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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Provides a unique and methodologically consistent treatment of various areas of fuzzy modeling and includes the results of mathematical fuzzy logic and linguistics This book is the result of almost thirty years of research on fuzzy modeling. It provides a unique view of both the theory and various types of applications. The book is divided into two parts. The first part contains an extensive presentation of the theory of fuzzy modeling. The second part presents selected applications in three important areas: control and decision-making, image processing, and time series analysis and forecasting. The authors address the consistent and appropriate treatment of the notions of fuzzy sets and fuzzy logic and their applications. They provide two complementary views of the methodology, which is based on fuzzy IF-THEN rules. The first, more traditional method involves fuzzy approximation and the theory of fuzzy relations. The second method is based on a combination of formal fuzzy logic and linguistics. A very important topic covered for the first time in book form is the fuzzy transform (F-transform). Applications of this theory are described in separate chapters and include image processing and time series analysis and forecasting. All of the mentioned components make this book of interest to students and researchers of fuzzy modeling as well as to practitioners in industry. Features: * Provides a foundation of fuzzy modeling and proposes a thorough description of fuzzy modeling methodology * Emphasizes fuzzy modeling based on results in linguistics and formal logic * Includes chapters on natural language and approximate reasoning, fuzzy control and fuzzy decision-making, and image processing using the F-transform * Discusses fuzzy IF-THEN rules for approximating functions, fuzzy cluster analysis, and time series forecasting Insight into Fuzzy Modeling is a reference for researchers in the fields of soft computing and fuzzy logic as well as undergraduate, master and Ph.D. students. Vilém Novák, D.Sc. is Full Professor and Director of the Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Czech Republic. Irina Perfilieva, Ph.D. is Full Professor, Senior Scientist, and Head of the Department of Theoretical Research at the Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Czech Republic. Antonín Dvorák, Ph.D. is Associate Professor, and Senior Scientist at the Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Czech Republic.
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Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
Acknowledgments
About the Companion Website
PART I: FUNDAMENTALS OF FUZZY MODELING
Chapter 1: What is Fuzzy Modeling
1.1 INDETERMINACY IN HUMAN LIFE
1.2 FUZZY MODELING: WITH AND WITHOUT WORDS
Chapter 2: Overview of Basic Notions
2.1 RELATIONS, FUNCTIONS, ORDERED SETS
2.2 FUZZY SETS AND FUZZY RELATIONS
2.3 ELEMENTS OF MATHEMATICAL FUZZY LOGIC
Chapter 3: Fuzzy If-Then Rules in Approximation Of Functions
3.1 RELATIONAL INTERPRETATION OF FUZZY IF-THEN RULES
3.2 APPROXIMATION OF FUNCTIONS USING FUZZY IF-THEN RULES
3.3 GENERALIZED MODUS PONENS AND FUZZY FUNCTIONS
3.4 TAKAGI–SUGENO RULES
Chapter 4: Fuzzy Transform
4.1 FUZZY PARTITION
4.2 THE CONCEPT OF F-TRANSFORM
4.3 DISCRETE F-TRANSFORM
4.4 F-TRANSFORM OF FUNCTIONS OF TWO VARIABLES
4.5 F
1
-TRANSFORM
4.6 METHODOLOGICAL REMARKS TO APPLICATIONS OF THE F-TRANSFORM
Chapter 5: Fuzzy Natural Logic and Approximate Reasoning
5.1 LINGUISTIC SEMANTICS AND LINGUISTIC VARIABLE
5.2 THEORY OF EVALUATIVE LINGUISTIC EXPRESSIONS
