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Industrial Applications of
Neuro-Fuzzy Networks
Prof. Dr. Rudolf Kruse
University of Magdeburg
Faculty of Computer Science
Magdeburg, Germany
rudolf.kruse@cs.uni-magdeburg.de
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Example: Continously Adapting Gear Shift Schedule in VW New Beetle
classification of driver / driving situation gear shift
by fuzzy logic computation
fuzzification inference defuzzifi- interpolation
machine cation
accelerator pedal
filtered speed of
determination
accelerator pedal
of speed limits
rule for shifting gear
number of sport
base into higher or selection
changes in factor [t]
lower gear
pedal direction
depending on
sport factor
sport factor [t-1]
März 2001 Rudolf Kruse
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Continously Adapting Gear Shift Schedule: Technical Details
Mamdani controller with 7 rules
Optimized program AG4
24 Byte RAM
702 Byte ROM
} on Digimat
Runtime 80 ms
12 times per second a new sport factor is assigned
How to generate knowledge automatically from data?
März 2001 Rudolf Kruse
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Learning from Examples (Observations, Databases)
Statistics: parameter fitting, structure
identification, inference method,
model selection
Machine Learning: computational learning (PAC
learning), inductive learning, learning
decision trees, concept learning, ...
Neural Networks: learning from data
Cluster Analysis: unsupervised classification
Learning Problem is transformed into an optimization problem.
How to use these methods in fuzzy systems?
März 2001 Rudolf Kruse
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Function Approximation with Fuzzy Rules
if x is large then y is large
y
output value
x
current input value
März 2001 Rudolf Kruse
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How to Derive a Fuzzy Controller Automatically from Observed Process Data
ut
• Function approximation
tp
ou
input
current input value
• Perform fuzzy cluster analysis of input-output data (FCM, GK, GG, ...)
• Project clusters
• Obtain fuzzy rules of the kind: „If x is small then y is medium“
März 2001 Rudolf Kruse
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Fuzzy Cluster Analysis
Classification of a given data set X = {x1, ..., xn} p into c
clusters.
Membership degree of datum xk to class i is uik.
Representation of cluster i by prototype vi p.
Formal: Minimisation of functional:
JX, U, v uik d2 v i, x k
c n
m
i1 k 1
under constraints n
u ik
0 for all i , ..., c
1
k 1
c
u ik
1 for all k , ..., n
1
März 2001
i1
Rudolf Kruse
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Simplest Algorithm: Fuzzy-c-Means (FCM)
d v i , x k v i x k
2 2
Iterative Procedure (with random initialisation of prototypes vi)
u
n
m
ik xk
1
ui k and vi k 1
u
1 n
d2 v i , x k
m
c m 1
d2 v , x
j 1
k 1
ik
j k
FCM is searching for equally large clusters in form of (hyper-)balls.
März 2001 Rudolf Kruse
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Examples
März 2001 Rudolf Kruse
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Fuzzy Cluster Analysis
Fuzzy C-Means: simple, looks for spherical
clusters of same size, uses Euclidean distance
Gustafson & Kessel: looks for hyper-ellipsoidal
clusters of same size, distance via matrices
Gath & Geva: looks for hyper-ellipsoidal clusters
of arbitrary size, distance via matrices
Axis-parallel variations exist that use diagonal
matrices (computationally less expensive and less
loss of information when rules are created)
März 2001 Rudolf Kruse
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Fuzzy Cluster Analysis with DataEngine
März 2001 Rudolf Kruse
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Construct Fuzzy Sets by Cluster Projection
u(x) Approximation by a
triangular fuzzy set
1
Convex hull of the discrete
degrees of membership
Connection of the discrete
degrees of membership
x
Projecting a cluster means to project the degrees of membership
of the data on the single dimensions: Histograms are obtained.
