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1
Introduction
In the real world,
fuzziness plays an essential role in human cognition. Fuzzy modeling
techniques make use of linguistic hedges as fuzzy set transformers,
which modify the shape of a fuzzy set's surface to cause a change in
the related truth membership function. Linguistic hedges play the
same role in fuzzy modeling as adverbs and adjectives do in
language: they both modify qualitative statements.
In Zadeh's fundamental
papers on fuzzy reasoning (4), (5), (6), Cox's book (2) and Jiang's
book (3), the linguistic hedges very, more or less, positively and
negatively were introduced and verified in a limited mathematical
model. There, their corresponding operations on a membership
function association, distribution and product were also introduced
and verified.
In Zadeh's fundamental
papers on fuzzy reasoning (4), (5), (6), Cox's book (2) and Jiang's
book (3), the operation of membership function association,
distribution, and product of very, more or less, positively and
negatively were introduced and verified in a limited mathematical
model.
Zadeh
interpreted very using the function f(x) = x^2.
Very acting on a membership function tall men results in a
membership function very tall men. The grade of membership of a man
in the tall men class is lowered in the very tall men class. Zadeh's
interpretation of very distributes over addition, intersection and
production (see (4)).
In some practical
applications a different function than Zadeh's may be needed; often
a specific data base requires transforming. A transormer can be
characterized piecemeal across linguistic hedges over parts of the
fuzzy region. An industrial application would be the qualification
of information from a bank database system.
Because the underlying
mechanics of a linguistic hedge are heuristic, both the nature of a
fuzzy surface and its degree of transformation are associated with
people's subjective judgement as to how the region should look, not
on mathematical theory of fuzzy surface topology operations.
Predicting future applications of such models is diffcult;
therefore, our investigation exceeds traditional definitions to
provide a loose framework. Section two presents a general
interpretation of linguistic hedges very, more or less, positively
and negatively. Section three presents a special software profram
which illustrates the potential practical applications of the new
general form of fuzzy hedges developed in this paper. The program is
to aid class instructors in curving students grades using a varity
of fuzzy hedges. The class instructors can analyze and fine tune the
grade distribution and save the desired results. A conclusion is
made on our new general fuzzy hedges.
2
Mathematical Modeling of Linguistic Hedges
In literature, a number
of other terms such as linguistic variable, linguistic qualifier,
modifier are used to describe fuzzy linguistic hedges (see (4), (1)
and (2)). Here we prefer to use hedge as in (4), (5) and (6).
Discussion will focus on the fuzzy linguistic hedges very, more or
less, positively and negatively.
2.1
Very
Zadeh in (4) gave a
mathematical interpretation of very by squaring the membership
function at each point in the fuzzy region:
A concentrator transforms a fuzzy set into a less
fuzzy set. In practical applications the exact effect of applying very to a
membership function differs. A general interpretation of very therefore is
reasonable to attempt, if one agrees with (4).
a concentrator reduces the grade of
membership of an element y in a fuzzy set A relatively small for
those y with high grade of membership in A but relatively large for
those y with low grade of membership in A.
Hence, a general concentrator very
can be
interpreted as a continuous function satisfying
where
f(y) is a function on the interval [0,
1] satisfying
It is clear that this definition is equivalent to
the above statement of a concentrator. Generally, a concentrator function is increasing but
may not be concave up. However, each concave up function could
be a concentrator function in some applications. For the function f(y)
in (3), for any y1
<= y2,
So f(y1)
<= f(y2)
and f(y) is increasing. It seems that every concentrator is a
concave up function and vice versa because of the geometric
structures of f(y) and a concave up function. However, a concave up
function f(y) that starts at (0, 0) and ends at (1, 1) satisfies
condition (3) from its geometric structure. There still exist
functions that are not concave up but satisfy condition (3). For
example,
It is, for example,
possible to use y^2,
y^3
and y^4
to describle the hedges very,
extremely and very very respectively-all as defined by Zadeh in (4),
(5) and (2). All the above hedges are concave up. The function f(y)
can have other forms such as yey-1.
The second derivative of f(y) is (2 + y)e^(y-1)
> 0. Hence, f(y) is concave up
and satisfies condition (3). The interpretation offered here of very
aims at encompassing the various uses that very may be put to in
different stituations. Take membership function Tall described by
Figure 1 shows the
effect of applying three hedges y^2,
y^4
and ye^y
- 1 to the membership function
Tall y.
Fig. 1.