5.3 INTERPRETATION OF FUZZY/LINGUISTIC IF-THEN RULES
5.4 APPROXIMATE REASONING WITH LINGUISTIC INFORMATION
Chapter 6: Fuzzy Cluster Analysis
6.1 BASIC NOTIONS
6.2 FUZZY CLUSTERING ALGORITHMS
6.3 THE ALGORITHM OF FUZZY c-MEANS
6.4 THE GUSTAFSON–KESSEL ALGORITHM
6.5 HOW THE NUMBER OF CLUSTERS CAN BE DETERMINED
6.6 CONSTRUCTION OF FUZZY RULES BASED ON FOUND CLUSTERS
PART II: SELECTED APPLICATIONS
Chapter 7: Fuzzy/Linguistic Control and Decision-Making
7.1 THE PRINCIPLE OF FUZZY CONTROL
7.2 FUZZY CONTROLLERS
7.3 DESIGN OF FUZZY/LINGUISTIC CONTROLLER
7.4 LEARNING
7.5 DECISION-MAKING USING LINGUISTIC DESCRIPTIONS
Chapter 8: F-Transform in Image Processing
8.1 IMAGE AND ITS BASIC PROCESSING USING F-TRANSFORM
8.2 F-TRANSFORM-BASED IMAGE COMPRESSION AND RECONSTRUCTION
8.3 F
1
-TRANSFORM EDGE DETECTOR
8.4 F-TRANSFORM-BASED IMAGE FUSION
8.5 F-TRANSFORM-BASED CORRUPTED IMAGE RECONSTRUCTION
Chapter 9: Analysis and Forecasting of Time Series
9.1 CLASSICAL VERSUS FUZZY MODELS OF TIME SERIES
9.2 ANALYSIS OF TIME SERIES USING F-TRANSFORM
9.3 TIME SERIES FORECASTING
9.4 CHARACTERIZATION OF TIME SERIES IN NATURAL LANGUAGE
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Part
Begin Reading
List of Illustrations
Chapter 1: What is Fuzzy Modeling
Figure 1.1 Fuzzy set modeling the meaning of “red color”.
Chapter 2: Overview of Basic Notions
Figure 2.1 Three partially ordered sets. (a) has two maximal and two minimal elements. (b) is a five-element chain. (c) is a lattice with the greatest element
e
and smallest
a
.
Figure 2.2 Schematic depiction of a possible fuzzy set of small people.
Figure 2.3 Convex and nonconvex fuzzy set.
Figure 2.4 Membership function of the fuzzy number “about ”.
Figure 2.5 Membership function of a fuzzy number: (a) with triangular membership function and (b) with Gaussian membership function.
Figure 2.6 Example of a fuzzy partition of the interval .
Figure 2.7 Trapezoidal fuzzy set which is extensional with respect to fuzzy relation and the Łukasiewicz t-norm .
Figure 2.8 Schematic depiction of fuzzy sets with respect to condition (2.73): the couples of fuzzy sets (a) and (b) fulfill it while (c) does not.
Chapter 3: Fuzzy If-Then Rules in Approximation Of Functions
Figure 3.1 Schematic depiction of a fuzzy function
F
represented by a fuzzy graph that approximates a crisp function
f
. Below
x
-axis and to the left of
y
-axis, membership functions of the fuzzy sets and , respectively, , are depicted.
Figure 3.2 Schematic depiction of a fuzzy relation , which interprets a disjunctive normal form approximating a function
f
.
Figure 3.3 Schematic depiction of a fuzzy relation , which interprets a conjunctive normal form approximating a function
f
.
Figure 3.4 COG defuzzification—the resulting value (the center of gravity) is denoted by the thick dot.
Figure 3.5 MOM defuzzification—the resulting value (the center of maximal values) is denoted by the thick dot.
Figure 3.6 DEE defuzzification—the resulting value for each type of fuzzy set (Z, M, or S) is denoted by the thick dot. The universe is represented by an interval .
Figure 3.7 (a) Fuzzy partition of input and output variables of the original fuzzy approximation. (b) Fuzzy partition of the input variable of the tuned fuzzy approximation. (c) Fuzzy partition of the output variable of the tuned approximation for Example 3.3.
Figure 3.8 Function for the original fuzzy approximation of for Example 3.3.
Figure 3.9 Function for the tuned fuzzy approximation of for Example 3.3.
Figure 3.10 A schematic depiction of fuzzy approximation of function
f
at point ().
Figure 3.11 Triangular fuzzy partitions for Example 3.5.
Figure 3.12 The fuzzy relations (a) and (b) from Example 3.5.