März 2001 Rudolf Kruse
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FCLUSTER: Tool for Fuzzy Cluster Analysis
März 2001 Rudolf Kruse
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Introduction
Building a fuzzy system requires
prior knowledge (fuzzy rules, fuzzy sets)
manual tuning: time consuming and error-prone
Therefore: Support this process by learning
learning fuzzy rules (structure learning)
learning fuzzy set (parameter learning)
Approaches from Neural Networks can be used
März 2001 Rudolf Kruse
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Learning Fuzzy Sets: Problems in Control
Reinforcement learning must be used to compute an error
value
(note: the correct output is unknown)
After an error was computed, any fuzzy set learning
procedures can be used
Example: GARIC (Berenji/Kedhkar 1992)
online approximation to gradient-descent
Example: NEFCON (Nauck/Kruse 1993)
online heuristic fuzzy set learning using a
rule-based fuzzy error measure
März 2001 Rudolf Kruse
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März 2001 Rudolf Kruse
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Neuro-Fuzzy Systems in Data Analysis
Neuro-Fuzzy System:
System of linguistic rules (fuzzy rules).
Not rules in a logical sense, but function
approximation.
Fuzzy rule = vague prototype / sample.
Neuro-Fuzzy-System:
Adding a learning algorithm inspired by neural
networks.
Feature: local adaptation of parameters.
März 2001 Rudolf Kruse
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Example: Prognosis of the Daily Proportional Changes of the DAX at
the Frankfurter Stock Exchange (Siemens)
Database: time series from 1986 - 1997
DAX Composite DAX
German 3 month interest rates Return Germany
Morgan Stanley index Germany Dow Jones industrial index
DM / US-$ US treasury bonds
Gold price Nikkei index Japan
Morgan Stanley index Europe Price earning ratio
März 2001 Rudolf Kruse
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Fuzzy Rules in Finance
Trend Rule
IF DAX = decreasing AND US-$ = decreasing
THEN DAX prediction = decrease
WITH high certainty
Turning Point Rule
IF DAX = decreasing AND US-$ = increasing
THEN DAX prediction = increase
WITH low certainty
Delay Rule
IF DAX = stable AND US-$ = decreasing
THEN DAX prediction = decrease
WITH very high certainty
In general
IF x1 is m1 AND x2 is m2
THEN y = h
WITH weight k
März 2001 Rudolf Kruse
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Classical Probabilistic Expert Opinion Pooling Method
DM analyzes each source (human expert, data +
forecasting model) in terms of (1) Statistical accuracy,
and (2) Informativeness by asking the source to asses
quantities (quantile assessment)
DM obtains a “weight” for each source
DM “eliminates” bad sources
DM determines the weighted sum of source outputs
Determination of “Return of Invest”
März 2001 Rudolf Kruse
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E experts, R quantiles for N quantities
each expert has to asses R·N values
stat. Accuracy:
R si
C 1 R 2 N I s, p ,
2
I s, p si ln
i 0 p
information score:
1 N R 1 pr 1
I lnvi, R 1 vi,o pr 1 ln
N i 1 r 1 vi,r vi,r 1
ce I e id ce
weight for expert e: we
E
e1ce I e id e ce
E
outputt= we outputt
e
e 1
T
roi = yt sign outputt
DM
t 1
März 2001 Rudolf Kruse
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Formal Analysis
Sources of information
R1 rule set given by expert 1
R2 rule set given by expert 2
D data set (time series)
Operator schema
fuse (R1, R2) fuse two rule sets
induce(D) induce a rule set from D
revise(R, D) revise a rule set R by D
März 2001 Rudolf Kruse
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Formal Analysis
Strategies:
fuse(fuse (R1, R2), induce(D))
revise(fuse(R1, R2), D)
fuse(revise(R1, D), revise(R2, D))
Technique: Neuro-Fuzzy Systems
Nauck, Klawonn, Kruse, Foundations of Neuro-Fuzzy
Systems, Wiley 97
SENN (commercial neural network environment, Siemens)
März 2001 Rudolf Kruse
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From Rules to Neural Networks
1. Evaluation of membership degrees
2. Evaluation of rules (rule activity)
x j 1 m c( ,js) xi
n r
ml: IR [0,1] ,
l D
3. Accumulation of rule inputs and normalization
kl m l x
NF: IR IR, x l 1 wl
n r
k j m j x
r
j 1
März 2001 Rudolf Kruse
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Neuro-Fuzzy Architecture
März 2001 Rudolf Kruse
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The Semantics-Preserving Learning Algorithm
Reduction of the dimension of the weight space
1. Membership functions of different inputs share their parameters,
e.g.
