lecture notes fuzzysets and techniques - cbajoakim/course/fuzzy/vt07/lectures/l15_4.pdfjoakim...

Fuzzy Setsand FuzzyTechniques

JoakimLindblad

Outline

L1: Intro

L1–3: Basics

L4: Constr.anduncertainty

L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic

L11: Defuzzi-fication

L12: Fuzzycontrol

L13: Fuzzyconnectedness

Fuzzy Sets and Fuzzy Techniques

Lecture 15 – Repetition

Joakim [email protected]

Centre for Image AnalysisUppsala University

2007-03-16

Joakim Lindblad, 2007-03-16 (1/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Lecture notes

http://www.cb.uu.se/~joakim/course/fuzzy/lectures.html



JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Topics of today

L1: Introduction, motivation

L1–3: Basics of fuzzy sets

L4: Constructing fuzzy sets, Uncertainty measures

L5: Fuzzy thresholding, Fuzzy c-means clustering,Some features of spatial fuzzy sets

L6–7: Distances on and between fuzzy sets

L8: Operations on fuzzy sets

L9: Fuzzy numbers and fuzzy arithmetics

L10: Fuzzy logic and approximate reasoning

L11: Defuzzification

L12: Fuzzy control

L13: Fuzzy connectedness



JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


L1: Introduction, motivation


http://www.cb.uu.se/~joakim/course/fuzzy/lectures.html


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


What will we learn in this course?Fuzzy Sets and Fuzzy Techniques

• The basics of fuzzy sets• How to define fuzzy sets• How to perform operations on fuzzy sets• How to extend crisp concepts to fuzzy ones• How to extract information from fuzzy sets

• The very basics of fuzzy logic and fuzzy reasoning

• We will look at some applications of fuzzy in• Image processing• Control systems• Machine intelligence / expert systems



JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


What is a fuzzy set?

Btw., what is a set? “... to be an element...”

A set is a collection of its members.

The notion of fuzzy sets is an extensionof the most fundamental property of sets.

Fuzzy sets allows a grading of to what extentan element of a set belongs to that specific set.



JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Why Fuzzy?

Precision is not truth.- Henri Matisse

So far as the laws of mathematics refer to reality, they are notcertain. And so far as they are certain, they do not refer toreality.

- Albert Einstein

As complexity rises, precise statements lose meaning andmeaningful statements lose precision.

- Lotfi Zadeh



JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Fuzzy is not just another name for probability.

The number 10 is not probably big!...and number 2 is not probably not big.

Uncertainty is a consequence ofnon-sharp boundaries between the notions/objects,

and not caused by lack of information.

Statistical models deal with random events and outcomes;fuzzy models attempt to capture and quantify nonrandomimprecision.



JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


What is a fuzzy set?Randomness vs. Fuzziness

Randomness refers to an event that may or may not occur.Randomness: frequency of car accidents.

Fuzziness refers to the boundary of a set that is not precise.Fuzziness: seriousness of a car accident.

Prof. George J. Klir



JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Using fuzzy techniques is

to avoid throwing away data early (by crisp decisions).

Joakim Lindblad, 2007-03-16 (10/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


L1–3: Basics of fuzzy sets

Joakim Lindblad, 2007-03-16 (11/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy sets

A fuzzy set of a reference set is a set of ordered pairs

F = {〈x , µF (x)〉 | x ∈ X},where µF : X → [0, 1].

Where there is no risk for confusion, we use the same symbolfor the fuzzy set, as for its membership function.

ThusF = {〈x ,F (x)〉 | x ∈ X},

where F : X → [0, 1].

To define a fuzzy set ⇔ To define a membership function

Joakim Lindblad, 2007-03-16 (12/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy sets

Continuous (analog) fuzzy sets

A : X → [0, 1]

Discrete fuzzy sets

A : {x1, x2, x3, ..., xs} → [0, 1]

Digital fuzzy sets

If a discrete-universal membership function can take only afinite number n ≥ 2 of distinct values, then we call this fuzzyset a digital fuzzy set.

A : {x1, x2, x3, ..., xs} → {0, 1n−1 ,

2n−1 ,

3n−1 , ...,

n−2n−1 , 1}

Joakim Lindblad, 2007-03-16 (13/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy sets of different types

The membership function may be vague in itself.

Fuzzy sets of type 2

A : X → F([0, 1])

Joakim Lindblad, 2007-03-16 (14/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy sets of different levels

Also the domain of the membership function may be fuzzy.

Fuzzy sets defined so that the elements of the universal set arethemselves fuzzy sets are called level 2 fuzzy sets.

A : F(X )→ [0, 1]

Joakim Lindblad, 2007-03-16 (15/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Basic concepts and terminology

The support of a fuzzy set A in the universal set X is a crispset that contains all the elements of X that have nonzeromembership values in A, that is,

supp(A) = {x ∈ X | A(x) > 0}

A crossover point of a fuzzy set is a point in X whosemembership value to A is equal to 0.5.

The height, h(A) of a fuzzy set A is the largest membershipvalue attained by any point. If the height of a fuzzy set isequal to one, it is called a normal fuzzy set, otherwise it issubnormal.

Joakim Lindblad, 2007-03-16 (16/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



An α-cut of a fuzzy set A is a crisp set αA that contains allthe elements in X that have membership value in A greaterthan or equal to α.

αA = {x | A(x) ≥ α}

A strong α-cut of a fuzzy set A is a crisp set α+A thatcontains all the elements in X that have membership value in Astrictly greater than α.

α+A = {x | A(x) > α}

We observe that the strong α-cut 0+A is equivalent to thesupport supp(A). The 1-cut 1A is often called the core of A.

Joakim Lindblad, 2007-03-16 (17/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



The ordering of the values of α in [0, 1] is inversely preservedby set inclusion of the corresponding α-cuts as well as strongα-cuts. That is, for any fuzzy set A and α1 < α2 it holds thatα2A ⊆α1 A.

All α-cuts and all strong α-cuts for two distinct families ofnested crisp sets.

The set of all levels α ∈ [0, 1] that represent distinct α-cuts ofa given fuzzy set A is called a level set of A.

Λ(A) = {α | A(x) = α for some x ∈ X}.

Joakim Lindblad, 2007-03-16 (18/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



A fuzzy set A defined on �

n is convex iff

A(λx1 + (1− λ)x2) ≥ min (A(x1),A(x2)) ,

for all λ ∈ [0, 1], x1, x2 ∈ �

n and all α ∈ [0, 1].

Or, equivalently, A is convex if and only if all its α-cuts αA, forany α in the interval α ∈ (0, 1], are convex sets.

Any property that is generalized from classical set theory intothe domain of fuzzy set theory by requiring that it holds in allα-cuts in the classical sense is called a cutworthy property.

Joakim Lindblad, 2007-03-16 (19/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Don’t forget to read in the book here!

Chapter 1.4 and Chapter 2.

Joakim Lindblad, 2007-03-16 (20/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Standard fuzzy set operations

A(x) = 1− A(x) − fuzzy complement(A ∩ B)(x) = min[A(x),B(x)] − fuzzy intersection(A ∪ B)(x) = max[A(x),B(x)] − fuzzy union

An equilibrium point of a fuzzy set is a point in X such thatA(x) = A(x). (Same as crossover point for standardcomplement.)

