# Incorporating Prior Knowledge in Fuzzy c-Regression Models

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```					 Incorporating Prior Knowledge in
Fuzzy c-Regression Models
— Application to System
Identification
Janos Abonyi, Robert Babuska,
Tibor Chovan, Ferenc Szeifert
University of Veszprém, Department of Process
Engineering, P.O. Box 158, H-8201, Hungary
www.fmt.vein.hu/softcomp
Outline
•   Fuzzy clustering
•   Fuzzy c-regression method
•   Application to system identification
•   What makes a good fuzzy system ?
•   Constrained fuzzy c-regression clustering
•   Discussion and Conclusion

Department of Process Engineering,   2
University of Veszprem,
Fuzzy Clustering
• CLUSTERS:
– Groups with uncertain boundaries
– Each item has some membership in all groups

• A method to classify items into groups upon
a SIMILARITY CRITERION.

Department of Process Engineering,   3
University of Veszprem,
Concept of fuzziness
A real-life example                                Warm room example

Wife: Do you love me?                     1

Husband (Boolean logician): Yes.                                           Fuzzy
Wife: How much?
                                     Crisp

0
10           15        20       25         30
Temperature [°C]

Department of Process Engineering,                         4
University of Veszprem,
Fuzzy c-regression clustering
y

x

Rule i:    IF x is Ai THEN y = i x

PROBLEM: HOW TO CALCULATE i and Ai ?

Department of Process Engineering,       5
University of Veszprem,
The original algorithm
• The regression models                                     yk  f i ( x k , i )

• The prediction error (distance)                          Ei,k (θi )  ( yk  fi (xk , θi ))2

c      N
• The objective function                                  Em (U,{θi }      i,k m Ei,k (θi )
i 1 k 1
• The algorithm:
0., Initialisation
1., arg min Em (U,{θi }                       (weighted LS)
θi
1
 i , k (l )                                     , 1  i  c, 1  k  N
2.
 j 1
c
Ei,k / E j,k 2 /(m1)

Department of Process Engineering,                             6
University of Veszprem,
Application to system identification
• NARX model
yk  f (x k )  f ( yk 1, , yk  n y 1,uk  nd , , uk  nu  nd 1, )

• Example: First-order Hammerstein system

u                          v                  0.1z 1      y
vu   2                 G( z ) 
1  0.9 z 1

rl : IF [uk , yk 1] is Al THEN            yk  al yk 1  bl uk  cl

Department of Process Engineering,                     7
University of Veszprem,
What is the problem?
1

0.8

0.6

vs and i   0.4

0.2

0

-0.2

-0.4
0   0.1   0.2   0.3   0.4   0.5    0.6   0.7   0.8   0.9   1
u(k)

a      b      c
0.8999 0.0824 -0.0131
0.8988 0.0794 -0.0025
0.8991 0.1382 -0.0420
Department of Process Engineering,                       8
University of Veszprem,
What makes a good fuzzy system?
Accuracy
Approximation or Classification error
Certainty degree
Local models/global models !!!

Transparency and Interpretability
Moderate number of rules
Distinguishability!!!
Normality
Coverage

Department of Process Engineering,   9
University of Veszprem,
Explicit use of prior knowledge:
 Constrained parameter estimation
• Rule-consequents define a convex region (polytope).

• This polytope can be constrained by global linear
constraints.

• The individual rule-consequents can be constrained
separately local linear constraints.

Knowledge: The rule consequents should be different
   relative linear constraints.
Department of Process Engineering,   10
University of Veszprem,
Graphical representation
rl : IF [uk , yk 1] is Al THEN                      yk   l ,1 yk 1   l ,2uk   l ,3
Global constraints

i,2
[3,1 ,3,2 ]

[4,1 ,4,2 ]

1,2 <4,2                                  [2,1 ,2,2 ]

Local constraints

[1,1 ,1,2 ]
Relative
constraints
i,1
Department of Process Engineering,                                11
University of Veszprem,
Quadratic programming
1 T       T 
min   H   d  

2            
  
• H and d contain the measured input-output data
•  and  represents the a priori knowledge based
constraints

Department of Process Engineering,   12
University of Veszprem,
Applied Constraints
• Upper and lower bounds:
nb

b        i
l

K min 
l              i 1
na
 K max
l

1   ail
i 1

• Inequality constraints for QP:
    na
l
nb
K   l
min
1   ai    bil  0
         
 i 1  i 1
    na
l
nb
K   l
max
1   ai    bil  0
         
 i 1  i 1

Department of Process Engineering,   13
University of Veszprem,
This is a good fuzzy model !
1

0.8

0.6

vs and i   0.4

0.2

0

-0.2

-0.4

-0.6

-0.8
0   0.1    0.2   0.3   0.4   0.5    0.6   0.7   0.8   0.9   1
u(k)

a      b      c
0.8996 0.0204 0.0151
0.8996 0.0408 -0.0025
0.8996 0.1560 -0.0590
Department of Process Engineering,                    14
University of Veszprem,
Conclusions
Expert                Mechanistic                       Measured
knowledge               knowledge                           data

Model       Parameter             Steady-state data      Dynamic data
structure    constraints              and model         and (local) model

Optimization

Model

Accuracy,             Interpretability,                Generalisation
performance              applicability                   (wide range)
Department of Process Engineering,                  15
University of Veszprem,

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