2010, Vol. 18, No. 5

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2010, Vol. 18, No. 5


A Framework for Clustering Categorical Time
-
Evolving Data, Cao, F.
,

Liang, J.
,

Bai, L.
,

Zhao, X.
,

Dang, C.,
Page(s): 872


882

A fundamental assumption often made in unsupervised learning is that the problem is static, i.e., the descr
iption of the classes does
not change with time. However, many practical clustering tasks involve changing environments. It is hence recognized that the

methods and techniques to analyze the evolving trends for changing environments are of increasing inter
est and importance.
Although the problem of clustering numerical time
-
evolving data is well
-
explored, the problem of clustering categorical
time
-
evolving data remains as a challenging issue. In this paper, we propose a generalized clustering framework for
categorical
time
-
evolving data, which is composed of three algorithms: a drifting
-
concept detecting algorithm that detects the difference
between the current sliding window and the last sliding window, a data
-
labeling algorithm that decides the most
-
approp
riate cluster
label for each object of the current sliding window based on the clustering results of the last sliding window, and a
cluster
-
relationship
-
analysis algorithm that analyzes the relationship between clustering results at different time stamps.
The
time
-
complexity analysis indicates that these proposed algorithms are effective for large datasets. Experiments on a real dataset
show that the proposed framework not only accurately detects the drifting concepts but also attains clustering results of
better
quality. Furthermore, compared with the other framework, the proposed one needs fewer parameters, which is favorable for spec
ific
applications.

Comparing Fuzzy, Probabilistic, and Possibilistic Partitions, Anderson, D. T.
,

Bezdek, J. C.
,

Popescu, M.
,

Keller, J. M., Page(s): 906


918

When clustering produces more than one candidate to partition a finite set of objects ${bf O}$, there are two approaches to
validation (i.e., selection of a “best” partition, and implicitly, a best value for $c$ , which
is the number of clusters in ${bf O}$). First,
we may use an internal index, which evaluates each partition separately. Second, we may compare pairs of candidates with each

other, or with a reference partition that purports to represent the “true” cluster
structure in the objects. This paper generalizes many
of the classical indices that have been used with outputs of crisp clustering algorithms so that they are applicable for cand
idate
partitions of any type (i.e., crisp or soft, with soft comprising the f
uzzy, probabilistic, and possibilistic cases). Space prevents
inclusion of all of the possible generalizations that can be realized this way. Here, we concentrate on the Rand index and it
s
modifications. We compare our fuzzy
-
Rand index with those of Campel
lo, Hullermeier and Rifqi, and Brouwer, and show that our
extension of the Rand index is $O$($n$), while the other three are all $O(n^{2})$. Numerical examples are given to illustrate

various facets of the new indices. In particular, we show that our indic
es can be used, even when the partitions are probabilistic or
possibilistic, and that our method of generalization is valid for any index that depends only on the entries of the classical

(i.e.,
four
-
pair types) contingency table for this problem.

2010, V
ol. 18, No.
4


Toward General Type
-
2 Fuzzy Logic Systems Based on zSlices, Wagner, C.
,

Hagras, H., Page(s): 637


660

Higher order fuzzy logic systems (FLSs), such as interval type
-
2 FLSs, have been shown to be very well suited to deal with the high
levels

of uncertainties present in the majority of real
-
world applications. General type
-
2 FLSs are expected to further extend this
capability. However, the immense computational complexities associated with general type
-
2 FLSs have, until recently, prevented
th
eir application to real
-
world control problems. This paper aims to address this problem by the introduction of a complete
representation framework, which is referred to as zSlices
-
based general type
-
2 fuzzy systems. The proposed approach will lead to
a sig
nificant reduction in both the complexity and the computational requirements for general type
-
2 FLSs, while it offers the
capability to represent complex general type
-
2 fuzzy sets. As a proof
-
of
-
concept application, we have implemented a zSlices
-
based
gene
ral type
-
2 FLS for a two
-
wheeled mobile robot, which operates in a real
-
world outdoor environment. We have evaluated the
computational performance of the zSlices
-
based general type
-
2 FLS, which is suitable for multiprocessor execution. Finally, we have
com
pared the performance of the zSlices
-
based general type
-
2 FLS against type
-
1 and interval type
-
2 FLSs, and a series of results
is presented which is related to the different levels of uncertainty handled by the different types of FLSs.

