# Soft Computing 1. Introduction by wuyunyi

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```									         Additional Material:
Mahalanobis Distance

Prof. Dr. Rudolf Kruse          1
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Interpretation of a Covariance Matrix
 A univariate normal distribution has the density function

 A multivariate normal distribution has the density function

Prof. Dr. Rudolf Kruse                                                       2
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Variance and Standard Deviation

 Univariate Normal/Gaussian Distribution
The variance/standard deviation provides information about the
height of the mode and the width of the curve.

Prof. Dr. Rudolf Kruse                                                         3
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Interpretation of a Covariance Matrix

 The variance/standard deviation relates the spread of the distribution to
the spread of a standard normal distribution
 The covariance matrix relates the spread of the distribution to the spread
of a multivariate standard normal distribution
 Example: bivariate normal distribution

 Question: Is there a multivariate analog of standard deviation?

Prof. Dr. Rudolf Kruse                                                              4
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Eigenvalue Decomposition

 Yields an analog of standard deviation.
 Let S be a symmetric, positive definite matrix (e.g. a covariance
matrix).

Prof. Dr. Rudolf Kruse                                                       5
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Eigenvalue Decomposition

Special Case: Two Dimensions

Prof. Dr. Rudolf Kruse                              6
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Eigenvalue Decomposition

Prof. Dr. Rudolf Kruse                              7
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Eigenvalue Decomposition

Prof. Dr. Rudolf Kruse                              8
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Eigenvalue Decomposition

Special Case: Two Dimensions

Prof. Dr. Rudolf Kruse                              9
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Cluster-Specific Distance Functions
The similarity of a data point to a prototype depends on their distance.
 If the cluster prototype is a simple cluster center, a general distance
measure can be defined on the data space.
In this case the Euclidean distance is most often used due to its rotation
invariance. It leads to (hyper-)spherical clusters.

 However, more flexible clustering approaches (with size and shape
parameters) use cluster-specific distance functions.
The most common approach is to use a Mahalanobis distance with a
cluster-specific covariance matrix.

The covariance matrix comprises shape and size parameters.
The Euclidean distance is a special case that results for

Prof. Dr. Rudolf Kruse                                                          10
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Neuro-Fuzzy Systems

Prof. Dr. Rudolf Kruse                          11
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Beispiel : Automatik-Getriebe
Aufgabe: Verbesserung des VWAutomatik-Getriebes
- keine zusätzlichen Sensoren
- individuelle Anpassung des Schaltverhaltens

Idee (1995):
Das Fahrzeug “beobachtet” und klassifiziert den Fahrer nach
Sportlichkeit
- ruhig, normal, sportlich  Bestimmung eines Sport-Faktors aus [0, 1]
- nervös                    Beruhigung des Fahrers

Testfahrzeug:
- verschiedene Fahrer, Klassifikation durch Experten (Mitfahrer)
- gleichzeitige Messungen:
 Geschwindigkeit,
 Position,
 Geschwindigkeit des Gaspedals,
 Winkel des Lenkrades, ... (14 Attribute).

Prof. Dr. Rudolf Kruse                                                    12
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Modellierung unscharfer Informationen mit Fuzzy-Mengen

fast genau 2             etwa zwischen               ungefähr 13
6 und 8
1

2                   6          8                    13

negativ            negativ     negativ   ungefähr   positiv    positiv   positiv
groß               mittel      klein      null      klein     mittel     groß

Prof. Dr. Rudolf Kruse                                                                                  13
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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]

Prof. Dr. Rudolf Kruse                                                                          14
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If X is positive small       and Y is positive small   then Z is positive small
1                          1                         1

x          X         y              Y                               Z
If      X is positive big   and Y is positive small   then        Z is positive big
1                          1                          1

x          X         y              Y                               Z
1

Eingabewerte: x und y
Stellwert: z                                                                             Z
defuzzifizierter Wert

Prof. Dr. Rudolf Kruse                                                                       15
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 Fuzzy-Regler mit 7 Regeln

AG 4
 Optimiertes Programm
 24 Byte RAM 
 auf Digimat
 702 Byte ROM

 Laufzeit 80 ms,
12 mal pro Sekunde wird ein neuer Sportfaktor bestimmt

 In Serie im VW Konzern

 Erlernen von Regelsystemen mit Hilfe
von Künstlichen Neuronalen Netzen,
Optimierung mit evolutionären Algorithmen

Prof. Dr. Rudolf Kruse                                              16
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Beispiel : Fuzzy Datenbank

TOP
MANAGEMENT

NACH-
FOLGER         TALENTBANK

MANAGEMENT

Nachfolger für Top-Management Positionen

Prof. Dr. Rudolf Kruse                                               17
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Prof. Dr. Rudolf Kruse   18
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Beispiel : Automatisiertes sensor-basiertes Landen

Prof. Dr. Rudolf Kruse                                   19
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Neuro-Fuzzy Systems
 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

Prof. Dr. Rudolf Kruse                                              20
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Prof. Dr. Rudolf Kruse   21
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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

