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VIERTL

Statistical Methods for
Fuzzy Data
Reinhard Viertl, Vienna University of Technology, Austria

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

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
Methods for
Fuzzy Data

Reinhard Viertl


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Statistical Methods for
Fuzzy Data


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Statistical Methods for

Fuzzy Data
Reinhard Viertl
Vienna University of Technology, Austria

A John Wiley and Sons, Ltd., Publication


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This edition first published 2011
© 2011 John Wiley & Sons, Ltd
Registered office
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sought.

Library of Congress Cataloging-in-Publication Data
Viertl, R. (Reinhard)
Statistical methods for fuzzy data / Reinhard Viertl.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-69945-4 (cloth)
1. Fuzzy measure theory. 2. Fuzzy sets. 3. Mathematical statistics. I. Title.
QA312.5.V54 2010
515 .42–dc22
2010031105
A catalogue record for this book is available from the British Library.
Print ISBN: 978-0-470-69945-4
ePDF ISBN: 978-0-470-97442-1
oBook ISBN: 978-0-470-97441-4
ePub ISBN: 978-0-470-97456-8
Typeset in 10/12pt Times by Aptara Inc., New Delhi, India


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Contents
Preface

xi

Part I FUZZY INFORMATION

1

1
1.1
1.2
1.3
1.4
1.5

Fuzzy data
One-dimensional fuzzy data
Vector-valued fuzzy data
Fuzziness and variability
Fuzziness and errors
Problems

3

3
4
4
4
5

2
2.1
2.2
2.3
2.4

Fuzzy numbers and fuzzy vectors
Fuzzy numbers and characterizing functions
Vectors of fuzzy numbers and fuzzy vectors
Triangular norms
Problems

7
7
14
16
18

3
3.1
3.2
3.3
3.4
3.5

3.6
3.7

Mathematical operations for fuzzy quantities
Functions of fuzzy variables
Addition of fuzzy numbers
Multiplication of fuzzy numbers
Mean value of fuzzy numbers
Differences and quotients
Fuzzy valued functions
Problems

21
21
23
25
25
27
27
28

Part II DESCRIPTIVE STATISTICS FOR FUZZY DATA

31

4
4.1
4.2
4.3
4.4


33
33
33
34
34

Fuzzy samples
Minimum of fuzzy data
Maximum of fuzzy data
Cumulative sum for fuzzy data
Problems


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CONTENTS

5
5.1

5.2
5.3
5.4

Histograms for fuzzy data
Fuzzy frequency of a fixed class
Fuzzy frequency distributions
Axonometric diagram of the fuzzy histogram
Problems

37
37
38
40
41

6
6.1
6.2
6.3
6.4

Empirical distribution functions
Fuzzy valued empirical distribution function
Fuzzy empirical fractiles
Smoothed empirical distribution function
Problems

43
43

45
45
47

7
Empirical correlation for fuzzy data
7.1 Fuzzy empirical correlation coefficient
7.2 Problems

49
49
52

Part III FOUNDATIONS OF STATISTICAL
INFERENCE WITH FUZZY DATA

53

8
8.1
8.2
8.3
8.4

Fuzzy probability distributions
Fuzzy probability densities
Probabilities based on fuzzy probability densities
General fuzzy probability distributions
Problems


55
55
56
57
58

9
9.1
9.2
9.3
9.4
9.5

A law of large numbers
Fuzzy random variables
Fuzzy probability distributions induced by fuzzy random variables
Sequences of fuzzy random variables
Law of large numbers for fuzzy random variables
Problems

59
59
61
62
63
64

10
10.1
10.2

10.3
10.4

Combined fuzzy samples
Observation space and sample space
Combination of fuzzy samples
Statistics of fuzzy data
Problems

65
65
66
66
67

Part IV CLASSICAL STATISTICAL INFERENCE
FOR FUZZY DATA

69

11 Generalized point estimators
11.1 Estimators based on fuzzy samples

71
71


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vii

11.2 Sample moments
11.3 Problems

73
74

12
12.1
12.2
12.3

Generalized confidence regions
Confidence functions
Fuzzy confidence regions
Problems

75
75
76

79

13
13.1
13.2
13.3

Statistical tests for fuzzy data
Test statistics and fuzzy data
Fuzzy p-values
Problems

81
81
82
86

Part V BAYESIAN INFERENCE AND FUZZY
INFORMATION

87

14
14.1
14.2
14.3

Bayes’ theorem and fuzzy information
Fuzzy a priori distributions
Updating fuzzy a priori distributions

Problems

91
91
92
96

15
15.1
15.2
15.3

Generalized Bayes’ theorem
Likelihood function for fuzzy data
Bayes’ theorem for fuzzy a priori distribution and fuzzy data
Problems