Joakim Lindblad, 2007-03-16 (21/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


De Morgan lattice/algebra

For standard fuzzy set operations, the law of contradiction

A ∩ A 6= ∅

and the law of excluded middle, are violated.

A ∪ A 6= X

Joakim Lindblad, 2007-03-16 (22/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Set inclusion

A ⊆ B iff A(x) ≤ B(x)∀x ∈ X

EqualityA = B iff A(x) = B(x)∀x ∈ X

Scalar cardinality

|A| =∑

x∈X

A(x)

Joakim Lindblad, 2007-03-16 (23/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Standard fuzzy intersection and fuzzy union of two fuzzy setsare cutworthy and strong cutworty. Due to associativity of minand max, any finite intersection/union. However, caution withinfinitely many intersections/unions.

Decomposition theorems

Each standard fuzzy set is uniquely represented by the family ofall its α-cuts, or by the family of all its strong α-cuts.

Joakim Lindblad, 2007-03-16 (24/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Extension principleAny given function f : X → Y induces two functions,

f : F(X )→ Y[f (A)](y) = sup

x |y=f (x)A(x)

and

f −1 : F(Y )→ X[f −1(B)](x) = B(f (x))

Strong cutworthinessFor any A ∈ F(X ), and a function f : X → Y , it holds that

f (A) =⋃

α∈[0,1]

f (α+A)

Joakim Lindblad, 2007-03-16 (25/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Constructing fuzzy sets,Uncertainty measures

Joakim Lindblad, 2007-03-16 (26/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Constructing fuzzy sets,Uncertainty measures

Methods of construction

• Direct methods and indirect methods

• One expert and multiple experts

Joakim Lindblad, 2007-03-16 (27/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Direct methods with one expert

• Define the complete membership function based on ajustifiable mathematical formula• Often based on mapping of directly measurable features of

the elements of X

• Exemplifying it for some selected elements of X andinterpolate (/extrapolate) MF in some way.• Expert of some kind

Joakim Lindblad, 2007-03-16 (28/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Direct methods with multipleexperts

The opinions of several experts need to be aggregated.

Example: Average (Probabilistic interpretation)

A(x) =1

n

n∑

i=1

ai (x)

Joakim Lindblad, 2007-03-16 (29/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Indirect methods

It may be easier/more objective to ask simpler questions to theexperts, than the membership directly.

Example: Pairwise comparisons

• Problem: Determine membership ai = A(xi )

• Extracted information: Pairwise relative belongingness,matrix P with pij ≈ ai

aj

Joakim Lindblad, 2007-03-16 (30/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Uncertainty measures

• Nonspecificity of crisp sets

• Nonspecificity of fuzzy sets

• Fuzziness of fuzzy sets

Joakim Lindblad, 2007-03-16 (31/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Nonspecificity of crisp setsHartley function

Hartley [1928] showed that a function

U(A) = c logb|A| ,

where |A| is the cardinality of A, and b > 1 and c > 0 areconstants, is the only sensible way to measure the amount ofuncertainty associated with a finite set of possible alternatives.

b = 2 and c = 1 → uncertainty measure in bits

U(A) = log2|A|

Relates to the nonspecificity inherent in each set.Larger sets correspond to less specific predictions.

Joakim Lindblad, 2007-03-16 (32/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Nonspecificity of fuzzy setsU-uncertainty

Generalized Hartley function

U(A) =1

h(A)

∫ h(A)

0log2|αA| dα

Weighted average of the Hartley function for all distinct α-cutsof the normalized counterpart of A.

Fuzzy sets that are equal when normalized have the samenonspecificity.

Joakim Lindblad, 2007-03-16 (33/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzziness of fuzzy sets

A measure of fuzziness is a function

f : F(X )→ �

+

For each fuzzy set A, f (A) expresses the degree to which theboundary of A is not sharp.

The following three requirements are essential

1 f (A) = 0 iff A is a crisp set

2 f (A) attains its maximum iff A(x) = 0.5 for all x ∈ X

3 f (A) ≤ f (B) when set A is “undoubtedly” sharper thanset B

a) A(x) ≤ B(x) when B(x) ≤ 0.5b) A(x) ≥ B(x) when B(x) ≥ 0.5

Joakim Lindblad, 2007-03-16 (34/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



One way to measure fuzziness of a set A is to measure thedistance between A and the nearest crisp set. Remaining is tochoose the distance measure.

Another way is to view the fuzziness of a set as the lack ofdistinction between the set and its complement. The less aset differs from its complement, the fuzzier it is. Also this path(which is the one we will take) requires a distance measure.

Joakim Lindblad, 2007-03-16 (35/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



A simple and intuitive distance measure is the Hammingdistance.

d(A,B) =∑|A(x)− B(x)|

The measure of fuzziness as the distance to the complement,then becomes

f (A) = d(X , X )− d(A, A)

=∑

(1− |A(x)− (1− A(x))|)

=∑

(1− |2A(x)− 1|)

Joakim Lindblad, 2007-03-16 (36/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Information gain?

Fuzziness and nonspecificity are distinct types of uncertaintyand totally independent of each other.

They are also totally different in their connections toinformation. When nonspecificity is reduced, we view this as again in information, regardless of any associated change infuzziness. The opposite, however, is not true.

A reduction of fuzziness is reasonable to consider as a gain ofinformation only if the nonspecificity also decreases or remainsthe same.

Joakim Lindblad, 2007-03-16 (37/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


L5: Fuzzy thresholding, Fuzzyc-means clustering, Some features

of spatial fuzzy sets

Joakim Lindblad, 2007-03-16 (38/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Thresholding

Thresholding and fuzzy thresholding of fuzzy sets, based ondifferent ways of measuring and minimizing fuzziness.

Joakim Lindblad, 2007-03-16 (39/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Thresholding

Membershipdistributions assigned using

a) Pal and Rosenfeld (1988)

b) Huang and Wang (1995)

c) Fuzzy c-means (Bezdek 1981) algorithms.

Joakim Lindblad, 2007-03-16 (40/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy c-means clusteringBezdek

Chapter 13.2

Algorithm

• make initial guess for cluster means

• iteratively• use the estimated means to assign samples to clusters• update means

• until there are no changes in means

Joakim Lindblad, 2007-03-16 (41/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy c-means clustering

• a partition of the observed set is represented by a c × nmatrix U = [uik ], where uik corresponds to themembership value (anything between 0 and 1!) of the kthelement (out of n), to the ith cluster (out of c)

• boundaries between subgroups are not crisp

• each element may belong to more than one cluster - its”overall” membership equals one

• objective function includes parameter controlling degree offuzziness

Joakim Lindblad, 2007-03-16 (42/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Features of fuzzy setsAggregating over α-cuts

The fuzzification principle

Given a function f : P(X )→ � .

We can extends this function to f : F(X )→ � ,using one of the following equations

f (A) =

∫ 1

0f (αA) dα, (1)

f (A) = supα∈(0,1]

[αf (αA)] (2)

Both these definitions provide consistency for the crisp case.

Joakim Lindblad, 2007-03-16 (43/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Features of fuzzy sets

The area of a fuzzy set A on X ⊆ � is

area(A) =

∫

XA(x) dx

=

∫ 1

0area(αA) dα

For a discrete fuzzy set, the area is equal to the cardinality ofthe set

area(A) = |A| =∑

X

A(x)

The perimeter of a fuzzy set A

perim(A) =

∫ 1

0perim(αA) dα

Joakim Lindblad, 2007-03-16 (44/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Features of fuzzy sets

Geometric moments:

The moment mp,q(A) of a fuzzy set A defined on X ⊂ �

2, is

mp,q(A) =

∫∫

X

A(x , y) xpy q dxdy .

for integers p, q ≥ 0.Remark: The area of a set is the m0,0 moment.

