Timing of fuzzy membership functions from datajultika.oulu.fi/files/isbn9514264355.pdf · Frantti, Tapio, Timing of fuzzy membership functions from data Department of Process and

TIMING OF FUZZY MEMBERSHIP FUNCTIONS FROM DATA

TAPIOFRANTTI

Department of Process and EnvironmentalEngineering, University of Oulu

OULU 2001

TAPIO FRANTTI

TIMING OF FUZZY MEMBERSHIP FUNCTIONS FROM DATA

Academic Dissertation to be presented with the assent ofthe Faculty of Technology, University of Oulu, for publicdiscussion in Kuusamonsali (Auditorium YB 210),Linnanmaa, on July 27th, 2001, at 12 noon.

OULUN YLIOP ISTO, OULU 2001

Copyright © 2001University of Oulu, 2001

Manuscript received 20 June 2001Manuscript accepted 20 June 2001

Communicated byProfessor Jan JantzenProfessor Janusz Kacprzyk

ISBN 951-42-6435-5 (URL: http://herkules.oulu.fi/isbn9514264355/)

ALSO AVAILABLE IN PRINTED FORMATISBN 951-42-6434-7ISSN 0355-3213 (URL: http://herkules.oulu.fi/issn03553213/)

OULU UNIVERSITY PRESSOULU 2001

Frantti, Tapio, Timing of fuzzy membership functions from data Department of Process and Environmental Engineering, University of Oulu, P.O.Box 4300, FIN-90014 University of Oulu, Finland 2001Oulu, Finland(Manuscript received 20 June 2001)

Abstract

In this dissertation the generation and tuning of fuzzy membership function parameters areconsidered as a part of the fuzzy model development process. The automatic generation and tuningof fuzzy membership function parameters are needed for the fast adaptation and tuning of fuzzymodels of various nonlinear dynamical systems. The developed methods are especially useful inautomatic fuzzy membership function generation and tuning when dynamic of application area is fastenough to exclude manual tuning. The fuzzy model development process and development methods,modelling environment and nature of application area as well as algorithm development parametersare extensively discussed, because each of them sets their own restrictions on the design parts andparameters used in the modelling. The developed methods have been applied in different kinds ofapplications (in forecasting the demand of signal transmission products, power control and codetracking of cellular phone system, fuzzy reasoning in radio resource functions of cellular phonesystems), where other approaches are either very difficult or too time consuming to implement. Theprofessional areas of the thesis are fuzzy modelling and control in telecommunications.

Keywords: fuzzy control, adaptivity, tuning, WCDMA

Acknowledgements

The work reported in this thesis was performed between the years 1994 and 2001at the Control Engineering Laboratory, University of Oulu and at the TechnicalResearch Center of Finland.

I am indebted to Professor Kauko Leiviska and to Professor Petri Mahonen forguiding this thesis and creating a possibility to make this work.

I wish to express my thanks to Professor Jan Jantzen and Professor JanuszKacprzyk for reviewing the thesis. Professor Heikki Koivo has been appointed asthe opponent for the public defence of this dissertation. I wish to thank him inadvance for doing this invaluable work.

Esko Juuso is acknowledged for many useful discussions. I am also grateful for DrMatti Verkasalo and Raimo Kosonen for giving the possibility to start this researchand to Kari Rikkinen for giving the possibility to continue it in an new environment.

I want to thank Dr Johannes Frantti for useful discussions, technical assistanceand support during the years. Dr Pekka Koskela is also acknowledged for thetechnical assistance, support and useful discussions. I want to express special thanksto Professor Petri Mahonen for the possibility to finalize this thesis and includedarticles. Without his decisive help and support in the critical moment the finalizingof this thesis would not be possible. The staff of former Nokia Telecommunications,Fixed Access Systems Operations and the staff of the Technical Research Center ofFinland is also acknowledged for a creative working environment. Especially I amgrateful to Jyrki Huusko for important technical assistance with Linux and LATEX.

This work was financially supported by Nokia Telecommunications and the Tech-nical Research Center of Finland. The foundation of Tekniikan Edistamissaatio alsosupported this research. I am grateful to all of them. I also acknowledge myparents and Xu Wei for supporting this thesis.

Oulu, June 2001 Tapio Frantti

List of original papers

The thesis is based on the following seven papers:

I Frantti T and Juuso E K (1996) An Adaptive, Hierarchical Fuzzy Logic Ad-visory Tool (FLAT) for Anticipating the Demand of Transmission Products.Proc. 4th International Conference on Soft Computing, Iizuka ’96, Fukuoka,Japan, 1:410-413.

II Frantti T and Mahonen P (2001) Fuzzy Logic Based Forecasting Model. Jour-nal of Engineering Applications of Artificial Intelligence, Elsevier Science,14(2):189-201.

III Frantti T (1997) Fuzzy Power Control for Mobile Radio Systems. Proc. Eu-ropean Symposium on Applications of Intelligent Technologies, Aachen, Ger-many, 2:1284-1288.

IV Frantti T and Mahonen P (2001) Adaptive Fuzzy Power Control for WCDMAMobile Radio Systems. Journal of Control Engineering Practice, Elsevier Sci-ence, in press.

V Frantti T and Mahonen P (2001) Fading Rate Based Adaptive Fuzzy PowerControl for WCDMA Cellular Systems. Proc. SOCO/ISFI2001 Fourth Interna-tional ISCS Symposia on Soft Computing and Intelligent Systems for Industry,Paisley, Scotland.

VI Frantti T and Mahonen P (2000) Fuzzy Channel Synchronisation Search Proce-dure of Direct Sequence Spread Spectrum Cellular Phone Systems. Proc. 6thInternational Conference on Soft Computing, Iizuka ’2000, Fukuoka, Japan,1:949-956.

VII Frantti T and Mahonen P (2001) Fuzzy Reasoning in WCDMA Radio ResourceFunctions, In: Zimmermann H J and Tselentis G (ed) Advances in Computa-tional Intelligence and Learning Methods and Applications. Kluver AcademicPress, Germany, in press.

These papers are referred to in the text by their Roman numerals.

Papers I and II report generated methods and results obtained during thedevelopment of a forecasting model for transmission products demand. Paper IIIreports methods and obtained results of the developed fuzzy power control model.Papers IV and V enlarge the developed methods and results of fuzzy power control.Paper VI reports developed methods and results obtained during the developmentof a fuzzy synchronisation search procedure. Paper VII presents applications offuzzy reasoning in WCDMA radio resource functions. In all papers the authorconstructed all models used in the research, developed the methods and verifiedthem via testing. Manuscripts were written together with the coauthors.

Symbols and abbreviations

General Symbols

A Minimum value of sample range; angle point of fuzzy distributionB Maximum value of sample range; angle point of fuzzy distributiond(·, ·) Euclidean distancee Error of received signal powere Crisp input valueE Extraction of extreme valuesF Fuzzy setfc Carrier frequencyi, j, k IndicesK AmplificationL Lowpass filtering; Lattice -fuzzy setLn Lukasiewicz Ln logicmax(x1, ..., xn) Maximum of x1, ..., xn

min(x1, ..., xn) Minimum of x1, ..., xn

n Integer numberN Length of data setNVI Normalised crisp input variable in the normalised input domainp, q Regions of sample rangeP(L) Set of all fuzzy subsets of Lpk

r Received powerpk

t Transmitted powerpsp

r Set point value of power levelRm Mamdani’s relationR Set of all real numbersr, s Parameters, (r, s) ∈ Rn

ra, rb Reference values A and B in a fuzzy distributionR(s) Input in the s-planeTp Sampling periodSF Scaling factoru Member of set

U Universe of discourseUcomb, Uindiv Compositional and individual based inferencev Velocity of terminalVI Input for scalingW Width of distributionX, X State variablesy(t), Y(s) Responses in the t-plane and s-planeY Universal setz Central point of clusterZ UpsamplingZ Set of all integer numbers

Special Symbols

α Minimum value of normalised domainAα α -cut of fuzzy set Aβ Maximum value of normalised domain∆e Change of error on received signal power∆p Power stepε Normalized domainλ Parameter, λ ∈ [0, 1]τ1, τ2 time constantsµF Membership grade function of fuzzy set F

