IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 2, APRIL 2016
489
Short Papers
Designing Fuzzy Sets With the Use of the Parametric Principle
of Justifiable Granularity
Witold Pedrycz and Xianmin Wang
Abstract—This study is concerned with a design of membership functions of fuzzy sets. The membership functions are formed in such a way
that they are experimentally justifiable and exhibit a sound semantics.
These two requirements are articulated through the principle of justifiable
granularity. The parametric version of the principle is discussed in detail.
We show linkages with type-2 fuzzy sets, which are constructed on a basis of type-1 fuzzy sets. Several experimental studies are reported, which
illustrate a behavior of the introduced method.
Index Terms—Coverage of data, interval-valued fuzzy set, membership
function determination, principle of justifiable granularity, specificity,
type-2 fuzzy set.
I. INTRODUCTION
Determining membership functions has been one of the essential research pursuits in fuzzy sets implicating the developments
of their fundamentals and supporting a realization of applied
aspects. There are several views at fuzzy sets: likelihood view,
random set view, and typicality view. One can refer to a number
of important studies in the area including results published in
[2], [4]–[6], [10], [17], [20], and [21]. Depending upon the particular view of membership functions one focuses on, we can
distinguish between two main classes of methods considered in
the problems of membership function estimation, namely:
1) Expert-driven approaches: Here, elicitation of membership functions is viewed as a process of less or more sophisticated knowledge
acquisition through an interaction with a domain expert. The analytic hierarchy process (AHP) [19] is one of the systematic ways
of constructing membership functions by offering a mechanism of
validation of consistency of pairwise evaluations provided by the
expert.
2) Data-driven approaches: In this category, we confine ourselves to
the elicitation of membership functions on a basis of organization
(structuralization) of data. Fuzzy clustering methods [15] are the
most visible scheme present in this category.
Manuscript received October 16, 2014; revised January 27, 2015 and March
30, 2015; accepted May 11, 2015. Date of publication July 8, 2015; date of
current version March 29, 2016. This work was supported by the Natural Sciences and Engineering Research Council of Canada and Canada Research Chair
Program. This work was also supported by the National Centre for Research,
Poland, under Grant UMO-2012/05/B/ST6/03068 and the National Natural Science Foundation of China under Grant 41372341.
W. Pedrycz is with the Department of Electrical and Computer Engineering,
University of Alberta, Edmonton, AB T6G 2R3, Canada, with the Department of Electrical and Computer Engineering, Faculty of Engineering, King
Abdulaziz University, Jeddah 21589, Saudi Arabia, and with the Systems Research Institute, Polish Academy of Sciences, Warsaw 01-447, Poland (e-mail:
wpedrycz@ualberta.ca).
X. Wang is with the Hubei Subsurface Multi-scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences,
Wuhan 430074, China (e-mail: wangxianmin781029@hotmail.com).
Digital Object Identifier 10.1109/TFUZZ.2015.2453393
Although these two classes exhibit advantages when supporting a buildup of fuzzy sets, they are not free from limitations
and eventual bias. There are a number of compelling reasons,
which permeate the essential way the technology of fuzzy sets
is positioned within a realm of their applications. The expertdriven technique could be general and might not be necessarily reflective of the experimental data for which these fuzzy
sets are constructed. This becomes particularly visible when
such fuzzy sets are a part of the ensuing fuzzy model. This
may happen because of the lack of experimental support behind
some membership functions. On the other hand, the data-driven
approaches may result in fuzzy sets that are not semantically
meaningful: Fuzzy clustering could produce some “crowded”
fuzzy sets whose meaning is not so apparent. Their further adjustments when optimizing the fuzzy model fuzzy sets become
a part of, could substantially hamper the interpretability facet of
the fuzzy sets and the overall model.
With the growing interest and visibility of type-2 fuzzy
sets, the issue of determination of their membership functions
(which are more elaborate than the plain numeric counterparts of
type-1 fuzzy sets) becomes more acute. A number of systematic studies devoted to reasoning, modeling, and prediction have
been growing steadily [3], [7], [22]; nevertheless, the specific
studies on the elicitation of the type-2 membership are still lacking. The existing methods in this area (see, e.g., [1] and [8]) are
data-driven and embedded into the complete fuzzy model rather
than the individual fuzzy sets.
