FLEA

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FLEA

• Prof. dr. ir. J. Hellendoorn

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Situation of more rules
R1 if A1 then B1
R2 if A2 then B2
R3 if A3 then B3

Rn if An then Bn
Suppose A, what is B?
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The two main inference rules
Mamdani
Gödel
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Global inference, Mamdani

• Ri if Ai(x) then Bi(y), given A(x)
• Mamdani
• B (y) A(x) o Rm

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Global inference, Gödel

• Ri if Ai(x) then Bi(y), given A(x)
• Gödel
• B (y) A(x) o Rm

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Different cases

• A(x) Ai(x)
• A(x) Ai(x)? Ai1(x)
• A(x) Ai(x)? Ai1(x)
• A(x) Ai(x)? Ak(x)
• A(x) Ai(x)? Ak(x)
• A(x) ? Ai(x)
• A(x) ? Ai(x)

• A(x) ?Ai(x)
• A(x) Ai½(x)
• A(x) unknown
• A(x) undefined

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Example Fuzzy Sets
x1, x2, x3, x4, x5, x6, x7
A1(x) (1.0, 0.8, 0.6, 0.4, 0.2, 0.0, 0.0)
A2(x) (0.0, 0.4, 0.8, 1.0, 0.8, 0.4, 0.0)
A3(x) (0.0, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 0.9, 0.7, 0.5, 0.3, 0.0, 0.0)
B2(y) (0.0, 0.5, 0.7, 1.0, 0.7, 0.5, 0.0)
B3(y) (0.0, 0.0, 0.3, 0.5, 0.7, 0.9, 1.0)
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Example Rule Base
R1 if A1(x) then B1(y)
R2 if A2(x) then B2(y)
R3 if A3(x) then B3(y)
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Meaning of rule 1, Mamdani
R1 if A1(x) then B1(y)
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Meaning of rule 1, Gödel
R1 if A1(x) then B1(y)
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Meaning of rule 2, Mamdani
R2 if A2(x) then B2(y)
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Meaning of rule 2, Gödel
R2 if A2(x) then B2(y)
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Meaning of rule 3, Mamdani
R3 if A3(x) then B3(y)
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Meaning of rule 3, Gödel
R3 if A3(x) then B3(y)
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Three Mamdani-rules together
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Global meaning, Mamdani
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Three Gödel-rules together
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Global meaning, Gödel
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Example Fuzzy Sets (repeat)
x1, x2, x3, x4, x5, x6, x7
A1(x) (1.0, 0.8, 0.6, 0.4, 0.2, 0.0, 0.0)
A2(x) (0.0, 0.4, 0.8, 1.0, 0.8, 0.4, 0.0)
A3(x) (0.0, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 0.9, 0.7, 0.5, 0.3, 0.0, 0.0)
B2(y) (0.0, 0.5, 0.7, 1.0, 0.7, 0.5, 0.0)
B3(y) (0.0, 0.0, 0.3, 0.5, 0.7, 0.9, 1.0)
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Case 1, A A1

• A A1 (1, 0.8, 0.6, 0.4, 0.2, 0, 0)
• Expect B B1 (1, 0.9, 0.7, 0.5, 0.3, 0, 0)
• B A1 o Rm (1, 0.9, 0.7, 0.6, 0.6, 0.5, 0.4)
• B A1 o Rg (1, 0.9, 0.7, 0.5, 0.3, 0, 0)
• Mamdani B ? Bi
• Gödel B ? Bi

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Case 2, A A1 ? A2

• A A1 ? A2 (1, 0.8, 0.8, 1, 0.8, 0.4, 0)
• Expect B B1 ? B2 (1, 0.9, 0.7, 1, 0.7, 0.5,
  0)
• B (A1?A2) o Rm (1, 0.9, 0.7, 1, 0.7, 0.6,
  0.6)
• B (A1?A2) o Rg (1, 0.9, 0.7, 1, 0.7, 0.4, 0)
• Mamdani B ? B1 ? B2
• Gödel B ? B1 ? B2

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Case 3, A A1 ? A2

• A A1 ? A2 (0, 0.4, 0.6, 0.4, 0.2, 0, 0)
• Expect B B1 ? B2 (0, 0.5, 0.7, 0.5, 0.3, 0,
  0)
• B (A1 ? A2) o Rm (0.6, 0.6, 0.6, 0.6, 0.6,
  0.5, 0.4)
• B (A1 ? A2) o Rg (0, 0.4, 0.6, 1, 0.5, 0.3,
  0)
• Mamdani B ? B1 ? B2
• Gödel B ? B1 ? B2