97
97
97
101

16
16.1
16.2
16.3

Bayesian confidence regions
Bayesian confidence regions based on fuzzy data
Fuzzy HPD-regions

Problems

103
103
104
106

17
17.1
17.2
17.3
17.4

Fuzzy predictive distributions
Discrete case
Discrete models with continuous parameter space
Continuous case
Problems

107
107
108
110
111

18
18.1
18.2
18.3
18.4

18.5

Bayesian decisions and fuzzy information
Bayesian decisions
Fuzzy utility
Discrete state space
Continuous state space
Problems

113
113
114
115
116
117


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CONTENTS


Part VI REGRESSION ANALYSIS AND FUZZY
INFORMATION

119

19
19.1
19.2
19.3
19.4
19.5

121
121
125
128
131
132

Classical regression analysis
Regression models
Linear regression models with Gaussian dependent variables
General linear models
Nonidentical variances
Problems

20 Regression models and fuzzy data
20.1 Generalized estimators for linear regression models based on the
extension principle

20.2 Generalized confidence regions for parameters
20.3 Prediction in fuzzy regression models
20.4 Problems

133
134
138
138
139

21
21.1
21.2
21.3
21.4
21.5
21.6
21.7

Bayesian regression analysis
Calculation of a posteriori distributions
Bayesian confidence regions
Probabilities of hypotheses
Predictive distributions
A posteriori Bayes estimators for regression parameters
Bayesian regression with Gaussian distributions
Problems

141
141

142
142
142
143
144
144

22
22.1
22.2
22.3
22.4

Bayesian regression analysis and fuzzy information
Fuzzy estimators of regression parameters
Generalized Bayesian confidence regions
Fuzzy predictive distributions
Problems

147
148
150
150
151

Part VII FUZZY TIME SERIES

153

23

23.1
23.2
23.3

Mathematical concepts
Support functions of fuzzy quantities
Distances of fuzzy quantities
Generalized Hukuhara difference

155
155
156
161

24 Descriptive methods for fuzzy time series
24.1 Moving averages
24.2 Filtering
24.2.1 Linear filtering
24.2.2 Nonlinear filters

167
167
169
169
173


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CONTENTS

ix

24.3 Exponential smoothing
24.4 Components model
24.4.1 Model without seasonal component
24.4.2 Model with seasonal component
24.5 Difference filters
24.6 Generalized Holt–Winter method
24.7 Presentation in the frequency domain

175
176
177
177
182
184
186

25
25.1
25.2

25.3
25.4

189
189
190
193
196

More on fuzzy random variables and fuzzy random vectors
Basics
Expectation and variance of fuzzy random variables
Covariance and correlation
Further results

26 Stochastic methods in fuzzy time series analysis
26.1 Linear approximation and prediction
26.2 Remarks concerning Kalman filtering

199
199
212

Part VIII APPENDICES

215

A1

List of symbols and abbreviations


217

A2

Solutions to the problems

223

A3

Glossary

245

A4

Related literature

247

References

251

Index

253



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Preface
Statistics is concerned with the analysis of data and estimation of probability distribution and stochastic models. Therefore the quantitative description of data is essential
for statistics.
In standard statistics data are assumed to be numbers, vectors or classical functions. But in applications real data are frequently not precise numbers or vectors, but
often more or less imprecise, also called fuzzy. It is important to note that this kind of
uncertainty is different from errors; it is the imprecision of individual observations
or measurements.
Whereas counting data can be precise, possibly biased by errors, measurement
data of continuous quantities like length, time, volume, concentrations of poisons,

amounts of chemicals released to the environment and others, are always not precise
real numbers but connected with imprecision.
In measurement analysis usually statistical models are used to describe data
uncertainty. But statistical models are describing variability and not the imprecision
of individual measurement results. Therefore other models are necessary to quantify
the imprecision of measurement results.
For a special kind of data, e.g. data from digital instruments, interval arithmetic
can be used to describe the propagation of data imprecision in statistical inference.
But there are data of a more general form than intervals, e.g. data obtained from
analog instruments or data from oscillographs, or graphical data like color intensity
pictures. Therefore it is necessary to have a more general numerical model to describe
measurement data.
The most up-to-date concept for this is special fuzzy subsets of the set R of real
numbers, or special fuzzy subsets of the k-dimensional Euclidean space Rk in the case
of vector quantities. These special fuzzy subsets of R are called nonprecise numbers in
the one-dimensional case and nonprecise vectors in the k-dimensional case for k > 1.
Nonprecise numbers are defined by so-called characterizing functions and nonprecise
vectors by so-called vector-characterizing functions. These are generalizations of
indicator functions of classical sets in standard set theory. The concept of fuzzy
numbers from fuzzy set theory is too restrictive to describe real data. Therefore
nonprecise numbers are introduced.
By the necessity of the quantitative description of fuzzy data it is necessary to
adapt statistical methods to the situation of fuzzy data. This is possible and generalized
statistical procedures for fuzzy data are described in this book.