Figure 3.13 A necessary position of fuzzy sets for correct generalized modus ponens. (a) The fuzzy sets and fulfill condition (3.60). (b) The fuzzy sets and violate condition (3.60) (because ).
Figure 3.14 Trapezoidal fuzzy partition on and the approximating function from Example 3.6.
Figure 3.15 A schematic depiction of the fuzzy approximating function using TS rules with .
Chapter 4: Fuzzy Transform
Figure 4.1 Example of the -uniform triangular fuzzy partition.
Figure 4.2 Example of -uniform cosine fuzzy partition.
Figure 4.3 Example of a fuzzy 2-partition with triangular basic functions.
Figure 4.4 Scheme of the F-transform. (a) The original function
f
. (b) Components of the direct F-transform depicted as big full circles and also the inverse F-transform , which is depicted by a broken line passing through the components. The fuzzy partition is depicted under the
x
-axis.
Figure 4.5 The function defined on the interval and its inverse F-transform. The fuzzy partition is uniform and formed by triangular-shaped basic functions. (a) contains 10 and (b) contains 20 basic functions. The F-transform components are marked by small circles.
Figure 4.6 Scheme of the F-transform of two variables. (a) A fuzzy partition and (b) one of the components with respect to the fuzzy partition constructed over both variables.
Figure 4.7 Scheme of the direct F
1
-transform . The components of the direct F-transform are schematically depicted as the corresponding lines. The fuzzy partition is depicted under the
x
-axis.
Figure 4.8 The function defined on the interval and its inverse F
1
-transform . The fuzzy partition is uniform and formed by triangular-shaped basic functions. (a) contains 10 and (b) contains 20 basic functions.
Chapter 5: Fuzzy Natural Logic and Approximate Reasoning
Figure 5.1 Fuzzy sets characterizing extensions of vague categories.
Figure 5.2 Left horizon () given by the observer , middle horizon () by the observer , and right horizon () by the observer (in a given context
w
). Note that the middle horizon spreads out to the left as well as to the right.
Figure 5.3 Modification of the shape of extensions of simple evaluative expression “
small
” for typical hedges
very
and
roughly
.
Figure 5.4 Shapes of the functions realizing horizon modification for various values of the parameters .
Figure 5.5 Scheme of the construction of extensions of pure evaluative predications.
Figure 5.6 Intensions of the pure evaluative expressions “very small”, “small”, “roughly small”, “medium”, and “big”. Because intension is a function, each context on the left side of the arrow are assigned extensions of the corresponding linguistic expressions (right side of the arrow).
Figure 5.7 Experimentally estimated extensions of selected pure evaluative predications in the context . The curves depict membership functions of extensions of “
small
” from the left and “
big
” from the right where . Finally, in the midsection, there are membership functions of the extensions “
medium
” for hedges with widening effect.
Figure 5.8 Demonstration of the behavior of PbLD on the basis of linguistic description from Example 5.8.
Left:
Three considered input values;
Right:
The resulting fuzzy sets derived from the corresponding rules together with marked output values obtained using the defuzzification DEE.
Figure 5.9 Schematic depiction of perceptions.
Figure 5.10 (a) The membership function characterizing
small values, provided that they are related with big ones
on the basis of the rule “” and given the antecedent value . The marked output value is result of the defuzzification DEE. (c) The membership function of the extension of
big
with marked value . (b) The membership function characterizing
big values, provided that they are related with small ones
on the basis of the rule “” and given . The value is obtained using the DEE defuzzification. (d) The membership function of the extension of
small
with marked value .
Figure 5.11 Results of the PbLD method on the basis of the same linguistic description as was used in Figure 5.10, where the defuzzification MOM was used instead of the DEE.
Figure 5.12 Demonstration of the perception-based logical deduction with the ordinary DEE defuzzification. In the upper-left corner is the input value (in the given context ), in the middle right is the fired rule and the corresponding fuzzy set evaluating the possible output values, provided that the rule holds. The lower right corner contains course of all the output values (after DEE defuzzification) for all the input values in the given context.
Figure 5.13 Demonstration of the same application of PbLD as in Figure 5.12 with the smooth DEE defuzzification.
