AN ALGORITHM FOR RULE GENERATION IN FUZZY EXPERT

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AN ALGORITHM FOR RULE GENERATION IN FUZZY
EXPERT SYSTEMS Dmitry A. Kropotov, Dmitry P.
Vetrov Russian Federation, Moscow Dorodnicyn
Computing Centre of the Russian Academy of
Sciences dkropotov_at_yandex.ru, vetrovd_at_yandex.ru
PRINCIPAL SCHEME OF FUZZY EXPERT SYSTEM
WHAT IS FUZZY EXPERT SYSTEM (FES)?
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. Fuzzy sets allow
to represent data in linguistic terms, i.e. to
use such notions as big, average, more or
less appropriate in mathematical deduction
inference. Fuzzy expert systems combine fuzzy
sets for data representation, expert knowledge
(set of inference rule of type if, then) and
fuzzy logic for information processing into one
elegant conception which can be effectively used
in tasks of function approximation, pattern
classification, forecasting, etc.
The knowledge base is a set of fuzzy inference
rules of type if, then. Such rules reflect
expert knowledge about particular research field
in terms of qualitative relations between
variables. For a example if average temperature
in May is normal and precipitation level is
average, then crop will be good . The knowledge
base is central part of FES.
Knowledge base
Defuzzificator produces the result value using
output fuzzy set.
Qualitative relations

Fuzzyfication
Output fuzzy set
Vector of features
Forecast
Controller of fuzzy calculation
Defuzzificator
Fuzzificator
Fuzzy sets
White Yes Zero
Red No One
Light rose Surely One third
Dark rose Unlikely Two thirds
White Yes Zero
Red No One
Fuzzy controller realizes the mechanism of fuzzy
inference and produces output fuzzy set using
input fuzzy sets and relations between input and
output variables. Calculations go under selected
fuzzy logic.
Fuzzificator transforms input crisp values into
their fuzzy representation.
In comparison with other expert systems, the ones
based on fuzzy logic provide continuous outputs
for continuous inputs. This means they are stable
to random fluctuations and inaccuracies in data.
From the other hand they have advantages over
classical systems of data-mining and knowledge
discovery as they operate with not the abstract
numerical relations but with linguistic rules
formulated in human language. The last allows the
expert to make changes in knowledge base and
provides deeper understanding of the hidden
process.
WHAT RULES TO GENERATE?
ALGORITHM FOR RULE GENERATION
We would like to generate the rules according to
the set of precedents (training sample) which
will contain significant information but not
noise. In order to do this, first establish two
main definitions. Representativeness of the rule
the rate of objects in the sumption of the
rule Effectiveness of the rule the rate of
objects from the sumption that satisfy the rule
itself.
First we form all possible rules of 1st order
(Set A) e.g.
Consider the following rule IF the ball is in
the basket THEN its color is red
Set A
Unrepresentative rules with
are excluded from further processing
New rule
The rule is neither effective nor representative
The rule is effective but not representative
The rule is representative but not effective
The rule is both representative and effective
Tolerable rules which have both high
effectiveness and representativeness are
collected in Set D. It is repository of ready
rules to be used in fuzzy expert system
Set C
GENERALIZATION CAPABILITIES
All the remained Rules are intersected In pairs
forming rules Of 2nd order
In order the system to perform good, we should
guarantee high generalization capability. In
other words our aim is to avoid noise rules and
to catch all significant regularities contained
in data. This can be done by adjusting the
thresholds for effectiveness and
representativeness. Theory of statistical
solutions allows to establish relations between
these thresholds. The higher representativeness
the less effectiveness is needed to be sure the
rule does not contain noise. Moreover it is
possible to establish lower effectiveness
threshold that separates the rules which would
not become tolerable under any intersections and
hence could be rejected in order to reduce the
time of rule generation.
Set D
Set B is formed by all potential rules of kth
order which are still representative but not
effective yet
Set B
New rule
All the remained rules that have common parts in
sumption are intersected forming rules of higher
order
The proposed algorithm allows to generate ALL
tolerable rules. Numerous experiments showed that
expert system exploiting such rules doesnt
suffer from overfitting. In other words
statistical determination of significance of each
rule provides good generalization. There is
another good property of such knowledge
generation algorithm. Because of special method
of counting effectiveness and representativeness
we may easily adapt correction methods such as
boosting.
Set E contains all potential rules of k1 order.
All such rules (if any) will form new Set B on
next iteration. If Set E is empty then the
process of rule generation is finished
Set E
Reweighting of objects
EXPERIMENTAL RESULTS
The fuzzy expert system with proposed algorithm
for generation of fuzzy rules was tested on both
classification and forecasting tasks. In
recognition case FES results were compared with
q-nearest neighbours (QNN), support vector
machines (SVM), committee of linear
classificators (LM), test algorithm (TA), linear
Fisher discriminant (LDF) and multi-layer
perceptron (MLP). In area of forecasting FES was
compared with multiple linear regression (MLR)
and MatLab fuzzy logic toolbox (FLT).
EXAMPLES OF GENERATED RULES FOR FOOTBALL TASK 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
The second task - prediction of magnetic
amplitude oscillations in accelerating cavity of
a klystron. The information about
amplitude oscillations is extremely useful
and helps to compensate oscillation
thus providing the stability of whole system.
The necessary data was taken from Hamburg linear
accelerator, DESY. The source information was
oscillations on other cavities within the
same klystron. FES results appeared to
be practically satisfactory there were obtained
high accuracy in oscillations direction and
appropriate accuracy with amplitude. MatLab FLT
results suffered from sufficient overtraining.
MLR provided low accuracy even with oscillations
direction.
The third task detection of
drug- addiction degree based on pupillogram
(reaction of pupil on light flare). The necessary
data was kindly provided by "Iritech inc.
In comparison with other classification techniques
there was obtained the following accuracy (2
classes, 18 features, 450/450 objects
in training/test set) FES 89 TA
79 QNN 85.2 LDF 81 SVM 85.7 MLP
84.6 LM 83
Drug Meter DM2010 device for pupillogram
capturing
The first task - to predict the places of teams
in Russian football championship for two
years according to the tournament table with
one column been removed. In the removed
column there was a number of scores won by
each team. There were 16 teams in the table.
FES forecast included only one violation of
monotony with respect to three violations
obtained by multiple linear regression. The sum
of squared deviation was 21.936 with FES and
38.056 with MLR.
The 9-cell niobium cavity for TESLA accelerator
(Hamburg, DESY)
The Thomson TH1801 multibean klystron