Remark: The centroid (centre of gravity) of a set is

(xc , yc) =

(m1,0(A)

m0,0(A),

m0,1(A)

m0,0(A)

)

Joakim Lindblad, 2007-03-16 (45/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Inter-relations

All the definitions listed above reduce to the correspondingcustomary definitions for crisp sets. However, someinter-relations which these notions satisfy in the crisp case, donot hold for the generalized (fuzzified) definitions.

For example: The isoperimetric inequality,

4πarea(µ) ≤ perim2(µ),

Joakim Lindblad, 2007-03-16 (46/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Estimation of features

As is well know, features of a continuous spatial shape S , canbe estimated from features of its digitization D(S).

The precision of such estimates is in general limited by thespatial resolution of the digital representation.

For object represented by digital spatial fuzzy sets, where themembership of a point indicates to what extent the pixel/voxelis covered by the imaged continuous (crisp) object, significantimprovements in precision of feature estimates can beobtained. Especially so, for small objects/limited resolution.

Joakim Lindblad, 2007-03-16 (47/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Estimation of features

Significant improvement in the precision offeature estimates can be achieved using a fuzzyapproach.

Exploiting fuzzy can provide an alternative toincreasing the spatial resolution of the image.

Joakim Lindblad, 2007-03-16 (48/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy features values

Why give crisp values of features?

Fuzzy feature values still in its infancy.

Joakim Lindblad, 2007-03-16 (49/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


L6–7: Distances on and betweenfuzzy sets

• Set to set distances

• (Point to set distances)

• Point to point distances

A mix of notions

• The objects that the distance is measured between (startand stop)

- crisp or fuzzy, point or set

• The space where a path between start and stop isembedded (spatial cost function)

- Unconstrained (Euclidean)- Constrained (geodesic/cost function)

• Output: Crisp (number) or fuzzy

Joakim Lindblad, 2007-03-16 (50/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


L6–7: Distances on and betweenfuzzy sets

Definition (Metric)

A metric is a positive function d : X → � such that

1 d(x , x) = 0 (reflexivity)

2 d(x , y) = 0⇒ x = y (separability)

3 d(x , y) = d(y , x) (symmetry)

4 d(x , z) ≤ d(x , y) + d(y , z) (triangular inequality)

Joakim Lindblad, 2007-03-16 (51/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Set to set distances

Distances between fuzzy sets

a) Membership focused (vertical)

b) Spatially focused (horizontal)

c) Mix of spatial and membership (tolerance)

d) Feature distances (low or high dimensional representations)

e) Morphological (mixed focus)

Joakim Lindblad, 2007-03-16 (52/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Membership focusedLp norm

“The functional approach”

The most common:Based on the family of Minkowski distances

dp(A,B) =

(∫

X|µA(x)− µB(x)|p dx

)1/p

, p ≥ 1 ,

dEssSup(A,B) = limp→∞

dp(A,B)

d∞(A,B) = supx∈X|µA(x)− µB(x)| .

Joakim Lindblad, 2007-03-16 (53/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Membership focusedLp norm

Discrete version:

dp(A,B) =

(n∑

i=1

|µA(xi )− µB(xi )|p)1/p

, p ≥ 1 ,

d∞(A,B) = maxi=1...n

(|µA(xi )− µB(xi )|) .

dp for p ≥ 1 are all metrics in the discrete case.

Normalized variants, divide with |X | or∑

u + v or similar.

Joakim Lindblad, 2007-03-16 (54/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Membership focusedSet operations approach

Tversky 1977, et al.

Figure: Representation of two objects that each contains its ownunique features and also contains common features.

Joakim Lindblad, 2007-03-16 (55/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Membership focusedSet operations approach

An important aspect of Tversky’s model is that similaritydepends not only on the proportion of features common to thetwo objects but also on their unique features.

Based on this and several other assumptions, Tversky derivedthe following relationship:

S(a, b) = θf (A ∩ B)− αf (A− B)− βf (B − A)

Here, S is an interval scale of similarity, f is an interval scalethat reflects the salience of the various features, and θ, α and βare parameters that provide for differences in focus on thedifferent components.

Joakim Lindblad, 2007-03-16 (56/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Spatially focused

• Nearest point

• Mean distance

• Hausdorff

Three (four) approaches:

• fuzzy distance

• weighting with membership

• morphological and integration of alpha-cuts

Joakim Lindblad, 2007-03-16 (57/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Spatially focusedHausdorff

Crisp:

dH(A,B) = max{supx∈A

infy∈B

d(x , y), supy∈A

infx∈B

d(x , y)}

= inf{r ∈ �

+ | A ⊆ Dr (B) ∧ B ⊆ Dr (A)}

where Dr (A) is the dilation of the set A by a ball of radius r

Dr (A) = {y ∈ X | ∃x ∈ A : d(x , y) ≤ r}

The Hausdorff distance between A and B is the smallestamount that A must be expanded to contain B and vice versa.

Is a metric on the set of nonempty compact sets.

Remark:Usually extended with: dH(A, ∅) =∞ and dH(∅, ∅) = 0

Joakim Lindblad, 2007-03-16 (58/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Spatially focusedHausdorff

Ralescu and Ralescu (1984)

dH1(A,B) =

∫ 1

0dH(αA,αB) dα,

dH∞(A,B) = supα>0

dH(αA,αB),

where dH is the Hausdorff distance between two crisp sets,

A serious problem is that the distance between two fuzzy sets Aand B is infinite if height(A) 6= height(B).

No good solution to that problem is found.

Joakim Lindblad, 2007-03-16 (59/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Feature distances“Pattern recognition approach”

Use of a feature representation of the observed sets as anintermediate step in the distance calculations.

The distance between sets A and B is then given in terms ofthe distance between their feature vectors.

Often global shape features are used (think shape matching,image retrieval).

Joakim Lindblad, 2007-03-16 (60/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Point to point distances

Distances between points in a fuzzy setDefining the cost of traveling along a path

Joakim Lindblad, 2007-03-16 (61/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Cost functionshoveling snow distance

Similar to grey weighted distances (Levi & Montanari ’70) putin a fuzzy framework (Saha ’02).

Define the distance along a path πi between points x and y inthe fuzzy set A

dA(πi (x , y)) =

∫

s∈πA(t) dt

The distance between points x and y in A isthe distance along the shortest path

dA(x , y) = infπ∈Π(x ,y)

dA(π)

Joakim Lindblad, 2007-03-16 (62/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Cost functionvariations

Membership as another dimensionintegrate the arc-length

Bloch 1995, Toivanen 1996:

dA(π) =

∫

s∈π

√1 +

(dA(t)

dt

)2

dt

Problem: Scale of membership relative to spatial distance

Joakim Lindblad, 2007-03-16 (63/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Constrained distance

Geodesic distance – shortest path within the set; not allowed togo out of the set

a path that descends the least in terms of membership

Joakim Lindblad, 2007-03-16 (64/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Connectedness, Rosenfeld 1979

Strength of a path – the strength of its weakest link

Strength of a link between two points defined by themembership function.

