The Graduate School of Turku Centre for Computer Science (TUCS)
offers a programme for gaining the Doctoral (Ph.D.) degree in Computer
Science. It is open for students from everywhere.
The teaching language of the school is English.
TUCS is situated in the Turku Technology Center.
The Computer Science departments of the three universities in Turku
are all located in this center, together with a number of other
university departments in Biology, Biochemistry, Physics and so on.
The center also houses a large number of high-tech companies in
Computer and Communication Technology, Electronics, Biotechnology
and Material Sciences.
The Technology Center provides a fertile research environment with
ample opportunities for collaboration between industry and the
academia.
There are several research groups at TUCS:
Programming Methodology,
Strategic Information Systems Planning,
Theory Group: Mathematical Structures in Computer Science,
ComputationalIntelligence for Business,
Intelligent Technologies,
Coding Theory,
Algorithmics,
Probabilistic Algorithms and Software Quality,
Information Systems Research
and 29 different courses for Spring 1996.
It is now the third semester that I deliver a [5 credits, 30 hours]
course on "Neural Fuzzy Systems (NFS)" at TUCS.
A description of my recent course can be found under the URL
http://www.abo.fi/~rfuller/nfs.html
Last Fall I had 25 students from three different universities.
The basic principles and problems with teaching Neural Fuzzy Systems
include
1. Students have different preliminary knowledge.
Usually they have a little knowledge in Neural Nets, and
"heard about" fuzzy logic.
2. The extension principle. During the first time I realized
that students are not able to understand it. I do not include
the extension principle in the course, and it does not matter.
3. The compositional rule of inference. At least 2 lectures
are necessary. To understand the meaning of sup-min composition
of a fuzzy relation and a fuzzy set is the key issue here.
Graphical and discrete case illustrations are very important.
4. Fuzzy rule-based systems. Unlike in crisp systems here
each rule is applicable and fires with a certain firing strength.
The key issues here are: degree of match, individual rule output,
overall system output, aggregation operators.
5. Fuzzy logic controllers are universal approximators. We
try to approximate a partially unknown input-output function.
Our knowledge is given in the form of fuzzy IF-THEN rules.
Explicit input-output functions for Tsukomato's and
Sugeno's reasoning mechanism should be displayed. Larsen's and
Mamdani's inference mechanisms should be illustrated by a
public domain software.
6. Supervised learning algorithms. It should be emphasized that
(similarly to fuzzy rule-based systems) we try
to approximate a partially unkown input-output function.
Our knowledge is given in the form of crisp input-output
training patterns. The goal is to minimize the error
between the computed and the desired outputs. We use
descent-type methods for minimization of the error
function. One lecture is needed to explain descent-type
methods for optimization.
7. Fuzzy neurons. One should explain that by using fuzzy-like
operations in computing the input and output of a neuron
we extend the approximation ability of standard error
backpropagation networks.
8. Hybrid neural nets (ANFIS). Several examples should be
introduced to show how the shape parameters of fuzzy numbers are
changing during the training process.
9. Fuzzy classification. We are given a crisp training set,
and we want to create a set of fuzzy IF-THEN rules
correctly classifying most of the items.
The key issues here are: Find an initial fuzzy partition of
input and output spaces.
Derive the initial fuzzy rules. Optimize the rule-base
with a hybrid neural net.
I always use the public domain software NEFCLASS
(from Detlef Nauck) to illustrate the process of learning.
10. The case study. Students should create a connectionist
model for the portfolio evaluation problem. For simplicity,
Tsukamoto's or Sugeno's fuzzy reasoning mechanism is
applied with linear membership functions.
My experience shows that students (ranging from chmical engineers
to business administrators) are primarily interested in possible
applications of NFS in their future jobs. Furthermore, public
domain softwares and demo versions of commercional products
are of great importance in demonstration of the principles of
"neuro-fuzzy thinking".
With my best wishes.
Robert Fuller