Lecture 2 fuzzy inference system

Lecture 2 fuzzy inference system

• 1.
  Computational Intelligence Parveen Malik AssistantProfessor School of Electronics Engineering KIIT University
• 2.
  Outline • Fuzzy Inferencesystem • Extension Principle • Linguistic Variable • Fuzzy Relations
• 3.
  Fuzzy Inference system(FIS) Data Classification Fuzzy Inference system Automatic Control Expert Systems Time-Series Prediction Pattern Recognition Robotics Decision Analysis
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  Fuzzy Inference system(FIS) Popular Computing frame work based upon • Fuzzy set theory • Fuzzy IF THEN Rules • Fuzzy Reasoning
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  Extension Principle • Extendscrisp domains of mathematical expression to fuzzy domains. • Generalizes a common point to point mapping of a function to a mapping between fuzzy sets. Suppose 𝑓 is a function from X to Y and A is fuzzy set on X defined as 𝑨 = 𝝁𝑨 𝒙𝟏 𝒙𝟏 + 𝝁𝑨 𝒙𝟐 𝒙𝟐 +⋯ 𝝁𝑨 𝒙𝒏 𝒙𝒏 Then the extension principle states that image of fuzzy set A under the mapping 𝑓 ∙ can be expressed as a fuzzy set B 𝑩 = 𝒇(𝑨) = 𝝁𝑨 𝒙𝟏 𝒚𝟏 + 𝝁𝑨 𝒙𝟐 𝒚𝟐 + ⋯ 𝝁𝑨 𝒙𝒏 𝒚𝒏 Where 𝒚𝟏= 𝒇 𝒙𝟏 , 𝒚𝟐= 𝒇 𝒙𝟐 so on
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  Fuzzy Relations • Binaryfuzzy relations are fuzzy sets in 𝑋 × 𝑌 which map each element in to membership grade 0 to 1. Definition 1.1 Binary fuzzy relation Let X and Y be two universe of discourse. Then ℜ = 𝑥, 𝑦 , 𝜇𝑅 𝑥, 𝑦 | 𝑥, 𝑦 ∈ 𝑋 × 𝑌 is a binary fuzzy relation in 𝑋 × 𝑌.
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  Fuzzy Relations Definition 1.2Max-min composition Let ℜ1 and ℜ1 be two fuzzy relations defined on 𝑋 × 𝑌 and 𝑌 × 𝑍 respectively. The max-min composition of ℜ1 and ℜ2 is a fuzzy set defined by 𝕽𝟏 ∘ 𝕽𝟐 = 𝒙, 𝒛 , 𝐦𝐚𝐱 𝒚 𝒎𝒊𝒏 𝝁𝕽𝟏 𝒙, 𝒚 , 𝝁𝕽𝟐 𝒚, 𝒛 |𝒙 ∈ 𝑿, 𝒚 ∈ 𝒀, 𝒛 ∈ 𝒁 or 𝝁𝕽𝟏∘𝕽𝟐 = 𝐦𝐚𝐱 𝒚 𝒎𝒊𝒏 𝝁𝕽𝟏 𝒙, 𝒚 , 𝝁𝕽𝟐 𝒚, 𝒛
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  Fuzzy Relations Definition 1.2Max product composition Let ℜ1 and ℜ1 be two fuzzy relations defined on 𝑋 × 𝑌 and 𝑌 × 𝑍 respectively. The max product composition of ℜ1 and ℜ2 is a fuzzy set defined by 𝕽𝟏 ∘ 𝕽𝟐 = 𝒙, 𝒛 , 𝐦𝐚𝐱 𝒚 𝝁𝕽𝟏 𝒙, 𝒚 ∗ 𝝁𝕽𝟐 𝒚, 𝒛 |𝒙 ∈ 𝑿, 𝒚 ∈ 𝒀, 𝒛 ∈ 𝒁 or 𝝁𝕽𝟏∘𝕽𝟐 = 𝐦𝐚𝐱 𝒚 𝝁𝕽𝟏 𝒙, 𝒚 ∗ 𝝁𝕽𝟐 𝒚, 𝒛
