to
ensure a reasonable density of subsets within the
primary space. Eshragh & Mamdani (1979)
proposed another linguistic computation. But the
computational complexity of their method for
linguistic processing is very high. The reason is that
the search procedure has two main phases. The first
phase is exhaustive and the second phase is heuristic.
The exhaustive phase takes care of trivial cases. That
is, if a given subset shows characteristics similar to
those of primary or negated primary subsets, then it
will be tested against appropriate types of primary
subsets for perfect match. If the exhaustive phase
proves unsuccessful, the heuristic phase is applied.
In this phase, the input is appropriately processed
and its segments are separated. This search process
is time-consuming. Degani & Bortolan (1988)
proposed another linguistic computation. It is mainly
useful for its use of clinically recognized linguistic
terms whose meaning is rather well established in
the medical community. Batyrshin & Wagbnknbcht
(2002) described the problem of a linguistic
description of dependencies in data by a set of rules
R
k
: “If X is T
k
then Y is S
k
” where T
k
’s are linguistic
terms like SMALL, BETWEEN 5 AND 7 describing
some fuzzy intervals A
k
, and S
k
’s are linguistic
terms like DECREASING and QUICKLY
INCREASING describing the slopes p
k
of linear
functions y
k
= p
k
x + q
k
approximating data on A
k
.
Their linguistic approach can be used for the
calculation of granular derivatives of functional and
statistical dependencies between linguistic variables
in rules with aforementioned constraints. Their
search approach for linguistic terms is time-
consuming when merging fuzzy intervals of the
partition obtained by the genetic algorithm and
retranslation for generating rules from fuzzy
partitions and linear approximation.
A key problem in the application of fuzzy set
theory to real time control, expert systems, natural
language understanding, etc., is devising relatively
fast methods. The proposed linguistic case indexing
and retrieval is efficiently obtained by using the
parameters of the SPMF. Moreover, in order to
avoid exploring the irrelevant linguistic values in the
process of CBAR in LA, we use the partitioning
concept that disjoints the linguistic variables used in
the process of CBAR in LA. These features enable
the proposed linguistic case indexing and retrieval to
be processed relatively fast compared to the previous
linguistic approaches (Batyrshin et al., 2002, Degani
& Bortolan, 1988, Eshragh & Mamdani, 1979,
Kowalczyk, 1998, Wenstop, 1976). From the
engineering viewpoint, it may be a valuable
advantage.
We briefly summarize the difference between the
proposed method and existing linguistic
approximation methods in Table 1.
Young (A
1
)
Very young (A
2
)
Observ. (A
′
)
A NEW CASE-BASED APPROXIMATE REASONING BASED ON SPMF IN LINGUISTIC APPROXIMATION
261

Table 1: Comparisons.
Attributes Existing methods Proposed method
Membership f
n
Ad-hoc SPMF
Method Ad-hoc CBAR
Complexity Complex Simple
5 CONCLUSIONS
A new CBAR based on SPMF in LA is proposed.
Compared to existing linguistic approximation
methods, the proposed LA is achieved by using only
the small number of parameters in SPMF. In
addition, the proposed linguistic case indexing and
retrieval utilize the partitioning concept that disjoints
the linguistic variables used in the process of CBAR
in LA. It can be used to avoid exploring the
irrelevant linguistic values in the process of CBAR
in LA. These features enable the proposed linguistic
case indexing and retrieval to be processed relatively
fast compared to the previous linguistic approaches.
It provides an efficient mechanism for LA within
linear time complexity. Thus, the proposed method
can be used to improve the speed of LA. In the
meantime, a key problem in the application of fuzzy
set theory to real time control, expert systems,
natural language understanding, etc., is devising
relatively fast methods. So, we propose a new CBAR
based on SPMF in LA. From the engineering
viewpoint, it may be a valuable advantage.
ACKNOWLEDGEMENTS
The author wishes to thank Prof. L. A. Zadeh,
University of California, Berkeley, for his
inspirational address on the perceptual aspects of
humans in the BISC (Berkeley Initiative in Soft
Computing) seminars and group meetings, and also
thank Prof. Il Kyeun RA, University of Colorado,
Denver, for his support.
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