Statistical methods for fuzzy data

Statistical Methods for Fuzzy Data Reinhard Viertl, Vienna University of Technology, Austria Statistical data are not always precise numbers, or vectors, or categories.. In this book th

Statistical Methods for

Fuzzy Data

Reinhard Viertl, Vienna University of Technology, Austria

Statistical data are not always precise numbers, or vectors, or categories

Real data are frequently what is called fuzzy Examples where this fuzziness

is obvious are quality of life data, environmental, biological, medical,

sociological and economics data Also the results of measurements can be

best described by using fuzzy numbers and fuzzy vectors respectively

Statistical analysis methods have to be adapted for the analysis of fuzzy

data In this book the foundations of the description of fuzzy data are

explained, including methods on how to obtain the characterizing function

of fuzzy measurement results Furthermore, statistical methods are then

generalized to the analysis of fuzzy data and fuzzy a-priori information

Key features:

•Provides basic methods for the mathematical description of fuzzy data,

as well as statistical methods that can be used to analyze fuzzy data

•Describes methods of increasing importance with applications in areas

such as environmental statistics and social science

•Complements the theory with exercises and solutions and is illustrated

throughout with diagrams and examples

•Explores areas such as quantitative description of data uncertainty and

mathematical description of fuzzy data

This book is aimed at statisticians working with fuzzy logic, engineering

statisticians, finance researchers, and environmental statisticians The book

is written for readers who are familiar with elementary stochastic models and

basic statistical methods

Statistical Methods for Fuzzy Data

Reinhard Viertl

Statistical Methods for

Fuzzy Data

Part I

FUZZY INFORMATION

Fuzzy information is a special kind of information and information is an omnipresentword in our society But in general there is no precise definition of information.However, in the context of statistics which is connected to uncertainty, a possi-ble definition of information is the following: Information is everything which hasinfluence on the assessment of uncertainty by an analyst This uncertainty can be ofdifferent types: data uncertainty, nondeterministic quantities, model uncertainty, anduncertainty of a priori information

Measurement results and observational data are special forms of information.Such data are frequently not precise numbers but more or less nonprecise, also called

fuzzy Such data will be considered in the first chapter.

Another kind of information is probabilities Standard probability theory is sidering probabilities to be numbers Often this is not realistic, and in a more general

con-approach probabilities are considered to be so-called fuzzy numbers.

The idea of generalized sets was originally published in Menger (1951) and theterm ‘fuzzy set’ was coined in Zadeh (1965)

Part II

DESCRIPTIVE

STATISTICS FOR

FUZZY DATA

In standard statistics – objectivist and Bayesian analysis – observations are assumed to

be numbers, vectors, or classical functions By the fuzziness of real data of continuousquantities it is necessary to adapt descriptive statistical methods to the situation offuzzy data This is done for the most frequently used descriptive methods in this part

In this part the basic mathematical concepts for statistical inference in the case offuzzy data as well as fuzzy probabilities are explained.

In standard statistical inference the combination of observations of a classicalrandom variable to form an element of the sample space is trivial Different from thatfor fuzzy data the combination is nontrivial because a vector of fuzzy numbers is not

a fuzzy vector

Therefore it is important to distinguish between observation space and samplespace

There are survey papers on different concepts concerned with statistical inference

for fuzzy data by Gebhardt et al (1997) and Taheri (2003).

Classical statistical inference is based on the assumption that stochastic

quanti-ties X have an underlying true probability model P0 In order to estimate P0

usually a family P of possible probability distributions is considered, i.e P =

{P : P possible distribution of X}.

In the case of parametric families{Pθ :θ ∈ } the generalization of estimators

to the situation of fuzzy data is necessary, i.e to construct estimations ˆθ∗for the trueparameterθ0of the underlying probability distribution P θ0of X

Next a generalization of confidence regions in the case of fuzzy data is given, andthe resulting fuzzy confidence regions are typical examples of fuzzy sets

Based on fuzzy samples also test statistics have to be adapted The values of a teststatistic in the case of fuzzy data become fuzzy numbers Therefore test decisions arenot as simple as in the standard situation of data in the form of numbers or vectors.The resulting fuzzy values of test statistics contain more information than in theclassical situation because fuzzy values of a test statistic can make it clear that moredata are needed in order to provide a well-based decision

Part V

BAYESIAN INFERENCE AND FUZZY

INFORMATION

In standard Bayesian statistics all unknown quantities are described by stochasticquantities and their probability distributions Consequently, so are the parametersθ

of stochastic models X ∼ f (·|θ) ; θ ∈  in the continuous case, and X ∼ p (·|θ) ;

θ ∈  in the discrete case The a priori knowledge concerning θ is expressed by a

probability distributionπ(·) on the parameter space , called a priori distribution.