In this study, we are concerned with a systematic determination of membership functions with the use of the principle
of justifiable granularity [13], [14]. Alluding to the discussion
above, a construction of membership functions is guided by the
two apparent criteria of forming information granules so that
they are experimentally justified and semantically sound. The
relevance of the two requirements is visible from the arguments
posed above, and an elimination of such possible shortcomings
outlined above was stressed in [18]. The principle of justifiable granularity is of a general nature as it supports building
descriptions of not only fuzzy sets but information granules,
in general. Here, we introduce a complete algorithm based on
a parametric version of this principle. Addressing the growing
needs to construct membership functions of type-2 fuzzy sets
(or type-2 information granules, in general), we propose a constructive and efficient way of building membership bounds of
interval-valued fuzzy sets adhering to the principle of justifiable granularity. Furthermore, it is demonstrated how type-2
fuzzy sets help alleviate limitations of type-1 fuzzy sets. The
1063-6706 © 2015 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution
requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
490
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 2, APRIL 2016
Fig. 1. Interval information granules A and B with their characterization with
regard to coverage of the data and specificity.
principle of justifiable granularity, which is one of the fundamentals of granular computing, leads to the construction of
information granules, especially fuzzy sets, whose relevance becomes crucial when used in system modeling or classification.
For instance, information granules are building blocks of discrimination mappings encountered in classification problems.
Here, each of the granule has to exhibit some properties (such
as being homogeneous with regard to its classification content)
so that it serves as a meaningful entity when used as a building
block of a classifier.
This paper is structured as follows. In Section II, we briefly
recall the principle of justifiable granularity and then introduce
its parametric version. In the following, we discuss two essential
augmentations including the use of weighted data and inhibitory
data. Section III is focused on the linkages with type-2 fuzzy
sets, where we show how to construct interval-valued fuzzy sets.
Experimental studies are presented in Section IV. Concluding
comments are drawn in Section V.
II. PRINCIPLE OF JUSTIFIABLE GRANULARITY
AND ITS PARAMETRIC EXTENSION
The principle of justifiability granularity [13] comes as one of
the underlying fundamentals of granular computing and is about
forming an information granule on a basis of some experimental
evidence (numeric data). The essence of the method can be
summarized as follows.
Given some numeric data X = {x1 , x2 , . . . , xN }, X ⊂ R,
construct an information granule G so that it satisfies two sound
and intuitively appealing requirements of sufficient experimental evidence and high specificity.
Let us express these requirements in a more descriptive fashion. First, by stating that the information granule should be
experimentally justified, we mean that it should be justified
(supported) by the available data. This way, one can envision
that the information granule “covers” (represents) enough experimental data and as such is supported by the existing experimental evidence. Second, the information granule should be specific
enough, which implies that the granule has to exhibit some tangible meaning (semantics). The more specific (less abstract) the
information granule is, the higher becomes its specificity. As
a simple example, consider a collection of 1-D data of age reported in a certain community. Our intent is to describe the data
(capture their essence) by a certain information granule, say an
interval. The two extreme situations are displayed in Fig. 1. The
first information granule A “covers” all the data; however, its
specificity is low. A does not convey any useful and actionable
knowledge—it becomes apparent that the data are distributed
between xm in and xm ax . B is located on the other extreme of the
scale—it is very specific, but it does “cover” just a single data
point. The example presented above clearly shows that to assess
the usefulness of the produced information granule (interval),
one has to look at the two fundamental aspects: to which extent
the information granule is justified (supported) by the available
experimental evidence and how specific the information granule is. The information granule of age formed on a basis of the
data has to embrace (cover) most of the data; hence, it has to be
experimentally justifiable. The data should match (be included)
the produced information granule. Yet making the information
granule too broad (striving for the high satisfaction of the coverage criterion) may easily result in the granule, which has no
meaning, and, equally important, does not substantially support
any actionable conclusion. For instance, an information granule
such as the interval [1, 110] years of age covers all experimental data; however, its specificity is practically nonexistent—we
cannot take any reasonable action as to eventual investment in
social services or education. On the other hand, a very detailed
(degenerated) information granule, say 45.44, is very specific;
however, it may not cover any data point. In statistics, it is well
known that an average (which is a degenerate information granule, type-0 information granule) viewed as a generic estimator
of the population does not come alone, but we always augment
it by a confidence interval (which in terms of our study, is an
interval-valued information granule itself).