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  Linguistic Variable If thevariable takes linguistic terms, it is called “Linguistic Variable” Definition: The linguistic variable defined by the following quintuple Linguistic variable = (x, T(x), U,G,M) Where x: name of variable T(x) : set of linguistic terms which can be a value of the variable U : Set of universe of discourse which defines the characteristics of the variable G : syntactic grammar which produces terms in T(x) M: Semantic rules which map terms in T(x) to fuzzy set in U
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  Linguistic Variable Example: Let usconsider a linguistic variable “X” whose name is “Age” X = (Age, T(Age), U,G,M) Where Age : name of variable X T(Age) : {young, very young, very very young,not very young…,} U : [ 0, 100] G(Age) : 𝑇𝑖+1 = {𝑦𝑜𝑢𝑛𝑔} ∪ 𝑣𝑒𝑟𝑦 𝑇𝑖 M(young): (𝑢, 𝜇𝑦𝑜𝑢𝑛𝑔(𝑢)|𝑢 ∈ [0,100] Where, 𝜇𝑦𝑜𝑢𝑛𝑔 𝑢 = ቐ 1 𝑖𝑓 𝑢 ∈ [0,25] 1 + 𝑢−25 5 −2 𝑖𝑓 𝑢 ∈ [25,100]
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  Parts of FuzzyLinguistic Terms 1. Fuzzy Predicate (Primary Term) : Expensive, old, rare, dangerous, good etc. 2. Fuzzy Modifier Very, likely, almost impossible, extremely unlikely etc. a) Fuzzy Truth qualifier : quite true, very true, more or less, mostly false etc. b) Fuzzy quantifier : many, few, almost, all, usually etc. Example : “ x is a man” → man(x) “y is P” → P(y) “Man” and “P” is crisp set Fuzzy predicate : Its definition contains ambiguity “ Z is expensive” “W is young” (expensive and young are fuzzy terms) Let us consider “x is P” is a fuzzy predicate 𝑷 𝒙 𝒊𝒔 𝒂 𝒇𝒖𝒛𝒛𝒚 𝒔𝒆𝒕 𝒂𝒏𝒅 𝝁𝑷(𝒙)𝒊𝒔 𝒎𝒆𝒎𝒃𝒆𝒓𝒔𝒉𝒊𝒑 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 Fuzzy Modifier : young with a membership function 𝝁𝒚𝒐𝒖𝒏𝒈. If we use very young, 𝝁𝒗𝒆𝒓𝒚 𝒚𝒐𝒖𝒏𝒈 = 𝝁𝒚𝒐𝒖𝒏𝒈 𝟐
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  Example: T = {true, very true, fairly, absolutely,…, absolutely false, fairly false, false} 𝝁𝒕𝒓𝒖𝒆 𝜸 = 𝜸 𝝁𝒗𝒆𝒓𝒚 𝒕𝒓𝒖𝒆 𝜸 = (𝝁𝒕𝒓𝒖𝒆 𝜸 )𝟐 𝝁𝒇𝒂𝒊𝒓𝒍𝒚 𝒕𝒓𝒖𝒆 𝜸 = (𝝁𝒕𝒓𝒖𝒆 𝜸 )𝟏/𝟐 𝝁𝒇𝒂𝒍𝒔𝒆 𝜸 = 𝟏 − 𝝁𝒕𝒓𝒖𝒆 𝜸 = 𝟏 − 𝜸 𝝁𝒗𝒆𝒓𝒚 𝒇𝒂𝒍𝒔𝒆 𝜸 = (𝝁𝒇𝒂𝒍𝒔𝒆 𝜸 )𝟐 𝝁𝒇𝒂𝒊𝒓𝒍𝒚 𝒇𝒂𝒍𝒔𝒆 𝜸 = (𝝁𝒇𝒂𝒍𝒔𝒆 𝜸 )𝟏/𝟐 𝝁𝒂𝒃𝒔𝒐𝒍𝒖𝒕𝒆𝒍𝒚 𝒕𝒓𝒖𝒆 𝜸 = ቊ 𝟏 𝒊𝒇 𝜸 = 𝟏 𝟎 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆 𝝁𝒂𝒃𝒔𝒐𝒍𝒖𝒕𝒆𝒍𝒚 𝒇𝒂𝒍𝒔𝒆 𝜸 = ቊ 𝟏 𝒊𝒇 𝜸 = 𝟎 𝟎 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆
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  Example (J.S.R Jhang): T(Age) = { young, not young, very young, not very young, middle aged, not middle aged,…, old, not old, very old, more or less old, not very old,…, not very young and not very old } Definition : Concentration and dilation of linguistic values Let A be a linguistic value characterized by a fuzzy set with membership function 𝝁𝑨 . . Then 𝐴𝑘 is interpreted as a modified version of the original linguistic value expressed as 𝑨𝒌 = න 𝑿 Τ 𝝁𝑨 𝒙 𝒌 𝒙
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  Concentration and dilation(J.S.R Jhang) : Definition : Concentration and dilation of linguistic values Let A be a linguistic value characterized by a fuzzy set with membership function 𝝁𝑨 . . Then 𝐴𝑘 is interpreted as a modified version of the original linguistic value expressed as 𝑨𝒌 = න 𝑿 Τ 𝝁𝑨 𝒙 𝒌 𝒙 Concentration is Expressed as 𝑪𝑶𝑵(𝑨) = 𝑨 𝟐 = 𝝁𝑨 𝒙 𝟐 - very Dilation is Expressed as 𝑫𝑰𝑳(𝑨) = 𝑨 𝟎.𝟓 = 𝝁𝑨 𝒙 𝟏/𝟐 - more or less
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  Examples (J.S.R Jhang): • More or less old = 𝐃𝐈𝐋 𝒐𝒍𝒅 = 𝒐𝒍𝒅 𝟎.𝟓 • Not young and not old = ¬ young ∩ ¬ old • Young but not young = young ∩ ¬ young𝟐 • Extremely old = 𝑪𝑶𝑵 𝑪𝑶𝑵 𝑪𝑶𝑵 𝑨 = (((𝒐𝒍𝒅)𝟐 )𝟐 )𝟐
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  Fuzzy IF THENRules IF x is A Then y is B R = A→ 𝑩 = 𝑨 × 𝑩 = ‫׬‬ 𝑿×𝒀 Τ 𝝁𝑨 𝒙 ෤ ∗ 𝝁𝑩 𝒚 (𝒙, 𝒚) Examples: • IF temp is High then pressure is low • IF pressure is high, then volume is small • If the road is slippery, then driving is dangerous • If a tomato s red, then it is ripe • If speed is high, then apply the brake a little Denoted by 𝑨 → 𝑩
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  Fuzzy IF THENRules IF x is A Then y is B Rules: R = A→ 𝑩 = 𝑨 𝒄𝒐𝒖𝒑𝒍𝒆𝒅 𝒘𝒊𝒕𝒉 𝑩 1. R = A → 𝑩 = 𝑨 × 𝑩 = ‫׬‬ 𝑿×𝒀 Τ 𝝁𝑨 𝒙 ˄𝝁𝑩 𝒚 𝒙, 𝒚 𝑴𝒂𝒎𝒅𝒂𝒏𝒊 2. R = A → 𝑩 = 𝑨 × 𝑩 = ‫׬‬ 𝑿×𝒀 Τ 𝝁𝑨 𝒙 𝝁𝑩 𝒚 (𝒙, 𝒚) Larsen (Algebraic Product) R = 𝑨 𝒆𝒏𝒕𝒂𝒊𝒍𝒔 𝑩 Rules: 1. 𝑹𝒂 = ¬𝑨 ∪ 𝑩 = ‫׬‬ 𝑿×𝒀 Τ 𝟏˄(𝟏 − 𝝁𝑨 𝒙 + 𝝁𝑩 𝒚 ) 𝒙, 𝒚 𝒁𝒂𝒅𝒆𝒉 𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 2. 𝑹𝒎𝒎 = ¬𝑨 ∪ (𝑨 ∪ 𝑩) = ‫׬‬ 𝑿×𝒀 Τ (𝟏 − 𝝁𝑨 𝒙 )˅(𝝁𝑨 𝒙 ˄𝝁𝑩 𝒚 ) 𝒙, 𝒚 𝒁𝒂𝒅𝒆𝒉 𝑴𝒊𝒏𝒎𝒂𝒙 𝒓𝒖𝒍𝒆