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PREFACE

There are also other approaches for the analysis of fuzzy data. Here an approach
from the viewpoint of applications is used. Other approaches are mentioned in Appendix A4.
Besides fuzziness of data there is also fuzziness of a priori distributions in
Bayesian statistics. So called fuzzy probability distributions can be used to model
nonprecise a priori knowledge concerning parameters in statistical models.
In the text the necessary foundations of fuzzy models are explained and basic statistical analysis methods for fuzzy samples are described. These include generalized
classical statistical procedures as well as generalized Bayesian inference procedures.
A software system for statistical analysis of fuzzy data (AFD) is under development. Some procedures are already available, and others are in progress. The available
software can be obtained from the author.
Last but not least I want to thank all persons who contributed to this work: Dr
D. Hareter, Mr H. Schwarz, Mrs D. Vater, Dr I. Meliconi, H. Kay, P. Sinha-Sahay
and B. Kaur from Wiley for the excellent cooperation, and my wife Dorothea for
preparing the files for the last two parts of this book.
I hope the readers will enjoy the text.
Reinhard Viertl
Vienna, Austria
July 2010


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Part I
FUZZY INFORMATION
Fuzzy information is a special kind of information and information is an omnipresent
word 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 possible definition of information is the following: Information is everything which has
influence on the assessment of uncertainty by an analyst. This uncertainty can be of
different types: data uncertainty, nondeterministic quantities, model uncertainty, and
uncertainty 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 considering probabilities to be numbers. Often this is not realistic, and in a more general
approach probabilities are considered to be so-called fuzzy numbers.
The idea of generalized sets was originally published in Menger (1951) and the
term ‘fuzzy set’ was coined in Zadeh (1965).


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1

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.

1.1

One-dimensional fuzzy data


Measurement results of one-dimensional continuous quantities are frequently idealized to be numbers times a measurement unit. However, real measurement results
of continuous quantities are never precise numbers but always connected with uncertainty. Usually this uncertainty is considered to be statistical in nature, but this
is not suitable since statistical models are suitable to describe variability. For a single measurement result there is no variability, therefore another method to model
the measurement uncertainty of individual measurement results is necessary. The
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, length
measurements, 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
© 2011 John Wiley & Sons, Ltd

Reinhard Viertl

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STATISTICAL METHODS FOR FUZZY DATA

Figure 1.1 Variability and fuzziness.

1.2

Vector-valued fuzzy data

Many statistical data are multivariate, i.e. ideally the corresponding measurement
results are real vectors (x1 , . . . , x k ) ∈ Rk . In applications such data are frequently
not 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 like
positions of ships on radar screens, space–time data, multivariate nonprecise data in
the form of vectors (x1∗ , . . . , x n∗ ) of fuzzy numbers xi∗ .

1.3

Fuzziness and variability

In statistics frequently so-called stochastic quantities (also called random variables)
are observed, where the observed results are fuzzy. In this situation two kinds of
uncertainty are present: Variability, which can be modeled by probability distributions, also called stochastic models, and fuzziness, which can be modeled by fuzzy
numbers and fuzzy vectors, respectively. It is important to note that these are two
different 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 the
subject of Section 1.4.

1.4

Fuzziness and errors

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


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FUZZY DATA

5

error, i.e.
y = x + e.
The error is considered as the realization of another stochastic quantity. These
kinds 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.

1.5

Problems

(a) Find examples of fuzzy numerical data which are not given in Section 1.1 and
Section 1.2.
(b) Work out the difference between stochastic uncertainty and fuzziness of individual observations.
(c) Make clear how data in the form of intervals are obtained by digital measurement
devices.
(d) What do X-ray pictures and data from satellite photographs have in common?


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2

Fuzzy numbers and fuzzy
vectors
Taking care of the fuzziness of data described in Chapter 1 it is necessary to have a
mathematical model to describe such data in a quantitative way. This is the subject
of Chapter 2.