Figure 5.14 Behavior of the monotone linguistic description from Table 5.2. (a) Perception-based logical deduction and (b) DNF–COG (i.e., Mamdani–Assilian) approximation method.
Figure 5.15 The behavior of DNF–COG (Mamdani–Assilian) method for the monotone linguistic description based on uniform triangular fuzzy partition (cf. Figure 5.1) consisting of (a) 7 triangles and (b) 19 triangles.
Figure 5.16 The behavior of PbLD and DNF–COG fuzzy approximation for the monotone linguistic description when some rules were dropped. (a) PbLD, linguistic description from Table 5.2. (b) DNF–COG fuzzy approximation, 7 triangles. (c) DNF–COG fuzzy approximation, 19 triangles.
Figure 5.17 The behavior of PbLD with the smooth DEE defuzzification on the basis of linguistic description from Table 5.2. (a) Full description and (b) rules 3, 5, 11, 14, 19, and 20 were dropped.
Figure 5.18 Bypassing the obstacle: (a) PbLD with the ordinary DEE defuzzification; (b) PbLD with the smooth DEE defuzzification; (c) Mamdani–Assilian DNF–COG approximation method. On
x
-axis is the distance of the obstacle, on
y
-axis is the turn of the steering wheel (negative values denote the leftward turn, positive values denote the rightward turn.)
Figure 5.19 Bypassing the obstacle using PbLD based on reduced linguistic description with rule No. 1 omitted. (a) Ordinary defuzzification DEE and (b) smooth defuzzification DEE.
Figure 5.20 Bypassing the obstacle. (a) DNF–COG fuzzy approximation method if a fuzzy partition of a context using triangular membership functions (b) is used. (c) The result of this method based on the reduced linguistic description with Rule 1 being omitted.
Figure 5.21 The result of the PbLD of the simple linguistic description of a function based on (a) ordinary DEE defuzzification and (b) smooth DEE defuzzification.
Figure 5.22 The result of the DNF–COG fuzzy approximation on the basis of the linguistic description of a simple function (a) using evaluative expressions; (b) using triangular fuzzy sets depicted in Figure (c).
Chapter 6: Fuzzy Cluster Analysis
Figure 6.1 Data for fuzzy cluster analysis in Example 6.2.
Figure 6.2 Approximation of clusters (depicted by bold dots) obtained using fuzzy c-means algorithm by trapezoidal membership functions of fuzzy sets , and .
Figure 6.3 Fuzzy approximation using generated Takagi–Sugeno rules. The original data are indicated by diamonds and the results of approximation by squares.
Chapter 7: Fuzzy/Linguistic Control and Decision-Making
Figure 7.1 The basic scheme of closed-loop feedback control.
Figure 7.2 Scheme of a knowledge base consisting of one or more linguistic descriptions, each of which characterizes some kind of local knowledge of the control strategy and the use of specific variables. Some of the variables in different linguistic descriptions can be identical.
Figure 7.3 Scheme of the relational fuzzy action unit.
Figure 7.4 Scheme of the linguistic fuzzy action unit.
Figure 7.5 Scheme of a fuzzy controller that may consist of several fuzzy action units (fuzzy agents). These units may be of different kinds (relational or linguistic). Of course, units can be joined in various ways; their joining is determined by the structure of the knowledge base. Besides the value , the input may also contain some other kinds of variables (denoted by ).
Figure 7.6 Simulation of fuzzy control of the process using general fuzzy PI controller realized by the linguistic fuzzy action unit. The linguistic context is , , and .
Figure 7.12 Simulation of control of the process using general fuzzy PI controller realized by the linguistic fuzzy action unit. The linguistic context is , , and .
Figure 7.13 Simulation of fuzzy control of the inverted pendulum by fuzzy PD controller using the linguistic fuzzy action unit. The linguistic context is , , and . Figure (a) shows the control without changing the context and (b) the control with continuous change of context.
Figure 7.14 Simulation of fuzzy control of the inverted pendulum by fuzzy PD controller using the linguistic fuzzy action unit in which smooth DEE defuzzification has been used in the PbLD method. The jump in the middle is caused by the input disturbance (its magnitude was about 17% of the control action).