If the parameterθ is continuous, the a priori distribution has a density function, called a priori density.

For discrete parameter space = {θ1, , θ m} the a priori distribution is a

dis-crete probability distribution with point probabilitiesπ(θ j), j = 1(1)m.

If the observed stochastic quantity X ∼ p (·|θ) ; θ ∈ {θ1, · · · , θ m} is also discrete,

i.e the observation space M X of X is countable with

M X = {x1, x2, }

and observationsx◦1, , x◦n of X are given [sample data D = (x◦1, , x◦n)], then theconditional distribution Pr
θ| D

of the parameter is obtained via Bayes’ formula in

the following way: The so-called a posteriori probabilities π(θ j |D) of the parameter

valuesθ j , given the data D= (x◦1, , x◦n), are obtained by

88 STATISTICAL METHODS FOR FUZZY DATA

This probability distribution on the parameter space = {θ1, , θ m} is called

a posteriori distribution of  θ, where  θ denotes the stochastic quantity describing the

uncertainty concerning the parameter

For continuous stochastic quantities X with parametric probability density f ( |θ)

and continuous parameter space ⊆ R k , i.e stochastic model

X ∼ f (·|θ) ; θ ∈ ,

the a priori distribution of θ is given by a probability density π(·) on the parameter

space This probability density is called a priori density which obeys



π(θ)dθ = 1.

For observed sample x1, , x n of X the so-called a posteriori density

π (·|x1, · · · , x n) is the conditional density of the parameter, given the data

In the case of independent observations x1, , x n the joint density

g(x1, , x n , θ) is given by the following product:

g (x1, , x n , θ) =

n

BAYESIAN INFERENCE AND FUZZY INFORMATION 89

The marginal density h(x1, , x n) is given by

This is called Bayes’ theorem.

Remark V.1: After the sample x1, , x n is observed these values are constants

Therefore h(x1, , x n) is a normalizing constant, and Bayes’ theorem can be written

in its short form

π (θ|x1, , x n)∝ π(θ) · l (θ; x1, , x n)where∝ stands for ‘proportional to π(θ) · l(θx1, , x n) up to a normalizing con-stant’

For more general data D – for example for censored data – the likelihood function

has to be adapted The general form of Bayes’ theorem is given by

π (θ|D) ∝ π(θ) ·  (θ; D) ∀θ ∈ .

Part VI

REGRESSION ANALYSIS AND FUZZY

INFORMATION

In regression analysis usually it is assumed that observed data values are real numbers

or vectors In applications where continuous variables are observed this assumption

is unrealistic Therefore data have to be considered as fuzzy Another kind of fuzzyinformation is present in Bayesian regression models Here the a priori distributions

of the parameters in regression models are typical examples of fuzzy information inthe sense of fuzzy a priori distributions

Both kinds of fuzzy information, fuzzy data as well as fuzzy probability butions for the quantification of a priori knowledge concerning parameters has to betaken into account This is the subject of this part

distri-Part VII

FUZZY TIME SERIES

Many time series are the results of measurement procedures and therefore the obtainedvalues are more or less fuzzy Also, many economic time series contain values withremarkable uncertainty In particular, development data and environmental time seriesare examples of time series whose values are not precise numbers If such data are

modeled by fuzzy numbers the resulting time series are called fuzzy time series.

Based on the results from Part I of this book it is necessary to consider time

series with fuzzy data In standard time series analysis the values x tof a time series

(x t)t ∈T , where T is the series of time points, i.e T = {1, 2, · · · , N}, are asumed to be

real numbers By the fuzziness of many data, especially all measurement data fromcontinuous quantities, it is necessary to consider time series whose values are fuzzy

numbers x t∗ Such time series (x t∗)t ∈T are called fuzy time series.