2.1

Fuzzy numbers and characterizing functions

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 characterizing
function ξ (·) 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 C δ (x ∗ ) := {x ∈ R : ξ (x) ≥ δ} is a finite union
of compact intervals, [aδ, j ; bδ, j ], i.e. C δ (x ∗ ) =


kδ

[aδ, j ; bδ, j ] = ∅.

j=1

(3) The support of ξ (·), defined by supp[ξ (·)] := {x ∈ R : ξ (x) > 0} is bounded.
The set of all fuzzy numbers is denoted by F(R).
For the following and for applications it is important that characterizing functions
can be reconstructed from the family (Cδ (x ∗ ); δ ∈ (0; 1]), in the way described in
Lemma 2.1.
Statistical Methods for Fuzzy Data
© 2011 John Wiley & Sons, Ltd

Reinhard Viertl

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STATISTICAL METHODS FOR FUZZY DATA

Lemma 2.1: For the characterizing function ξ (·) of a fuzzy number x ∗ the following
holds true:
ξ (x) = max δ · ICδ (x ∗ ) (x) : δ ∈ [0; 1]

∀x ∈ R

Proof: For fixed x0 ∈ R we have
δ · ICδ (x ∗ ) (x0 ) = δ · I{x:ξ (x)≥δ} (x0 ) =

δ
0

for ξ (x 0 ) ≥ δ
for ξ (x 0 ) < δ.

Therefore we have for every δ ∈ [0;1]
δ · ICδ (x ∗ ) (x0 ) ≤ ξ (x0 ),
and further
sup δ · ICδ (x ∗ ) (x0 ) : δ ∈ [0; 1] ≤ ξ (x0 ).
On the other hand we have for δ0 = ξ (x0 ):
δ0 · ICδ0 (x ∗ ) (x0 ) = δ0 and therefore
sup δ · ICδ (x ∗ ) (x0 ) : δ ∈ [0; 1] ≥ δ0 which implies
sup δ · ICδ (x ∗ ) (x0 ) : δ ∈ [0; 1] = max δ · ICδ (x ∗ ) (x0 ) : δ ∈ [0; 1] = δ0 .
Remark 2.1: In applications fuzzy numbers are represented by a finite number of
δ-cuts.
Special types of fuzzy numbers are useful to define so-called fuzzy probability
distribution. These kinds of fuzzy numbers are denoted as fuzzy intervals.

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 by FI (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 ]. = {x 0 }. Therefore
precise data are specialized fuzzy numbers.
In Figure 2.2 the δ-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.


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FUZZY NUMBERS AND FUZZY VECTORS

9


Figure 2.1 Characterizing functions of fuzzy intervals.
Using these functions the characterizing function ξ (·) of an LR-fuzzy interval is
defined by:
⎧
m−s−x
⎪
⎪
L
⎪
⎪
⎨
l
ξ (x) = 1
⎪
⎪
m−s−x
⎪
⎪
⎩R
r

for

x
for

m−s ≤ x ≤m+s

for

x > m + s,

where m, s, l, r are real numbers obeying s ≥ 0, l > 0, r > 0. Such fuzzy numbers
are denoted by m, s, l, r LR .
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; ∞).


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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
⎧ x −m +s +l
⎪

⎪
⎪
⎪
l
⎨
1
ξ (x) = m + s + r − x
⎪
⎪
⎪
⎪
r
⎩
0

for m − s − l ≤ x < m − s
for m − s ≤ x ≤ m + s
for m + s < x ≤ m + s + r
otherwise.

In Figure 2.3 the shape of a trapezoidal fuzzy number is depicted.
The δ-cuts of trapezoidal fuzzy numbers can be calculated easily using the socalled pseudo-inverse functions L −1 (·) and R −1 (·) which are given by
L −1 (δ) = max {x ∈ R : L(x) ≥ δ}
R −1 (δ) = max {x ∈ R : R(x) ≥ δ}
Lemma 2.2: The δ-cuts Cδ (x ∗ ) of an LR-fuzzy number x ∗ are given by
Cδ (x ∗ ) = [m − s − l L −1 (δ); m + s + r R −1 (δ)] ∀δ ∈ (0, 1].


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11

Figure 2.3 Trapezoidal fuzzy number.
Proof: The left boundary of Cδ (x ∗ ) is determined by min {x : ξ (x) ≥ δ}. By the
definition of LR-fuzzy numbers for l > 0 we obtain
min {x : ξ (x) ≥ δ} = min x : L
= min x :

m−s−x
l

≥ δ and x < m − s

m−s−x
≤ L −1 (δ) and x < m − s
l

= min x : x ≥ m − s − l L −1 (δ) and x < m − s
= m − s − l L −1 (δ).
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 situations
procedures 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.
ξ (x) :=

h(x)
∀x ∈ R,
max {h(x) : x ∈ R}

and ξ (·) is the characterizing function of the fuzzy observation.
For light points on a computer screen the function h(·) is given on finitely many
pixels x 1 , . . . , x N with intensities h(xi ), i = 1(1)N . In order to obtain the characterizing function ξ (·) we consider the discrete function h(·) defined on the finite set
{x1 , . . . , x N }.