It becomes evident that these two requirements are in conflict, and the formation of the information granule is a result
of achieving a sound compromise between them. The principle
of justifiable granularity addresses this problem by producing a
certain optimization problem. The realized construct is accomplished in two steps: First, a numeric representative of the data
is being formed, and around it, built is an information granule.
The granule can be sought as an elastic band, which could be adjusted (stretched or contracted) so that the two requirements can
be met to the significant extent [13]. We again emphasize that
the principle of justifiable granularity is of a general character
and produces information granules in terms of sets (intervals),
fuzzy sets, rough sets, etc.
In this study, we consider a parametric principle of justifiable
granularity and focus on information granules in the form of
fuzzy sets. We also extend it to deal with two generalizations of
the generic problem:
1) Incorporation of weights of data (different levels of contribution of
data to the realization of information granule) that are the membership grades associated with the data.
2) Involvement of inhibitory experimental evidence (viz., data that
have to be excluded for the constructed information granules).
The experimental data X are given. Its numeric representative
m is formed (either by taking average, median, or their weighted
versions). The parametric version of the membership function
is given in terms of some bounded functions (left- and righthand parametric representations) f and g (see Fig. 2). These
functions are continuous and have bounded support. Their form
is specified in advance (say in the form of triangular, parabolic,
and trapezoidal membership functions). The bounds a and b
have to be determined (optimized).
The bounds a and b are optimized separately. Here, we discuss
the calculations for the upper bound b. They are guided by the
criteria of coverage and specificity:
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 2, APRIL 2016
491
partial results over [0,1] yielding
Sp =
0
Fig. 2. Construction of membership function. (a) Use of data to be covered.
(b) Use of weighted data. (c) Use of weighted data and inhibitory data (marked
by shaded squares).
Coverage: The criterion of coverage is expressed by computing a sum of membership grades of the data contained in the
fuzzy set (σ-count), that is,
f (xk ).
(1)
cov =
x k :x k ∈[m ,b]
One can generalize
this measure by taking any nondecreasing
function Ф, say x k :x k ∈[m ,b] Φ[f (xk )].
Specificity is expressed by quantifying how detailed (specific)
the fuzzy set is. As discussed in [23], a specificity measure Sp(.)
exhibits several self-evident properties: Sp({x}) = 1, Sp(U ) =
0, if X1 ⊂ X2 , Sp(X1 ) ≥ Sp(X2 ), where U is a universe of
discourse over which all fuzzy sets are defined. In case of an
interval [m, b], the specificity is computed as Sp([m, b]) =
1 − |b − m|/range where range = |xm ax − m| with xm ax being the largest element in the dataset. For the fuzzy set described
by f (we consider only the right-hand side of the membership
function), one can determine the corresponding cut and integrate
1
1−
|m − bα |
dα
range
(2)
where bα = f −1 (α). For instance, for the triangular membership function (f) with the modal value m and upper bound b,
the resulting specificity computed by (2) is equal to 1 – 0.5|m –
b|/range.
One has to stress that the proposed construct is of general
character with regard to the parametric form of the membership functions. They could be of any form, and typically, one
assumes that the support of such fuzzy sets is finite. In case of infinite support and as such a situation occurs in case of Gaussian
membership functions, one has to consider these segments of
membership functions where the membership grades are above
some minimal (practically sound) threshold and consider only
those data in the development of the membership function.
As the two design objectives (coverage and specificity) are in
conflict, the optimized performance index is taken as a product
of them, namely Q(b) = cov∗ Sp and bopt = arg maxb Q(b).
The optimization of the lower bound a is realized in the same
way, where the fuzzy set is now described by some function g.
Alluding to the concise formulation of the problem presented
above, some additional comments could be helpful, especially
in the context of the existing methodology and practice of constructing fuzzy sets. First, it has to be stressed that we are
concerned with a formation of a single fuzzy set describing
a collection of available experimental numeric data, not a family of fuzzy sets. As such, the produced fuzzy set can be used
as a descriptor (identifier) of the numeric data and serve as its
abstraction to be further used in data analysis, reasoning, and
modeling. In contrast, in clustering and fuzzy clustering, we are
concerned with a formation of a family of fuzzy sets and not
a single fuzzy descriptor of the data. Obviously, fuzzy clusters
can be further processed using the principle of justifiable granularity. The multidimensional data belonging to a single cluster
are used to construct a membership function of a fuzzy set in a
1-D space in the presence of weighted 1-D data with the weights
being the membership grades coming from fuzzy clustering.