Figure 7.15 Simulation of fuzzy control of the process . The linguistic context was automatically learned to , , and . Figure (a) shows the control without changing the context and (b) the control with continuous change of the context. In the middle of both graphs, one can see the reaction of the controller to an input disturbance (its magnitude was 70% of the set-point).
Figure 7.16 Manual control of the process represented by the differential equation .
Figure 7.17 Simulation of fuzzy control of the process using the linguistic fuzzy action unit with a linguistic description learned on the basis of monitoring of the manual control (see Figure 7.16).
Figure 7.18 Example of linguistic control of one aluminum smelting furnace in a randomly chosen time slot.
Figure 7.20 Hierarchy of linguistic descriptions for choosing the best house.
Chapter 8: F-Transform in Image Processing
Figure 8.1 (a) The original reference image. (b) Its reconstruction after simple F-transform compression with PSNR = 32 dB. (c) Its reconstruction using the advanced F-transform image compression algorithm, PSNR = 37 dB.
Figure 8.2 Two reconstructions of the circle image after applying the advanced F-transform compression. (a) Without preservation of sharp edges, the compression ratio is 0.013 and PSNR is 27 dB. (b) With preservation of sharp edges, the compression ratio is 0.031 and PSNR cannot be measured.
Figure 8.3 (a,c) Original images. (b,d) Edges found using the F
1
-transform.
Figure 8.4 Four inputs for the image “Castle”. Each of them is blurred on some part.
Figure 8.5 Three fusing algorithms applied to the image “Castle”. (a) SA, (b) CA, (c) ESA.
Figure 8.6 Types of corruption. (a) Drawing, (b) noise, (c) text, and (d) scratch.
Figure 8.7 The image “Girl” (the author of this drawing is Renáta Doležalová). (a) The original. (b) Corrupted by text. (c), (d) Reconstructions using inverse F-transform computed on a sequence of triangular
h
-uniform fuzzy partitions with increasing
h
.
Figure 8.10 The image “House”. (a) Its corruption by text and (b) its reconstruction.
Chapter 9: Analysis and Forecasting of Time Series
Figure 9.1 Example of a time series.
Figure 9.2 Scheme of the structure of time series.
Figure 9.3 F-transform of the artificial time series , obtained by values of the four sine members and the noise (formula (9.17)). The dashed line depicts the original trend-cycle given by the data. After application of the F-transform to , we obtain approximation of the trend-cycle. This is depicted by the solid line. One can see that it is almost identical with the original .
Figure 9.4 Extraction of the (a) trend-cycle and (b) trend of a time series using the F-transform.
Figure 9.5 Scheme of the division of time series for analysis and forecast.
Figure 9.6 Scheme of the trend-cycle forecast. The predicted components in this Figure are and . The fuzzy partition is depicted upside-down below the
x
-axis. The dotted lines denote its prolongation to cover also the testing part.
Figure 9.7 Analysis and forecast of the time series containing monthly gasoline demand in Ontario in 1960–1975. (a) The whole time series. (b) Its detail over validation and testing sets. The lower dot-and-dash line is estimation of the trend-cycle using the F-transform, the upper one is its forecast. The real data are black, the forecast is gray.
Figure 9.8 Analysis and forecast of the monthly time series. (a) The whole time series and (b) its detail over validation and testing sets. The Figure contains clearly depicted trend-cycle estimated using F-transform. The real data are black, the forecast is gray.
Figure 9.9 Automatically generated linguistic evaluations of monthly trend of inflation measure over 10 years. Evaluation of trend in the marked areas is the following: Slot 1:
clear increase
, Slot 2:
fairly large decrease
, Slot 3:
stagnating
, Slot 4:
significant increase
and Slot 5:
huge decrease
.
Figure 9.10 Demonstration of the evaluation of trend of various parts of a complicated time series. Trend of the whole series is
stagnating
. Slot 1 (time 23-32):
clear decrease
, Slot 2 (time 70-127):
negligible decrease
, Slot 3 (time 92-115):
small increase
, Slot 4 (time 116-127):
fairly large decrease
.