Abbreviations

BER Bit error rateDS Direct SequenceFAM Fuzzy Associative MemoryFCM Fuzzy C -meansFEC Forward error correctionFLAT Fuzzy Logic Advisory ToolFMLE Fuzzy Maximum Likelihood EstimationFMPC Fuzzy Mobile Power ControlFPI Fuzzy proportional integral controllerFSSM Fuzzy Synchronisation Search Modelhgt Height of fuzzy setKT Knowledgetron AlgorithmPI Proportional integral controllerSIR Signal to interference ratioSNR Signal to noise ratioSOM Self organizing mapUMTS Universal Mobile Telecommunication SystemUTRA UMTS Terrestrial Radio AccessWCDMA Wideband Code Division Multiple Access

Contents

AbstractAcknowledgementsList of original papersSymbols and abbreviationsContents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1 Research problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2 Research assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Scope of applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Fuzzy model development . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Design parts and parameters of the fuzzy model . . . . . . . . . . . . 202.3 Testing and tuning of parameters . . . . . . . . . . . . . . . . . . . . 222.4 Generation of membership functions . . . . . . . . . . . . . . . . . . 23

2.4.1 On-line and off-line approaches . . . . . . . . . . . . . . . . . 232.4.2 Preprocessing of source data . . . . . . . . . . . . . . . . . . 242.4.3 Automatic generation of fuzzy membership functions . . . . . 24

3 Design parts of the fuzzy model . . . . . . . . . . . . . . . . . . . . . . . . 293.1 Design limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Normalisation of input domain . . . . . . . . . . . . . . . . . . . . . 303.3 Fuzzification of input variables . . . . . . . . . . . . . . . . . . . . . 313.4 Fuzzy inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.1 Main parameters . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.2 Data base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4.3 Rule base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Defuzzification part . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Testing and tuning of parameters . . . . . . . . . . . . . . . . . . . . 38

4 Generation of membership functions . . . . . . . . . . . . . . . . . . . . . 40

4.1 Preprocessing and analysis of data . . . . . . . . . . . . . . . . . . . 404.2 Fuzzy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Average based division of a fuzzy distribution . . . . . . . . . 434.2.2 Cumulative distribution based division of a fuzzy distribution 44

4.3 Curve fitting method . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 Design parameters in the automatic generation of fuzzy membership

functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4.1 Period length of source data . . . . . . . . . . . . . . . . . . . 504.4.2 Feature extraction . . . . . . . . . . . . . . . . . . . . . . . . 604.4.3 Division of the physical domain and overlapping of labels . . 61

4.5 Illustrative example of the modelling problem - power control of mo-bile terminal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6 Evaluation of the developed methods . . . . . . . . . . . . . . . . . . 664.7 Adaptation of the membership functions . . . . . . . . . . . . . . . . 73

5 Compact rule base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 Introduction to the papers . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.1 Paper I, An adaptive, hierarchical fuzzy logic advisory tool (FLAT)for anticipating the demand of transmission products . . . . . . . . . 80

6.2 Paper II, Fuzzy logic based forecasting model . . . . . . . . . . . . . 816.3 Paper III, Fuzzy power control for mobile radio systems . . . . . . . 816.4 Paper IV, Adaptive fuzzy power control for WCDMA mobile radio

systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.5 Paper V, Fading rate based adaptive fuzzy power control for

WCDMA cellular systems . . . . . . . . . . . . . . . . . . . . . . . . 826.6 Paper VI, Fuzzy decision making in synchronisation procedure of

DS/WCDMA based cellular mobile phone radio system . . . . . . . 826.7 Paper VII, Fuzzy reasoning in WCDMA radio resource functions . . 83

7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.1 Why the generation of fuzzy membership functions? . . . . . . . . . 857.2 Effects on the application model . . . . . . . . . . . . . . . . . . . . 867.3 Results achieved in real-world applications . . . . . . . . . . . . . . . 87

8 Directions for further research . . . . . . . . . . . . . . . . . . . . . . . . . 919 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Appendix 1

1 Introduction

Fuzzy set theory was first presented by Lotfi Zadeh in his seminal paper ”FuzzySets” in Information and Control 1965 Zadeh (1965). Fuzzy logic was developedlater from fuzzy set theory to reason with uncertain and vague information and torepresent knowledge in an operationally powerful form. The first fuzzy laboratoryapplication (fuzzy control of steam engine) was created in the mid of 1970’s Mam-dani (1975) and the first industrial application at the beginning of 1980’s (fuzzycontrol of cement kiln) ∅stergaard & Holmblad (1982).

Since then the number of fuzzy applications have grown rapidly, especially inJapan, where ”fuzzy” is very popular word. This, on the other hand, has led to anunnecessary profileration of fuzzy applications. There are many applications wherethere is no need to apply fuzzy logic, at least from the methodological point of view.

The main reason to develop fuzzy logic from fuzzy set theory was to form aconceptual framework for linguistically represented knowledge. As an example,consider a fuzzy set F, which is the subset of the universe of discourse U. Thetransition between the membership and non-membership of F is continuous (ordiscrete approximation of it) and if F is the set of small mobile phone models -some of the phones are definitely small, whereas some of them definitely are not.Some models are small in some degree and at the same time they possibly belongto some other fuzzy set in another degree. The interval [0,1] offers a convenientnormalised representation for degrees of memberships.

Sets of the conventional set theory are called crisp sets in order to distinguishthem from fuzzy sets. The characteristic function of a crisp set, µU (u), assigns adiscrete value (in binary case usually either 0 or 1) to each element in the (universe ofdiscourse, U ), i.e., it discriminates members and non-members of the crisp set underconsideration (then for each element u of U, either u ∈ U or u /∈ U). In order toenlarge two-valued logic to include a third discrete truth value, e.g, indeterminablewe have three-valued logic. In the same way for each positive integer number n ≥ 2,the n-valued logic can be defined, e.g, Lukasiewicz Ln logic (Giles 1977, di Nola1999). Continuing in an analog way an infinite-valued logic is defined. Its truthvalues are all real values between the unit interval [0,1] if the membership functionvalues are normalised between the zero and one (standard Lukasiewicz logic L1)(µU (u) ∈ R if the real number field is used). On the other hand, we might also have

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µU (u) ∈ Z, if the discrete integer number field is used instead of the real numberfield.

Multi-valued logic forms the kernel of fuzzy logic, i.e., fuzzy logic is actually anextension of multi-valued logic. It offers a base for approximate reasoning with un-certain and vague propositions. Fuzzy logic allows the use of fuzzy predicates (whatkind of phenomenon, e.g., ’small’ mobile), fuzzy quantifiers (quantifying presuffix,e.g., ’all’ mobiles), fuzzy truth values (in what degree something is truth, e.g., ’quite’small mobile) and other kinds of fuzzy modifiers (e.g., ’very’ small mobile).

The characteristic function can be generalised in fuzzy set theory so that thevalues assigned to the elements of the universal set fall within a prespecified range,indicating the membership grade of these elements of the set in question. In fuzzyset theory it is not necessary that either u ∈ F or u /∈ F (F is a fuzzy set). This kindof generalisation has been represented in many theories Rescher (1969), but theyhave not the same kind of intuitive nature as fuzzy set theory has. The generalisedfunction used is called amembership function and the set defined with the aid of it isa fuzzy set, respectively. The mathematical representation of the fuzzy membershipfunctions, fuzzy sets and standard operations of the fuzzy set theory are presentedin the Appendix A.

1.1 Research problem

Since the pioneering applications of fuzzy set theory and fuzzy logic in the middle of1970’s the industrial as well as research institutes’ interest has grown rapidly, espe-cially in recent years Driankov et al. (1994). This has lead to an increasing numberof industrial and theoretical applications, but also into easy and user friendly en-vironments for fuzzy model development. Even if these development environmentsare aimed for easy and quick model development most of them are not capableof the automatic generation of fuzzy membership functions. Instead they offer amanual way to create or estimate membership functions, which practically excludesautomatic adaptive tuning via membership functions for applications with very fastoscillations.