With regard to the validation of the resulting fuzzy set, the
validity of the construct is implied by the sound criteria used
in the principle of justifiable granularity. This way of building
a fuzzy set assures us that it retains its meaningful features of
experimental legitimacy and semantic soundness. One can contrast here the common approach when fuzzy sets are constructed
altogether with the formation of the fuzzy model, and their formation is not guided by any directly optimized performance
index but the overall performance index of the model. Not surprisingly, this does not necessarily assure us that the fuzzy sets
formed this way are really meaningful in the sense of the criteria
outlined above.
There might be a situation that the optimized performance
index Q(b) or Q(a) assumes quite low values. This stipulates that
in building fuzzy sets, we encounter some strongly conflicting
requirements that cannot be easily reconciled when developing
a single fuzzy set. In such situations, original data are split into
492
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 2, APRIL 2016
some sets (e.g., using clustering), and the principle of justifiable
granularity is applied to the individual subsets. In this manner,
a hierarchy of fuzzy sets is being formed.
There are two useful augmentations of the above construct
[see Fig. 2(b) and (c)].
Weighted data: The data points come with their corresponding
weights wk assuming values in [0,1] and identifying their relevance [see Fig. 2(b)]. Constructing the fuzzy set in the presence
of the weighted data (x1 , w1 ) (x2 , w2 ) . . . (xN , wN ) requires
a prudent refinement of the coverage criterion. Here, we do
the counting by considering the impact of each data point expressed with regard to the value of f (xk ) and wk and taking their
minimum, namely min(f (xk ), wk ). The coverage criterion now
reads as
min(f (xk ), wk ).
(3)
Cov =
x k :x k ∈[m ,b]
The specificity criterion is computed in the same way as
before.
The weights of the data could come from fuzzy clustering.
Consider the ith cluster, viz., the ith row of the partition matrix
produced by the fuzzy c-means (FCM) algorithm. The elements
of this row are treated as the weights of the corresponding data,
w1 , w2 , .., wN . Similarly, when it comes to the inhibitory data,
their weights are formed on a basis of a function of the membership values located in the rows different from the ith one.
Inhibitory data: Along with the original data, there are some
weighted data (y1 , z1 ), (y2 , z2 ), . . . , (yM , zM ) with weights zi
located in the unit interval, which are of inhibitory nature, viz.,
the fuzzy set constructed should exclude them from its content.
The higher the weight zk , the stronger the inhibitory nature of
the data. This type of data can be encountered in classification
problems; the elements not belonging to a class the information
granule is focused on should be excluded from it. The presence
of the inhibitory data implies changes to the coverage criterion, which now consists of the two components (again, we are
concerned with xk and yk ):
cov = max(0,
min(f (xk ), wk )
x k :x k ∈[m ,b]
−γ
min(f (xk ), zk ))
(4)
where the inhibitory data are coming with some discount coefficient γ, γ ≥ 0. This coefficient is used to impact an influence
of the inhibitory data; the higher the value of this parameter, the
more significant is the resulting information granule impacted
by the inhibitory data. Obviously, if we assume excessively high
values of γ, this may lead to the zero value of the coverage measure, thus preventing us from building a fuzzy set at all. To
account for a different number of data in these two sets, one can
modify (4) as follows:
min(f (xk ), wk )
Cov = max(−,
x k :x k ∈[m ,b]
x k :x k ∈[m ,b]
III. BUILDING MEMBERSHIP FUNCTIONS OF TYPE-2 FUZZY SETS
The principle of justifiable granularity can be used to build
type-2 membership functions and, more precisely, intervalvalued membership functions. The underlying idea is motivated
by a suitable (optimized) allocation of information granularity, where information granularity is regarded as an essential
design asset [12], [13], [16], [24]. Given that a type-1 fuzzy
set has been provided, we generalize it to the type-2 information granule, more specifically interval-valued fuzzy set (type-2
information granule).
Let us consider a membership function of type-1 fuzzy set
f, which has been formed by running the principle of justifiable granularity. The elevation of type-1 fuzzy set to the
interval-valued membership function is motivated by the intent to “cover” the experimental data by introducing intervals
of membership grades. To do so, we admit two membership
functions f − and f + such that f ≥ f − and f + ≥ f . If one assumes that the original membership function f is convex, then
the bounds of the interval-valued set can be considered convex.