Mathematically a fuzzy time series is a mapping from the index set T to the set F(R) of fuzzy numbers The generalization of time series analysis techniques to the

situation of fuzzy values should fulfill the condition that it specializes to the classicaltechniques in the case of real valued time series

In this part, first the necessary mathematical techniques are introduced which areused to analyze fuzzy time series Then descriptive methods of time series analysis aregeneralized for fuzzy valued time series Finally, stochastic methods of time seriesanalysis are generalized for fuzzy time series This Part is based on the doctoraldissertation of Hareter (2003)

Appendix A2 consists of solutions to the problems given in the different chapters.

If needed further details can be obtained from the author

Appendix A3 is a glossary of the most important terms used throughout the book

In Appendix A4 related literature is presented which is not referenced in the text

It gives also hints to approaches to the analysis of fuzzy data which are different tothe approach in this book

Fuzzy data

All kinds of data which cannot be presented as precise numbers or cannot be precisely

classified are called nonprecise or fuzzy Examples are data in the form of linguistic

descriptions like high temperature, low flexibility and high blood pressure Also,precision measurement results of continuous variables are not precise numbers but

always more or less fuzzy.

Measurement results of one-dimensional continuous quantities are frequently ized to be numbers times a measurement unit However, real measurement results

ideal-of continuous quantities are never precise numbers but always connected with certainty Usually this uncertainty is considered to be statistical in nature, but this

un-is not suitable since statun-istical models are suitable to describe variability For a gle measurement result there is no variability, therefore another method to modelthe measurement uncertainty of individual measurement results is necessary The

sin-best up-to-date mathematical model for that are so-called fuzzy numbers which are

described in Section 2.1 [cf Viertl (2002)]

Examples of one-dimensional fuzzy data are lifetimes of biological units, lengthmeasurements, volume measurements, height of a tree, water levels in lakes and rivers,speed measurements, mass measurements, concentrations of dangerous substances

in environmental media, and so on

A special kind of one-dimensional fuzzy data are data in the form of intervals

[a; b]⊆ R Such data are generated by digital measurement equipment, because they

have only a finite number of digits

Statistical Methods for Fuzzy Data Reinhard Viertl

© 2011 John Wiley & Sons, Ltd

3

4 STATISTICAL METHODS FOR FUZZY DATA

Figure 1.1 Variability and fuzziness.

Many statistical data are multivariate, i.e ideally the corresponding measurement

results are real vectors (x1, , x k)∈ Rk

In applications such data are frequentlynot precise vectors but to some degree fuzzy A mathematical model for this kind of

data is so-called fuzzy vectors which are formalized in Section 2.2.

Examples of vector valued fuzzy data are locations of objects in space likepositions of ships on radar screens, space–time data, multivariate nonprecise data in

the form of vectors (x1∗, , x∗

n ) of fuzzy numbers x i∗

In statistics frequently so-called stochastic quantities (also called random variables)are observed, where the observed results are fuzzy In this situation two kinds ofuncertainty are present: Variability, which can be modeled by probability distribu-tions, also called stochastic models, and fuzziness, which can be modeled by fuzzynumbers and fuzzy vectors, respectively It is important to note that these are twodifferent kinds of uncertainty Moreover it is necessary to describe fuzziness of data

in order to obtain realistic results from statistical analysis In Figure 1.1 the situation

is graphically outlined

Real data are also subject to a third kind of uncertainty: errors These are thesubject of Section 1.4

In standard statistics errors are modeled in the following way The observation y of

a stochastic quantity is not its true value x, but superimposed by a quantity e, called

FUZZY DATA 5error, i.e.

y = x + e.

The error is considered as the realization of another stochastic quantity Thesekinds of errors are denoted as random errors

For one-dimensional quantities, all three quantities x, y, and e are, after the

experiment, real numbers But this is not suitable for continuous variables because

the observed values y are fuzzy.

It is important to note that all three kinds of uncertainty are present in real data.Therefore it is necessary to generalize the mathematical operations for real numbers

to the situation of fuzzy numbers

In order to model one-dimensional fuzzy data the best up-to-date mathematical model

is so-called fuzzy numbers

Definition 2.1: A fuzzy number x∗ is determined by its so-called characterizingfunctionξ(·) which is a real function of one real variable x obeying the following:

(1) ξ : R → [0; 1].