In this thesis the research problem is formulated as follows:

How to develop, as a part of fuzzy modelling process, fast and appropriate automaticgeneration methods of fuzzy membership functions so that automatic tuning of fuzzymodel via membership functions in highly dynamic nonlinear systems or systemswith high number of inputs or high dimensionality of input data is effective andcorrect? The fuzzy modelling methods and design of them are out of the scope ofthe thesis.

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1.2 Research assumptions

From the systems engineering and developers point of view we argue that the infor-mation extracted from data has a very important effect on fuzzy modelling. Espe-cially it holds in an automatic generation of fuzzy membership functions as a partof fuzzy model development process and tuning via membership functions in orderto quickly update fuzzy models with changing circumstances without overhead ona rule base.

For example, fuzzy power control of a mobile transmitter in a cellular networkneeds adaptive estimation and tuning of membership functions in order to avoid arule base with overhead. The behavior of the signal envelope is ’self-similar’ (theshape of the signal envelope remains the same, but the scale changes depending onthe transmission power level). Therefore, the increase of variables’ resolution, withthe number of labels instead of membership functions adaptation, increases the sizeof the rule base. Moreover, because the characteristic time-scale for adaptation isin the order of milliseconds, manual tuning is also excluded.

From this point of view we can summarize our research assumptions as follows:

Fast adaptation and tuning of fuzzy models in an environment with fast dynamicchanges or a large number of different physical domains with continuous changes ora high dimensionality of input data needs a specific automatic generation algorithmfor fuzzy membership functions as a part of fuzzy model development.

1.3 Hypothesis

We consider that the generation of fuzzy membership functions is a part of thefuzzy model development process. Especially the automatic generation of fuzzymembership functions is needed for the fast adaptation and tuning of fuzzy models.Therefore, the development process, methods, environment and application areaof the fuzzy model as well as algorithm development related matters have to beunderstood and taken into consideration in the fast adaptation and tuning process.Each of them sets their own restrictions to the design part and parameters usedin the modelling. From that point of view, the research problem should be con-sidered from many aspects and the research hypothesis can be formulated as follows:

In order to model nonlinear systems in the above mentioned application areas,automatic generation and tuning of fuzzy membership functions is needed for thefuzzy models fast adaptation into new circumstances dynamically.

Fuzzy model development process and development methods can be specified toinclude restrictions set by design parts and design parameters of the model. Themodelling environment and nature of application area include restrictions originatedfrom the development tools and natural environment. Algorithm development pa-rameters have their own restrictions, which should be identified and selected to

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support the automatic procedures.This hypothesis is supported by showing several applications where other ap-

proaches are either very difficult or too time consuming to implement.

1.4 Scope of applicability

In this dissertation we concentrate on the generation and tuning of fuzzy member-ship functions as a part of fuzzy model design. We use high level programminglanguages such as C or C++ and IDL and Matlab for model implementation. Thedeveloped methods and algorithms are applicable in embedded systems and com-puter controlled environments or processes. Especially, they are applicable andpractical for time critical applications where online adaptation is needed, such asterminal power control in cellular mobile phone systems (Papers III-V), synchro-nisation search of cellular phone systems (Paper VI) and radio resource functionsparameters estimation of cellular phone systems (Paper VII). Another successfulapplication area seems to be decision support systems with a large number of inputvariables (Papers I and II) and different multidimensional classification problemsMahonen & Frantti (2000).

We considered here only fuzzy systems combined with conventional numericalmethods. This does not mean that developed algorithms are not suitable, for in-stance, in neuro-fuzzy applications or other kind of hybrid methods with fuzzy logic.The advantage of our methods compared to the conventional modelling and tuningmethods is that the model adapts itself into new circumstances dynamically, whichmakes it possible to model nonlinear dynamic systems.

1.5 Results

In this work a general development model Isomursu (1995) was enlarged to includethe preprocessing and analysis of data fo generation of membership functions andgeneration and tuning of membership function for adaptation of the fuzzy model intochanging circumstances phases as their own separate phases. Therefore we considerhere that the fuzzy model development process includes the following phases:

I Preprocessing and Analysis of Data

II Normalization Phase

III Membership Function Generation Phase

IV Fuzzyfication Phase

V Reasoning Phase

VI Defuzzification Phase

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VII Denormalization Phase

VIII Representation of Results Phase

We considered that appropriate division of fuzzy modelling clarifies the design pro-cesses of fuzzy membership function generation and tuning. Three kinds of auto-matic generation methods for fuzzy membership functions were developed:

1. average based division of fuzzy distribution

2. cumulative distribution based division of fuzzy distribution and

3. curve fitting method.

The developed methods generate membership functions from online time series data.They were found to be useful especially in time critical applications, decision supportsystems with a large number of inputs and multidimensional classification problems.

1.6 Outline of the dissertation

In the first chapter, the research problem was stated including some backgroundintroduction. In chapter two, the state of the art is reviewed from the automaticfuzzy membership function generation and fuzzy model tuning point of view. Inchapter three, design parts of the fuzzy model development process are discussed. Inchapter four, on-line estimation and tuning methods for membership functions foran environment with fast dynamic changes or a large number of different physicaldomains with continuous changes are presented and considered as a model designpart. Chapter five describes the concept of compact rule base. In chapter six, anintroduction to the papers is presented. In chapter seven discussion on the researchresults and developed methods is given followed by direction of future researchin chapter eight. Conclusions on the results and developed methods are given inchapter nine.

2 Related work

In many applications the membership functions and the set of fuzzy rules are basedon an operator’s and/or control engineer’s knowledge or heuristics. However, thereare many circumstances, where the fuzzy membership functions are difficult orimpossible to generate and/or update in this way. This turns up, for example, insituations where fast adaptation of a model according to system dynamics is needed.

Many kinds of procedures for the generation of membership functions from datacan be found from the literature, (Kim et al. 1996, Chen et al. 1995, Kim & Russell1993, Iokibe 1994). However, most of them are not suitable for the automaticgeneration of membership functions from on-line data nor are they suitable forfuzzy model on-line adaptation via the tuning of membership functions. Mostlythe procedures are used to find out features from data in order to make manualmembership function generation easier. Therefore they were not applicable for thecases considered here (Papers I-VII).

2.1 Fuzzy model development

Next we explore the general procedure of the fuzzy logic model. We refine it byincreasing two design parts in order to take into consideration the requirements ofautomatic generation and tuning of membership functions from process data. Laterwe consider the preprocessing of data and generation and tuning of membershipfunctions to be well-defined design parts of fuzzy modelling.

A well-known method for dealing with large and complex problems is to de-compose the problem into smaller and simpler parts. This can be applied to fuzzymodelling, too. A brief introduction to the development tasks of a fuzzy logic modelis given below (a more thorough discussion of fuzzy model development is found,for example, from Isomursu (1995)):

- Knowledge acquisition. A variety of methods can be used for knowledge acqui-sition beginning from simple interviews to the complex data mining of processdata.

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- Tuning. The knowledge base as well as the data base needs to be adjusted toreach the desired performance level. This procedure is usually quite compli-cated and time consuming.

- Proving optimality and stability. Despite the research done in the area (Tong1976, Tong 1978, Braae 1979a, Braae 1979b, de Glas 1984) this is still a prob-lem, especially in critical applications.

- Testing. The specific nature of fuzzy logic makes the testing of the fuzzy modelmore difficult than a conventional model. This is because several rules mayfire at the same time with different grades of membership. They might evencontradict if the rule base is not well-enough verified. Moreover the controlarea itself might be vague in nature and cause testing problems.

As mentioned in Isomursu (1995), the tuning of the knowledge base (rules) anddata base (membership functions) is often a non-trivial and time consuming part offuzzy modelling. In this dissertation we focus on the estimation and tuning of thedata base, i.e., the generation and tuning of membership functions. The proposedmethodology is closely connected with other modelling phases, especially knowledgeacquisition and testing. Optimality and stability are not proven, because the aim ofthe developed model has been in all cases to show that the systems works better thancompeting design alternatives. This kind of view is supported also in (Mamdani1993, Isomursu 1995). .