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  Fuzzy Reasoning Fuzzy Reasoning( Approximate Reasoning) is an inference procedure that derives conclusions from a set of fuzzy IF THEN rules and Known Facts. Modus Ponens Premise 1 (Fact ) : x is a Premise 2 (Rule ) : If x is a then y is b Consequences (Conclusion) : y is b Forward Inference Modus Tollens Premise 1 (Fact ) : y is b Premise 2 (Rule ) : If x is a then y is b Consequences (Conclusion) : x is a Backward Inference
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  Fuzzy Reasoning Generalized ModusPonens (Approximate Reasoning) Premise 1 (Fact ) : x is 𝐀′ R(x) Premise 2 (Rule ) : If x is 𝐀 then y is B R(x,y) Consequences (Conclusion) : y is 𝐁′ R(y) Example: Premise 1 (Fact ) : If tomato is more or less red R(x) Premise 2 (Rule ) : If tomato is red then it is ripe R(x,y) Consequences (Conclusion) : it is more or less ripe R(y) 𝐁′ = 𝐀′ ∘ (𝑨 → 𝑩) R 𝒚 = R 𝒙 ∘ R 𝒙, 𝒚 R 𝒚 = R 𝒙 ∘ R 𝒙, 𝒚
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  Fuzzy Reasoning Single Rulewith Multiple Antecedents Premise 1 (Fact ) : 𝑥 𝑖𝑠 𝐀′ 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝐁′ 𝐀′ × 𝐁′ Premise 2 (Rule ) : If 𝑥 𝑖𝑠 𝑨 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝑩 then 𝑍 𝑖𝑠 𝑪 𝑨 × 𝑩 → 𝑪 Conclusion : Z is 𝐂′ 𝐂′ = 𝐀′ × 𝐁′ · 𝑨 × 𝑩 → 𝑪 Example: Premise 1 (Fact ) : If tomato is more or less red Premise 2 (Rule ) : If tomato is red then it is ripe Conclusion : it is more or less ripe 𝐂′ = 𝐀′ × 𝐁′ ∘ 𝑨 × 𝑩 → 𝑪
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  Fuzzy Reasoning Multiple Rulewith Multiple Antecedents Premise 1 (Fact ) : 𝑥 𝑖𝑠 𝐀′ 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝐁′ 𝐀′ × 𝐁′ Premise 2 (Rule 1) : If 𝑥 𝑖𝑠 𝑨𝟏 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝑩𝟏 then 𝑍 𝑖𝑠 𝑪𝟏 𝑹𝟏= 𝑨𝟏 × 𝑩𝟏→ 𝑪𝟏 Premise 2 (Rule 1) : If 𝑥 𝑖𝑠 𝑨𝟐 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝑩𝟐 then 𝑍 𝑖𝑠 𝑪𝟐 𝑹𝟐= 𝑨𝟐 × 𝑩𝟐→ 𝑪𝟐 Conclusion : Z is 𝐂′ 𝐂′ = 𝐀′ × 𝐁′ ∘ ( 𝑹𝟏∪ 𝑹𝟐) 𝐂′ = 𝐀′ × 𝐁′ ∘ ( 𝑹𝟏∪ 𝑹𝟐)