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Case 3, A A1 ? A2 normalized

• A A1 ? A2 (0, 0.7, 1, 0.7, 0.3, 0, 0)
• Expect B B1 ? B2 (0, 0.8, 1, 0.8, 0.3, 0,
  0)
• B (A1 ? A2) o Rm (0.7, 0.7, 0.7, 0.8, 0.7,
  0.5, 0.4)
• B (A1 ? A2) o Rg (0, 0.7, 0.7, 0.7, 0.3, 0,
  0)
• Mamdani B ? B1 ? B2
• Gödel B ? B1 ? B2

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Case 4, A A1 ? A3

• A A1 ? A3 (1, 0.8, 0.6, 0.4, 0.6, 0.8, 1)
• Expect B B1 ? B3 (1, 0.9, 0.7, 0.5, 0.7,
  0.9, 1)
• B (A1?A3) o Rm (1, 0.9, 0.7, 0.6, 0.7, 0.9,
  1)
• B (A1?A3) o Rg (1, 0.9, 0.7, 0.5, 0.7, 0.9,
  1)
• Mamdani B ? B1 ? B3
• Gödel B ? B1 ? B3

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Case 5, A A1 ? A3

• A A1 ? A3 (0, 0, 0.2, 0.4, 0.2, 0, 0)
• Expect B B1 ? B3 (0, 0, 0.3, 0.5, 0.3, 0,
  0)
• B (A1 ? A3) o Rm (0.4, 0.4, 0.4, 0.4, 0.4,
  0.4, 0.4)
• B (A1 ? A3) o Rg (0, 0, 0.3, 0.4, 0.3, 0, 0)
• Mamdani B ? B1 ? B3
• Gödel B ? B1 ? B3

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Case 6, A ? A1

• A (1, 0.6, 0.4, 0.2, 0, 0, 0) ? (1, 0.8, 0.6,
  0.4, 0.2, 0, 0)
• Expect B ? B1 (1, 0.9, 0.7, 0.5, 0.3, 0, 0)
• B A o Rm (1, 0.9, 0.7, 0.5, 0.4, 0.4, 0.2)
• B A o Rg (1, 0.9, 0.7, 0.5, 0.3, 0, 0)
• Mamdani B ? B1
• Gödel B B1

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Case 7, A ? A1

• A (1, 1, 0.8, 0.6, 0.4, 0.2, 0) ? (1, 0.8,
  0.6, 0.4, 0.2, 0, 0)
• Expect B ? B1 (1, 0.9, 0.7, 0.5, 0.3, 0, 0)
• B A o Rm (1, 0.9, 0.7, 0.8, 0.7, 0.5, 0.4)
• B A o Rg (1, 1, 0.7, 0.6, 0.4, 0.2, 0)
• Mamdani B ? B1
• Gödel B ? B1

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Case 8, A ?A1

• A ?A1 (0, 0.2, 0.4, 0.6, 0.8, 1, 1)
• Expect B? B1 (0, 0.1, 0.3, 0.5, 0.7, 1, 1)
• B A o Rm (0.4, 0.5, 0.7, 0.8, 0.7, 0.9, 1)
• B A o Rg (0, 0.2, 0.4, 0.6, 0.7, 1, 1)
• Mamdani B ? ? B1
• Gödel B ? ? B1

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Case 9, A A1½

• A A1 ½ (1, 0.64, 0.36, 0.16, 0.04, 0, 0)
• Expect B B1 ½ (1, 0.81, 0.49, 0.25, 0.09, 0,
  0)
• B A o Rm (1, 0.9, 0.7, 0.5, 0.4, 0.4, 0.2)
• B A o Rg (1, 0.9, 0.7, 0.5, 0.3, 0, 0)
• Mamdani B ? B1 ½
• Gödel B ? B1 ½

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Case 10, A unknown

• A unknown (1, 1, 1, 1, 1, 1, 1)
• Expect B unknown (1, 1, 1, 1, 1, 1, 1, 1)
• B A o Rm (1, 0.9, 0.7, 1, 0.7, 0.9, 1)
• B A o Rg (1, 1, 0.7, 1, 0.7, 1, 1)
• Mamdani B ? unknown
• Gödel B ? unknown