(2) ∀δ ∈ (0; 1] the so-called δ-cut Cδ (x∗) := {x ∈ R : ξ(x) ≥ δ} is a finite union

of compact intervals, [a δ, j ; b δ, j ], i.e C δ (x∗ =
k δ

j=1[a δ, j ; b δ, j]= ∅

(3) The support ofξ(·), defined by supp[ξ(·)] := {x ∈ R : ξ(x) > 0} is bounded.

The set of all fuzzy numbers is denoted byF(R).

For the following and for applications it is important that characterizing functions

can be reconstructed from the family (C δ (x∗); δ ∈ (0; 1]), in the way described in

Lemma 2.1

Statistical Methods for Fuzzy Data Reinhard Viertl

© 2011 John Wiley & Sons, Ltd

7

8 STATISTICAL METHODS FOR FUZZY DATA

Lemma 2.1: For the characterizing functionξ(·) of a fuzzy number x∗the followingholds true:

Definition 2.2: A fuzzy number is called a fuzzy interval if all its δ-cuts are non-empty

closed bounded intervals

In Figure 2.1 examples of fuzzy intervals are depicted

The set of all fuzzy intervals is denoted byFI(R)

Remark 2.2: Precise numbers x0∈ R are represented by its characterizing function

I {x0}(·), i.e the one-point indicator function of the set {x0} For this characterizing

function the δ-cuts are the degenerated closed interval [x0; x0] = {x0} Therefore

precise data are specialized fuzzy numbers

In Figure 2.2 theδ-cut for a characterizing function is explained.

Special types of fuzzy intervals are so-called LR- fuzzy numbers which are defined

by two functions L : [0; ∞) → [0; 1] and R : [0, ∞) → [0, 1] obeying the following:

(1) L( ·) and R(·) are left-continuous.

(2) L( ·) and R(·) have finite support.

(3) L( ·) and R(·) are monotonic nonincreasing.

FUZZY NUMBERS AND FUZZY VECTORS 9

Figure 2.1 Characterizing functions of fuzzy intervals.

Using these functions the characterizing functionξ(·) of an LR-fuzzy interval is

A special type of LR-fuzzy numbers are the so-called trapezoidal fuzzy numbers, denoted by t∗(m , s, l, r) with

L(x) = R(x) = max {0, 1 − x} ∀x ∈ [0; ∞).

10 STATISTICAL METHODS FOR FUZZY DATA

Figure 2.2 Characterizing function and a δ-cut.

The corresponding characterizing function of t∗(m , s, l, r) is given by

In Figure 2.3 the shape of a trapezoidal fuzzy number is depicted

Theδ-cuts of trapezoidal fuzzy numbers can be calculated easily using the called pseudo-inverse functions L−1(·) and R−1(·) which are given by

so-L−1(δ) = max {x ∈ R : L(x) ≥ δ}

R−1(δ) = max {x ∈ R : R(x) ≥ δ}

Lemma 2.2: Theδ-cuts C δ (x∗) of an LR-fuzzy number x∗are given by

C δ (x∗ = [m − s − l L−1(δ); m + s + r R−1(δ)] ∀δ ∈ (0, 1].

FUZZY NUMBERS AND FUZZY VECTORS 11

Figure 2.3 Trapezoidal fuzzy number.

Proof: The left boundary of C δ (x∗) is determined by min{x : ξ(x) ≥ δ} By the

definition of LR-fuzzy numbers for l > 0 we obtain

The proof for the right boundary is analogous

An important topic is how to obtain the characterizing function of fuzzy data.There is no general rule for that, but for different important measurement situationsprocedures are available

For analog measurement equipment often the result is obtained as a light point

on a screen In this situation the light intensity on the screen is used to obtain the

characterizing function For one-dimensional quantities the light intensity h(·) is

normalized, i.e

andξ(·) is the characterizing function of the fuzzy observation.

For light points on a computer screen the function h(·) is given on finitely many

pixels x1, , x N with intensities h(x i), i = 1(1)N In order to obtain the

charac-terizing functionξ(·) we consider the discrete function h(·) defined on the finite set

{x , , x }

12 STATISTICAL METHODS FOR FUZZY DATA

Let the distance between the points x1< x2< < x Nbe constant and equal to

x Defining a function η(·) on the set {x1, , x N} by

η(x i) := h(x i)

max{h(x i ) : i = 1(1)N} for i = 1(1)N,

the characterizing functionξ(·) is obtained in the following way:

Based on the functionη(·) the values ξ(x) are defined for all x ∈ R by

2

.max{η(x i−1), η(x i)} for x = x i+x

2

.max{η(x N−1), η(x N)} for x = x N−x

Remark 2.3:ξ(·) is a characterizing function of a fuzzy number.