The connectedness of two points x and y in A –the strength of the strongest path between x and y

cA(x , y) = supπ∈Π(x ,y)

inft∈π

A(t)

Joakim Lindblad, 2007-03-16 (65/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Bloch and Maıtre 1995:

d(x , y) = infπ∈ΠcA

(x ,y)

∫π ds

cA(x , y)

where cA(x , y) is the strength of connectedness of points x andy , and ΠcA

(x , y) is the set of path contained within the α-cutcAA.

Does not provide triangle inequality.

Joakim Lindblad, 2007-03-16 (66/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


L8: Operations on fuzzy sets

Joakim Lindblad, 2007-03-16 (67/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


slide

A(x) = 1− A(x) − fuzzy complement(A ∩ B)(x) = min[A(x),B(x)] − fuzzy intersection(A ∪ B)(x) = max[A(x),B(x)] − fuzzy union

Joakim Lindblad, 2007-03-16 (68/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Properties of the standardoperations

• They are generalizations of the corresponding (uniquelydefined!) classical set operations.

• They satisfy the cutworthy and strong cutworthyproperties. They are the only ones that do.

• The standard fuzzy intersection of two sets contains (isbigger than) all other fuzzy intersections of those sets.

• The standard fuzzy union of two sets is contained in (issmaller than) all other fuzzy unions of those sets.

Joakim Lindblad, 2007-03-16 (68/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Aggregation operators

Aggregation operators are used to combine several fuzzy sets inorder to produce a single fuzzy set.

Associative aggregation operations

• (general) fuzzy intersections - t-norms

• (general) fuzzy unions - t-conorms

Non-associative aggregation operations

• averaging operations - idempotent aggregation operations

Joakim Lindblad, 2007-03-16 (69/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy complementsAxiomatic requirements

Ax c1. c(0) = 1 and c(1) = 0. boundary condition

Ax c2. For all a, b ∈ [0, 1], if a ≤ b, then c(a) ≥ c(b). monotonicity

c1 and c2 are called axiomatic skeleton for fuzzy complements

Ax c3. c is a continuous function.

Ax c4. c is involutive, i.e., c(c(a)) = a, for each a ∈ [0, 1].

Joakim Lindblad, 2007-03-16 (70/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


GeneratorsIncreasing generators

• Increasing generatoris a strictly increasing continuous function g : [0, 1]→ R,such that g(0) = 0.

• A pseudo-inverse of increasing generator g is defined as

g (−1) =

0 for a ∈ (−∞, 0)g−1(a) for a ∈ [0, g(1)]1 for a ∈ (g(1),∞)

• An example:

g(a) = ap , p > 0

g (−1)(a) =

0 for a ∈ (−∞, 0)

a1p for a ∈ [0, 1]

1 for a ∈ (1,∞)

Joakim Lindblad, 2007-03-16 (71/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Generating fuzzy complements

Theorem

(First Characterization Theorem of Fuzzy Complements.)

Let c be a function from [0, 1] to [0, 1]. Then c is a(involutive) fuzzy complement iff there exists an increasinggenerator g such that, for all a ∈ [0, 1]

c(a) = g−1(g(1)− g(a)).

Theorem

(Second Characterization Theorem of Fuzzy Complements.)

Let c be a function from [0, 1] to [0, 1]. Then c is a(involutive) fuzzy complement iff there exists an decreasinggenerator f such that, for all a ∈ [0, 1]

c(a) = f −1(f (0)− f (a)).Joakim Lindblad, 2007-03-16 (72/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy intersectionsAxiomatic requirements

For all a, b, d ∈ [0, 1],

Ax i1. i(a, 1) = a. boundary condition

Ax i2. b ≤ d implies i(a, b) ≤ i(a, d). monotonicity

Ax i3. i(a, b) = i(b, a). commutativity

Ax i4. i(a, i(b, d)) = i(i(a, b), d). associativity

Axioms i1 - i4 are called axiomatic skeleton for fuzzyintersections.

If the sets are crisp, i becomes the classical (crisp)intersection.

Joakim Lindblad, 2007-03-16 (73/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy intersectionsAdditional (optional) requirements

For all a, b, d ∈ [0, 1],

Ax i5. i is a continuous function. continuity

Ax i6. i(a, a) ≤ a. subidempotency

Ax i7. a1 < a2 and b1 < b2 implies i(a1, b1) < i(a2, b2).strict monotonicity

Note:

The standard fuzzy intersection, i(a, b) = min[a, b], is the only

idempotent t-norm.

Joakim Lindblad, 2007-03-16 (74/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy intersectionsExamples of t-norms frequently used

• Drastic intersection

i(a, b) =

8<:

a if b = 1b if a = 10 otherwise

• Bounded differencei(a, b) = max[0, a + b − 1]

• Algebraic producti(a, b) = ab

• Standard intersectioni(a, b) = min[a, b]

• imin(a, b) ≤ max(0, a + b − 1) ≤ ab ≤ min(a, b).

• For all a, b ∈ [0, 1], imin(a, b) ≤ i(a, b) ≤ min[a, b].

Joakim Lindblad, 2007-03-16 (75/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy intersectionsHow to generate t-norms

Theorem

(Characterization Theorem of t-norms) Let i be a binaryoperation on the unit interval. Then, i is an Archimedeant-norm iff there exists a decreasing generator f such that

i(a, b) = f (−1)(f (a) + f (b)), for a, b ∈ [0, 1].

Joakim Lindblad, 2007-03-16 (76/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy unionsAxiomatic requirements

For all a, b, d ∈ [0, 1],

Ax u1. u(a, 0) = a. boundary condition

Ax u2. b ≤ d implies u(a, b) ≤ u(a, d). monotonicity

Ax u3. u(a, b) = u(b, a). commutativity

Ax u4. u(a, u(b, d)) = u(u(a, b), d). associativity

Axioms u1 - u4 are called axiomatic skeleton for fuzzy unions.They differ from the axiomatic skeleton of fuzzy intersections only

in boundary condition.

For crisp sets, u behaves like a classical (crisp) union.

Joakim Lindblad, 2007-03-16 (77/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy unionsAdditional (optional) requirements

For all a, b, d ∈ [0, 1],

Ax u5. u is a continuous function. continuity

Ax u6. u(a, a) ≥ a. superidempotency

Ax u7. a1 < a2 and b1 < b2 implies u(a1, b1) < u(a2, b2).strict monotonicity

Note:

Requirements u5 - u7 are analogous to Axioms i5 - i7.

The standard fuzzy union, u(a, b) = max[a, b], is the only idempotent

t-conorm.

Joakim Lindblad, 2007-03-16 (78/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Combinations of set operationsDe Morgan laws and duality of fuzzy operations

De Morgan laws in classical set theory:

A ∩ B = A ∪ B and A ∪ B = A ∩ B.

The union and intersection operation are dual with respect tothe complement.

De Morgan laws for fuzzy sets:

c(i(A,B)) = u(c(A), c(B)) and c(u(A,B)) = i(c(A), c(B))

for a t-norm i , a t-conorm u, and fuzzy complement c.

Notation: 〈i , u, c〉 denotes a dual triple.

Joakim Lindblad, 2007-03-16 (79/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Dual triples - Six theorems (1)

TheoremThe triples 〈min,max, c〉 and 〈imin, umax, c〉 are dual with respect to anyfuzzy complement c.