Next we proceed to the design parts of the fuzzy model by decomposing it intoappropriate modules with their own characteristic parameters. We have added twoextra modules compared to more traditional fuzzy model division by including datapreprocessing and membership function generation phases.

2.2 Design parts and parameters of the fuzzy model

The principal fuzzy model design parts are grouped in Driankov et al. (1994) asfollows:

- Normalization. The input values of the physical domain are normalised to someappropriate interval.

- Fuzzification. Representation of input values in the fuzzy domain.

- Reasoning. Interpretation of the output from fired rules.

- Defuzzification. Crisp representation of the fuzzy output.

- Denormalisation. Normalised output value is scaled back to the physical do-main.

- Representation. Representation of the model result(s).

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We claim that the information extracted from data has an important effect on afuzzy model design in the above mentioned applications areas. Furthermore we statethat the exploitation of expert knowledge and system engineering knowhow in thetraditional way is either very difficult or impossible on an ad hoc basis. Thereforewe need a new kind of approach to the fuzzy model design when fast adaptationand tuning of the fuzzy model is needed. We consider also that specific automaticalgorithms are then necessary. In our approach we include the preprocessing andanalysis of data and generation of membership functions phases as own parts of thedesign process (Fig. 2.1).

Figure 2.1. Design parts of fuzzy model.

Different design parameters include the chosen parameters used in the various designparts. Isomursu 1995 approaches the grouping of design parameters by the followingclassification Lee (1990):

- Fuzzyfication. Fuzzification strategy and the interpretation of the fuzzifier.

- Data Base. Normalisation, fuzzy partition of the input and output spaces,choice of type and completeness of fuzzy membership functions.

- Rule Base. Choice of input and output variables, source and derivation of fuzzyrules, type of rules, continuity, consistency, interactivity and completeness ofrules.

- Decision Making Logic. Definition of fuzzy relations, interpretation of the lin-guistic terms, definition of a compositional operator and inference mechanism.

- Defuzzification. Defuzzification strategy and the interpretation of the defuzzi-fier.

A review done in this area can also be found in Driankov et al. (1994). In ourapproach to fuzzy model design, we include parameters of an adaptive fuzzy par-tition of the input space in the data base group. These parameters are due to thegeneration and tuning of fuzzy membership functions. Isomursu 1995 also presentsthe following factors to affect the selection of design parameters:

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- intelligibility

- used development methodology

- computational efficiency

- properties of the target environment

- personal preferences of the designer.

We will discuss design parts and parameters in more detail in chapter 3.

2.3 Testing and tuning of parameters

Testing of parameters is extremely important with respect to the correctness, sta-bility and performance analysis of a fuzzy model. Certain specific internal featuresof fuzzy models cause difficulties to test runs. In contrast to conventional expertsystems, a number of rules in fuzzy systems can operate at the same time withdifferent grades of membership. Furthermore the problems to which fuzzy logic isapplied are often themselves more vague and uncertain in nature.

Tuning, the modification of one or more of a design parameters Isomursu (1995),is usually started by modifying membership functions. The second phase is to editthe rule base, number of adjectives, connectives or rule weights or to introduce newvariables. The defuzzification method may also be changed or modified Raju et al.(1991). Pedrycz 1993 also lists scaling factors of the input and output variables asmodifiable parameters.

Tuning methods can be divided into on-line and off-line methods, depending onwhether the design parameters are tuned while the model is running or afterwards.The on-line tuning mechanism often consists of two modules Isomursu (1995):

- performance evaluator, and

- adaptation algorithm.

The performance evaluator detects changes in identification measures of a processmodel or in a specific process variable and informs the adaptation algorithm tochange the design parameters accordingly. The adaptation algorithm can be basedon a wide variety of techniques. For example Halme and Handroos (1994) presentedan automatic tuning method based on a genetic algorithm. Nomura et al. (1991)presented learning of fuzzy rules based on the gradient descent method. Usuallyonly one or a few of design parameters are tuned because of the unpredictable sumeffect of several parameters changes Isomursu (1995).

In off-line tuning mathematical methods are used to match the model’s resultsto a given set of monitoring data as closely as possible. These methods include atleast Isomursu (1995):

- variations of least-square methods

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- gradient descent algorithms

- fuzzy clustering and

- neural net approaches.

In order to increase the performance of a fuzzy model, the tuning algorithm requirescareful analysis of parameters. In tuning of fuzzy membership functions the shape,number and width of membership functions, the length of data series for member-ship function generation and tuning period (time interval between the membershipfunctions tuning) are considered here as tuning parameters.

2.4 Generation of membership functions

2.4.1 On-line and off-line approaches

Now we briefly discuss the generation of membership functions from data. Weconsider it as one of the design parts of fuzzy model development. In chapters 4and 7 we return to the subject.

Membership function generation occurs either during the operation of a model(on-line) or before operation (off-line). It is usually performed off-line; on-linegeneration is not included in traditional fuzzy models. In off-line generation, resultsusually depend on the analysis of data, tuning of a model and skills of an operatorand system engineer.

However, there are many circumstances where the generation of fuzzy member-ship functions is difficult or impossible with the traditional way where a model-basedapproach may be more appropriate. Systems with very fast dynamics, systems withlarge number of state variables and multidimensional classification problems seem tobe appropriate application areas for the automatic generation of fuzzy membershipfunctions for a continuous adaptation.

We consider on-line generation as a separate design part, where different kindsof methods can be applied to data in order to generate membership functions.Human side expertise (operator knowledge) is taken into consideration during themodelling phase. The on-line generation and tuning of membership functions is themain subject of this dissertation and therefore we discuss it throughout the thesis.

We group the design parameters of fuzzy membership functions as follows:

- fuzzy partitioning strategy of input and output space

- choice of the type of fuzzy membership functions

- source data related parameters, e.g., period length of the source data used inthe generation of membership functions.

Fuzzy partitioning strategy and choice of fuzzy membership functions are discussedin more detail in chapter 5. Data related parameters are shortly introduced in thenext section and in more detail in chapters 3 and 4.

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2.4.2 Preprocessing of source data

Preprocessing of source data is performed to generate a suitable data distributionfor the automatic generation and tuning of membership functions. Preprocessingmethods depend on the source data at hand. Typical preprocessing methods areupsampling, interpolation, various kind of filtering and downsampling of sourcedata.

The upsampling of zeros (fill gaps in time series), for example, is performed inorder to achieve correct time dependency of data values if, e.g. zero values, aredownsampled (filtered off) from a time series in the data restoring phase. Forexample in the model presented in Papers I and II the time series consist of thediscrete data values. The zero values are filtered off, i.e. downsampled, in orderto define realistic magnitude distribution. Later on the zero values are set back totime series, i.e. upsampled. Interpolation of a time series by, e.g., spline interpola-tion is applied to ’generate’ data between sample points (Paper II). Extreme valuefiltering (removing the largest and smallest values) might be used to avoid too widea distribution of grade of membership (Paper II). Lowpass filtering can be used tocalculate the moving average of sequential values in order to remove noise (PapersI-VII).

2.4.3 Automatic generation of fuzzy membership functions

The representation theorem (Kosko 1992) presents that any nonlinear function canbe represented by fuzzy sets and fuzzy rules. However, it does not tell how manyfuzzy sets and rules are needed to achieve the defined accuracy. Therefore we haveto mathematically and linguistically describe systems, analyse data at hand, andtune models. For that reason we think that the automatic generation and tuningof membership functions is key to fuzzy model development and tuning in order tofind an accurate and practical modelling approach.In clustering methods data samples are partitioned into groups or classes and the

classes’ central points are defined (Yoshinari et al. 1993, Babuska & Verbruggen1994, Zhao et al. 1994, Kaymak & Babuska 1995). The location of classes (clus-ters) central points can be defined by using the c -means clustering algorithm (moredetails about the c -means can be found from (Kosanovic 1994, Bezdek et al. 1984,Zimmermann 1992, Mahonen & Frantti 2000). When a reasonable approximation ofcentral points has been done, membership functions are estimated using an appro-priate distance metric. For example Kosanovic (1994), the existence of two clustershas been presumed and positioned at ”reasonable” position with the samples. Thegrade of memberships µi are defined by calculating a similarity relationship basedon an Euclidean distance d(·, ·) between samples xi and the cluster’s central pointzi. Then

µi =1

1 + d2(xi, zi). (2.1)

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The values are then normalised in order to make the sum of membership valuesequal to one, i.e.,

∑

i

µi(xi) = 1. (2.2)

Difficulties arise in determining the proper number of clusters and how to includethe dynamic properties of a system into fuzzy models (cluster validity problem). Ithas been largely studied by a number of researchers.