Figure 9.11 Demonstration of dimensionality reduction of a time series using F-transform with . (a) is the original time series (consisting of 186 points). (b) is the reduced time series consisting only of the components .
Figure 9.12 Example of perceptionally important points found in a time series from Figure 9.11 using the outlined method. The points are marked by thick vertical lines. They clearly correspond to areas of more radical change.
List of Tables
Chapter 5: Fuzzy Natural Logic and Approximate Reasoning
Table 5.1 Possible experimentally estimated values of the parameters determining selected linguistic hedges
Table 5.2 Monotone linguistic description
Chapter 7: Fuzzy/Linguistic Control and Decision-Making
Table 7.1 The linguistic description learned on the basis of monitoring of manual control from Figure 7.16
Table 7.2 Input data of four selected houses (from 20 found on the Internet)
Chapter 8: F-Transform in Image Processing
Table 8.1 Basic characteristics of the three fusion algorithms applied to the image “Castle”. The resolution is 1120840
Table 8.2 RMSE values for three kinds of corruption
INSIGHT INTO FUZZY MODELING
Vilém Novák, Irina Perfilieva, and Antonín Dvořák
University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, the Czech Republic
Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
Names: Novák, Vilém, 1951- author. Perfilieva, Irina, 1953- author.
Dvořák, Antonín, 1970 - author.
Title: Insight into fuzzy modeling / Vilém Novák, Irina Perfilieva, and Antonín Dvořák, University of Ostrava, Czech Republic.
Description: Hoboken, New Jersey : John Wiley & Sons, Inc., 2016. | Includes bibliographical references and index.
Identifiers: LCCN 2015040313 (print) | LCCN 2015047241 (ebook) | ISBN 9781119193180 (hardback) | ISBN 9781119193197 (pdf) | ISBN 9781119193203 (epub)
Subjects: LCSH: Simulation methods. | Fuzzy mathematics. | Fuzzy systems–Mathematical models. | BISAC: TECHNOLOGY & ENGINEERING/Electronics / General.
Classification: LCC T57 .N68 2016 (print) | LCC T57 (ebook) | DDC 511.3/13–dc23
LC record available at http://lccn.loc.gov/2015040313
To our children David, Vitalik, Martin, Anna, and Jaroslav
Preface
Fuzzy modeling is a special branch of mathematical modeling that has two goals: (i) to construct models based on information that can be given not only in numbers but also, imprecisely, usually in a form of expressions of natural language; (ii) to construct models with less computational demands, which are more robust, that is, little sensitive to changes in the input data.
In comparison with classical models, the fuzzy ones are closer to human way of thinking. For example, when processing images, classical methods work with single pixels. People, however, do not see pixels but larger and usually imprecisely delineated parts of the image. This is the main reason why it is so difficult to develop methods that are as powerful as the human eye. It happens quite often that some objective measure says that a given image is good but the human eye sees it differently and says “no”.
The fuzzy modeling methods are developed with the idea to capture the way how people grasp and manipulate available information. Therefore, fuzzy models make it possible to solve highly nontrivial problems. On one side, these problems come from areas where mathematics has not yet (or very little) contributed, for example, psychology, geography, or geology. The fuzzy modeling methods, however, are successful also in solving classical problems, such as control, time series forecasting, image processing, or classical mathematical problems such as approximation of functions, solution of differential equations, or signal processing. In all cases, these methods manifest the above-mentioned properties — less computational demands and robustness.
Our book is specific from several points of view. First of all, the reader will find in it a consistent and well-established notation and terminology. Furthermore, we made maximum efforts to explain the basic ideas of the presented methods and to avoid misleading terminology occurring in some older books (e.g., we avoid erroneous terms such as “Mamdani implication”).
One of the main contributions of our book is that it contains original results obtained both in the theory as well as in the applications. These results cover especially newly developed theories and also original and, in our opinion, more precise explanation of older known results (this concerns especially relational interpretation of the widely used fuzzy IF-THEN rules).