TheoremGiven a t-norm i and an involutive fuzzy complement c, the binaryoperation u on [0, 1], defined for all a, b ∈ [0, 1] by

u(a, b) = c(i(c(a), c(b)))

is a t-conorm such that 〈i , u, c〉 is a dual triple.

TheoremGiven a t-conorm u and an involutive fuzzy complement c, the binaryoperation i on [0, 1], defined for all a, b ∈ [0, 1] by

i(a, b) = c(u(c(a), c(b)))

is a t-norm such that 〈i , u, c〉 is a dual triple.Joakim Lindblad, 2007-03-16 (80/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Dual triples - Six theorems (2)

TheoremGiven an involutive fuzzy complement c and an increasing generator g ofc, the t-norm and the t-conorm generated by g are dual with respect to c.

TheoremLet 〈i , u, c〉 be a dual triple generated by an increasing generator g of theinvolutive fuzzy complement c. Then the fuzzy operations i , u, c satisfy thelaw of excluded middle, and the law of contradiction.

TheoremLet 〈i , u, c〉 be a dual triple that satisfies the law of excluded middle andthe law of contradiction. Then 〈i , u, c〉 does not satisfy the distributivelaws.

Joakim Lindblad, 2007-03-16 (81/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Aggregation operationsDefinition

Aggregations on fuzzy sets are operations by which severalfuzzy sets are combined in a desirable way to produce a singlefuzzy set.

Definition

Aggregation operation on n fuzzy sets (n ≥ 2) is a functionh : [0, 1]n → [0, 1].

Applied to fuzzy sets A1,A2, . . . ,An, function h produces an aggregatefuzzy set A, by operating on membership grades to these sets for eachx ∈ X :

A(x) = h(A1(x),A2(x), . . . ,An(x)).

Joakim Lindblad, 2007-03-16 (82/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Axiomatic requirements

Ax h1 h(0, 0, . . . , 0) = 0 and h(1, 1, . . . , 1) = 1. boundary conditions

Ax h2 For any (a1, a2, . . . , an) and (b1, b2, . . . , bn), such that ai , bi ∈ [0, 1]and ai ≤ bi for i = 1, . . . , n,

h(a1, a2, . . . , an) ≤ h(b1, b2, . . . , bn).

h is monotonic increasing in all its arguments.

Ax h3 h is continuous.

Ax h4 h is a symmetric function in all its arguments; for any permutation pon {1, 2, . . . , n}

h(a1, a2, . . . , an) = h(ap(1), ap(2), . . . , ap(n)).

Ax h5 h is an idempotent function; for all a ∈ [0, 1]

h(a, a, . . . , a) = a.

Joakim Lindblad, 2007-03-16 (83/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Averaging operations

• If an aggregation operator h is monotonic and idempotent (Axh2 and Ax h5), then for all (a1, a2, . . . , an) ∈ [0, 1]n

min(a1, a2, . . . , an) ≤ h(a1, a2, . . . , an) ≤ max(a1, a2, . . . , an).

• All aggregation operators between the standard fuzzyintersection and the standard fuzzy union are idempotent.

• The only idempotent aggregation operators are those betweenstandard fuzzy intersection and standard fuzzy union.

Idempotent aggregation operators are called averagingoperations.

Joakim Lindblad, 2007-03-16 (84/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Averaging operations

Generalized means:

hα(a1, a2, . . . , an) =

(aα1 + aα2 + · · ·+ aαn

n

) 1α

,

for α ∈ R, and α 6= 0, and for α < 0 ai 6= 0.

• Geometric mean: For α→ 0,

limα→0

hα(a1, a2, . . . , an) = (a1 · a2 · · · · · an)1n ;

• Harmonic mean: For α = −1,

h−1(a1, a2, . . . , an) =n

1a1

+ 1a2

+ · · ·+ 1an

;

• Arithmetic mean: For α = 1,

h1(a1, a2, . . . , an) =1

n(a1 + a2 + . . . an).

Functions hα satisfy axioms Ax h1 - Ax h5.

Joakim Lindblad, 2007-03-16 (85/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Averaging operationsDo we need more than standard operations?

Intersection: No positive compensation (trade-off)between the memberships of the fuzzy sets observed.

Union: Full compensation of lower degrees of membershipby the maximal membership.

In reality of decision making, rarely either happens.

(non-verbal) “merging connectives” → (language) connectives

{’and’, ’or’,...,}.

Aggregation operations called compensatory and are neededto model fuzzy sets representing to, e.g., managerial decisions.

Joakim Lindblad, 2007-03-16 (86/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


An Application: Fuzzymorphologies

Morphological operations

• Mathematical morphology is completely based on settheory. Fuzzification started in 1980s.

• Basic morphological operations are dilation and erosion.Many others can be derived from them.

• Dilation and erosion are, in crisp case, dual operationswith respect to the complementation: D(A) = c(E (cA)).

• In crisp case, dilation and erosion fulfil a certain number ofproperties.

Main construction principles:double integration over all combinations of α-cuts;fuzzification of set operations.

Joakim Lindblad, 2007-03-16 (87/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


L9: Fuzzy numbers and fuzzyarithmetics

Joakim Lindblad, 2007-03-16 (88/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Interval numbers

An interval number, representing an uncertain real number

A = [a1, a2] = {x | a1 ≤ x ≤ a2, x ∈ � }

For intervals A and B, and operator ∗ ∈ {+,−, ·, /}we define

A ∗ B = {a ∗ b | a ∈ A, b ∈ B}Division, A/B, is not defined when 0 ∈ B.

The result of an arithmetic operation on closed intervals isagain a closed interval.

Joakim Lindblad, 2007-03-16 (89/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Interval numbers

For closed intervals A = [a1, a2] and B = [b1, b2], the fourarithmetic operations are defined as follows (equivalent withdefinition on previous slide)

A + B = [a1 + b1, a2 + b2]

A− B = A + B− = [a1 − b2, a2 − b1]

A · B = [min(a1b1, a1b2, a2b1, a2b2),max(a1b1, a1b2, a2b1, a2b2)]

and, if 0 /∈ [b1, b2]A/B = A · B−1 = [a1, a2] · [ 1

b2, 1

b1]

= [min( a1b1, a1

b2, a2

b1, a2

b2),max( a1

b1, a1

b2, a2

b1, a2

b2)].

Joakim Lindblad, 2007-03-16 (90/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy numbers and fuzzy intervals

A fuzzy number is a fuzzy set on �

A : � → [0, 1]

such that

(i) A is normal (height(A) = 1)

(ii) αA is a closed interval for all α ∈ (0, 1]

(iii) The support of A, Supp(A) = 0+A, is bounded

Joakim Lindblad, 2007-03-16 (91/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy numbers and fuzzy intervals

Theorem (4.1)

Let A ∈ F( � ). Then, A is a fuzzy number iff there exists aclosed interval [a, b] 6= ∅ such that

A(x) =

1 for x ∈ [a, b]l(x) for x ∈ (−∞, a)r(x) for x ∈ (b,∞)

where l : (−∞, a)→ [0, 1] is monotonic non-decreasing,continuous from the right, and l(x) = 0 for x < ω1

and r : (b,∞)→ [0, 1] is monotonic non-increasing, continuousfrom the left, and r(x) = 0 for x > ω2.