Babuska et al. (1995) have been used the fuzzy c-means (FCM) algorithm todivide data into a fuzzy partitioning matrix. Elements of the matrix are degrees offuzzy membership of data points in the fuzzy clusters. Fuzzy membership functionsare obtained from the clusters by projection either onto the axes of the featurespace or onto axes spanned by eigenvectors Kosanovic et al. (1995). There arevarious techniques to improve this, for example the prototypes can be defined aslinear subspaces (lines, planes) instead of points. Furthermore, the distance measurecan be defined in various ways to quantify the distance of a point from the linearsubspace Bezdek (1981).

In Kosanovic et al. (1995) the fuzzy c-means and fuzzy maximum likelihoodestimation (FMLE) algorithms are used in the estimation of fuzzy sets for signalanalysis. The FCM algorithm tries to place cluster center points within the featurespace where dispersion of the samples is the smallest Kosanovic et al. (1995). TheFMLE algorithm, on the other hand, is based on maximum likelihood estimationof the parameters. Each cluster is assumed to have a different geometry that ismeasured by an exponential distance measure. The exponential distance measureis used to calculate the a posteriori probability. Using the a posteriori probabilityinstead of µik (grade of membership of the kth feature vector i the ith cluster) in theFCM algorithm Zimmerman (1992), the fuzzy modification of the FMLE can beobtained. Results achieved by comparing FCM and FMLE algorithms during theestimation of fuzzy sets derived from discrete-time quasi-stationary Gaussian signalsare presented in Kosanovic et al. (1995). Interested reader finds more informationabout the clustering methods, for example from (Bezdek 1981, Gath & Geva 1989,Rivera et al. 1990, Roubens 1987, Xie & Beni 1991).

Kim et al. (1996) have presented neural network model for part of speech tag-ging problem where network connections are used as membership functions. Themembership functions are updated according to the changes of the neural net-work connections. Therefore, the membership functions are not presented explicitly.Chwan-Hwa (1995) used a neural fuzzy model to classify waveforms with differentmagnitudes, frequencies, noises and positions of spikes and chops in a cycle of asine wave. The system learns membership functions from a training set composedof waveforms with prescribed features. The membership functions are updated ei-ther expanding or reducing the width of it by increasing or decreasing a learningcoefficient, respectively.

Koene (1998) has studied knowledge extraction from a trained neural networkin terms of rules using the Knowledgetron Algorithm (KT) by Fu (1994). Theintention is to provide insight into the extraction of uncertainty already containedin the trained network. In other words, the uncertainty originated with data is

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hidden in the network weight patterns and activations instead of considering it asan extra variable with input data. Therefore, the rules extracted from the neuralnetwork are attached a rule weight interpreted as membership function values. Thesignificant rule selection per neuron for an input pattern is possible by selecting therule with the highest degree of membership value.

Isik and Farrukh (1993) have presented a self-organizing neural network controllerthat automatically obtains membership functions and fuzzy rules from input andoutput data of a system Isik & Farrukh (1993). The membership functions are,however, embedded within the self-organizing map (SOM). Therefore they are notvery easily usable.

Khan (1993) presented a method, which uses neuro-fuzzy technology to selectfuzzy rules and membership functions from training data. The method forms fuzzyassociative memory (FAM) from training data. This is, however, aimed at smallapplications. This kind of approach has also been shown to have stability androbustness problems Khan (1993).

Chen et al. (1995) have presented an algorithm, in which complicated input-output relationships are decomposed into the accumulation of simpler input-outputrelationships. The membership functions and rules are generated by a three layerfeedforward neural network. The first layer performs fuzzification, the second layerperforms rules generation for the single input multi-output case and the third layerperforms rules. Fuzzification values are simply the first layer’s weight coefficients.The rules in the second layer are the sum of the second layer’s weight coefficientsand the fuzzy rules of the whole system are aggregated together by summing them.

In Kim & Russell (1993) automatic generation of fuzzy membership functionsand fuzzy rules are considered using inductive reasoning for the partitioning prob-lem. Partitioning is performed by the entropy minimisation principle in order toclassify samples. The main idea in the entropy minimisation analysis is the deter-mination of the quantity of information in a given data set der Lubbe & Hoeve(1997). In the membership functions generation, a threshold line is drawn betweentwo classes of sample data. A threshold value for samples in the range [A,B] is de-fined by writing an entropy equation for regions p = [A,x] and q = [x,B]. By movinga value x between A and B the entropy is calculated for each value of x. The valueof x that gives the minimum entropy is the optimal threshold value. The left sideof the sample space is the negative side and the right side is the positive side. Thesecondary threshold values are obtained from both side areas resulting in threefuzzy sets. Each threshold value separates two classes in each area, therefore, e.g.,7 fuzzy sets are achieved with three partitions. Inductive learning based methodis also presented in Ross (1995) for the extraction of fuzzy rules and membershipfunctions from data without prior knowledge.

In Nishimori et al. (1994) the tuning of fuzzy membership functions using alearning procedure (steepest gradient method) is proposed in the driving controlsystem of a model car, where three kinds of kinetic equations of the car are used.The membership functions in the driving control system have a restriction that theadded degree of membership of any variable has to be one. The steepest descentmethod is carried out for a tuning of membership functions.

In Chwan-Hwa & Chihwen (1995) a knowledge acquisition algorithm is proposed

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to automatically generate fuzzy rules and fuzzy membership functions for fuzzyassociative memory. It is done by the following steps:

1. Input feature vectors are selected from the given input data.

2. Fuzzy rule base format is defined and a pair of input-output data is checked tosee if the previously generated rules can classify the pair correctly. If not then anew rule is generated by using this pair and membership functions are tuned usingstatistical training law to obtain a correct output.

Moreover, orthogonal least squares Wang (1994) and evidential reasoningKosanovic et al. (1995) based methods exist for the extraction of fuzzy rules andmembership functions from data without prior knowledge. In Iokibe (1994) theautomatic fuzzy membership function generation method based on the performancefunction has also been mentioned.

The methods reported in the literature are not appropriate for time critical ap-plication areas considered here. Moreover, we are also focused on to keep rule basestable and small with continuous update of the membership functions and thereforenew methods were developed for the time critical applications and applications withthe large number of input variables. Existing methods, for example fuzzy c-means,were used on other application areas like in Mahonen & Frantti (2000). In Mahonen& Frantti (2000) 14 dimensional data vectors were classified into 2 categories withthe fuzzy c-means based classification algorithm. The automatic method of clas-sification via grade of memberships was used in order to avoid laborious manualdefinition of grade of memberships and to demonstrate that quite a reliable auto-matic classification is possible with a simple fuzzy method. The automatic tuningof grade of membership will be very essential if the length of the data set increases,for example, a couple of decades larger than the example data set used here.

3 Design parts of the fuzzy model

Our primary target in modelling dynamic systems with fast changes, a large numberof inputs, or a high dimensional input/output space is to find out on-line member-ship function estimation and tuning method. The method has to be complete,robust and simple enough to model nonlinear behavior of the system under consid-eration, naturally together with other design parts of the fuzzy model, especiallywith the rule base. We consider the design process of the fuzzy model as a wholeand tailor automatic generation of membership function method based on this pointof view. Therefore in this chapter the design parts of the fuzzy model are discussedwith details representing each part as a section. However, the membership functiongeneration part as well as data analysis and preprocessing phases are postponed tothe next chapter, because of their central role in this dissertation. Effects of thegeneration of membership functions on application models are discussed separatelyin a later chapter.

