Computational Intelligence - August 2012 - 51

shoulder, to triangular to right shoulder. These extreme variations resulted
in the value of the overall centroid
varying by less than 1%. More research
is need to understand this result in the
context of a working system and compare the result to a similar experiment
which does not make use of the alphaplane representation.
2.2.2. Learning Algorithms
Given the variety of forms and representations available to construct type-2
fuzzy sets an intuitive and appealing
approach is learning the membership
functions from data. Robert John's early
work looked how an ANFIS like
approach might be applied to general
type-2 fuzzy systems. Mendel et al
showed how to compute derivatives for
interval type-2 fuzzy systems and how
to perform supervised learning for such
systems using gradient descent type
algorithms [16, 21]. Hani Hagras et al
[27, 11] have used evolutionary computation techniques to successfully tune
interval type-2 fuzzy systems in a range
of engineering applications. Almaraashi
et al are conducting ongoing work [3, 2]
into the use of simulated annealing for
tuning type-2 fuzzy systems.
In any work which is learning type2 sets it is necessary to define a set of
parameters which define the form of the
fuzzy set. Learning algorithms tune
these parameterized type-2 fuzzy systems, as happens in the majority of
tuned type-1 systems. The choice of
parameters used to define a type-2 set
restricts the form the membership function can take and restricts the form each
secondary membership function can
take. There are three approaches which
can be taken to parameterize a general
type-2 fuzzy set:
❏ No slicing. The membership function is defined by a set of parameters
across three dimensions. In practice
this is often done with a curried
function, a function which it self
returns a function i.e., n Au : X "
[0, 1] " [0, 1] .
❏ Vertical slicing. The membership
function is defined in two parts; a
principal membership function and a

Given the variety of forms and representations
available to construct type-2 fuzzy sets an intuitive
and appealing approach is learning the membership
functions from data.
secondary membership function for
each point on the principal membership function. The principal membership function is a type-1 function
and make use of standard type-1
parameterization. Each secondary
membership function is also a type-1
function and makes use of standard
type-1 parameterization.
❏ Alpha-plane/Z-slicing. The
membership function is sliced into a
number of interval type-2 fuzzy sets.
Each of these interval type-2 fuzzy
sets can be defined using appropriate parameters.
The research challenge for learning
type-2 systems is to investigate which
learning algorithms tune the sets efficiently for a given problem. Another
important question is which parameter
model used to define a type-2 fuzzy
set gives the right level of flexibility
without raising the level of computational complexity of learning prohibitively high. Information measures for
type-2 fuzzy sets are in their infancy
[13, 31] but are likely to inform research in this area.
2.3. Applications

2.3.1. Application Areas
Type-2 fuzzy system have shown
improved performance over type-1 systems in engineering applications where
high levels of uncertainty are present
[10, 5]. This improved performance is
not guaranteed. There has been a great
deal of debate about the performance of
type-2 systems, largely centered around
engineering applications. This debate is
in danger of missing a key objective for
type-2 fuzzy systems, effectively modelling of linguistic uncertainties. The type2 community have demonstrated in a
range of applications that it is possible to
construct a type-2 systems which gives

improved performance over a type-1
system. Perhaps a worthy research questions is: does an application exist which
a type-2 fuzzy system can do which a
type-1 system cannot? In our opinion
whilst this killer application does not
exist then the debate of the usefulness of
type-2 will continue.
The majority of the applications of
type-2 reported in the literature are in
areas where type-1 has previously been
applied. The ambition of researchers
working on type-2 applications should
be to identify areas where type-2 systems haven't worked and investigate
these areas. Areas where there are high
levels of uncertainty, particularly linguistic uncertainty, would appear to be a
good place to begin. These areas include
medical decision making, scheduling
and timetabling and complex geographical information systems.
2.3.2. Performance Comparisons
Claims made about applications of
type-2 fuzzy systems need be substantiated by hard evidence. In an increasing number of papers in the field,
rigorous qualitative analysis is being
used to support claims being made
about type-2 systems. Clearly a thorough analysis of experimental results is
the best way of supporting claims
made in a piece of research, and
vital when performing comparisons.
However, the other aspect of any
experiment is the way in which the
experiment is setup to ensure like is
being compared with like.
When comparing a type-1 fuzzy
system with a type-2 fuzzy system the
question of system equality is a complex one. The simplistic answer is to
use systems with the same number of
sets and rules [4, 5]. When this ap proach has been taken the type-2 system often gives a better performance

AUGUST 2012 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE

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