For digital measurement displays the results are decimal numbers with a finite

number of digits Let the resulting number be y, then the remaining (infinite many)

decimals are unknown The numerical information contained in the result is an interval

[x; x], where x is the real number obtained from y by taking the remaining decimals all to be 0, and x is the real number obtained from y by taking the remaining decimals

all to be 9 The corresponding characterizing function of this fuzzy number is the

indicator function I [x;x ](·) of the closed interval [x; x].

Therefore the characterizing functionξ(·) is given by its values

FUZZY NUMBERS AND FUZZY VECTORS 13

If the measurement result is given by a color intensity picture, for examplediameters of particles, the color intensity is used to obtain the characterizing function

ξ(·) Let h(·) be the color intensity describing the boundary then the derivative h (·)

of h(·) is used, i.e

ξ (x) := h (x)

max{|h (x) | : x ∈ R} ∀x ∈ R.

An example is given in Figure 2.4

For color intensity pictures on digital screens a discrete version for step functionscan be applied

Let {x1, , x N } be the discrete values of the variable and h(x i) be the color

intensities at position x i as before, andx the constant distance between the points

x i Then the discrete analog of the derivative h (·) is given by the step function η(·)

Figure 2.4 Construction of a characterizing function.

14 STATISTICAL METHODS FOR FUZZY DATA

which is constant in the intervals

ξ (x) := η (x)

max{η (x) : x ∈ R} for all x ∈ R.

The functionξ(·) is a characterizing function of a fuzzy number.

For multivariate continuous data measurement results are fuzzy too In the idealized

case the result is a k-dimensional real vector (x1, , x k) Depending on the problemtwo kinds of situations are possible

The first is when the individual values of the variables x i are fuzzy numbers x i∗

Then a vector (x1∗, , x∗

k) of fuzzy numbers is obtained This vector is determined

by k characterizing functions ξ1(·), , ξ k(·) Methods to obtain these characterizing

functions are described in Section 2.1

The second situation yields a fuzzy version of a vector, for example the position

of a ship on a radar screen In the idealized case the position is a two-dimensional

vector (x , y) ∈ R2 In real situation the position is characterized by a light point on

the screen which is not a precise vector The result is a so-called fuzzy vector, denoted

as (x , y)∗.

The mathematical model of a fuzzy vector is given in the following definition,

using the notation x = (x1, , x k)

Definition 2.3: A k-dimensional fuzzy vector x∗is determined by its so-called characterizing functionξ(, ,) which is a real function of k real variables x1, , x k

vector-obeying the following:

(1) ξ : R k→ [0; 1]

(2) The support ofξ(, ,) is a bounded set.

(3) ∀δ ∈ (0; 1] the so-called δ-cut C δx∗

:=x∈ Rk:ξx

≥ δis non-empty,bounded, and a finite union of simply connected and closed bounded sets

The set of all k-dimensional fuzzy vectors is denoted by F(R k)

In Figure 2.5 a vector-characterizing function of a two-dimensional fuzzy vector

is depicted

Remark 2.4: There are different definitions of fuzzy vectors The above definition

seems to be best for applications

FUZZY NUMBERS AND FUZZY VECTORS 15

1.0

0.8

0.6 0.4 0.2 0.0 –4

–4

–2

–2 0

Figure 2.5 Vector-characterizing function.

For statistical inference specialized fuzzy vectors are important, the so-called

fuzzy k-dimensional intervals.

Definition 2.4: A k-dimensional fuzzy vector is called a k-dimensional fuzzy interval

if allδ-cuts are simply connected compact sets.