The effect of applying very hedges y2 (dotted line), y4 (dashed
line) and yey-1 (dash-dot line) to Tall (solid line)
2.2 More or less
Fuzzy set linguistic
hedges more or less, somewhat, rather and quite are dilutors with
the same meaning (see (4), (5), (6) and (2)). We will be using more
or less throughout this paper. It is the complement of the
linguistic hedge very, originally defined by Zadeh in (4),
¿From the Zadeh's
interpretations of more or less and very It is clear that the
linguistic hedges more or less (very) and very (more or less) have
same meaning. A general version of the dilutor (5) is given in Zadeh
(4) and (2). This is done by replacing 0.5 in (5) with a positive
fractional (1/n),
The linguistic hedge
more or less defined by Zadeh has the opposite effect of his
linguistic hedge very. In a similar fashion to the way very is
generalized to produce (2), the authors propose that dilutor more or
less is generalized in line with following statement.
A dilutor increases
the grade of membership of an element y in a fuzzy set A relatively
large for those y with low grade of membership in A but relatively
small for those y with high grade of membership in A.
This can be
interpretated mathematically as
where f(y) is a
continuous function over the fuzzy region satisfying
The function f(y)
satisfying condition (8) may be increasing/decreasing over one or
more intervals within the fuzzy region. For example, the function
satisfies condition
(8). It is increasing on intervals [0, 1/4] and [1/2, 1] but
decreasing on interval [1/4, 1/2]. Also, it is interesting to note
that this function is not concave down. However, a concave down
function f(y) that starts at (0, 0) and ends at (1, 1) satisfies
condition (8) due to its geometric structure.
The hedge y^0.5
originally defined by Zadeh in
(4) and its extension y^1/n
in (4) and (2) are concave down
functions. This can be verified by checking that their second
derivatives are less or equal to 0. The effect of applying different
types of the linguistic hedge more or less to membership function
Tall is given in Figure 2.
Fig. 2.
The effect of applying more or less hedges y^1/3 (dash-dot line),
y^3/5 (dashed line) and ye1-y (dotted line) to Tall (solid line)
If DIL is used to
denote the dilutor defined in (7), it is clear that
and
However, the
distribution of DIL over the product operation does not hold for a
general dilutor. But, the distribution of DIL over the product
operation holds for any function of the form y^1/n
(n > 1).
2.3 Positively
In (4), a formula
corresponding to positively, but not to the term positively, is used
as a contrast intensifier.
However, the term
positively as a linguistic hedge is used in (3). In this paper,
positively is used as a general contrast intensifier. Zadeh in (4)
provided a basic statement of the effect of this contrast
intensifier:
a contrast
intensifier increases all the values of a truth membership function
which are above 0.5 and
diminishes those which are below 0.5.
Based on the statement of a contrast intensifier we
mathematically model a contrast intensifier as
where f(y) is a
continuous function on the interval [0, 1] satisfying the condition
that f(y) <= y when y < 0.5, f(y) <= y when y > 0.5 and f(0.5) =
0.5. An example of this type of functions is
For n = 2 in (11), the
resulting contrast intensifier is the original contrast intensifier
defined by Zadeh in (4). Figure 3 shows the effect of such contrast
intensifiers (n = 2, 4) applied to membership function Tall.
Analytically, for a general contrast intensifier a condition f(1 -
y) = 1 - f(y) should be imposed on the function f(y) to achieve
sharper contrast intensification. There exist such good functions
other than (11) which constitute legitimate contrast intensifiers.
2.4
Negatively
The effect of applying
negatively to a membership function should be opposite of the effect
of applying positively to the same function. In line with the effect
Fig. 3. The
effect of applying positively and negatively (n = 2, 4) to Tall
of applying positively
to a membership function, the basic effect of applying negatively
can be stated as follows.
Negatively hedge
increases all the values of a truth membership function which are
above 0.5 and diminishes those which are below 0.5.
Based on the statement
above negatively could be defined mathematically in the most general
sense as
where f(y) is a
continuous function on the interval [0, 1] satisfying the condition
that f(y) >= y when y<0.5, f(y) <= y when y > 0.5 and f(0.5) = 0.5.
An example of this type of functions is
Figure 3 also shows the
effect of applying such type of hedges negatively (n = 2, 4, 5) to
membership function Tall. Also, for a hedge negatively a condition
f(1-y) = 1-f(y) should be imposed on the function f(y) to achieve
sharper contrast fuzzification. There still such good functions
other than (13) which constitute legitimate constrast fuzzifiers.
3 Application
3.1 Introduction
To illustrate the
potential practical applications of fuzzy hedges, especially the new
general form of hedges developed in this paper, a special software
program has been developed.
The purpose of this
program is to aid in curving students grades using a varity of fuzzy
hedges. The program gives the user (e. g. class instructor) the
ability to open a spreadsheet containing the student numbers and
final grades. The user can then analyze and fine tune the grade
distribution and save the results back to the spreadsheet.
