An Introduction to Fuzzy Sets Theory - unige.it · Fuzzy sets An Introduction to Fuzzy Sets Theory Francesco Masulli DIBRIS - University of Genova, ITALY & S.H.R.O. - Sbarro Institute

IntroductionFuzzy sets

An Introduction to Fuzzy Sets Theory

Francesco Masulli

DIBRIS - University of Genova, ITALY&

S.H.R.O. - Sbarro Institute for Cancer Research and Molecular MedicineTemple University, Philadelphia, PA, USA

email: [email protected]

ML-CI 2016

Francesco Masulli Introduction to Fuzzy Sets Theory


Outline

1 Introduction

2 Fuzzy sets



IntroductionFrom: B. Kosko, Scientific American 1993

Computers do not reason as brain do. Computers"reason" when they manipulate precise facts that havebeen reduced to strings of zeros and ones and statementsthat are either true or false. The human brain can reasonwith vague assertions or claims that involve uncertaintiesor value judgments: "The air is cool", or "That speed isfast" or "She is young". Unlike computers, humans havecommon sense that enables them to reason in a worldwhere things are inly partially true.Fuzzy logic is a branch of machine intelligence that helpscomputers paint gray, commonsense pictures of anuncertain world. Logicians in the 1920s first broached itskey concept: everything is a matter of degree.




Fuzzy logic manipulates such vague concepts as "warm"or ’still dirty’ and so helps engineers to build airconditioners, washing machines and other devices thatjudge how fast they should operate or shift from one settingto another even when the criteria for making thosechanges are hard to define.When mathematicians lack specific algorithms that dictatehow a system should respond to inputs, fuzzy logic cancontrol or describe the system by using "common sense’rules that refer to indefinite quantities.




No known mathematical model can back up atruck-and-trailer rig from a parking lot to a loading dockwhen the vehicle starts from a random spot. Both humansand fuzzy systems can perform this nonlinear guidancetask by using practical but imprecise rules such as "If thetrailer turns a little to the left, then turn it a little to the right."Fuzzy systems often glean their rules from experts. Whenno expert gives the rules, adaptive fuzzy systems learn therules by observing how people regulate real systems.



IntroductionApplications of Fuzzy Sets Theory (from somewhere??)

Machine Systems Human-Based Systems Human/Machine Systemspicture/voice recognition human reliability model medical diagnosisChinese character recognition cognitive psycology inspection data processingnatural language understanding thinking/beaviour models trnafusion consultationintelligent robots sensory investigations expert systemscrop recognition public awareness investigation CAIprocess control risk assesment CAEproduction management enviromental assesment optimization planningcar/train opration human relation structures personnel managementsafety/maintenance systems demand trend models development planningbreakdown diagnosis social pychology equipement diagnosticselectric power systems operations category analysis quality evaluationfuzzy controllers insurance systemshome electrical appliance control human interfacesautomatic operation management decision-making

multiporpouse decision-makingknowledege basesdata bases




A recent wave of commercial fuzzy products, most of themfrom Japan, has popularized fuzzy logic.In 1980 the contracting firm of F. L Smidth & Company inCopenhagen first used fuzzy system to oversee theoperation of a Cement kiln.In 1983 Hitaci turned over control of a subway in Sendai,Japan. to a fuzzy system.Since then, Japanese companies have used fuzzy logic todirect hundreds of household appliances and electronicsproducts.




Applications for fuzzy logic extend beyond control systems.Recent theorems show that in principle fuzzy logic can heused to model any continuous system, be it based inengineering or physics or biology or economics.Investigators in many fields may find that fuzzy,commonsense models are more useful or accurate thanare standard mathematical ones.



IntroductionA brief hystory

1910-1913 Bertrand Russell & Alfred North Whitehead:"Principia Mathematica"1920 Jan Łukasiewicz (Polish) multi-modal logic: paper"On Three-valued Logic"1937 Max Black - vague sets: paper "Vagueness: Anexercise in logical analysis"1965 Lotfali (Lofti) Askar Zadeh (born in Azerbaijan in1921): paper "Fuzzy sets" (Information and Control. 1965;8: 338–353). Zadeh applies the Łukasiewicz logic to eachelement of a set and proposes a complete algebra forfuzzy sets.



Fuzzy SetsCrisp sets

Fuzzy sets are sets whose elements have degrees ofmembership.Fuzzy sets were introduced by Lotfi A. Zadeh (1965) as anextension of the classical notion of set.In classical set theory, the membership of elements in a setis assessed in binary terms according to a bivalentcondition an element either belongs or does not belong tothe set.



Fuzzy SetsExample

In classical sets theory, the set H of real numbers from 150to 200 is:

H = {r ∈ < | 150 ≤ r ≤ 200}The indicator function (or characteristic function) µH(r)gives the membership of each element of the universe < toh:

µH(r) ={

1 if 150 ≤ r ≤ 2000 otherwise



Fuzzy SetsCrisp sets

Fuzzy set theory permits the gradual assessment of themembership of elements in a set; this is described with theaid of a membership function valued in the real unit interval[0,1].Fuzzy sets generalize classical sets, since the indicatorfunctions of classical sets are special cases of themembership functions of fuzzy sets, if the latter only takevalues 0 or 1.Classical bivalent sets are in fuzzy set theory usuallycalled crisp sets.The fuzzy set theory can be used in a wide range ofdomains in which information is incomplete or imprecise,such as bioinformatics.