A location of these two functions is controlled by a parameter ε
assuming values from [0,1] such that the functions f − and f +
are move away from f [see Fig. 3(a)]. In a descriptive way, by
changing the value of ε, we cause a “sliding” effect of f − and
f + , which with the increase of ε makes the functions to become
more remotely positioned from f.
As before, there are two measures involved in the construction
of the information granule, namely coverage and specificity. As
the functions f − = and f + form a certain band (interval) of
membership grades produced for each xk , we count the number
of cases where the membership grade (weight) wk falls within
the interval [f − (xf ), f + (xk )]. The higher the value of ε, the
higher the coverage of the membership values, cov. Having this
in mind, the coverage measure for the membership function f
decreasing in Ω+ = [m, xm ax ], cov, is defined as follows:
cov = card{xk ∈ Ω+ |wk ∈ [f −1 (xk ), f + (xk )]}.
x k :x k ∈[m ,b]
− γN/M
The optimization is carried out in the same way as discussed
before.
min(f (xk ), zk )).
(5)
(6)
It quantifies an ability of the interval-valued fuzzy set formed
in this manner to “cover” (represent) experimentally available
membership grades. An example plot of the coverage versus
ε is displayed in Fig. 3(b). It is noticeable that this relationship is a nondecreasing function of ε. Furthermore, its shape
could imply a choice of a suitable level of granularity (ε); any
“knee” point suggests a feasible value of ε, beyond which further increase of the level of information granularity does not
translate into a substantial increase of the coverage. The inclusion predicate used in (6) is Boolean returning 0 or 1. A
useful generalization is a multivalued version of the original
binary inclusion. We consider a multivalued inclusion predicate incl(w, z) defined on a basis of a continuous t-norm,
incl(w, z) = sup{z ∈ [0, 1]|wty ≤ z}, w, z ∈ [0, 1]. The predicate returns a degree to which w is included in z. For instance,
for the minimum (t = min) and algebraic product (t = ∗), one
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 2, APRIL 2016
493
Fig. 4. Linear segment of membership function (dotted line) along with the
lower and upper linear bounds. Shown are numeric data to be covered (black
dots) and inhibitory data (open dots) and their weights.
function f):
1
sp = 1 −
range
= 1−
Fig. 3. (a) Membership functions f − and f + of interval-valued fuzzy set. (b)
Relationship between coverage treated as a function of the spread of the bounds
of the interval-valued fuzzy sets. (c) Specificity–coverage characteristics.
has
incl(w, z) =
incl(w, z) =
1,
if w ≤ z
z,
otherwise
1,
if w ≤ z
z/w
otherwise.
(7)
The coverage formula (6) is now rewritten in the following
form:
min[incl(f −1 (xk ), wk ), incl(wk , f + (xk ))].
cov =
k :x k ∈Ω +
(8)
The specificity is concerned with the type-2 fuzzy construct, which entails the difference between the upper and lower
bounds, which in this case yields the following expression (the
formula applies to the decreasing portion of the membership
b+
f (z)dz −
b
+
m
A(FOU)
.
range
f −1 (z)dz
m
(9)
range = |xm ax − m|. In fact, the specificity measure is related
to an area of the footprint of uncertainty (FOU), namely A(FOU).
Note that the specificity of interval-valued fuzzy set is a decreasing function of ε. The relationship between ε and the specificity
depends upon the form of f − and f + . Furthermore, one can form
a coverage–specificity plot [see Fig. 3(c)], in which we plot the
coverage provided by the interval-valued fuzzy set and its specificity. It is a redrawn dependence visualized in Fig. 3(b), and this
clearly emphasizes the competitive nature of the two criteria and
leads to an informed selection of ε0 producing a sound selection. The allocation of information granularity could be realized
in a more advanced manner by admitting a refined allocation of
information granularity. In other words, we consider two translation levels δ1 and δ2 controlling a position of the translated
bounds around b with the requirement that δ1 + δ2 =ε. This is a
constraint to retain an overall balance of the level of granularity
of the construct and, at the same time, admit more flexibility to
the construct (in comparison with these two values δ1 and δ2
being equal).