Joakim Lindblad, 2007-03-16 (92/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Arithmetics on fuzzy numbers

Moving from interval numbers, we can define arithmetics onfuzzy numbers based on two principles:

1 Cutworthiness (thanks to inclusion monotonicity ofintervals)

α(A ∗ B) =αA ∗αB

in combination withA ∗ B =

⋃

α∈(0,1]

α(A ∗ B)

2 or the extension principle

(A ∗ B)(z) = supz=x∗y

min [A(x),B(x)]

Joakim Lindblad, 2007-03-16 (93/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Theorem (4.2)

Let ∗ ∈ {+,−, ·, /}, and let A, B denote continuous fuzzynumbers. Then, the fuzzy set A ∗ B defined by the extensionprinciple (prev. slide) is a continuous fuzzy number.

Lemma

(A ∗ B)(z) = supz=x∗y

min [A(x),B(x)] ⇒ α(A ∗ B) =αA ∗αB

So the two definitions are equivalent for continuous fuzzynumbers. (The proof is built on continuity.)

Joakim Lindblad, 2007-03-16 (94/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


MIN and MAX operators

MIN(A,B)(z) = supz=min(x ,y)

min [A(x),B(x)] ,

MAX(A,B)(z) = supz=max(x ,y)

min [A(x),B(x)]

Again, for continuous fuzzy numbers, this is equivalent with adefinition based on cutworthiness.

α(MIN(A,B)) = MIN(αA,αB),α(MAX(A,B)) = MAX(αA,αB), ∀α ∈ (0, 1].

Where, for intervals [a1, a2], [b1, b2]MIN([a1, a2], [b1, b2]) = [min(a1, b1),min(a2, b2)],

MAX([a1, a2], [b1, b2]) = [max(a1, b1),max(a2, b2)].

Joakim Lindblad, 2007-03-16 (95/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


MIN and MAX operators

Figure: Comparison of the operators MIN, min, MAX, and max.

Joakim Lindblad, 2007-03-16 (96/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Not all fuzzy numbers are comparable (only partial order).However, values of linguistic variables are often defined byfuzzy numbers that are comparable.

For example:

very small ¹ small ¹ medium ¹ large ¹ very large

Joakim Lindblad, 2007-03-16 (97/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Linguistic variables

When fuzzy numbers are connected to linguistic concepts, suchas very small, small, medium, and so on, and interpreted in aparticular context, the resulting constructs are usually calledlinguistic variables.

A linguistic variable is fully characterized by a quintuple〈v ,T ,X , g ,m〉, in which v is the name of the variable, T isthe set of linguistic terms of v that refer to the base variablewhose values range over a universal set X , g is a syntacticrule (a grammar) for generating linguistic terms, and m is asemantic rule that assigns to each linguistic term t ∈ T itsmeaning, m(t), which is a fuzzy set on X(i.e., m : T → F(X )).

Joakim Lindblad, 2007-03-16 (98/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Linguistic variables

Figure: An example of a linguistic variable.

Joakim Lindblad, 2007-03-16 (99/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Interval equationsEquation A + X = B

A + X = B

Let X = [x1, x2].

Then [a1 + x1, a2 + x2] = [b1, b2] follows immediately.

Clearly: x1 = b1 − a1 and x2 = b2 − a2.

Since X must be an interval, it is required that x1 ≤ x2.

That is, the equation has a solution iff b1 − a1 ≤ b2 − a2.

Then X = [b1 − a1, b2 − a2] is the solution.

Joakim Lindblad, 2007-03-16 (100/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy equations

The solution to a fuzzy equation can be obtained by solving aset of interval equations, one for each nonzero α in the level setΛ(A) ∪ Λ(B).

The equation A + X = B has a solution iff

(i) αb1 −αa1 ≤α b2 −αa2 for every α ∈ (0, 1], and

(ii) α ≤ β impliesαb1 −αa1 ≤β b1 −βa1 ≤β b2 −βa2 ≤α b2 −αa2.

If a solution αX exists for every α ∈ (0, 1] (property (i)),and property (ii) is satisfied, then the solution X is given by

X =⋃

α∈(0,1]

αX

Joakim Lindblad, 2007-03-16 (101/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy equationsEquation A · X = B

Similarly as A + X = B

The equation A · X = B has a solution iff

(i) αb1/αa1 ≤αb2/

αa2 for every α ∈ (0, 1], and

(ii) α ≤ β implies αb1/αa1 ≤βb1/

βa1 ≤βb2/βa2 ≤αb2/

αa2.

If the solution exists, it has the form

X =⋃

α∈(0,1]

αX

where αX = [αb1/αa1,

αb2/αa2].

Again, X = B/A is not a solution of the equation.

Joakim Lindblad, 2007-03-16 (102/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Linear programmingAn example

Minimize z = x1 − 2x2

Subject to 3x1 − x2 ≥ 12x1 + x2 ≤ 60 ≤ x2 ≤ 20 ≤ x1

The feasible set, i.e.,the set of vectors x thatsatisfy all constraints, isalways a convexpolygon (if bounded).

Figure: An example of a classical linearprogramming problem.

Joakim Lindblad, 2007-03-16 (103/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy linear programming

In many practical situations, it is not reasonable to require thatthe constraints or the objective function are specified in crispprecise terms.

The most general case of fuzzy linear programming growsrather complex, and is not discussed in the book.

A realistic special case to provide the feeling:

The situation where the right-hand-side vector and theconstraint matrix are expressed by fuzzy triangular numbers.

Simple membership functions allows transformation of theproblem

Joakim Lindblad, 2007-03-16 (104/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


L10: Fuzzy logic and approximatereasoning

Joakim Lindblad, 2007-03-16 (105/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Classical logic: A brief overviewLogic functions

Logic function assigns a truth value to a combination of truthvalues of its variables:

f : {true, false}n → {true, false}

2n choices of n arguments → 22nlogic functions of n variables.

Joakim Lindblad, 2007-03-16 (106/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Classical logic: A brief overviewLogic functions of two variables

v2 1 1 0 0 Function Adoptedv1 1 0 1 0 name symbol

ω1 0 0 0 0 Zero function 0ω2 0 0 0 1 NOR function v1 ↓ v2

ω3 0 0 1 0 Inhibition v1 > v2

ω4 0 0 1 1 Negation v2

ω5 0 1 0 0 Inhibition v1 < v2

ω6 0 1 0 1 Negation v1

ω7 0 1 1 0 Exclusive OR v1 ⊕ v2

ω8 0 1 1 1 NAND function v1|v2

ω9 1 0 0 0 Conjunction v1 ∧ v2

ω10 1 0 0 1 Equivalence v1 ⇔ v2

ω11 1 0 1 0 Assertion v1

ω12 1 0 1 1 Implication v1 ⇐ v2

ω13 1 1 0 0 Assertion v2

ω14 1 1 0 1 Implication v1 ⇒ v2

ω15 1 1 1 0 Disjunction v1 ∨ v2

ω16 1 1 1 1 One function 1Joakim Lindblad, 2007-03-16 (107/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Classical logic: A brief overviewLogic primitives

We can express all the logic functions of n variables by usingonly a small number of simple logic functions. Such a set is acomplete set of logic primitives.

Examples:{negation, conjunction, disjunction},{negation, implication}.