Fuzzy SetsExample

Let we define the fuzzy sets F of real numbers that areclose to 175 by meas the following membership function:

µF (r) =1√2π

e−12 (r−175)2



Fuzzy Sets

Definition (Fuzzy set)

A fuzzy set A in X is a set of ordered pairs

A = {( x , µA(x) ) | x ∈ X}.µA is called the membership function, µA : X → M, where M is themembership space where each element of X is mapped to.

If M = {0, 1}, A is a crisp set.



Fuzzy SetsMembership functions and probabilities




Suppose you had been in the desert for a week without a drinkand you came upon two bottles




C could contain, say, swamp water. That is membership of0.91 means that the contents of C are fairly similar toperfectly potable liquids (e.g., pure water).The probability that A is potable = 0.91 means that over along run of experiments, the contents of A are expected tobe potable in about 91% of the trials; in the other 9% thecontents will be hydrochloric acid.







While it is of great intellectual interest to establish theproper connections between FL and probability, this authordoes not believe that doing so will change the ways inwhich we solve problems, because both probability and FLshould be in the arsenal of tools used by engineers[Mendel, 1995].



Fuzzy SetsMemberships

Defined by the user, as it is need by the problem to model



Clustering methods and fuzzy sets

Jim Bezdek (1981) introduced the concept of hard andfuzzy partition in order to extend the notion of membershipof pattern to clusters.The motivation of this extension is related to the fact that apattern often cannot be thought of as belonging to a singlecluster only. In many cases, a description in which themembership of a pattern is shared among clusters isnecessary.



Fuzzy SetsFuzzy singleton

Definition (Fuzzy singleton)

The singleton is a fuzzy set A associated to a crisp number x0.Its membership function is

µA(x) =

{1 if x = x00 otherwise



Fuzzy SetsExample 1

A real estate agent wants to classify the houses offering tocustomers.Let X = {1,2,3, ...,10} the number of bedrooms. The fuzzyset A "kind of comfortable home for a family of 4" can bedefined as

A = {(1,0.2), (2,0.5), (3,0.8), (4,1.0), (5,0.7), (6,0.3)}.



Fuzzy SetsExample 2

A = "real numbers considerably larger than 10"

A = {(x , µA(x))|x ∈ X}

where

µA(x) ={

0 x ≤ 10(1 + (x − 10)−2)−1 x > 10.

gnuplot> set xrange [10:50]gnuplot> plot (1+(x-10)**(-2))**(-1)



Fuzzy SetsNotation

A = µA(x1)/x1 + µA(x2)/x2 + · · · =n∑

i=1

µA(xi)/xi if X discrete

A =

∫XµA(x)/x if X continuous

Note: in this context, the meaning of the symbols +∑ ∫

isunion of elements



Fuzzy SetsNormal fuzzy set

Definition (Normal fuzzy set)

A normal⇐⇒ supxµA(x) = 1

NOTE: if A is not normal, it can be normalized:

µA(x) −→µA(x)

supxµA(x)



Fuzzy SetsExample 3

A = "integers close to 10"

A = .1/7 + .5/8 + .8/9 + 1/10 + .8/11 + .5/12 + .1/13



Fuzzy SetsExample 4

A = "real numbers close to 10"

A =

∫R

11 + (x − 10)2

/x

gnuplot> set xrange [0:20]gnuplot> plot 1/(1+(x-10)**2)



Fuzzy Sets

Definition (Support of a fuzzy set S(A))

The support S(A) of a fuzzy set is a crisp set containing allelements of S(A) with µA(x) > 0,i.e.,

S(A) = { x | µA(x) > 0, x ∈ X}.

Definition (α-level set (or α-cut))The α-level set Aα of the fuzzy set A is a crisp set containingthe elements of A with membership degree at least α, i.e.,

Aα = { x | µA(x) ≥ α, x ∈ X}.



Fuzzy Sets

Definition (Strong α-level set (or Strong α-cut))

The strong α-level set Aα of the fuzzy set A is a crisp setcontaining the elements of A with membership degree greaterthan α, i.e.,

Aα = { x | µA(x) > α, x ∈ X}.

A0 support ; A1 nucleusFrancesco Masulli Introduction to Fuzzy Sets Theory


Fuzzy SetsFuzzy number

Definition (Fuzzy number)A fuzzy number F in a continuous universe U , e.g., a real line,is a fuzzy set F in U which is normal and convex

Example:

µF (r) =1√2π

e−12 (r−175)2

,



Fuzzy SetsAlgebra of fuzzy sets

Definition (Union of fuzzy sets)Let A e B be two fuzzy sets in X. The union of A and B is thefuzzy set D = A ∪ B with membership function

µD(x) = max{µA(x), µB(x)} ∀x ∈ X .