As an example, let us consider a decreasing portion of a linear
membership function (see Fig. 4), which is now augmented by
the two new membership functions (bounds) such that when ε
changes, the point at which f − (x) attains zero moves in between
modal value and b. The same sliding effect is realized for b+ .
The intensity of this sliding effect is impacted by the value
of ε. The detailed formulas read as b+ = b + ε(xm ax –b) and
b− = b–ε(b–m). Apparently, when ε = 0, b− and b+ collapse
to b. If ε = 1, then b− = m and b+ = xm ax .
The coverage is determined using (6) or (8) when considering
the generalized version of the inclusion predicate.
The A(FOU) becomes a linear function of ε, A(FOU) =
1/2b+ –1/2b− = 1/2ε(xm ax –m), and the specificity is computed as sp = 1–A(FOU)/range = 1–1/2ε.
The choice of the classes of membership functions to which
f and g belong to could be an interesting design alternative
supporting an overall optimization. For instance, admitting that
the area under curve (AUC) for the plot in Fig. 3(b) serves as
494
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 2, APRIL 2016
Fig. 5. Plots of 1-D data and their weights: data to be covered shown by black
dots, and inhibitory data marked by open dots.
a viable optimization criterion, we can decide upon the class
of membership functions and choose the one for which AUC
attains its maximal value.
One can note that in the design of type-2 fuzzy set (intervalvalued fuzzy set, to be more specific), we established a twophase process: First, type-1 fuzzy set has been formed, and
then, it was extended to interval-valued construct in an attempt
to cover data. In contrast, the existing methods of building type2 fuzzy sets are realized in a single shot usually resulting in
a far more challenging optimization problem involving more
parameters to be estimated at the same time. Building type-2
fuzzy sets could be accomplished by combining the proposed
algorithm with one of the efficient schemes of representation of
fuzzy sets [9], [11], [25], [26].
IV. ILLUSTRATIVE EXAMPLES
In this section, we cover some examples illustrating how the
principle of justifiable granularity is realized for numeric data. In
all cases, the constructed fuzzy sets are described by triangular
membership functions T(x; a, m, b) with a and b standing for the
lower and upper bound of the fuzzy set and m being its modal
value.
Synthetic data: We consider 1-D data illustrated in Fig. 5.
Some of the data are of an inhibitory nature.
We build a fuzzy set by starting with a determination of the
numeric representative (modal value of the membership function) computed as a weighted average, whose value is 1.53. We
consider here an auxiliary parameter ε, where γ is computed in
the form γ = (N/M )∗ ξ. Given that the number of the inhibitory
data (M) is different from the number of data used to construct
fuzzy set (N), this helps us keep the balance between these two
types of data. The plots of the performance index Q determined
for the bounds of the triangular membership function obtained
for selected values of ξ are included in Fig. 6.
The corresponding bounds obtained here visualize an expected phenomenon: Higher values of γ (emphasizing the impact of inhibitory data) give rise to a more specific fuzzy set: ξ
= 0.0: a = 1.1, b = 4.9, T(x; 1.1, 1.53, 4.9); ξ = 1.0: a = 1.1, b
= 2.7, T(x; 1.1, 1.53, 2.7); ξ = 2.0: a = 0.9, b = 2.7, T(x; 0.9,
1.53, 2.7).
Fig. 6. Plots of the performance index versus the bounds of the triangular
membership function; shown are results obtained for selected values of ξ: (a)
Q versus b, (b) Q versus a; black circles −ξ = 0.0; open circles −ξ = 1.0;
triangles- ξ = 2.0.
Machine Learning Repository (http://archive.ics.uci.edu/
ml/): In these two examples, we consider data coming from
this repository, namely Boston housing (housing) and energy
efficiency (energy). Initially, the data are clustered using FCM
with the fuzzification coefficient p = 2. The obtained membership grades are regarded as weighs associated with the corresponding data points. The prototypes are regarded as numeric
representatives of the fuzzy sets. For the ith cluster, the weights
wk are coming from the corresponding row of the partition matrix U = [uik ], namely uik , k = 1, 2, . . . , N . As the inhibitory
weights, we take the maximal values of membership located in
the remaining rows, namely maxj =1,2,..c,j = i uj k . The size of the
inhibitory data is the same as the data to be covered; hence, we
use the parameter γ. In light of the available information, the
constructed fuzzy sets can be viewed as granular (rather than
numeric) prototypes being capable of offering a more comprehensive description of the data; cf., [14].