A special focus is placed on two original theories: the fuzzy natural logic, which, besides others, includes mathematical model of special parts of linguistic semantics, and the theory of fuzzy transform. None of these theories has been yet published in the book form. Our aim is to demonstrate the power of both theories and their increasing potential in applications. We therefore split the book into two parts.
The first part contains extensive presentation of the theory of fuzzy modeling. Let us remark that the development of this theory is in large extent motivated by necessity to solve concrete problems.
The second part presents selected applications in three important areas: control and decision-making, image processing, and time series analysis and forecasting. It is interesting that the latter application combines both above-mentioned theories.
Recall that the book is focused on fuzzy modeling — the essential part of soft computing. Therefore, methods based on fuzzy set theory are preferred. We deliberately omitted other well-known and important theories that are usually mentioned in connection with soft computing, namely, the theory of artificial neural nets is missing because it is deeply elaborated in numerous specialized books and papers that we suggest to read. We think that peripheral presentation of this theory in one chapter of the book focused on fuzzy modeling would be more confusing than beneficial. For the same reasons, we omitted optimization techniques such as evolutionary algorithms or particle swarm optimization that can also be found in numerous specialized books and papers.
We want to thank our colleagues for their help when preparing this book: Martin Dyba, Michal Holčapek, Petr Hurtík, Viktor Pavliska, Marek Vajgl, Radek Valášek and Pavel Vlašánek. Last but not least, we also thank the workers of John Wiley & Sons publishing house.
Vilém Novák, Irina Perfilieva, and Antonín DvořákOstrava, Czech RepublicAugust 2015
Acknowledgments
We want to thank to the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070) and to the NPU II project LQ1602 “IT4Innovations excellence in science” provided by the MŠMT for their support.
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/novak/fuzzy/modeling
The website includes:
executable files with programs realizing methods described in the book
manuals to these programs
selected demonstration problems
updates
PART IFUNDAMENTALS OF FUZZY MODELING
This part consists of six chapters in which we first explain the role of indeterminacy in human life, which led to the development of special mathematical theories such as probability theory, fuzzy set theory, and fuzzy logic. In Chapter 2, we give an overview of the basic notions of the latter two theories. The main contribution of this part is contained in the three subsequent chapters in which we explain the theory of fuzzy IF-THEN rules, show that they have two possible interpretations enabling different kinds of applications, and explain the theory of fuzzy transform. A strong accent is put to the description of the model of semantics of a special class of linguistic expressions that are used on many places in this book. The theoretical part is finished by brief description of the main principles of fuzzy cluster analysis.
Chapter 1What is Fuzzy Modeling
1.1 INDETERMINACY IN HUMAN LIFE
Fuzzy modeling is a group of special mathematical methods that make it possible to include in the model imprecise or vaguely formulated expert information that is often characterized using natural language. The developed models (we call them fuzzy models) are very successful because they provide solution in situations when traditional mathematical models fail—either due to their non-adequacy, or due to their inability to utilize the full available information.
Note that the idea to include imprecise information in our models contradicts to what has always been required: as high precision as possible. There is, however, a good reason for doing it, namely, we face a discrepancy between relevance and precision. The so-called principle of incompatibility formulated by L. A. Zadeh in [149] says the following:
As a complexity of system increases, our ability to make absolute, precise, and significant statements about the system's behavior diminishes. At some moment, there will be trade-off between precision and relevance. Increase in precision can be gained only through decrease in relevance; increase in relevance can be gained only through the decrease in precision.
For example, from the description of an enterprise in several sentences, we may learn about its main activity, size, total number of its employees, its business successes, and problems. But we will know nothing about individual people, specific machines, and their parts. To describe everything in detail, we would need much more sentences, numbers, tables, and so on. But then the amount of information exponentially increases. We would thus learn more, but any detail would concern only a small part of the enterprise. The requirement to describe the whole enterprise in full detail would lead to a big pile of thick books that, however, nobody would be able to read. And if yes, to understand the content, he/she would need natural language, which means that he/she would have to return to imprecise characterization. Otherwise, he/she would be lost in the abundance of irrelevant details.
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
Lesen Sie weiter in der vollständigen Ausgabe!