An Interval Type
-
2 F
uzzy
-
Neural Network With Support
-
Vector Regression for Noisy Regression Problems,
Chia
-
Feng Juang
,

Ren
-
Bo Huang
,

Wei
-
Yuan Cheng, Page(s): 686


699

This paper proposes an interval type
-
2 fuzzy
-
neural network with support
-
vector regression (IT2FNN
-
SVR) for
noisy regression
problems. The antecedent part in each fuzzy rule of an IT2FNN
-
SVR uses interval type
-
2 fuzzy sets, and the consequent part is of the
Takagi
-
Sugeno
-
Kang (TSK) type. The use of interval type
-
2 fuzzy sets helps improve the network's noise res
istance. The network
inputs may be numerical values or type
-
1 fuzzy sets, with the latter being used for further improvements in robustness. IT2FNN
-
SVR
learning consists of both structure learning and parameter learning. The structure
-
learning algorithm is

responsible for online rule
generation. The parameters are optimized for structural
-
risk minimization using a two
-
phase linear SVR algorithm in order to endow
the network with high generalization ability. IT2FNN
-
SVR performance is verified through compari
sons with type
-
1 and type
-
2
fuzzy
-
logic systems and other regression models on noisy regression problems.

Type
-
2 Fuzzy Sets as Functions on Spaces, Aisbett, J. Rickard, J.T. Morgenthaler, D.G., Page(s): 841
-

844

For many readers and potential authors, typ
e
-
2 (T2) fuzzy sets might be more readily understood if expressed by the use of standard
mathematical notation and terminology. This paper, therefore, translates constructs associated with T2 fuzzy sets to the lang
uage
of functions on spaces. Such translat
ions may encourage researchers in different disciplines to investigate T2 fuzzy sets, thereby
potentially broadening their application and strengthening the underlying theory.

2010, Vol. 18, No.
3


On Finity, Countability, Cardinalities, and Cylindric Ext
ensions of Type
-
2 Fuzzy Sets in Linguistic
Summarization of Databases, Niewiadomski, A., Page(s): 532
-

545

The discussion in this paper is closely related to our idea of
a type
-
2 linguistic summary of a database

presented by Niewiadomski,
which is intende
d to be an efficient tool of knowledge discovery from large databases. In that approach, we put emphasis on
algorithms and applications of type
-
2 fuzzy sets in summarizing databases. Hence, we have implicitly assumed and considered
linguistic expressions r
epresented by sets in discrete (or even finite) universes of discourse. Now, in this paper, we discuss more
thoroughly some of the properties and formal aspects of both discrete and continuous type
-
2 fuzzy sets that represent the elements
of linguistic sum
maries, i.e., quantifiers, summarizers, and/or qualifiers. We underline differences between type
-
2 fuzzy sets in finite
and in infinite (usually continuous) universes of discourse and, henceforth, between their cardinalities and similar measures
, which
are

used in evaluating the goodness (quality) of type
-
2 linguistic summaries. In addition, we define new imprecision measures of
linguistic expressions represented by type
-
2 fuzzy sets, and propose a definition of the cylindric extension of a type
-
2 fuzzy set
.
Finally, we apply these new concepts to generalize algorithms of type
-
2 linguistic summarization and propose some new features in
comparison to our previous approach.

FSVM
-
CIL: Fuzzy Support Vector Machines for Class Imbalance Learning, Batuwita, R.
,