m dax m cdax
stable stable
2. Membership functions of the same input variable are not allowed to pass
each other, they must keep their original order,
e.g.
m decreasing m stable m increasing
Benefits: the optimized rule base can still be interpreted
the number of free parameters is reduced
März 2001 Rudolf Kruse
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Return-on-Investment Curves of the Different Models
Validation data from March 01, 1994 until April 1997
März 2001 Rudolf Kruse
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A Neuro-Fuzzy System
is a fuzzy system trained by heuristic learning
techniques derived from neural networks
can be viewed as a 3-layer neural network with fuzzy
weights and special activation functions
is always interpretable as a fuzzy system
uses constraint learning procedures
is a function approximator (classifier, controller)
März 2001 Rudolf Kruse
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Learning Fuzzy Rules
Cluster-oriented approaches
=> find clusters in data, each cluster is a rule
Hyperbox-oriented approaches
=> find clusters in the form of hyperboxes
Structure-oriented approaches
=> used predefined fuzzy sets to structure the
data space, pick rules from grid cells
März 2001 Rudolf Kruse
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Hyperbox-Oriented Rule Learning
y
Search for hyperboxes in
the data space
Create fuzzy rules by
projecting the hyperboxes
Fuzzy rules and fuzzy
sets are created at the
same time
x Usually very fast
März 2001 Rudolf Kruse
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Hyperbox-Oriented Rule Learning
y y y y
x x x x
Detect hyperboxes in the data, example: XOR function
Advantage over fuzzy cluster anlysis:
No loss of information when hyperboxes are represented as
fuzzy rules
Not all variables need to be used, don‘t care variables can be
discovered
Disadvantage: each fuzzy rules uses individual fuzzy sets,
i.e. the rule base is complex.
März 2001 Rudolf Kruse
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Structure-Oriented Rule Learning
y
Provide initial fuzzy sets for
large
all variables.
The data space is partitioned
by a fuzzy grid
medium
Detect all grid cells that
contain data (approach by
Wang/Mendel 1992)
Compute best consequents and
small
select best rules (extension by
Nauck/Kruse 1995,
x NEFCLASS model)
small medium large
März 2001 Rudolf Kruse
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Structure-Oriented Rule Learning
Simple: Rule base available after two cycles through the
training data
1. Cycle: discover all antecedents
2. Cycle: determine best consequents
Missing values can be handled
Numeric and symbolic attributes can be processed at the
same time (mixed fuzzy rules)
Advantage: All rules share the same fuzzy sets
Disadvantage: Fuzzy sets must be given
März 2001 Rudolf Kruse
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Learning Fuzzy Sets
Gradient descent procedures
only applicable, if differentiation is possible, e.g.
for Sugeno-type fuzzy systems.
Special heuristic procedures that do not use
gradient information.
The learning algorithms are based on the idea of
backpropagation.
März 2001 Rudolf Kruse
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Learning Fuzzy Sets: Constraints
Mandatory constraints:
Fuzzy sets must stay normal and convex
Fuzzy sets must not exchange their relative
positions (they must not „pass“ each other)
Fuzzy sets must always overlap
Optional constraints
Fuzzy sets must stay symmetric
Degrees of membership must add up to 1.0
The learning algorithm must enforce these
constraints.
