Fuzzy Systems

1
Fuzzy Systems

• Alternative operators for AND and OR
• A series of alternative operators to min and max
  have been proposed.
• They arise from both theoretical and practical
  consideration.
• Sometimes referred to as compensatory operators
  i.e. they compensate for a strict application of
  minimum (for AND), maximum (for OR) and negation.
  
• They can be less sensitive to extreme differences
  in set membership
• Some examples
• algebraic product for set intersection,
  algebraic sum for set union
• eg if A(x) a and B(x) b THEN (A AND B)(x)
  ab and (A OR B)(x) a b -ab
• algebraic mean for set intersection
• eg if A(x) a and B(x) b THEN (A AND B)(x)
  (ab)/2

2
Fuzzy Systems

• Look at an example
• A(x) 0.1 and B(x) 0.9 THEN
• (A AND B)(x) 0.1 if min is being used
• (A AND B)(x) 0.09 if algebraic product is
  being used
• (A AND B)(x) 0.5 if algebraic mean is being
  used

3
Fuzzy Systems

• What effect might these have on a mandami fuzzy
  system?
• Let us return to the stock market advisory system
  example
• price (-10)
  decreasing (0.25), stable (0.75), increasing
  (0)
• trading_volume(300) light (0.5),
  moderate (0.5), heavy (0)
• 1 decreasing0.25 AND heavy0 -gt sell
  0
• 2 decreasing0.25 AND moderate0.5 -gt
  sell0.125 (using algebraic product)
• 3 stable0.75 AND moderate0.5 -gt
  hold0.375 (using algebraic product)
• 4 stable0.75 AND light0.5 -gt
  hold0.375 (using algebraic product)
• 5 increasing0 AND light0.5 -gt
  hold0
• 6 increasing0 AND moderate0.5 gt
  buy0
• 7 increasing0 AND heavy0 -gt buy0
• 8 decreasing0.25 AND light0.5 -gt
  hold0.125 (using algebraic product)
• 9 stable0.75 AND heavy0 -gt sell0
• maximum membership of market_order in the sell
  set is 0.125, in the
• hold set is 0.375 and in the buy set is 0.

4
Fuzzy Systems

• Output set (sell OR hold OR buy) becomes

5
Fuzzy Systems

• De-fuzzifying using a 20 point method centroid
  approximation

Output -84.375/5.25 -16
sell about 16 shares' . cf with sell about 28
shares
6
Fuzzy Systems

• The use of rule strengths
• In some application areas you wish to indicate
  that some rules are more important than others
• A rule strength' measured on a scale 0..1
  could be assigned to each rule
• The strength modifies the truth value of the
  rule consequent i.e. the RHS
• In our example we may wish to weaken' the
  effect of rules that use price is stable i.e.
  rules 3,4 and 9 by assigning a rule strength of
  0.9 to each. Using price -10 and trading_volume
  300 the affected rules become
• 3 stable0.75 AND moderate0.5 -gt
  hold0.90.5 0.45
• 4 stable0.75 AND light0.5 -gt
  hold0.90.50.45
• 9 stable0.75 AND heavy0 -gt
  sell0.90 0
• The maximum membership in hold is now 0.45 (sell
  is still 0.25 and buy is still 0)

7
Fuzzy Systems

• The final result may be affected
• Output set (sell OR hold OR buy)

8
Fuzzy Systems

• and a 20 point centroid approximation gives

Output -190.625/6.425 -30
i.e sell about 30 shares
9
Fuzzy Systems

• Another example
• The washing machine control example
• The system attempts to replicate human decision
  making. The selection of an appropriate washing
  cycle time depends on the softness' and size of
  the washing load.

10
Fuzzy Systems
softness
control system
washing cycle
quantity
11
Fuzzy Systems

• Input variable softness has four linguistic
  values hard, soft, normal hard, normal soft

12
Fuzzy System

• The input variable, quantity has three
  linguistics values small, medium and large

13
Fuzzy Systems

• The output variable washing cycle has four
  linguistic values delicate, light, normal and
  strong

14
Fuzzy Systems

• The rule base can be represented as a rule matrix

15
Fuzzy Systems

• Evaluate the washing cycle for a 20 kg washing
  load with a softness measure of 17.
• Fuzzifying inputs
• Quantity 20 small (0.5), medium (0.5), large
  (0)
• Softness 17 hard (0), soft (0.375), normal
  hard (0.125), normal soft (0.625)
• Fire rule base

16
Fuzzy Systems

• Evaluate the washing cycle for a 20 kg washing
  load with a softness measure of 17.
• Quantity 20 small (0.5), medium (0.5), large
  (0)
• Softness 17 hard (0), soft (0.375), normal
  hard (0.125), normal soft (0.625)
• Fire rule base Max membership in the output sets
  are shown

17
Fuzzy Systems

• The output set is

18
Fuzzy Systems

• De-fuzzify using an approximate centroid method

Washing cycle 491.875/9.5 52
19
Fuzzy Systems

• Non-linear fuzzy set parameters
• Fuzzy sets need not be linear (i.e. triangular
  or trapezoid), they could be defined using a
  suitable non-linear function
• For example
• A gaussian function

20
Fuzzy Systems

• Non-linear fuzzy set parameters
• Fuzzy sets need not be linear (i.e. triangular
  or trapezoid), they could be defined using a
  suitable non-linear function
• For example
• A bell function

21
Fuzzy Systems

• Non-linear fuzzy set parameters
• Fuzzy sets need not be linear (i.e. triangular
  or trapezoid), they could be defined using a
  suitable non-linear function
• For example
• A sigmoidal function

22
Fuzzy Systems

• The fuzzy sets for price in the stock market
  example could defined using non-linear functions
  i.e.

23
Next time

• An alternative fuzzy model the Sugeno approach