3.1 Design limitations

The application areas (dynamic systems with fast changes) set their own require-ments and limitations. Especially in wireless telecommunications (Papers III-VIIand) one has to take into consideration limitations of bandwidth, available process-ing power, memory, available length of word (quantization noise), signal propagationdelays and different distortions of the signal in the transmission medium (temporalnoise, multipath fadings, Doppler frequency shifts, multiuser interference, weather,movements around and near the transmitter or receiver). Moreover, the signal trans-formations into and out of the air-interface set their own limitations (quantizationnoise, transformation time delays, processing time delays).

Bandwidth is a limited resource, which can not be compensated for by increasingthe processing power on terminals, although by using some special methods it is pos-sible to increase spectral efficiency (like using optimal pulse shapes). Propagationdelays also belong to physical limitations, which in the power control application

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(Papers III-V) set their own requirements. Closed loop power control must reactto signal fading in real time. If this can not be done, it is better not to control it(this is the case when the speed of the terminal is more than about 27 m/s). Inorder to interpret power control commands as fast as possible they are sent in thebeginning of slots of frames or a separate pilot symbol transmission channel is used.Because of the speed we also can not use any kind of forward error correction (FEC)codes and interleaving, therefore the bit error ratio (BER) is between 5-10 % (asa reference the BER in fiber optic transmission is mostly about 10−10). Therefore,we must optimise signal processing algorithms carefully. Moreover, the implementa-tion of very effective processors into the portable wireless terminal without carefulalgorithm design leads easily to cooling problems. The consumption of power isa very limiting factor, too. In practice, this also means very careful algorithmsdesign and optimisation of pipelines to the processor by hand. Furthermore, wehave to take into consideration the fact that we are mostly operating on integernumbers. The increase of the complexity of the communication systems with mul-tiple parallel activities also very effectively limit the available processing time. Forthese reasons portable handphones and other kind of portable wireless terminalshave two processors, one for applications and the user interface and the other fortelecommunications.

In the other application areas considered here (Papers I-II), processing powerand time delays are not primary limiting factors. Instead of that, we have to takeinto consideration limitations which are due to a large number of inputs and limitedamount of data available.

3.2 Normalisation of input domain

Normalisation or scaling of the physical input domain maps physical values of thesystem state variables into a normalised universe of discourse (normalised domain)

ε = [α, β]. (3.1)

The traditional choice of ε, α and β are ε = [-6,6]. Nowadays, the larger areasof [-100,100] and [-4096,4096] are used. The scale transformation could be doneboth for discrete and continuous domains. It is simply done by performing themultiplication between the scaling factor, SF, and the input variable, VI as follows:

NVI= SF× VI (3.2)

where NVI is a normalised (scaled) crisp input variable in the normalised inputdomain. The scaling factor SF is achieved by dividing the width of the selectednormalised domain by the width of the physical domain:

SF =width(selected normalised domain)

width(physical domain)(3.3)

where width means the subtraction of maximum and minimum values of the domain.

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The scaling factor has a significant role with respect to the model’s performanceand stability. It is, in some cases, a source of possible instabilities, oscillations anddeteriorated damping effects Driankov et al. (1994). The advantage of normali-sation is that different parts of the fuzzy model can now be defined on a commondomain, which results in more general models.

In the developed FMPC (Fuzzy Mobile Power Control, Paper III - later in thePapers IV and V normalisation was not used anymore) model we used the nor-malised domain, ε = [−15, 15]. The ε = [−15, 15] was chosen because of the verylimited processing times in mobile phones signal processors. This is especially truein the power control, where we even have not time for the forward error correctionand interleaving of the received bits, but instead we have to accept higher bit errorrate because of the fastness. Moreover, we had only 5 bits available for the powercontrol commands. The input variables in the FMCP model were:

- error of received signal power, e

- change of error on received signal power, ∆e = ei − ei−1

The corresponding physical domains were:

- e ∈ [−15dB,+15dB]

- ∆e ∈ [−7dB,+7dB]

In the other reported models such as FLAT (Papers I and II) and FSSM (PaperVI) normalisation was not used. We noticed as a disadvantage, that even if thenormalisation is a linear operation it increases quantization noise, which might, onthe other hand, lead to instabilities in the tuning phase of membership functions.This naturally depends on whether SF > 1 or SF < 1. However, in the mobileenvironment we considered to process with one to one relation between incomingvalues and values used in the fuzzy decision making in order to minimise processingtime of digital signal processor and therefore we considered to give up from thenormalisation.

3.3 Fuzzification of input variables

In the fuzzification part a normalised or non-normalised domain’s crisp input vari-ables are fuzzified by converting them into a fuzzy membership. This makes themconsistent with the fuzzy set representation of the system’s state variables in thefuzzy rule’s antecedent part. The crisp value is fuzzified by selecting the fired labelor labels with corresponding degrees of membership, as graphically presented inFigure 3.1. Membership functions are stored in a data base of the model.

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grade of membership

1.0

0.3

0.7

crisp input

Figure 3.1. The graphical representation of the fuzzification procedure.

3.4 Fuzzy inference

3.4.1 Main parameters

The inference, reasoning or rule firing phase includes a knowledge base and aninference engine of fuzzy models. The knowledge base consists of a data base anda rule base. The data base has information for a proper functioning of the nor-malisation, fuzzification, rule base, defuzzification and denormalisation phases. Itsinformation includes fuzzy membership functions and physical domains with theirscaled counterparts. The data base is discussed in more detail in section 3.3.2. Inthe rule base fuzzy reasoning rules (production rules) describing system behaviorare presented either in a linguistic or numerical form. The rule base is discussed insection 3.3.3.

The inference engine performs the control of reasoning. It is performed by firingthe rules, whose rule-antecedent parts have grade of membership values greaterthan a threshold value (usually zero), i.e., in the inference engine, fuzzy results areinferred from the memberships of fuzzy sets with the aid of the knowledge base.The inference engine includes different kinds of parameters (here we have followedthe terminology used by Driankov et al. (1994)), e.g.:

- type of fuzzy relations

- type of interaction and union operators

- type of firing ( i.e., selection between individual rule based inference and com-position based inference)

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- rule base division related parameters (number of hierarchical levels in reasoningand number of parts in each reasoning level).

Selection of the type of fuzzy relations affects essentially the meaning of fuzzy rules.It is a system dependent process. The literature and experience have pointed outthat only some guidelines for it could be given. The feedback is achieved fromresults, i.e., this is tuning by changing the type of relations.

Selection of the type of interaction and union operators is a system dependentprocess, too.

In composition based inference all rules are combined into one relation to explic-itly represent the connection between fuzzy propositions and then fired with fuzzyinput. In individual based inference each rule is individually fired with crisp inputto implicitly represent a fuzzy relation and then combined into one overall fuzzyset. However, using Mamdani’s implication, the end result is same on both typeof inferences, whereas using, e.g., the Godel implications these results differ. Weconsider, based on experience, that the choice of type of firing is also an applicationbased procedure.

The number of hierarchical levels and parts describes the construction of a rulebase by separating the rule base into several phases, which are connected with eachother (see Figure 3.2). The advantage of hierarchical rule base is that the numberof rules not increase exponentially with the number of input variables as in theconventional rule base. Thus we can avoid the exponential explosion of the rulebase when the number of input variables increases. In the FLAT-model (Papers Iand II), where the number of input variables was as high as eight, the rule base wasdivided into seven separate hierarchical parts at three different levels each of themhaving 49 rules and the reasoning was done in three sequential phases, as describedin Figure 3.2. Totally we had 7 × 49 = 343 rules instead of the 77 = 823543 rules.In the case of 4 input variables we would have 3 × 7 = 147 rules instead of the73 = 343 rules.

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demand forecasted

weighted averagevalue of earlier demand

by a customerdemand forecasted

trend of earlierdemand

phase 2 output 1

phase 2 output 2

phase 3 output

phase 1 output 4

phase 1 output 3phase 1 output 2

phase 1 output 1

Input Values Phase 1 Output Values Phase 2 Output Value

weighted average value of orderstrend of orders

weighted average of

trend of earlier earlier forecasts

forecasts

by a market expert

part 1

part 4

part 3

part 2

part 1

part 2

part 1

Figure 3.2. Hierarchical tree structure of the FLAT’s rule base.