Prof. Dr. Rudolf Kruse                                                                     22
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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

Prof. Dr. Rudolf Kruse                                                         23
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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 determines the weighted sum of source outputs

 Determination of “Return of Invest”
Prof. Dr. Rudolf Kruse                                     24
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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   lnvi, 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
e1ce  I e  id e ce 
E

 outputt=  we  outputt
e
e 1
T
 roi =  yt   sign outputt
DM
                 
Prof. Dr. 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

Prof. Dr. Rudolf Kruse                                    26
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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)

Prof. Dr. Rudolf Kruse                                          27
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Neuro-Fuzzy Architecture

Prof. Dr. Rudolf Kruse                              28
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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

Prof. Dr. Rudolf Kruse                                                               29
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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

Prof. Dr. Rudolf Kruse                                                          30
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Return-on-Investment Curves of the Different Models

Validation data from March 01, 1994 until April 1997

Prof. Dr. Rudolf Kruse                                                                   31
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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.

Prof. Dr. Rudolf Kruse                                            32
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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)

Prof. Dr. Rudolf Kruse                                                      33
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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

Prof. Dr. Rudolf Kruse                                       34
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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

Prof. Dr. Rudolf Kruse                                                  35
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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.

Prof. Dr. Rudolf Kruse                                                   36
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Structure-Oriented Rule Learning
y
large

Provide initial fuzzy sets for
all variables.
The data space is partitioned
medium

by a fuzzy grid
Detect all grid cells that
contain data (approach by
Wang/Mendel 1992)
small

Compute best consequents
and select best rules
x (extension by Nauck/Kruse
1995, NEFCLASS model)

small   medium   large
Prof. Dr. Rudolf Kruse                                                         37
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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

Prof. Dr. Rudolf Kruse                                             38
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Learning Fuzzy Sets

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.

Prof. Dr. Rudolf Kruse                                               39
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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.

Prof. Dr. Rudolf Kruse                                                         40
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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.

Prof. Dr. Rudolf Kruse                                             41
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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.

Prof. Dr. Rudolf Kruse                                                    42
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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

Prof. Dr. Rudolf Kruse                                                   43
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NEFCLASS: Neuro-Fuzzy Classifier

Output variables (class labels)

Unweighted connections

Fuzzy rules

Fuzzy sets (antecedents)

Input variables (attributes)

Prof. Dr. Rudolf Kruse                                                 44
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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

Prof. Dr. Rudolf Kruse                                  45
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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
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.

Prof. Dr. Rudolf Kruse                                                                      46
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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

Prof. Dr. Rudolf Kruse                                                      47
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2. Training Step: Rule Base

Algorithm:                                      Variations:
for (all patterns p) do                         Fuzzy rule bases can
find antecedent A,                      also be created by
such that A( p) is maximal;             using prior
if (A  L) then add A to L;             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;
end;
Select a rule base from B;

Prof. Dr. Rudolf Kruse                                                  48
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Selection of a Rule Base

e
Pe rformanc of a Rule :
• Order rules by
performance.
 
N
  1
1
Pr 
c
Rr x p , with   • Either select
N             p 1                         the best r rules or
the best r/m rules per
class.
0 if class(x p )  con( Rr ),               • r is either given or is

c                                              determined automatically
such that all patterns are
1 otherwise.
                                              covered.

Prof. Dr. Rudolf Kruse                                                    49
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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

Prof. Dr. Rudolf Kruse                                                      50
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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
c1        c2                                             ad   
2

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

Prof. Dr. Rudolf Kruse                                                                      51
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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
antecedent            b  f  E  c  a   sgn(x  b)
fuzzy set.            a   f  E  b  a   b
c  f  E  c  b   b

Prof. Dr. Rudolf Kruse                                                            52
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Training of Fuzzy Sets
y

large
initial fuzzy set
m(x)

medium
reduce            enlarge
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.

Prof. Dr. Rudolf Kruse                                                               53
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Training of Fuzzy Sets
Algorithm:
Variations:
repeat
for (all patterns) do                Adaptive learning rate
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

Prof. Dr. Rudolf Kruse                                             54
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Constraints for Training Fuzzy Sets

 Valid parameter values
 Non-empty intersection of           1
 Keep relative positions
2
 Maintain symmetry
 Complete coverage
3
to 1 for each element)

Correcting a partition after
modifying the parameters

Prof. Dr. Rudolf Kruse                                                   55
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4. Training Step: Pruning
Goal: remove variables, rules and fuzzy sets, in order to
improve interpretability and generalisation.

Prof. Dr. Rudolf Kruse                                              56
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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);

Prof. Dr. Rudolf Kruse                                                          57
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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

Prof. Dr. Rudolf Kruse                                                             58
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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%

Prof. Dr. Rudolf Kruse                                                               59
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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

Prof. Dr. Rudolf Kruse                                                            60
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NEFCLASS-J

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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

Prof. Dr. Rudolf Kruse                              62
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http://fuzzy.cs.uni-magdeburg.de

Prof. Dr. Rudolf Kruse                            63
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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.

