fuzzy logic, fuzzy inference - retis labretis.sssup.it/~giorgio/slides/neural/colla-fuzzy.pdf ·...

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Valentina Colla

Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems with applications

2014

Valentina CollaScuola Superiore Sant'Anna

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Fuzzy logic

Fuzzy inference

Neuro-fuzzy systems

Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems

Outline

Scuola Superiore Sant'Anna

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Basic conceptsOn fuzzy sets and fuzzy logic

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Fuzzy logic emerged as a consequence ofthe 1965 proposal of fuzzy set theory byLotfi Zadeh.

→ Fuzzy logic tries to overcome thecriticalities encountered by standard logic(based on the two truth values True andFalse) when describing human reasoning

→ Fuzzy logic uses the extends the conceptsof membership to the whole interval between0 (False) and 1 (True) to describe humanreasoning.


Introduction


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The crisp sets approach

Collections of sets from an Universe U in which every set is supposed to be included


Crisp sets and fuzzy sets


Universe Any set A (i.e. the blue balls set) can be identified by a characteristic function

A

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The crisp vision of the set of numbers in [5;9]

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Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems Fuzzy sets in practice

It's a problem of age


Many sets have more than an either-or criterion for membership.

The age problem. Who is young?A one year old baby will clearly be a member of the set, and a 100years old person will not be a member of this set

→ what about people at the age of 20, 30, or 40 years?

young

0 100

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It is simple to verify the stateman

Water temperature is 27°C

What about the following one?

Is this a pleasant summer day?




→ temperature not too high

→ little bit windy

→ sea temperature close to 24°C

Deals with uncertainty, qualitative, non-objective information

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The fuzzy sets approach

Elements belong to a given set A with a certain degree.

Characteristic functions are substituted by membershipfunctions valued in [0, 1].

A fuzzy subset A of X is defined by its membership functionassigning to every element x of X its degree ofmembership to A




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A description of the fuzzy set of real numbers close to 7 could be given by the following figure:

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A human-like representation of the concepts related to people age.

Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems Fuzzy sets in practice

It's a problem of age


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Popular membership functions


Triangular

Trapezoidal

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Gaussian


Popular membership functions


Bell-shaped

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Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems Some nomenclature

About MFs


Example: the average membership function

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Standard sets theory: basic operations


Working with sets


How to extend to fuzzy sets?

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In standard set theoryThrough the characteristic functions:

Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems Operations on fuzzy sets

Union


Set A → A

Set B → B

Characteristic functions suggest the answer

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Union


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Intersection


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Complement


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Other functions T(·,·) [0,1]x[0,1]→[0,1] can be used to performintersection between fuzzy sets. Some basic requirements:

→ commutativity

→ associativity

→ to be non-decreasing

Objects with such properties are called t-norms.


More on intersection


xxxxTxxT FSSFFSSF )(),()(),(

xxxxxTTxxTxT QFSQSFQSFQSF ][][)(,)(),()(),(),(

xx QSQF xxTxxTthenxxXxIf QSQFSF ,,

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Lukasiewicz

Product (usual product)

Godel

Drastic

Frank

Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems Popular t-norms

Intersection


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The concept of membership function allows us to define fuzzysystems in natural language. Direct correspondence betweenlinguistic variables and fuzzy sets.

Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems Linguistic variables

And fuzzy sets


Fuzzy set Linguistic variable

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The fuzzy logic invalidate two of the strongholds of the classical logic:

1. Law (or principle) of the excluded third (or Tertium non datur in Latin):

A or not(A) = TRUE

2. Principle of contradiction (or principium contradictionis in Latin):

A and not(A) = FALSE


Breaking the rules...