März 2001 Rudolf Kruse
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Different Neuro-Fuzzy Approaches
ANFIS (Jang, 1993)
no rule learning, gradient descent fuzzy set learning, function approximator
GARIC (Berenji/Kedhkar, 1992)
no rule learning, gradient descent fuzzy set learning, controller
NEFCON (Nauck/Kruse, 1993)
structure-oriented rule learning, heuristic fuzzy set learning, controller
FuNe (Halgamuge/Glesner, 1994)
combinatorical rule learning, gradient descent fuzzy set learning, classifier
Fuzzy RuleNet (Tschichold-Gürman, 1995)
hyperbox-oriented rule learning, no fuzzy set learning, classifier
NEFCLASS (Nauck/Kruse, 1995)
structure-oriented rule learning, heuristic fuzzy set learning, classifier
Learning Fuzzy Graphs (Berthold/Huber, 1997)
hyperbox-oriented rule learning, no fuzzy set learning, function approximator
NEFPROX (Nauck/Kruse, 1997)
structure-oriented rule learning, heuristic fuzzy set learning, function approx.
März 2001 Rudolf Kruse
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Example: Medical Diagnosis
Results from patients tested for breast cancer
(Wisconsin Breast Cancer Data).
Decision support: Do the data indicate a malignant or a benign
case?
A surgeon must be able to check the classification for
plausibility.
We are looking for a simple and interpretable classifier:
knowledge discovery.
März 2001 Rudolf Kruse
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Example: WBC Data Set
699 cases (16 cases have missing values).
2 classes: benign (458), malignant (241).
9 attributes with values from {1, ... , 10}
(ordinal scale, but usually interpreted as a numerical
scale).
Experiment: x3 and x6 are interpreted as nominal
attributes.
x3 and x6 are usually seen as „important“ attributes.
März 2001 Rudolf Kruse
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Applying NEFCLASS-J
Tool for developing Neuro-Fuzzy Classifiers
Written in JAVA
Free version for research available
Project started at Neuro-Fuzzy Group of University of
Magdeburg, Germany
März 2001 Rudolf Kruse
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NEFCLASS: Neuro-Fuzzy Classifier
Output variables (class labels)
Unweighted connections
Fuzzy rules
Fuzzy sets (antecedents)
Input variables (attributes)
März 2001 Rudolf Kruse
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NEFCLASS: Features
Automatic induction of a fuzzy rule base from data
Training of several forms of fuzzy sets
Processing of numeric and symbolic attributes
Treatment of missing values (no imputation)
Automatic pruning strategies
Fusion of expert knowledge and knowledge obtained
from data
März 2001 Rudolf Kruse
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Representation of Fuzzy Rules
Example: 2 Rules
c1 c2 R1: if x is large and y is small, then class is c1.
R2: if x is large and y is large, then class is c2.
The connections x R1 and x R2
R1 R2
are linked.
small
large
large The fuzzy set large is a shared weight.
x y
That means the term large has always the
same meaning in both rules.
März 2001 Rudolf Kruse
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1. Training Step: Initialisation
Specify initial fuzzy partitions for all input variables
y
large
c1 c2
medium
small x
x y
small medium large
März 2001 Rudolf Kruse
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2. Training Step: Rule Base
Algorithm:
Variations:
for (all patterns p) do
find antecedent A, Fuzzy rule bases can
such that A( p) is maximal; also be created by using
if (A L) then add A to L; prior knowledge, fuzzy
end; cluster analysis, fuzzy
decision trees, genetic
for (all antecedents A L) do algorithms, ...
find best consequent C for A;
create rule base candidate R = (A,C);
Determine the performance of R;
Add R to B;
end;
Select a rule base from B;
März 2001 Rudolf Kruse
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Selection of a Rule Base
e
Pe rformanc of a Rule :
• Order rules by performance.
• Either select
N
1
1 the best r rules or
Pr
c
Rr x p , with the best r/m rules per class.
N p 1
• r is either given or is
determined automatically such
that all patterns are covered.
0 if class(x p ) con( Rr ),
c
1 otherwise.