Prof. Dr. Rudolf Kruse                                                     64
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Information Mining
 Information mining is the non-trivial process of identifying valid,
novel, potentially useful, and understandable information and patterns
in heterogeneous information sources.

 Information sources are
 data bases,
 expert background knowledge,
 textual description,
 images,
 sounds, ...

Prof. Dr. Rudolf Kruse                                                     65
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Information Mining

Problem       Information   Information Modeling                   Evaluation   Deployment
Understanding Understanding Preparation

Determine         Collect Initial   Select Infor-   Select         Evaluate     Plan
Problem           Information       mation          Modeling       Results      Deployment
Objectives                                          Technique

Assess            Describe          Clean Infor-    Generate Test Review        Plan Moni-
Situations        Information       mation          Design        Process       toring and
Maintenance

Determine         Explore           Construct In- Build Model      Determine    Produce Final
Information       Information       formation                      Next Steps   Results
Mining Goals
Verify              Integrate In-   Assess Model                Review
Produce Project Information         formation                                   Project
Plan            Quality
Format Infor-
mation

Prof. Dr. Rudolf Kruse                                                                           66
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SEURO

UZZY
Example: Line Filtering

 Extraction of edge segments (Burns‟ operator)
 Production net:
edges  lines  long lines  parallel lines  runways

Prof. Dr. Rudolf Kruse                                                      67
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UZZY
Example: Line Filtering

 Problems
 extremely many lines due to distorted images
 long execution times of production net

Prof. Dr. Rudolf Kruse                                                    68
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SEURO

UZZY
Example: Line Filtering

 Only few lines used for runway assembly                                            d

 Approach:                                                               ow
left wind                d
 Extract textural features of lines
dow
right win
 Identify and discard superfluous lines        ient

Prof. Dr. Rudolf Kruse                                                                    69
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UZZY
Example: Line Filtering

 Several classifiers:
 minimum distance, k-nearest neighbor, decision trees, NEFCLASS
 Problems: classes are overlapping and extremely unbalanced
 Result above with modified NEFCLASS:
 all lines for runway construction found
 reduction to 8.7% of edge segments

Prof. Dr. Rudolf Kruse                                                        70
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UZZY
Surface Quality Control: the 2 Approaches
 Today’s Approach
The current surface quality control is done
manually          an experienced worker treats the
exterior surfaces with a grindstone. The experts
classify surface form deviations by means of
linguistic descriptions.
Cumbersome – Subjective - Error Prone Time
Consuming

 The Proposed Approach
Our Approach is based on the digitization of the exterior body panel surface
with an optical measuring system.
We characterize the form deviation by mathematical properties that are close
to the subjective properties that the experts used in their linguistic description.
Prof. Dr. Rudolf Kruse                                                       71
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UZZY
Topometric 3-D measuring system

Triangulation and Gratings Projection

0      P(x,y)                              φn
1                    b(x,y)       (x,y)
0                             b
0
z(x,y)
Miniaturized
Pixel                                             Projection
coding                                             Technique
(Grey Code
z                  Phase shift)
y
x

    High Point Density
    Fast Data Collection
    Measurement Accuracy
    Contact less and Non-destructive

Prof. Dr. Rudolf Kruse                                                                      72
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S EURO

UZZY
Data Processing
• Approximation by a         • Difference               • Colour-Coded
Polynomial Surface                                        Visualization
z(x,y)

Dz(x,y)

˜ (x,y)
z
˜ (x,y)
z

3-D Data                                                 Detection of
Post-Processing                                       Features Analysis
Acquisition                                              Form Deviation

• 3-D-Point Cloud
• Feature Calculation
Form Deviation
• Classification (Data-Mining)
z(x,y)

Prof. Dr. Rudolf Kruse                                                                                        73
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UZZY
Color Coded Visualization
Result of Grinding

Prof. Dr. Rudolf Kruse                                          74
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UZZY
3D Visualization of Local Surface Defects
Uneven Surface                               Press Mark
(several sink marks in series or adjoined)   (local smoothing of (micro-)surface)

Sink Mark                              Waviness
(slight flat based depression inward)   (several heavier wrinklings in series)

Prof. Dr. Rudolf Kruse                                                        75
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UZZY
Data Characteristics
 We analysed 9 master pieces with a total number of 99 defects
 For each defect we calculated 42 features
 The types are rather unbalanced
 We discarded the rare classes
 We discarded some of the extremely correlated features (31 features
left)
 We ranked the 31 features by importance
 We use stratified 4-fold cross validation for the experiment.
line
w aviness
draw line
flat area
sink mark
press mark

Prof. Dr. Rudolf Kruse
uneven surface

0   10   20   30   40   50
76
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UZZY
Application and Results

The Rule Base for NEFCLASS

Classification Accuracy

NBC      DTree    NN      NEFCLASS    DC
Train Set             89.0%    94.7%    90%         81.6%   46.8%
Test Set              75.6%    75.6%   85.5%        79.9%   46.8%

Prof. Dr. Rudolf Kruse                                            77
NF
SEURO

UZZY

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