3.2 Concept
Figure 4 demonstrates the usage of the Fuzzy Markx
program. Figure 5 is the main screen generated from the Fuzzy Markx
program. Figure 6 demonstrates the buttons for a user to use the
Fuzzy Mark program. Figure 7 is the grade scheme obtained by the
user.
. The Spreadsheet
Initially, the user
must open a spreadsheet containing the grades of students in the
classroom. The spreadsheet should be formatted in the following
manner. The first row should contain columns headings only. The
first and second columns must contain the name and student numbers
of the students. The third column must contain the final grades of
the students formatted as a percentage and excluding the '%' symbol.
.
Grade Scheme
Initially the user must
also decide on a grade scheme for the course. This is where the
instructor can decide which percentage grades are assigned to which
letter grades. If no grade scheme is chosen, a default scheme is
automatically used.
.
Viewing Distribution
Once the user has
loaded the grades and adjusted the grade scheme as desired, the
results are ready to be viewed. Depending on whether the instructor
is satisfied with the distribution of grades, he or she may wish to
adjust the fuzzy hedge before saving and continuing.
.
Adjusting Fuzzy Hedge
The user has a choice
of selecting between four different fuzzy hedges, each with its own
characteristics. The severity of the curve (i. e. the value of n)
can also be adjusted and fine tuned until the grade distribution
satisfies the instructor.
Fig. 4. Overall
Algorithm of FuzzyMarkx
.
Saving Results
Once the user is
satisfied with the grade distribution, the curved grade of each
student can be saved (appended) back to the spreadsheet.
3.3
Description
. Overview
The Grade Display Area
lists the grade distribution, displayed as letter grades, the number
of students receiving each letter grade and the percentage of the
class receiving each grade. The Graph Area displays the function
being applied to students grades. There are four functions that the
user can apply are under the Graph Area. These functions are, very,
more or less, positively and negatively. The function variable n, is
adjusted using the Function Variables Area.
The variable n can be
adjusted by typing it into a text box, or by dragging a sliding bar.
It is not possible to enter a value n that is smaller than 1 using
either method. The resulting curve can be seen in the Graph Area.
The slide bar's, minimum, and maximum values, as well as the number
of discrete steps between min and max can be adjusted for finer
tuning when using the slide bar. Inappropriate entries in these
fields results in the program reverting them to a default value.
. Menu
The user can use Open
from the File menu to load the excel format file into the program.
When the user is satisfied with the grade distribution, the user can
use the Save also from the File menu to save the adjusted grades
back to the spreadsheet. The Grade Scheme in the edit menu, is used
to map numeric grades to a letter grade. Adjusting the grade scheme
can be done directly or by loading a previously saved grade scheme
from a *.1st file.
Fig. 5.
The Main Screen
Fig. 6. The File
and Edit Menus
Fig. 7.
Altering the Grade Scheme
3.4
Summary
This program is
excellent for understanding and applying fuzzy hedges to list of
data. It is specially designed for altering school grade
distributions. It is a very simple, and easy to use program, allows
the user to view the new grade and as well as giving a graphical
representation of the function being applied.
4
Conclusion
This paper discusses
existing definitions of commonly used fuzzy linguistic hedges and
expose their limitations. New, and most general definitions and
formulae are developed. Specific configurations of the general
hedges are presented, charted, and discussed. A practical software
program of applying these fuzzy hedges has been developed to aid
class instructors in curving students grades. It is hoped these
general formulae will provide more versatility to both fuzzy
theorists as well as application engineers.
Acknowledgements
The authors would like
to thank Mr. Denton Reed (an undergraduate student at the ECE Dept.
of Concordia University) for his help in the development and testing
of FuzzyMarkx- the program.
References
[1] E. Cox, Effectively
Using Fuzzy Logic and Fuzzy Expert System Modeling-in Theory and
Practice, Proceedings of the First Intl. Conf. on artificial
Intelligence Applications on Wall Street, IEEE Computer Society
Press, Los Alamitos, CA., 1992, 194-199.
[2] E. Cox, The Fuzzy
System Handbook, second edition, AP Professional, New York, 1998.
[3] J.S.R. Jiang, C-T.
Sun and E. Mizutani, Neuro-Fuzzy and Soft Computing-A Computational
Approach to Learning and Machine Intelligence, Prentice Hall, 1997.
[4] L. A. Zadeh, A
Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges, Journal of
Cybernetics, Vol. 2, No. 3, 1972, 4-34.
[5] L. A. Zadeh, The
Concept of a Linguistic Variable and its Applications to Approximate
Reasoning, Information Sciences, Vol. 8, 1975, 199-249 (Part I),
301-357 (Part II), Vol. 9, 1975, 43-80 (Part III).
[6] L. A. Zadeh, PRUF-a
meaning representation language for natural language,Int. J.
Man-Machine Stud., Vol. 10, 1978, 395-480.
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