März 2001 Rudolf Kruse
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Rule Base Induction
NEFCLASS uses a modified Wang-Mendel procedure
y
large
c1 c2
medium
R1 R2 R3
small x
x y
small medium large
März 2001 Rudolf Kruse
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Computing the Error Signal
Error Signal Fuzzy Error ( jth output):
E j sgn(d ) 1 (d ) , with d t j o j
2
c1 c2 ad
d
and : 0, 1, (d ) e max
(t : correct output, o : actual output)
R1 R2 R3
Rule Error:
x y Er r 1 r Econ( Rr ) , with 0 1
März 2001 Rudolf Kruse
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3. Training Step: Fuzzy Sets
x a
b a if x [a, b)
Example:
triangular c x
membership m a ,b,c : [0,1], m a ,b,c ( x) if x [b, c]
function. c b
0 otherwise
m ( x) if E 0
f
1 m ( x) otherwise
Parameter
updates for an
antecedent b f E c a sgn(x b)
fuzzy set. a f E b a b
c f E c b b
März 2001 Rudolf Kruse
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Training of Fuzzy Sets
y
large
initial fuzzy set
m(x)
reduce enlarge
medium
0.85
0.55
small
0.30
x
x
small medium large
Heuristics: a fuzzy set is moved away from x (towards x)
and its support is reduced (enlarged), in order to
reduce (enlarge) the degree of membership of x.
März 2001 Rudolf Kruse
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Training of Fuzzy Sets
Algorithm:
Variations:
repeat
for (all patterns) do • Adaptive learning rate
accumulate parameter updates; • Online-/Batch
accumulate error; Learning
end;
modify parameters; • optimistic learning
until (no change in error); (n step look ahead)
local Observing the error on
minimum a validation set
März 2001 Rudolf Kruse
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Constraints for Training Fuzzy Sets
• Valid parameter values
• Non-empty intersection of 1
adjacent fuzzy sets
• Keep relative positions
2
• Maintain symmetry
• Complete coverage
(degrees of membership add up to
3
1 for each element)
Correcting a partition after
modifying the parameters
März 2001 Rudolf Kruse
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4. Training Step: Pruning
Goal: remove variables, rules and fuzzy sets, in order to
improve interpretability and generalisation.
März 2001 Rudolf Kruse
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Pruning
Algorithm: Pruning Methods:
repeat 1. Remove variables
select pruning method; (use correlations, information
gain etc.)
repeat
execute pruning step; 2. Remove rules
train fuzzy sets; (use rule performance)
if (no improvement) 3. Remove terms
then undo step; (use degree of fulfilment)
until (no improvement); 4. Remove fuzzy sets
(use fuzziness)
until (no further method);
März 2001 Rudolf Kruse
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WBC Learning Result: Fuzzy Rules
R1: if uniformity of cell size is small and bare nuclei is fuzzy0 then benign
R2: if uniformity of cell size is large then malignant
März 2001 Rudolf Kruse
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WBC Learning Result: Classification Performance
Predicted Class
malign benign not sum
classified
malign 228 (32.62%) 13 (1.86%) 0 (0%) 241 (34.99%)
benign 15 (2.15%) 443 (63.38%) 0 (0%) 458 (65.01%)
sum 243 (34.76%) 456 (65.24%) 0 (0%) 699 (100.00%)
Estimated Performance on Unseen Data (Cross Validation)
NEFCLASS-J: 95.42% NEFCLASS-J (numeric): 94.14%
Discriminant Analysis: 96.05% Multilayer Perceptron: 94.82%
C 4.5: 95.10% C 4.5 Rules: 95.40%
März 2001 Rudolf Kruse
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WBC Learning Result: Fuzzy Sets
uniformity of cell size
sm lg
1.0
0.5
0.0
1.0 2.8 4.6 6.4 8.2 10.0
bare nuclei
1.0
0.5
0.0
1.0 2.8 4.6 6.4 8.2 10.0
März 2001 Rudolf Kruse
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NEFCLASS-J
März 2001 Rudolf Kruse
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Resources
Detlef Nauck, Frank Klawonn & Rudolf Kruse:
Foundations of Neuro-Fuzzy Systems
Wiley, Chichester, 1997, ISBN: 0-471-97151-0
Neuro-Fuzzy Software (NEFCLASS, NEFCON, NEFPROX):
http://www.neuro-fuzzy.de
Beta-Version of NEFCLASS-J:
http://www.neuro-fuzzy.de/nefclass/nefclassj
März 2001 Rudolf Kruse
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Conclusions
Neuro-Fuzzy-Systems can be useful for knowledge discovery.