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Inferencewith fuzzy sets and fuzzy rules

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Fuzzy inference


A fuzzy inference system achieves approximated conclusions on the basis of a set of approximate premises through a set of if-then rules and an inference engine

Standard (canonical) rulesIF (x1 is A1) AND (x2 is A2) AND … (xN is AN) THEN (y is Bj)Sub-casesPartial rules

IF (x1 is A1) AND … AND (xm is Am) THEN (y is Bj) con m<NOR rules

IF (x1 is A1) AND … AND (xm is Am) OR (xm+1 is Am+1) AND … AND (xN is AN) THEN (y is Bj) con m<NEquivalent to two standard rules

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Rules set is complete if for each x in Ux, one rule at least is active

Rules set is consistent if no rules with the same premise butdifferent consequent exist

Mamdami rules:IF x1 is A1 and x2 is A2 and . . . and xN is AN THENy1 is B1 and y2 is B2 and . . . and yM is BM

Takagi-Sugeno rules:IF x1 is A1 and x2 is A2 and . . . and xN is AN THENy = fk (x1 , x2 , . . . , xN )


Rules


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If x is A then y is B


Fuzzy inference


→ A and B are fuzzy sets defined on the universe X and Y (x ∈ X and y ∈ Y)

→ the fuzzy rule defines a relation R on the space X × Y ;→ a fuzzy relation is a fuzzy set defined on several domains:

R = X × Y × Z × ...μ R (x, y , z, . . .)

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Fuzzy inference


Example or relation derived from a rule:

R = A × Bμ R (x, y ) = min (μ A (x), μ B (y ))

Mamdani implication

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Fuzzy inference


Although Mamdani inference type is widely used, it is not theonly one (as for the other logic operators used for fuzzy sets).

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Given the rule if x is A then y is B corresponding to the fuzzy relation RWhere A and B are fuzzy sets defined on the universe X and Y and x ∈ X , y ∈ Y

How can I produce fuzzy reasoning?

In classical logic:

Modus ponens

Premise: if (x is A) then (y is B)Antecedent: x is A→ Consequent: y is B


Fuzzy inference


Modus Tollens

Premise: if (x is A) then (y is B)Antecedent: y is not BConsequent: x is not A

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Standard modus ponens cannot be used in the fuzzy logic (it works iff the premise is exactly the same as the antecedent of the IF-THEN rule).

Generalized modus ponens allows an inference when the fact is only similar but not equal to it.

Generalized modus-ponensPremise: if (x is A) then (y is B)Antecedent: x is A'Consequent: y is B'

B' (y ) = max (min (μA' (x), μR (x, y)))[max-min composition; max-prod is an alternative]


Fuzzy inference


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Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems Fuzzy inference

One rule, one atom


if (x is A) then (y is C)Calculating the resulting fuzzy set

Modus ponens

Where

Hence

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One rule, two atoms


if (x1 is A) and (x2 is B) then (y is C)Calculating the resulting fuzzy set

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When more rules are involved in the inferencesystem, the fuzzy sets resulting as consequentsof each rule must be combined through the so-called aggregation.

Several aggregation operators exist. Mostcommon are min and max


Aggregation


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Aggregation


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Defuzzification is the process of producing a quantifiable result in fuzzylogic. A fuzzy quantity is converted into a crisp one (a single number).

Defuzzifier is the system implementing the defuzzification. Several methodsare implemented for defuzzification.


Defuzzification


Centroid

Min of maximum

Mean of maximum

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To sum up...


Fuzzyfication

Input variablesCrisp

Rule base

Defuzzyfication

Fuzzy operators Aggregation

Output variablesCrisp

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→ based on natural language→ can exploit human knowledge and experience→ flexible, immediate, easy→ robustness to unreliable data→ can model complex non linear functions→ can be integrated to adaptive systems (neural networks)→ can be integrated to standard control techniques


Some Pros


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→ implementing systems with many inputs and outputs is difficult→ requires knowledge on the relation between I/O (memberships, rules)→ membership functions parameters should be carefully tuned→ do not exploit any eventual numerical data


Some Cons


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The problemDecide the amount of the tip on the basis of→ food quality→ service

In crisp terms a 1-10 rating for each criterium is used

Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems Example

Determining the Tip


The rules1. If service is BAD or food is AWFUL then tip is LOW2. if service is GOOD then tip is AVERAGE3. if service is EXCELLENT or food is DELICIOUS the tip is HIGH

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Which fuzzy sets? Just use natural language (and rules)

Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems Tip example

Step 1: fuzzyfication


The rules1. If service is BAD or food is AWFUL then tip is LOW2. if service is GOOD then tip is AVERAGE3. if service is EXCELLENT or food is DELICIOUS the tip is HIGH

Service → BAD, GOOD, EXCELLENTFood → AWFUL, DELICIOUS

Tip → LOW, AVERAGE, HIGH

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Step 1: fuzzyfication


Tip

Service Food

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Step 2: Fuzzy inference


Mamdani implication, max aggregation, centroid defuzzyfication

Sample inference on rule 3

3. if service is EXCELLENT or food is DELICIOUS the tip is HIGH

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Step 3: aggregation


1. If service is BAD or food is AWFUL then tip is LOW2. if service is GOOD then tip is AVERAGE3. if service is EXCELLENT or food is DELICIOUS the tip is HIGH

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Alternative: Mean of maximumTip is 25.1


Step 4: defuzzyfication


Centroid (Center Of Gravity - COG)Tip is 21.5

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Neuro-fuzzy networks

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Motivation


Neuro Fuzzy Systems(aka Fuzzy neural networks)

Artificial neural networkslearning and

connectionist structure

Fuzzy logicHuman-like reasoning

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Genealogy


Neuro Fuzzy Systems(aka Fuzzy neural networks)

Artificial neural networksData exploitation

Training capability

Fuzzy logicfuzzy sets, a linguistic model

IF-THEN fuzzy rules

Universal approximator

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Critical: determining FISparameters in an efficient mannerby using the sole humanexperience

Approach: use experimental datato tune FIS parameters andminimize prediction error


Basic idea


Data

Fis

Learningalgorithm

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Optimal parameters

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ANFIS Artificial Neuro-Fuzzy Inference Systems

→ are a class of adaptive networks that arefuncionally equivalent to fuzzy inference systems.

→ represent Sugeno e Tsukamoto fuzzymodels.

→ use a hybrid learning algorithm


ANFIS


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Fuzzy logic, fuzzy inferenceAnd neuro-fuzzy systems ANFIS

Sugeno model


Assume that the fuzzy inference system has two inputs x and y and oneoutput z .

A simple first-order Sugeno fuzzy model has rules as the following:

• Rule1:If x is A1 and y is B1 , then f1 = p1 x + q1 y + r1• Rule2:If x is A2 and y is B2 , then f2 = p2x + q2y + r2

RECALLMamdami rules:IF x1 is A1 and x2 is A2 and . . . and xN is AN THEN y1 is B1 and y2 is B2 and . . . and yM is BM

Takagi-Sugeno rules:IF x1 is A1 and x2 is A2 and . . . and xN is AN THEN y = fk (x1 , x2 , . . . , xN )

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Output of the Sugeno model (for a given x, y):


Sugeno model


1111 ryqxpf

2222 ryqxpf

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Corpo della slide


ANFIS Structure


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Layer 1

→ Ol,i is the output of the ith node of the layer l .

Every node i in this layer returns the membership degree of the input crisp variable with respect to the associated fuzzy set

Typical membership function:


ANFIS Structure


Parameters: ai, bi, ciPremise parameters

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Layer 2

Every node in this layer performs product of all incoming signals:→ O2,i = w i = μAi(x) · μ Bi(y), i = 1, 2→ Each node represents the fire strength of the rule→ Any other T-norm operator that performs the AND operator canbe used

Layer 3

Every node returns the normalized firing strength for thecorresponding rule


ANFIS Structure


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Layer 4

Every node i in this layer is an adaptive node which returns

Introducing three more parameters (called consequent parameters) → pi, qi, ri

Layer 5

Overall output is calculated


ANFIS Structure


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The ANFIS can be trained by a hybrid learning Algorithm derived(and very similar) from the ANN back-propagation

As back-propagation, it works in two passes:• In the forward pass the algorithm uses least-squares method toidentify the consequent parameters on the layer 4.• In the backward pass the errors are propagated backward and thepremise parameters are updated by gradient descent.


Training algorithm


[email protected]

Thank you for Your attention!

fuzzy logic, fuzzy inference - retis labretis.sssup.it/~giorgio/slides/neural/colla-fuzzy.pdf ·...

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