Housing dataset: The housing data comprises 506 data points,
each of them described by 14 attributes. For this dataset, we
form three fuzzy sets for the house price. We consider one
of the prototypes formed there, namely m = 22.35. The plots
of the data along with their weights for the three fuzzy sets
to be constructed are displayed in Fig. 7. It is noticeable that
the inhibitory data overlap quite visibly the data to be used to
support the constructed information granule.
By running the principle of justifiable granularity, the maximized performance index Q is shown in a series of figures (see
Fig. 8); both the optimization of the lower and upper bounds are
reported on separate plots.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 2, APRIL 2016
Fig. 7. Data along with their weights used for the construction of three fuzzy
sets: (a) data to be covered and (b) inhibitory data.
495
Fig. 9. Performance index for the three fuzzy sets versus the bounds: (a) Q(b)
and (b) Q(a). The results are reported for γ = 0.0 (gray dots) and γ = 0.4
(black dots).
The obtained triangular fuzzy sets we obtained for γ equal
to 0.0 and 0.4 are given as T(x; 20.70, 22.35, 26.20) and T(x;
22.00, 22.35, 26.20), respectively.
Energy efficiency dataset: The dataset concerns data coming as a result of assessing the heating load and cooling load
requirements of buildings (energy efficiency) as a function of
building parameters. There are 768 data with eight variables.
We form a fuzzy set describing energy consumption. As before, a triangular fuzzy set is formed for one of the prototypes
equal to m = 31.38. The values of the performance index Q are
displayed in Fig. 9.
Subsequently, the obtained triangular fuzzy sets obtained for
the inhibition coefficient γ equal to 0.0 and 0.4 are given as T(x;
30.45, 31.38, 32.07) and T(x; 31.28, 31.38, 31.80), respectively.
V. CONCLUSION
Fig. 8. Performance index for a fuzzy set versus the bounds (a and b). The
results are reported for (a) γ = 0.0 and (b) γ = 0.4.
The approach presented in this study underlines the fact that
any information granule builds on a basis of experimental data
and, at the same time, incorporates domain knowledge supplied
by the designer (coming here in the form of the predefined
type of membership function). The criteria of coverage and
specificity are the essential components well reflecting the nature of the build-up of information granules. When contrasting
this method with the existing approaches, we note that each of
them build upon different conceptual settings. The expert-driven
496
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 2, APRIL 2016
constructs (such as the AHP approach) focus on a systematic
involvement of domain knowledge; however, quite commonly,
there are no explicit performance indexes invoked. The dataoriented techniques are not guided by any available mechanisms
of domain knowledge.
Fuzzy sets formed by the principle of justifiable granularity retain their semantics (which is originally captured by the
coverage and specificity criteria). This stands in contrast with
the construction of fuzzy sets, which is a part of the optimization process of fuzzy models, and because of this, the semantics of fuzzy sets cannot be guaranteed. The construction of
type-2 fuzzy sets is realized sequentially on a basis of the already formed type-1 construct. This prevents us from more demanding optimization required to build type-2 from scratch (as
usually reported in the literature).
To cast the study in the general setting of system modeling
with fuzzy sets (where fuzzy sets are directly used), one could
remark that the membership functions formed this way can form
a starting point of all modeling constructs and are eventually
further adjusted to optimize the model.
REFERENCES
[1] H. Cao, L. Jia, G. Si, and Y. Zhang, “A clustering-analysis-based membership functions formation method for fuzzy controller of ball mill pulverizing system,” J. Process Control., vol. 23, no. 1, pp. 34–43, Jan. 2013.
[2] J. L. Chameau and J. C. Santamarina, “Membership functions I: Comparing methods of measurement,” Int. J. Approx. Reason., vol. 1, no. 3,
pp. 287–301, Jul. 1987.
[3] B. I. Choi and F. C. H. Rhee, “Interval type-2 fuzzy membership function
generation methods for pattern recognition,” Inf. Sci., vol. 179, no. 13,
pp. 360–377, Jun. 2009.
[4] M. R. Civanlar and H. J. Trussell, “Constructing membership functions
using statistical data,” Fuzzy Set. Syst., vol. 18, no. 1, pp. 1–13, Jan. 1986.