Furthermore, testing of completeness, consistency, continuity and interaction ofthe set of rules is a crucial phase in the inference engine development process.Completeness of rules means that all kinds of situations of system behavior aretaken into consideration, i.e., all kinds of combinations of input variables results inan appropriate output value,

∀x, x : hgt(OUT (x, x)) > 0 (3.4)

where x and x denotes the state variables and hgt means the height of the fuzzyset (hgt(A) = supu∈XµA(u)). At least from the software point of view, a rule baseshould be absolutely complete even if certain regions in the input domain are notof interest. This can be done easily for example by widening the utmost labels intoinfinity.

The rule base is consistent if it does not contain any contradiction1. It can beformulated as in Driankov et al. (1994): A set of rules is inconsistent if thereare at least two rules with the same rule-antecedent and different rule-consequent.Continuity means that neighboring rules have no output fuzzy sets with an emptyintersection. Definitions of neighboring rules are given for example in Driankovet al. (1994) as follows: two rules are neighbors, if their cells are neighbors inmatrix representation of a rule base. An interaction of a set of rules is definedmany ways in the literature. In Driankov et al. (1994) it is stated that a set offuzzy rules interacts if composition based inference does not equal individual basedinference

Ucomb �= Uindiv (3.5)1In the literature it is also defined like continuity below.

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where the output fuzz sets Ucomb and Uindiv are

Ucomb = e ◦ Rm and (3.6)

Uindiv =⋃

k

˜CLU(k)

(3.7)

where e is a crisp input. The relation describing the meaning of a whole set offuzzy rules is

Rm =n⋃

k=1

R(k)m (3.8)

where R(k)m is a Mamdani’s interpretation of the rule k, n represents the number of

rules under consideration, m refers to the Mamdani’s relation and ˜CLU(k)

denotes aclipped fuzzy set k. In other words, a set of fuzzy rules interacts if composition basedinference does not equal with individual rule based inference. It can immediatelybe seen that Mamdani’s relation guarantees that rules do not interact.

3.4.2 Data base

The parameters of a data base represent:

- the physical domains and these normalised counterparts

- fuzzy membership functions

- quantization information and

- tracking information of a model’s calculation phases and functions.

The data base has interfaces to the source data, data preprocessing phase, mem-bership functions generation phase, rule base and defuzzification phase as well asnormalisation and denormalisation phases. In other words, it interacts with alldesign parts of the fuzzy model.

The parameters of physical domains represent the system’s functional dimen-sions. The right sizes of these are significant. If they are chosen too large, theinaccuracies of the model increase, and if they are chosen too narrow, the modelworks on a limited scale. These problems arise at least while generating the mem-bership functions from data. The selection of the minimum and maximum valuesafter filtering easily leads to either too large or too narrow a physical domain andfalsify domain division into a set of labels.

The type and representation formalism of fuzzy membership functions are essen-tial things on the model’s tuning process in order to make it work correctly. The

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parameters of membership functions inform the required type of membership func-tions. The representation formalism describes the representation of the membershipfunctions on a computer. In the Papers I-VII, a parametric, functional description ofmembership functions is used. The membership functions are stored with minimaluse of memory and they are efficiently modifiable for real-time requirements.

The performance of the fuzzy model is very dependent on quantization infor-mation. The wrong kind of quantization easily causes instability and oscillationproblems as was observed in the tuning phase of the FMPC-model (Paper III).

Tracking information of the model’s calculation is essential in analysing the sys-tem’s behavior afterwards.

3.4.3 Rule base

The fuzzy rule base represents at the linguistic level the functioning of a modeledsystem in the form of production rules or linguistic equations. The most importantparameters of it are

- the representation formalism of the rules

- the choice of process state and process output variables

- the contents of rule-antecedent and the rule-consequent

- the determination of the set of rules.

The fuzzy rules can be represented either in linguistic or numeric form. The advan-tage of the linguistic form is descriptiveness whereas maintenance and the handlingof rules might be disadvantages. The advantages of numerical representation for-malism are the compact rule base and fast computation with an easier rule updatingprocesses. The lack of descriptiveness and some flexibility limitations are the worstdisadvantages. The former can be compensated by representing rules also in lin-guistic form by building an interpretation module to transform numerical rules intolinguistic form, when needed as was done in the FLAT application (Papers I andII). The flexibility limitations such as ’the sum of the degree of membership func-tion values has to be one’ (this is due to linguistic equation approach) are trade-offsbetween the simplicity and the accuracy of the model and can be affected by thedesigner(s).

The choice of process variables, which represent the contents of the rule-antecedents and rule-consequents, is essential in order to achieve proper functioningof fuzzy systems. The contents of rule-antecedents and rule-consequents are mostlydefined by interviewing system experts.

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3.5 Defuzzification part

The defuzzification phase converts the set of fuzzy output values into a crisp valueand maps it via denormalisation, if needed (normalisation done earlier), onto aphysical domain. The most important parameters of the defuzzification phase are

- defuzzification operator and

- number of labels.

The number of labels simply tells how many separate parts the physical domain isdivided into. The defuzzification operator defines the used defuzzification method.The most common defuzzification operators are

- center of area

- center of sums

- center of largest area

- first of maxima

- last of maxima

- middle of maxima

- height defuzzification.

These procedures are described for example in Driankov et al. (1994). In the testingphase of the FLAT -application (Papers I and II) all of these methods were carefullytested in order to find best method for the application area. Finally the center ofarea method was finally selected. The other reported applications also utilized thecenter of area method.

The disadvantage of the center of area method is that it is computationally quitecomplex and slow. The center of sums method is its faster version. The difference isthat instead of taking the union of fired fuzzy sets, it takes the sum of them. Thuspossible overlapping areas are reflected more than once in this method.

The center of largest area is the modified center of area or gravity method appliedto the largest convex fuzzy area. Its performance compared to the center of areamethod depends on how easily the largest convex area is found compared to thedefinition of the weight point from the total fired area.

The first, middle and last of maxima values are computationally effective meth-ods. The correctness of these are heavily dependent on the used type of membershipfunctions and the number of labels. The middle of maxima doesn’t oscillate so easilyas the first and last of maxima.

Height defuzzification is the ’accelerated’ version of the center of sums method.Its disadvantage is that it doesn’t consider either the support or shape of fuzzy sets.

The defuzzification methods should satisfy some criteria listed below Driankovet al. (1994)

38

- robustness

- disambiguity

- plausibility

- low computational complexity and

- weight counting.

Robustness means a measure or extent of the ability of a system to continue tofunction despite the existence of faults, disturbances or noise. In the ambiguitycase, the defuzzification method can not choose between choices. Plausibility isdefined so that every defuzzified control output which has a horizontal componentapproximately in the middle of the support has a vertical component with a highdegree of membership in the support. The criteria of computational complexity isdefined with the aid of computational operations needed to define the defuzzifiedvalue. In weight counting the weight values of different fired sets, e.g. the sumof grades of membership if the same set is fired several times, are considered bydefining the final output of defuzzification.

3.6 Testing and tuning of parameters

In the testing of fuzzy systems, parameters are chosen to describe directly the criticalfunctions of the model. Tuning is usually a difficult and time consuming process.It can be performed by modifying the model’s design parameters either off-line oron-line. Using on-line tuning the model includes a so-called performance evaluatorand adaptation algorithm. In off-line tuning mathematical methods are used tomatch the output of the model with the given set of monitoring data Pedrycz(1993). In the next chapter we discuss on-line tuning of membership functions withmore detail.

4 Generation of membership functions

Fuzzy set theory and fuzzy logic provide a general method for handling uncertainand vague information. They are unavoidable parts of many real-world applications(Tsai et al. 1994, Frantti & Mutka 1997). However, in many cases feature extrac-tion is the crucial point for the success of application development. Therefore, inthe following sections, we consider in more detail data preprocessing and analysismethods and proceed via it to the developed fuzzy membership function generationmethods.

4.1 Preprocessing and analysis of data

The generation of membership functions is performed in the FLAT (Papers I andII), FMPC (Papers III-V) and FSSM (Paper VI) models, directly from preprocessedand possibly normalised online data. We performed the preprocessing of data byapplying the following kinds of methods:

1. upsampling

2. interpolating and

3. filtering.