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Case 11, A undefined

• A undefined (0, 0, 0, 0, 0, 0, 0)
• Expect B undefined (0, 0, 0, 0, 0, 0, 0)
• B A o Rm (0, 0, 0, 0, 0, 0, 0)
• B A o Rg (0, 0, 0, 0, 0, 0, 0)
• Mamdani B undefined
• Gödel B undefined

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Global inference

• Ri if Ai(x) then Bi(y), given A(x)
• Mamdani
• Gödel
• B (y) A(x) o Rm or B (y) A(x) o Rg

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Local inference

• Ri if Ai(x) then Bi(y), given A(x)
• Mamdani Bi(y) A(x) o Rim
• B(y) ?i Bi(y)
• Gödel Bi(y) A(x) o Rig
• B(y) ?i Bi(y)

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Example Fuzzy Sets (repeat)
x1, x2, x3, x4, x5, x6, x7
A1(x) (1.0, 0.8, 0.6, 0.4, 0.2, 0.0, 0.0)
A2(x) (0.0, 0.4, 0.8, 1.0, 0.8, 0.4, 0.0)
A3(x) (0.0, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 0.9, 0.7, 0.5, 0.3, 0.0, 0.0)
B2(y) (0.0, 0.5, 0.7, 1.0, 0.7, 0.5, 0.0)
B3(y) (0.0, 0.0, 0.3, 0.5, 0.7, 0.9, 1.0)
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Example Rule Base (repeat)
R1 if A1(x) then B1(y)
R2 if A2(x) then B2(y)
R3 if A3(x) then B3(y)
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Mamdani-local, A A1(x)
x1, x2, x3, x4, x5, x6, x7
A A1(x) (1.0, 0.8, 0.6, 0.4, 0.2, 0.0, 0.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 0.9, 0.7, 0.5, 0.3, 0.0, 0.0)
B2 (y) (0.0, 0.5, 0.6, 0.6, 0.6, 0.5, 0.0)
B3 (y) (0.0, 0.0, 0.3, 0.4, 0.4, 0.4, 0.4)
B(y) ?i Bi(y) (1.0, 0.9, 0.7, 0.6, 0.6,
0.5, 0.4)
Compare to A o Rm (1.0, 0.9, 0.7, 0.6, 0.6,
0.5, 0.4)
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Gödel-local , A A1(x)
x1, x2, x3, x4, x5, x6, x7
A A1(x) (1.0, 0.8, 0.6, 0.4, 0.2, 0.0, 0.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 0.9, 0.7, 0.5, 0.3, 0.0, 0.0)
B2 (y) (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
B3 (y) (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
B(y) ?i Bi(y) (1.0, 0.9, 0.7, 0.5, 0.3,
0.0, 0.0)
Compare to A o Rg (1.0, 0.9, 0.7, 0.5, 0.3,
0.0, 0.0)
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Mamdani-local, A A1(x) ? A2(x)
x1, x2, x3, x4, x5, x6, x7
A A1(x) ? A2(x) (1.0, 0.8, 0.8, 1.0, 0.8,
0.4, 0.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 0.9, 0.7, 0.5, 0.3, 0.0, 0.0)
B2 (y) (0.0, 0.5, 0.7, 1.0, 0.7, 0.5, 0.0)
B3 (y) (0.0, 0.0, 0.3, 0.5, 0.6, 0.6, 0.6)
B(y) ?i Bi(y) (1.0, 0.9, 0.7, 1.0, 0.7,
0.6, 0.6)
Compare to (A1(x) ? A2(x)) o Rm (1.0, 0.9,
0.7, 1.0, 0.7, 0.6, 0.6)
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Gödel-local, A A1(x) ? A2(x)
x1, x2, x3, x4, x5, x6, x7
A A1(x) ? A2(x) (1.0, 0.8, 0.8, 1.0, 0.8,
0.4, 0.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 1.0, 1.0, 1.0, 0.8, 0.4, 0.4)
B2 (y) (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
B3 (y) (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
B(y) ?i Bi(y) (1.0, 1.0, 1.0, 1.0, 0.8,
0.4, 0.4)
Compare to (A1(x) ? A2(x)) o Rg (1.0, 0.9,
0.7, 1.0, 0.7, 0.4, 0.0)
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Mamdani-local , A? A1(x)
x1, x2, x3, x4, x5, x6, x7
A? A1(x) (1.0, 0.6, 0.4, 0.2, 0.0, 0.0, 0.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 0.9, 0.7, 0.5, 0.3, 0.0, 0.0)
B2 (y) (0.0, 0.5, 0.7, 0.4, 0.4, 0.4, 0.0)
B3 (y) (0.0, 0.0, 0.2, 0.2, 0.2, 0.2, 0.2)
B(y) ? i Bi(y) (1.0, 0.9, 0.7, 0.5, 0.4,
0.4, 0.2)
Compare to A o Rm (1.0, 0.9, 0.7, 0.5, 0.4,
0.4, 0.2)
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Gödel-local , A? A1(x)
x1, x2, x3, x4, x5, x6, x7
A? A1(x) (1.0, 0.6, 0.4, 0.2, 0.0, 0.0, 0.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 0.9, 0.7, 0.5, 0.3, 0.0, 0.0)
B2 (y) (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
B3 (y) (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
B(y) ? i Bi(y) (1.0, 0.9, 0.7, 0.5, 0.3,
0.0, 0.0)
Compare to A o Rg (1.0, 0.9, 0.7, 0.5, 0.3,
0.0, 0.0)
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Mamdani-local , A? A1(x)
x1, x2, x3, x4, x5, x6, x7
A? A1(x) (1.0, 1.0, 0.8, 0.6, 0.4, 0.2, 0.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 0.9, 0.7, 0.5, 0.3, 0.0, 0.0)
B2 (y) (0.0, 0.5, 0.7, 0.8, 0.7, 0.5, 0.0)
B3 (y) (0.0, 0.0, 0.3, 0.4, 0.4, 0.4, 0.4)
B(y) ? i Bi(y) (1.0, 0.9, 0.7, 0.8, 0.7,
0.5, 0.4)
Compare to A o Rm (1.0, 0.9, 0.7, 0.8, 0.7,
0.5, 0.4)
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Gödel-local , A? A1(x)
x1, x2, x3, x4, x5, x6, x7
A? A1(x) (1.0, 1.0, 0.8, 0.6, 0.4, 0.2, 0.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 1.0, 0.8, 0.6, 0.4, 0.2, 0.0)
B2 (y) (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
B3 (y) (1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)
B(y) ? i Bi(y) (1.0, 1.0, 0.8, 0.6, 0.4,
0.2, 0.0)
Compare to A o Rg (1.0, 1.0, 0.7, 0.6, 0.4,
0.2, 0.0)
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Inference with crisp input

