FUZZY LOGIC IS

1
ONE APPROACH TO FUZZY EXPERT SYSTEMS
CONSTRUCTION. Dmitry A. Kropotov, Dmitry P.
Vetrov. Russia, Moscow, 119991 Dorodnicyn
Computing Centre of the Russian Academy of
Sciences.
FUZZY LOGIC IS Fuzzy logic is based on the
theory of fuzzy sets, proposed by Zadeh in 1965.
The main idea of this theory is generalization of
classical sets theory for the continuous case.
This means that the objects may partly belong to
the sets with different degree of membership. In
other words fuzzy set can be associated with
their membership function
. It is easy to define the generalized logical
operations, like, for example, conjunction and
disjunction, over the fuzzy sets as minimum and
maximum of membership functions. Fuzzy
implication can be defined by different ways, but
in present report, we use Mamdani scheme by
taking minimum of membership functions from the
sumption and using it as a belonging degree to
the result set of the rule.
Crisp set Between 2 and 5
Fuzzy sets
Qualitative relations
Knowledge base
Feature values
Mamdanis implication scheme for the rule IF
Average temperature is Normal AND Precipitations
are Average THEN Crop is Good
Fuzzy set Approx. between 2 and 5
Fuzzy set
Forecast
A GOOD WAY OF CONSTRUCTING EXPERT SYSTEMS. The
main advantage of fuzzy logic is its ability to
operate in natural terms for a user. By defining
linguistic variables, one may construct fuzzy
rules. Fuzzy expert system works by interpreting
input data as linguistic variables, implicating
the necessary fuzzy rules and then defuzzyfying
the result. But THERE APPEAR TWO BOTTLENECKS IN
BUILDING FUZZY EXPERT SYSTEMS
Fuzzy rules
Defuzzyficator
Fuzzyficator
Principal scheme of fuzzy expert system.
HOW TO OBTAIN THE SHAPES OF FUZZY SETS It is
usually difficult to form the exact shapes of
fuzzy sets for the expert. It is not obvious how
fuzzy they should be. In order to solve this
problem we use so-called (a, b)-parameterization.
Let the shapes of membership functions belong to
the parameterized family of isosceles trapeziums.
The location of fuzzy set is defined by the
approximate borders which are
part of the partition of numerical axis, that can
be easily assigned by the user. In this case
aparameter can be interpreted as a crossing
degree of the sets, and bparameter shows the
fuzziness of set. To reduce the number of
coefficients to be optimized, we use an
assumption that pair (a, b) is responsible for
the properties of the whole feature rather than a
single set. In this case we may find the
coefficients by solving optimization task on the
learning sample.
Representation - the rate of objects in the
sumption of the rule.
(a,b) parameterization of fuzzy sets shapes
Effectiveness - the rate of objects from the
sumption that satisfy the rule.
AND HOW TO GET THE NECESSARY FUZZY RULES. In
many areas one hasnt enough knowledge about the
process being researched in order to form
linguistic rules. If there are some precedents
with known forecasts, the necessary rules may be
generated automatically. The proposed algorithm
is based on the two notions representation and
effectiveness of the rule. The first shows the
rate of objects being involved in the
consideration, while the second shows the rate of
objects that satisfy the rule. The more
representation and effectiveness are, the better
is the rule. In fact, we want to increase the
effectiveness at least to some threshold, holding
the representation above the predefined level. To
do this, we fuse (restrict) several rules to one
of higher order, by conjuncting their sumptions,
until we exhaust the set of potential rules.
During such process the representation becomes
lower, but the effectiveness may become higher.
All rules that have both representation and
effectiveness higher than corresponding
thresholds are accepted rules that are not
enough representative are rejected and all other
rules are used for further fusion
Each rule can be represented as a point in
effectiveness/representation plane.
SUCH SYSTEM CAN BE USED EITHER FOR
FORECASTING To forecast continuous values, we
find (a, b) coefficients by using least squares
method. In fact we just minimize the sum of
squared deviations from the correct answer. And
to calculate the forecasted variable according to
the given number of fuzzy rules, use the centre
of gravity defuzzification method. This mode was
used to predict the places of football teams in
Russian Championship according to the tournament
table (won scores were excluded) and for
forecasting magnetic field oscillations in
cavities of accelerating klystrons (DESY,
Hamburg). The results received by described
method (program ExSys) were compared with linear
regression for football and Matlab fuzzy logic
toolbox for cavities. It appeared that in some
cases fuzzy logic works better than linear
regression even for such simple and obviously
linear tasks like finding the place of the team
by the tournament table. As for Matlab toolbox,
the system tends to overfit to the learning data
even after the use of independent precedents,
which had to prevent overtraining.