An example of a two-dimensional fuzzy interval is given in Figure 2.6

Lemma 2.3: The vector-characterizing functionξ(, ,) of a fuzzy vector x∗can bereproduced by itsδ-cuts in the following way:

The proof is similar to the proof of Lemma 2.1

Again it is important how to obtain the vector-characterizing function of a fuzzyvector There is no general rule for this but some methodology

For two-dimension fuzzy vectors x∗ = (x, y)∗given by light intensities the characterizing function ξ(·) of x∗ is obtained from the values h(x , y) of the light

vector-intensity by

ξ (x, y) := h (x , y)

max

h (x , y) : (x, y) ∈ R2 ∀ (x, y) ∈ R2.

16 STATISTICAL METHODS FOR FUZZY DATA

Figure 2.6 Fuzzy two-dimensional interval.

If only fuzzy values of the components x i of a k-dimensional vector (x i , , x k)

are available, i.e x i∗ with corresponding characterizing function ξ i(·), i = 1(1)n,

these characterizing functions can be combined into a vector-characterizing function

of a k-dimensional fuzzy vector using so-called triangular norms Details of this are

explained in Section 2.3

A vector (x1∗, , x∗

k ) of fuzzy numbers x i∗is not a fuzzy vector For the generalization

of statistical inference, functions defined on sample spaces are essential Therefore it

is basic to form fuzzy elements in the sample space M × × M, where M denotes

the observation space of a random quantity These fuzzy elements are fuzzy vectors

By this it is necessary to form fuzzy vectors from fuzzy samples This is done byapplying so-called triangular norms, also called t-norms

Definition 2.5: A function T : [0; 1]× [0; 1] → [0; 1] is called a t-norm, if for all

x , y, z, ∈ [0; 1] the following conditions are fulfilled:

(1) T (x , y) = T (y, x) commutativity.

(2) T (T (x , y), z) = T (x, T (y, z)) associativity.

(3) T (x , 1) = x.

(4) x ≤ y ⇒ T (x, z) ≤ T (y, z).

FUZZY NUMBERS AND FUZZY VECTORS 17Examples of t-norms are:

(a) Minimum t-norm

Remark 2.5: For statistical and algebraic calculations with fuzzy data the minimum

t-norm is optimal For the combination of fuzzy components of vector data in someexamples the product t-norm is more suitable

For more details on mathematical aspects of t-norms see Klement et al (2000).

Based on t-norms the combination of fuzzy numbers into a fuzzy vector is

pos-sible For two fuzzy numbers x∗and y∗with corresponding characterizing functions

ξ(·) and η(·) a fuzzy vector x∗= (x, y)∗is given by its vector-characterizing function

ξ(·) The values ξ(x, y) are defined based on a t-norm T by

For the minimum t-norm theδ-cuts of the combined fuzzy vector are very easy

to obtain This is shown in Proposition 2.1

Proposition 2.1: For k fuzzy numbers x∗1, , x∗

k with characterizing functions

ξ1(·), , ξ k(·) the δ-cuts Cδ (x∗) of the fuzzy vector x∗, combined by the mum t-norm are the Cartesian products of theδ-cuts C δ (x i∗) of the fuzzy numbers

18 STATISTICAL METHODS FOR FUZZY DATA

Figure 2.7 Combination of two fuzzy numbers.

Proof: Letξ(., ,.) be the vector-characterizing function of x∗ Then forδ ∈ (0; 1]

theδ-cut C δ (x∗) obeys

In Figure 2.7 this is explained graphically

Remark 2.7: The minimum t-norm is also the natural t-norm for the generalization

of algebraic operations to the system of fuzzy numbers

(a) Draw a graph of a characterizing function and construct five δ-cuts for this

characterizing function, withδ = 0.2, 0.4, 0.5, 0.7 and 0.9.

FUZZY NUMBERS AND FUZZY VECTORS 19(b) Let the color intensity for a one-dimensional quantity be given by the followingfunction:

Find the characterizing function of the fuzzy boundary between low colorintensity and maximal color intensity using the method described in Section 2.1.(c) Calculate theδ-cuts of a fuzzy vector with vector-characterizing function

16 STATISTICAL METHODS FOR FUZZY DATA< /p>

Figure 2.6 Fuzzy two-dimensional interval.

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14 STATISTICAL METHODS FOR FUZZY DATA< /p>

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ξ (x) := η (x)

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FUZZY NUMBERS AND FUZZY VECTORS 17Examples of t-norms are:

(a) Minimum t-norm

Remark 2.5: For statistical and algebraic calculations with fuzzy data the