Definition (Intersection of fuzzy sets)Let A e B be two fuzzy sets in X. The intersection of A and B isthe fuzzy set D = A ∩ B with membership function

µD(x) = min{µA(x), µB(x)} ∀x ∈ X .




Definition (Complement of a fuzzy set)Let A be a fuzzy set in X. The complement of A in X is the fuzzyset Ã with membership function

µA(x) = 1− µA(x) ∀x ∈ X .



Fuzzy SetsTriangolar norms, t-norms (FUZZY AND)



Fuzzy SetsTriangolar co-norms (FUZZY OR)



Fuzzy SetsGeometrical interpretation [Kosko, 1991]

Fuzzy hypercube

A = {(1, .2), (2, .7)}

Principle of Non-Contradiction: A ∩ A′ = ∅Principle of Excluded Middle: A ∪ A′ = U




Fuzzy hypercube

A = {(1, .2), (2, .7)}





Fuzzy hypercube

A = {(1, .2), (2, .7)}




Aggregation operations

MulticriteriaMulti-expertInformation Fusion




Several fuzzy sets are combined in a desirable way toproduce a single fuzzy set.An aggregation operation on n fuzzy sets (n ≥ 2) is definedas a functionh : [0,1]n → [0,1].When applied to fuzzy sets A1,A2, . . . ,An defined on X ,function h produces an aggregate fuzzy set A by operatingon the membership grades of these sets for each x ∈ X .Thus,µA(x) = h(µA1(x), µA2(x), . . . , µAn (x)) ∀x ∈ X .




Axioms expressing the essence of the notion of aggregation:1 h(0,0, . . . ,0) = 0 and h(1,1, . . . ,1) = 1 (boundary

condition).2 ∀ pair of < a1,a2, . . . ,an > and < b1,b2, . . . ,bn > of n-ples

such that ai ,bi ∈ [0,1]∀i ∈ Nn, if ai ≤ bi∀i ∈ Nn, thenh(a1,a2, . . . ,an) ≤ h(b1,b2, . . . ,bn);i.e., h is monotonic increasing in all its arguments.

3 h is a continuous function.




Additional axioms (context depending):1 h is a symmetric function in all its arguments, i.e.,

h(a1,a2, . . . ,an) = h(ap(1),ap(2), . . . ,ap(n))for any permutation p on Nn.

2 h is a idempotent function, i.e.,h(a,a, . . . ,a) = a∀a ∈ [0,1].




Valid aggregation operations:Fuzzy intersections and unionsGeneralized meanshα(a1,a2, . . . ,an) =

(aα1 +aα2 +···+aαn

n

)1/α

α = −1: harmonic meanh−1(a1,a2, . . . ,an) = n

1a1

+ 1a2

+···+ 1an

α = 1: aritmentic meanh1(a1,a2, . . . ,an) = 1

n (a1 + a2 + · · ·+ an)




Valid aggregation operations:

Ordered Weighted Averanging (OWA) operations [Yager,1988]

weighting vectorw =< w1,w2, . . . ,wn > wi ∈ [0,1] ∀i ∈ Nn and

n∑i=1

wi = 1




Ordered Weighted Averanging (OWA) operations [Yager,1988]

the OWA operation associated with w is the function:hw(a1,a2, . . . ,an) = w1b1 + w2b2 + . . . + wnbnwhere bi for any i ∈ Nn is the i-th largest element ina1,a2, . . . ,an.Vector < b1,b1, . . . ,bn > is a permutation vector of vector< a1,a2, . . . ,an > in wich the elements are ordered: bi ≥ bjfor any pair i , j ∈ NnExample: w =< .3, .1, .2, .4 >hw(.6, .9, .2, .7) = .3× .9 + .1× .7 + .2× .6 + .4× .2 = .54




NOTE: Ordered Weighted Averanging (OWA) operations[Yager, 1988]

w =< 1/n,1/n, . . . ,1/n >→ hw arithmetic meanlower boundw? =< 0,0, . . . ,1 >→ hw? = min(a1,a2, . . . ,an)

upper boundw? =< 0,0, . . . ,1 >→ hw? = max(a1,a2, . . . ,an)



Fuzzy SetsLinguistic variable



Fuzzy SetsLinguistic variable

Definition (Linguistic variable)A linguistic variable is characterized by a quintuple(x ,U,T (x),G,M) in which

x is the name of variable;U is the universe of discourse;T (x) is the term set of x , that is, the set of names oflinguistic values of x with each value being a fuzzy numberdefined on U;G is a syntactic rule for generating the names of values ofx ;M is a semantic rule for associating each value with itsmeaning.



Fuzzy SetsLinguistic variable and biomedical knowledge

age = {very young, young,middle,old , very old}blood glucose level = {slightly increased , increased ,significantly increased , strongly increased}.insulin doses = {none, low ,medium,high}


Documents

An Introduction to Fuzzy Sets Theory - unige.it · Fuzzy sets An Introduction to Fuzzy Sets Theory Francesco Masulli DIBRIS - University of Genova, ITALY & S.H.R.O. - Sbarro Institute