Joakim Lindblad, 2007-03-16 (108/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Classical logic: A brief overviewLogic formulae

Definition

1. If v is a logic variable, then v and v are logic formulae;

2. If v1 and v2 are logic formulae, then v1 ∧ v2 and v1 ∨ v2

are also logic formulae;

3. Logic formulae are only those defined (obtained) by thetwo previous rules.

Joakim Lindblad, 2007-03-16 (109/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy propositions

The range of truth values of fuzzy propositions is not only{0, 1}, but [0, 1].The truth of a fuzzy proposition is a matter of degree.

Classification of fuzzy propositions:

• Unconditional and unqualified propositions“The temperature is high.”

• Unconditional and qualified propositions“The temperature is high is very true.”

• Conditional and unqualified propositions“If the temperature is high, then it is hot.”

• Conditional and qualified propositions“If the temperature is high, then it is hot is true.”

Joakim Lindblad, 2007-03-16 (110/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Linguistic hedges (modifiers)

• Linguistic hedges are linguistic terms by which otherlinguistic terms are modified.

“Tina is young is true.”“Tina is very young is true.”“Tina is young is very true.”“Tina is very young is very true.”

• Fuzzy predicates and fuzzy truth values can be modified.Crisp predicates cannot be modified.

• Examples of hedges: very, fairly, extremely.

Joakim Lindblad, 2007-03-16 (111/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Modifiers

Strong modifier reduces the truth value of a proposition.Weak modifier increases the truth value of a proposition (byweakening the proposition).

One commonly used class of modifiers is

hα(a) = aα, for α ∈ R+ and a ∈ [0, 1].

For α < 1, hα is a weak modifier.Example: H : fairly ↔ h(a) =

√a.

For α > 1, hα is a strong modifier.Example: H : very ↔ h(a) = a2.

h1 is the identity modifier.

Joakim Lindblad, 2007-03-16 (112/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy quantifiers

• Absolute quantifiers:“about 10”; “much more than 100”, ...

• Relative quantifiers:“almost all”; “about half”, ...

Examples:p: “There are about 3 high-fluent students in the group.”

q: “Almost all students in the group are high-fluent.”

Joakim Lindblad, 2007-03-16 (113/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy propositionsUnconditional and unqualified propositions

The canonical formp : ν is F

ν is a variable on some universal set VF is a fuzzy set on V that represents a fuzzy predicate

(e.g., low, tall, young, expensive...)

The degree of truth of p is

T (p) = F (v), for v ∈ ν.

T is a fuzzy set on V . Its membership function is derived form themembership function of a fuzzy predicate F .

The role of a function T is to connect fuzzy sets and fuzzy propositions.

In case of unconditional and unqualified propositions, the identity function

is used.

Joakim Lindblad, 2007-03-16 (114/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy propositionsUnconditional and qualified propositions

The canonical form

p : ν is F is S (truth qualified proposition)

where ν is a variable on some universal set V ,F is a fuzzy set on V that represents a fuzzy predicate,and S is a fuzzy truth qualifier.

To calculate the degree of truth T (p) of the proposition p, weuse:

T (p) = S(F (v))

Joakim Lindblad, 2007-03-16 (115/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy propositionsConditional and unqualified propositions

The canonical form

p : If X is A, then Y is B,

where X ,Y are variables on X ,Y respectively,and A,B are fuzzy sets on X ,Y respectively.

Alternative form:〈X ,Y〉 is R

where R(x , y) = J (A(x),B(x)) is a fuzzy set on X × Yrepresenting a suitable fuzzy implication.

Joakim Lindblad, 2007-03-16 (116/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy propositionsConditional and qualified propositions

The canonical form

p : If X is A, then Y is B is S

where X ,Y are variables on X ,Y respectively,A,B are fuzzy sets on X ,Y respectively,and S is a truth qualifier.

Joakim Lindblad, 2007-03-16 (117/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy implicationsDefinition(s)

A fuzzy implication J of two fuzzy propositions p and q is afunction of the form

J : [0, 1]× [0, 1]→ [0, 1],

which for any truth values a = T (p) and b = T (q) defines thetruth value J (a, b) of the conditional proposition

“if p, then q”.

Fuzzy implications as extensions of the classical logic implication:

Crisp implication a⇒ b Fuzzy implication J (a, b)(S) a ∨ b u(c(a), b)(R) max{x ∈ {0, 1} | a ∧ x ≤ b} sup{x ∈ [0, 1] | i(a, x) ≤ b}(QL) a ∨ (a ∧ b) u(c(a), i(a, b))(QL) (a ∧ b) ∨ b u(i(c(a), c(b)), b)

Joakim Lindblad, 2007-03-16 (118/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy implicationsHow to select fuzzy implication

Look at Table 11.2 , Table 11.3, and Table 11.4(pp. 315-317).

One good choice:

Js(a, b) =

{1 a ≤ b0 a > b

One frequently used implication: ÃLukasiewicz

Ja(a, b) = min[1, 1− a + b]

Joakim Lindblad, 2007-03-16 (119/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Inference rules

Fuzzy inference rules are basis for approximate reasoning.

As an example, three classical inference rules(Modus ponens, Modus Tollens, Hypothetical syllogism)

are generalized by using compositional rule of inference

For a given fuzzy relation R on X ×Y , and a given fuzzy set A′

on X , a fuzzy set B ′ on Y can be derived for all y ∈ Y , so that

B ′(y) = supx∈X

min[A′(x),R(x , y)].

In matrix form, compositional rule of inference is

B′ = A′ ◦ R

Joakim Lindblad, 2007-03-16 (120/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Inference rulesExample: Generalized modus ponens

Rule: If X is A, then Y is BFact: X is A′

Conclusion: Y is B ′

In this case,R(x , y) = J [A(x),B(y)]

andB ′(y) = sup

x∈Xmin[A′(x),R(x , y)].

Joakim Lindblad, 2007-03-16 (121/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Multiconditional approximatereasoning

General schema is of the form:

Rule 1: If X is A1, then Y is B1

Rule 2: If X is A2, then Y is B2

. . .Rule n: If X is An, then Y is Bn

Fact: X is A′

Conclusion: Y is B ′

A′,Aj are fuzzy sets on X ,

B ′,Bj are fuzzy sets on Y , for all j .

Joakim Lindblad, 2007-03-16 (122/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Method of interpolation

Most common way to determine B ′ is by usingmethod of interpolation.

Step 1. Calculate the degree of consistency between the given factand the antecedent of each rule.Use height of intersection of the associated sets:

rj (A′) = h(A′ ∧ Aj ) = supx∈X

min[A′(x),Aj (x)].

Step 2. Truncate each Bj by the value rj (A′) and determine B ′ asthe union of truncated sets:

B ′(y) = supj∈ � n

min[rj (A′),Bj (y)], for all y ∈ Y .

Note that interpolation method is a special case of the composition ruleof inference, with

R(x , y) = supj∈ � n

min[Aj (x),Bj (y)]

where then B ′(y) = supx∈X min[A′(x),R(x , y)] = (A′ ◦ R)(y).Joakim Lindblad, 2007-03-16 (123/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Method of interpolation-Example

Joakim Lindblad, 2007-03-16 (124/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Joakim Lindblad, 2007-03-16 (125/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Defuzzification is a process that maps a fuzzy set to a crisp set.

Approaches:

• Defuzzification to a point.

• Defuzzification to a set.

• Generating a good representative of a fuzzy set.

• Recovering a crisp original set.