Interpretability enables plausibility checks and improves
acceptance.
(Neuro-)Fuzzy systems exploit tolerance for sub-optimal
solutions.
Neuro-fuzzy learning algorithms must observe constraints in
order not to jeopardise the semantics of the model.
Not an automatic model creator, the user must work with the
tool.
Simple learning techniques support explorative data analysis.
März 2001 Rudolf Kruse
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Download NEFCLASS-J
Download the free version of NEFCLASS-J at
http://fuzzy.cs.uni-magdeburg.de
März 2001 Rudolf Kruse
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Fuzzy Methods in Information Mining: Examples
here: Exploiting quantitative and qualitative
information
Fuzzy Data Analysis (Projects with Siemens)
Information Fusion (EC Project)
Dependency Analysis (Project with Daimler/Chrysler)
März 2001 Rudolf Kruse
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Analysis of Daimler/Chrysler Database
Database: ~ 18.500 passenger cars
> 100 attributes per car
Analysis of dependencies between special equipment and
faults.
Results used as a starting point for technical experts looking
for causes.
März 2001 Rudolf Kruse
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Learning Graphical Models
data
+
prior information
A B
Inducer C local models
März 2001 Rudolf Kruse
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The Learning Problem
known structure unknown structure
A B A B
C C
complete data Statistical Parametric Discrete Optimization over
Estimation (closed from eq.): Structures (discrete search):
A B C statistical parameter fitting, likelihood scores,
<a4, b3, c1> ML Estimation, MDL
Bayesian Inference, ... Problem:
<a3, b2, c4> search complexity heuristics
incomplete data Parametric Optimization: Combined Methods:
(missing values, EM, structured EM
hidden variables,...) gradient descent, ... only few approaches
Problems:
A B C
criterion for fit?
<a4, ?, c1> new variables?
<a3, b2, ?> local maxima?
fuzzy values?
März 2001 Rudolf Kruse
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Possibility Theory
fuzzy set induces possibility
A sup m
A
mcloudy
2
0 55, 60
1
3
axioms
0 0
50 65 85 100
1
A B max A , B
A B min A , B
März 2001 Rudolf Kruse
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General Structure of (most) Learning Algorithms for Graphical Models
Use a criterion to measure the degree to which a
network structure fits the data and the prior
knowledge
(model selection, goodness of hypergraph)
Use a search algorithm to find a model that
receives a high score by the criterion
(optimal spanning tree, K2: greedy selection of
parents, ...)
März 2001 Rudolf Kruse
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Measuring the Deviation from an Independent Distribution
Probability- and Information-based Measures
information gain *
identical with mutual information
information gain ratio *
g-function (Cooper and Herskovits)
minimum description length
gini index *
Possibilistic Measures
expected nonspecificity
specificity gain
specificity gain ratio
(Measures marked with * originated from decision tree learning)
März 2001 Rudolf Kruse
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Data Mining Tool Clementine
März 2001 Rudolf Kruse
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Analysis of Daimler/Chrysler Database
electrical air con- type of type of slippage
roof top ditioning engine tyres control
faulty faulty faulty
battery compressor brakes
Fictituous example:
There are significantly more faulty batteries, if both
air conditioning and electrical roof top are built
into the car.
März 2001 Rudolf Kruse
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