• In this case
• where mA(x0) 1and mA(x0) 0 if x ? x0
• For example

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Global inference

• Ri if Ai(x) then Bi(y), given A(x)
• Mamdani
• Gödel
• B (y) A(x) o Rm or B (y) A(x) o Rg

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Local inference

• Ri if Ai(x) then Bi(y), given A(x)
• Mamdani
• Gödel

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A is crisp, global

• A (0.0, 0.0, 1.0, 0.0, 0.0, 0, 0)
• Mamdani global B A o Rm (0.6, 0.6, 0.7,
  0.8, 0.7, 0.5, 0.2)
• Gödel global B A o Rg (0.0, 0.0, 0.7, 0.5,
  0.3, 0.0, 0.0)

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A local, Mamdani
x1, x2, x3, x4, x5, x6, x7
A (0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (0.6, 0.6, 0.6, 0.5, 0.3, 0.0, 0.0)
B2 (y) (0.0, 0.5, 0.7, 0.8, 0.7, 0.5, 0.0)
B3 (y) (0.0, 0.0, 0.2, 0.2, 0.2, 0.2, 0.2)
B(y) ? i Bi(y) (0.6, 0.6, 0.7, 0.8, 0.7,
0.5, 0.2)
Compare to A o Rm (0.6, 0.6, 0.7, 0.8, 0.7,
0.5, 0.2)
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A local, Gödel
x1, x2, x3, x4, x5, x6, x7
A (0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0)
y1, y2, y3, y4, y5, y6, y7
B1(y) (1.0, 1.0, 1.0, 0.5, 0.3, 0.0, 0.0)
B2 (y) (0.0, 0.5, 0.7, 1.0, 0.7, 0.5, 0.0)
B3 (y) (0.0, 0.0, 1.0, 1.0, 1.0, 1.0, 1.0)
B(y) ?i Bi(y) (0.0, 0.0, 0.7, 0.5, 0.3,
0.0, 0.0)
Compare to A o Rg (0.0, 0.0, 0.7, 0.5, 0.3,
0.0, 0.0)
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Conclusion

• In the case of fuzzy inputs
• Gödel gives different results for global and
  local inference
• Mamdani gives similar results for global and
  local inference
• In the case of crisp inputs
• Mamdani and Gödel give similar results for global
  and local inference