Joakim Lindblad, 2007-03-16 (126/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Defuzzification to a pointExamples of common methods

• Centre of gravity (Set of real numbers)

COG(A) =

Pxmaxxmin

x · A(x)Pxmax

xminA(x)

.

• Mean of maxima (Set of real numbers)

MeOM(A) =

Px∈core(A) x

|core(A)| .

• Centre of area (COA)COA(A) is the value that minimizes the expression

ŕŕŕŕŕŕ

COA(A)X

x=inf(X )

A(x)−sup(X )X

x=COA(A)

A(x)

ŕŕŕŕŕŕ.

Joakim Lindblad, 2007-03-16 (127/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Defuzzification to a setA.k.a. Averaging procedures

• α-cutschosen at various levels α.

• Average α-cutsbased on an integration of set-valued function,called Kudo-Aumann integration.

• Feature distance minimizationfind the crisp set at the minimal feature distance tothe given fuzzy set.

Joakim Lindblad, 2007-03-16 (128/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Defuzzification to a setAverage α-cuts

Let a fuzzy set A be given by a membership functionµ : R → [0, 1].

• Sets F (w) are α cuts, Aα of the fuzzy set A, for α ∈ [0, 1];• Selectors are ϕ(α) = inf Aα and ϕ(α) = sup Aα.

Then, the average α-cut of A is

Aµ =

[∫

[0,1]inf Aαdα,

∫

[0,1]sup Aαdα

].

Joakim Lindblad, 2007-03-16 (129/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Definition

An optimal defuzzification D(A) of a fuzzy set A on a referenceset X , with respect to the distance d , is

D(A) ∈ {C ∈ P(X ) | d(A,C ) = minB∈P(X )

[d(A,B)]} . (3)

Use a feature distance containing both local and globalfeatures. Find minimum using heuristic search methods, e.g.,simulated annealing.

Joakim Lindblad, 2007-03-16 (130/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


L12: Fuzzy control

Joakim Lindblad, 2007-03-16 (131/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Conventional control system

PID Control (Proportional-Integral-Derivative)

The PID controller is the workhorse of the process industries.

Output = bias + KPε + KI

∫ t

0ε dt + KD

dε

dt

Joakim Lindblad, 2007-03-16 (132/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy logic control

Methodology first developed by Mamdani in 1975 used tocontrol a steam plant. Based on work by Zadeh (1973) onfuzzy algorithms for complex systems and decision processes.

In a manner analogous to conventional control systems,inputs of a system are mapped to outputs using fuzzylogic rather than differential equations.

Joakim Lindblad, 2007-03-16 (133/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Motivation

• Can be used for systems that are difficult or impossible tomodel mathematically.

• Can also be applied to processes that are too complex ornonlinear to be controlled with traditional strategies.

• In fact, a detailed and precise mathematical description isnot always necessary for optimized operation of anengineering process.

• Human operators often are capable of managing complexsituations of a plant without knowing anything aboutdifferential equations.

• Such a rule based system can be used to define acontroller that emulates the heuristic rule-of-thumbstrategies of an expert.

Joakim Lindblad, 2007-03-16 (134/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


MotivationUseful cases

1 The control processes are too complex to analyze by conventionalquantitative techniques.

2 The available sources of information are interpreted qualitatively,inexactly, or uncertainly.

Advantage of Fuzzy logic control

• Flexible

• Universal approximator

• Easy to understand

• Powerful – yet simple

• Linguistic control

• linguistic terms – human knowledge

• Tolerant of imprecision / Robust control• more than 1 control rules - an error of a rule is not fatal• limited trust in input data

• Parallel or distributed control

• multiple fuzzy rules - complex nonlinear systemJoakim Lindblad, 2007-03-16 (135/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Motivation

Disadvantages

• More complex than PID

• More parameters to tune

• Un-mathematical (stability?)

Joakim Lindblad, 2007-03-16 (136/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Fuzzy control

Four main components

1 The fuzzification interface : transforms input crisp valuesinto fuzzy values

2 The knowledge base : contains a knowledge of theapplication domain and the control goals.

3 The decision-making logic : performs inference for fuzzycontrol actions

4 The defuzzification interface : provides a crisp controlaction out

Joakim Lindblad, 2007-03-16 (137/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Five steps...

How to build a fuzzy controller in five easy steps. . .

1 Partition input and output spaces:Select meaningful linguistic states for each variable andexpress them as appropriate fuzzy sets.

2 Fuzzification of input:Introduce a fuzzification function for each input variableto express the associated measurement uncertainty.

3 Formulate a set of inference rules:If ε = A and dε

dt = B, then C .

4 Design an inference engine:Use method of interpolation (Lecture 10).

5 Select a suitable defuzzification method (Lecture 11).

Joakim Lindblad, 2007-03-16 (138/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Joakim Lindblad, 2007-03-16 (139/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Hanging-togetherness natural grouping of voxels constitutingan object a human viewer readily sees in a display of the sceneas a Gestalt in spite of intensity heterogeneity.

Basic idea:

Compute global hanging-togetherness from localhanging-togetherness.

Joakim Lindblad, 2007-03-16 (140/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Strength of a path – the strength of its weakest link

A. Rosenfeld 1979Strength of a link between two points defined by themembership function.

J. K. Udupa and S. Samarasekera 1996Strength of a link between two points defined by affinity

The connectedness of two points x and y in A –the strength of the strongest path between x and y

cA(x , y) = supπ∈Π(x ,y)

inft∈π

A(t)

Joakim Lindblad, 2007-03-16 (141/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Fuzzy spel adjacency is a reflexive and symmetric fuzzyrelation α in � n and assigns a value to a pair of spels (c, d)based on how close they are spatially.

Fuzzy spel affinity is a reflexive and symmetric fuzzy relation κin � n and assigns a value to a pair of spels (c, d) based on howclose they are spatially and intensity-based-property-wise (localhanging-togetherness).

µκ(c, d) = h(µα(c, d), µ(c), µ(d), c, d)

The fuzzy κ-connectedness assigns a value to a pair of spels (c,d) that is the maximum of the strengths of connectednessassigned to all possible paths from c to d (globalhanging-togetherness).

Joakim Lindblad, 2007-03-16 (142/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol


Components of fuzzy affinity

Fuzzy spel adjacency µα(c, d) indicates the degree of spatialadjacency of spels The homogeneity-based componentµψ(c, d) indicates the degree of local hanging-togetherness ofspels due to their similarities of intensities Theobject-feature-based component µϕ(c, d) indicates thedegree of local hanging-togetherness of spels with respect tosome given object feature

Example:

µκ =1

2µα(µψ + µϕ)

Joakim Lindblad, 2007-03-16 (143/144)


JoakimLindblad

Outline

L1: Intro

L1–3: Basics


L5: Features

L6–7:Distances

L8: Setoperations

L9: Fuzzynumbers

L10: Fuzzylogic


L12: Fuzzycontrol



Computation – A graph search problem

Dynamic programming solution (think distance transform orlevel sets computation)

Practical usage examples:

• Seed foreground (one or multiple seeds), threshold at somelevel of fuzzy connectedness.

• Seed different regions and let them compete (relative fc,iterated relative fc).

Joakim Lindblad, 2007-03-16 (144/144)

lecture notes fuzzysets and techniques - cbajoakim/course/fuzzy/vt07/lectures/l15_4.pdfjoakim...

Documents