Russian football championship. Sum of squared
deviations for linear regression is 38.056, for
ExSys is 21.936.
Oscillations of magnetic field amplitude. Sum of
squared deviations for MatLab is 0.405, for ExSys
is 0.274.
Melanoma 48/32 objects, 33 features, 3
classes. Phoneme 2200/1404 objects, 6 features,
2 classes. Liver 170/150 objects, 8 features, 7
classes.
OR CLASSIFICATION. If one needs to predict the
value which belong to the finite set (i.e.
classification task), the quality functional is
just the rate of misclassified objects. And as
defuzzification method, we use defuzzification by
mode. In other words the object is classified to
the set with the maximal value of membership
function. The system in such mode was tested on
several tasks and was compared with numerous
recognition methods linear Fisher discriminant
(LDF), q-nearest neighbors (QNN), test algorithm
(TA), committee of hyperplanes (LM), support
vector machines (SVM) and multilayer perceptron
(MLP). The results of work are presented in the
table.
Results of classifications
THEORY OF STATISTICAL SOLUTIONS IS USED TO FIND
THE NECESSARY THRESHOLDS AND PREVENT
OVERFITTING. Its clear that changing
representation threshold, we may thus regulate
the number of discovered fuzzy rules and hence
the time of rule generation. Unfortunately we
cant do the same directly with the effectiveness
threshold. Making it too low forces computer to
generate a lot of parasitic rules, which
contain no useful information. As a result the
expert system suffers from overtraining and
degrades greatly. To avoid this, the apparatus of
mathematical statistics was applied. This allowed
to define the effectiveness threshold by finding
the upper bound of confidence interval after
fixing the rate of parasitic rules we would
like to exclude. By varying the level of
confidence we get different thresholds and may
regulate the number of discovered rules. To
reduce the time of rule generation, the lower
effectiveness bound is calculated. If the
effectiveness of the rule if less than this
bound, it cannot be raised to the demanded level
without decreasing representation value too low
(lower than corresponding threshold). With these
remarks, the effectiveness/representation plane
becomes as shown on the figure.
Fuzzy Expert System
Low and upper bounds for effectiveness
THE PROPOSED SYSTEM CAN BE USED IN MULTIPLE
MODES Depending on what the expert can do, the
system described above, may work in different
modes. When there is no prior information about
the task, all necessary steps can be done
automatically. The user only needs to input the
number of fuzzy sets to be generated for each
variable and to give them names. If needed,
expert can define the approximate borders of sets
himself. The rules are either generated according
to the learning table or/and entered by the
expert. He can also change the sumptions or
weights of some rules if necessary. The shapes of
sets are either set or found as a result of
optimization procedure. The modes, that system
support, may vary from the most autonomous way,
to the case, when all fuzzy sets and rules are
known for the expert. In the last case system
acts as classical fuzzy controller.
Fuzzy rules
Feature partition
Fuzzy sets
Auto generation
Auto partitioning
Optimization
Manual partitioning
Manual input
Manual definition
Different modes of ExSys
NOT ONLY FOR FORECASTING, BUT ALSO FOR
UNDERSTANDING THE NATURE OF PROCESS. Using rule
generation option, we may not only use the
discovered rules for further forecasting, but
also examine them for understanding the nature of
the process being researched. Here are some rules
extracted from the football tournament table.
Such knowledge can be extremely useful if some
of known features can be managed. Knowing the
influence they have on the hidden value, we may
effectively control the process by adjusting the
features that are available for changing.
IF Dropped goals are Not many AND Losses are Few
THEN Rank is High IF Wins are Not few AND Scored
goals are Many AND Draws are Some THEN Rank is
Very high IF Scored goals are Few THEN Rank is
Low IF Dropped goals are Not few AND Draws are
Many THEN Rank is Medium
Examples of generated fuzzy rules for football
ranks predictions