Pal
ade, V., Page(s):
558
-

571

Support vector machines (SVMs) is a popular machine learning technique, which works effectively with balanced datasets. Howev
er,
when it comes to imbalanced datasets, SVMs produce suboptimal classification models. On the other h
and, the SVM algorithm is
sensitive to outliers and noise present in the datasets. Therefore, although the existing class imbalance learning (CIL) meth
ods can
make SVMs less sensitive to class imbalance, they can still suffer from the problem of outliers a
nd noise. Fuzzy SVMs (FSVMs) is a
variant of the SVM algorithm, which has been proposed to handle the problem of outliers and noise. In FSVMs, training example
s are
assigned different fuzzy
-
membership values based on their importance, and these membership
values are incorporated into the SVM
learning algorithm to make it less sensitive to outliers and noise. However, like the normal SVM algorithm, FSVMs can also su
ffer
from the problem of class imbalance. In this paper, we present a method to improve FSVMs
for CIL (called FSVM
-
CIL), which can be
used to handle the class imbalance problem in the presence of outliers and noise. We thoroughly evaluated the proposed FSVM
-
CIL
method on ten real
-
world imbalanced datasets and compared its performance with five exis
ting CIL methods, which are available for
normal SVM training. Based on the overall results, we can conclude that the proposed FSVM
-
CIL method is a very effective method
for CIL, especially in the presence of outliers and noise in datasets.

2010, Vol. 18,

No.
2


Fuzzy Clustering With Viewpoints, Pedrycz, W.
,
Loia, V.
,

Senatore, S., Page(s): 274
-

284

In this study, we introduce a certain knowledge
-
guided scheme of fuzzy clustering in which domain knowledge is represented in the
form of so
-
called viewpoints
. Viewpoints capture a way in which the user introduces his/her point of view at the data by identifying
some representatives, which, being treated as externally introduced prototypes, have to be included in the clustering process
. More
formally, the viewp
oints (views) augment the original, data
-
based objective function by including the term that expresses distances
between data and the viewpoints. Depending upon the nature of domain knowledge, the viewpoints are represented either in a pl
ain
numeric format

(considering that there is a high level of specificity with regard to how one establishes perspective from which the
data need to be analyzed) or through some information granules (which reflect a more relaxed way in which the views at the da
ta
are being
expressed). The detailed optimization schemes are presented, and the performance of the method is illustrated through
some numeric examples. We also elaborate on a way in which the clustering with viewpoints enhances fuzzy models and
mechanisms of decision

making in the sense that the resulting constructs reflect the preferences and requirement that are present
in the modeling environment.

A Novel Hierarchical
-
Clustering
-
Combination Scheme Based on Fuzzy
-
Similarity Relations, Mirzaei, A.
,

Rahmati, M., Page
(s): 27
-

39

Clustering
-
combination methods have received considerable attentions in recent years, and many ensemble
-
based clustering
methods have been introduced. However, clustering
-
combination techniques have been limited to Ã
‚¿flat¿ clustering
comb
ination, and the combination of hierarchical clusterings has yet to be addressed. In this paper, we address and formalize the

conce
pt of hierarchical
-
clustering combination and introduce an algorithmic framework in which multiple hierarchical clusterings
c
ould be easily combined. In this framework, the similarity
-
based description matrices of input hierarchical clusterings are
aggregated into a transitive consensus matrix in which the final hierarchy could be formed. Empirical evaluation, by using po
pular
a
vailable datasets, confirms the superiority of combined hierarchical clustering introduced by our method over the standard (s
ingle)
hierarchical
-
clustering methods.

2010, Vol. 18, No.
1


Fuzzy PCA
-
Guided Robust
k
-
Means Clustering, Honda, K.
,

Notsu, A.
,

Ic
hihashi, H., Page(s): 67
-

79

This paper proposes a new approach to robust clustering, in which a robust k
-
means partition is derived by using a noise
-
rejection
mechanism based on the noise
-
clustering approach. The responsibility weight of each sample for
the k
-
means process is estimated
by considering the noise degree of the sample, and cluster indicators are calculated in a fuzzy principal
-
component
-
analysis (PCA)
guided manner, where fuzzy PCA
-
guided robust k
-
means is performed by considering responsibil
ity weights of samples. Then, the
proposed method achieves cluster
-
core estimation in a deterministic way. The validity of the derived cluster cores is visually
assessed through distance
-
sensitive ordering, which considers responsibility weights of samples
. Numerical experiments
demonstrate that the proposed method is useful for capturing cluster cores by rejecting noise samples, and we can easily asse
ss
cluster validity by using cluster
-
crossing curves.