[5] H. Diskhant, “About membership functions estimation,” Fuzzy Set. Syst.,
vol. 5, no. 2, pp. 141–147, Mar. 1981.
[6] E. Hisdal, “Are grades of membership probabilities?,” Fuzzy Set. Syst.,
vol. 25, no. 3, pp. 325–348, Mar. 1988.
[7] C. Hwang and F. C. H Rhee, “Uncertain fuzzy clustering: interval type2 fuzzy approach to C-means,” IEEE Trans. Fuzzy Syst., vol. 15, no. 1,
pp. 107–120, Feb. 2007.
[8] R. Hosseini, S. D. Qanadli, S. Barman, M. Mazinani, T. Ellis, and
J. Dehmeshki, “An automatic approach for learning and tuning Gaussian
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
interval type-2 fuzzy membership functions applied to lung CAD classification system,” IEEE Trans. Fuzzy Syst., vol. 20, no. 2, pp. 224–234, Apr.
2012.
F. Liu, “An efficient centroid type-reduction strategy for general type2 fuzzy logic system,” Inf. Sci., vol. 178, no. 9, pp. 2224–2236, May
2008.
S. Medasani, J. Kim, and R. Krishnapuram, “An overview of membership
function generation techniques for pattern recognition,” Int. J. Approx.
Reason., vol. 19, no. 3, pp. 391–417, Oct. 1998.
J. M. Mendel, F. Liu, and D. Zhai, “Alpha-plane representation for Type-2
fuzzy sets: Theory and applications,” IEEE Trans. Fuzzy Syst., vol. 17,
no. 5, pp. 1189–1207, Oct. 2009.
W. Pedrycz, “Granular computing—The emerging paradigm,” J. Uncertain Syst., vol. 1, no. 1, pp. 38–61, 2007.
W. Pedrycz, Granular Computing: Analysis and Design of Intelligent
Systems. Boca Raton, FL, USA: CRC Press/Francis & Taylor, 2013.
W. Pedrycz and W. Homenda, “Building the fundamentals of granular
computing: A principle of justifiable granularity,” Appl. Soft Comput.,
vol. 13, no. 10, pp. 4209–4218, Oct. 2013.
W. Pedrycz, Knowledge-Based Fuzzy Clustering. New York, NY, USA:
Wiley, 2005.
W. Pedrycz and A. Bargiela, “An optimization of allocation of information
granularity in the interpretation of data structures: toward granular fuzzy
clustering,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 3,
pp. 582–590, Jun. 2012.
W. Pedrycz, “Why triangular membership functions?” Fuzzy Sets Syst.,
vol. 64, no. 1, pp. 21–30, May. 1994.
W. Pedrycz and G. Vukovich, “On elicitation of membership functions,” IEEE Trans. Syst., Man, Cybern. A, Syst. Humans, vol. 32, no. 6,
pp. 761–767, Nov. 2002.
T. L. Saaty, The Analytic Hierarchy Process. New York, NY, USA:
McGraw-Hill, 1975.
V. Torra, “Interpreting membership functions: A constructive approach,”
Int. J. Approx. Reason., vol. 20, no. 3, pp. 191–207, Mar. 1999.
L. B. Turksen, “Measurement of membership functions and their acquisitions,” Fuzzy Sets Syst., vol. 40, no. 1, pp. 5–38, Mar. 1991.
C. Ulu, M. Güzelkaya, and I. Eksin, “Granular type-2 membership functions: A new approach to formation of footprint of uncertainty in type-2
fuzzy sets,” Appl. Soft Comput., vol. 13, no. 8, pp. 3713–3728, Aug. 2013.
R. R. Yager, “Ordinal measures of specificity,” Int. J. Gen. Syst., vol. 17,
no. 1, pp. 57–72, 1990.
J. T. Yao, A. V. Vasilakos, and W. Pedrycz, “Granular computing: Perspectives and challenges,” IEEE Trans. Cybern., vol. 43, no. 6, pp. 1977–1989,
Dec. 2013.
C. Wagner and H. Hagras, “Toward general type-2 fuzzy logic systems
based on zSlices,” IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 637–660,
Aug. 2012.
H. Wu and J. M. Mendel, “Uncertainty bounds and their use in the design
of interval type-2 fuzzy logic systems,” IEEE Trans. Fuzzy Syst., vol. 10,
no. 5, pp. 622–639, Oct. 2002.