The upsampling of zeros is performed in order to achieve the correct time depen-dency of data values. The source data of FLAT, for example, includes only non-zero time series values, therefore non demand periods have to be included into thetime-series as explicit zero values of demand. In FMPC and FSSM upsampling isnot needed because the sampling of signals itself is a time dependent process andtherefore produces the correct correspondence between the sample time and samplevalue.

Interpolation is applied to ”generate” data between sample points. With inter-polation one should be naturally very careful. On one hand, it causes linear depen-dency (linear interpolation) between data points but on the other hand it shows

41

the direction of changes earlier. The spline or polynomial fit is usually adequate aslong as the interpolation is not done for too long a time period.

After interpolation the extreme values are removed. Filtering is used to avoida too widely distributed grade of membership. The extreme values, minimum andmaximum, increase the scale of the time series and therefore the width of the socalled fuzzy distribution and fuzzy membership functions (see Figure 4.1). Thetime series is then fed into a lowpass filter in order to calculate the moving averageof sequential values.

In principle the fuzzy distribution of an original time series describes the pos-sibility of events in magnitude more correctly than the fuzzy distribution of anupsampled time series. Therefore the time dependency of events is considered byusing probability values determined from the original time series. For example inFLAT (Papers I and II), probability values are applied to final results in order todetermine the certainty level of the demand. If the probability is very low (≤0.2) thedemand forecast is not considered. However, the forecasted demand value with lowprobability value is taken into consideration upon determination of the probabilityof next week forecast as a realized demand value. In this way we can anticipate thedemand instead of updating the probability distribution via the realized demandafterwards.

4.2 Fuzzy distribution

In this section we describe the concept of fuzzy distribution. The fuzzy distribution(see Figure 4.1) is used in the automatic generation of fuzzy membership functionsgeneration in two separate methods described later in sections 4.2.1 and 4.2.2. Itconsists of the following values defined from the time series

- minimum value

- average of minimum and mean value (= angle point A)

- average of mean and maximum value (= angle point B)

- maximum value.

42

normalised gradeof membership

1.0

0.0

min max

quantity

angle point A angle point Breference value

W1 W2 W3

Figure 4.1. Fuzzy distribution.

From the fuzzy distribution a weight point value is determined as a ”referencevalue”. The reference value might be simply an arithmetic mean, weighted average,medium or mode value of a preprocessed data set.

The width values of a fuzzy distribution can be defined as follows

W = ‖min− ra‖ + ‖ra − rb‖ + ‖rb −max‖ = ‖min−max‖ (4.1)

where W is the width of the distribution, min is the minimum value of the distri-bution, max is the maximum value of the distribution, ra is reference value A (=angle point A in Fig. 4.1), rb is reference value B (= angle point B in Fig. 4.1).

The height of the distribution is normalised to 1.0. The subwidths can be definedas

W1 = ‖min− ra‖ (4.2)

W2 = ‖ra − rb‖ (4.3)

W3 = ‖rb −max‖ (4.4)

where W1 is the width of the first triangle, W2 is the width of the right-angledparallelogram, and W3 is the width of the second triangle.

The width values describe the shape of the fuzzy distribution (Fig. 4.1). Thelarger Wi is, the broader the fuzzy distribution is. The triangle inequality

W (Z) ≥ (W (↓ LZ) ≥ (W (↓ LZE) (4.5)

holds (Z is the upsampled and unfiltered time series, ↓ means downsampled (zerovalues removed) and L the lowpass filtered time series whereas E means extraction of

43

extreme values). The downsampling has been done in FLAT’s source data (PapersI and II), because if demand is realised it is realised with non-zero values. Theextraction of extreme values decreases diversity, but it also decreases the informationcontent of the distribution.

The fuzzy distribution is supposed to describe the distribution of time series val-ues on the physical domain. Therefore, the determination of membership functionsfrom the above mentioned distribution has been done by dividing the describeddistribution into several parts depending on the number of linguistic levels underconsideration. In this thesis two possible ways to achieve this division are described,namely average based division (section 4.2.1) and cumulative distribution based di-vision (section 4.2.2).

4.2.1 Average based division of a fuzzy distribution

In this method, the division starts from the middle of the distribution. Dependingon the number of labels, division points are defined symmetrically on both sides ofthe reference value. The number of division points is:

number of division points on one side =2 × (number of labels)− 4

2(4.6)

Division points are defined, for example, in the case of five labels (six extra points),in the following way (Fig. 4.2):

1. define mean value of samples between angle point A and reference value =division point 1

2. define mean value of samples between division point 1 and reference value =division point 2

3. define mean value of samples between angle point A and division point 1 =division point 3

4. define mean value of samples between angle point B and reference value =division point 4

5. define mean value of samples between division point 4 and reference value =division point 5

6. define mean value of samples between angle point B and division point 4 =division point 6

44

normalised gradeof membership

1.0

0.0quantity

reference valuedivision point 1division point 2

division point 3 division point 4division point 5

division point 6

Figure 4.2. Division of a fuzzy distribution.

When the number of divisions approaches the number of samples the division pointsform like the original distribution. We can then state that the performed division ofthe physical domain of a time series describes linguistic areas of it, if the resolutionof the division is high enough.

4.2.2 Cumulative distribution based division of a fuzzydistribution

Another possible way to define division points is to determine two discrete densitydistributions of time series values. One is determined from the values betweenminimum value and the reference value (distribution 1), the other from the valuesbetween the reference value and maximum value (distribution 2). The cumulativedistributions are determined from the density distributions. The division points aredefined in the following way from the cumulative distributions. We show the fivelabel case (Fig. 4.3):

1. define value 0.25 of vertical axis of distribution 1 and find corresponding valuefrom horizontal axis = division point 1




45



1.0

0.5

0.25

0.75

division point 2 (5) division point 3 (6)division point 1 (4)

Figure 4.3. Division points from cumulative distribution.

As an illustration consider the following example. Assume that the cornerpoints offuzzy distribution are: minimum = 5.1, angle point A = 5.9, reference value = 7.5,angle point B = 9.4 and maximum = 10.0. In order to find density distributions thevalues between the minimum and reference values (interval A [5.1-7.5]) are selectedfor the density distribution 1 and values between reference value and maximumvalue for the density distribution 2 (interval B [7.5-10.0]) (see Figures 4.4 and4.5, where distributions are defined with 10 classes). The density distributions aredefined by dividing the intervals A and B, for example, into 10 classes and definingthe share of the values in each of them. This is done by summing the number ofsamples in each class and dividing by the total number of samples in the interval.The cumulative distribution is achieved from the density values by summing themfrom left to right, as shown in Figures 4.4 and 4.5.

46

class 1 class 2 class 3 class 4 class 5 class 6 class 7 class 8 class 9 class 10

0.10

0.20

0.20

0.40

0.60

0.80

1.0

left density distribution

left cumulative distribution

minimum value = 5.1 reference value = 7.5

5.8 6.4 6.9

interval A

Figure 4.4. Left density (above) and cumulative (below) distributions.

Also in this method, when the number of divisions approach the number of samplesin the time series, the division points form the like original distribution. Thereforewe can argue that the method is suitable for the division of a time series intolinguistic areas if resolution is high enough.

47

class 1 class 2 class 3 class 4 class 5 class 6 class 7 class 8 class 9 class 10

0.10

0.20

0.20

0.40

0.60

0.80

1.0

right density distribution

right cumulative distribution

reference value = 7.5 maximum value = 10.0

8.0 8.5 9.2

interval B

Figure 4.5. Right density (above) and cumulative (below) distributions.

4.3 Curve fitting method

Fuzzy systems are known to offer a generic function structure that is capable ofapproximating arbitrary continuous nonlinear functions defined on a compact setTan & Vandewalle (1997). A number of papers have shown that fuzzy systemsare capable of universal approximation (Kosko 1992, Wang & Mendel 1992, Ying1994). Kosko (1992) states that every kind of non-linear function can be represented

48

with the aid of fuzzy membership functions and fuzzy rules, but it does not tellthe number of membership functions and rules needed. In the above mentionedmembership function generation methods the right number of labels as well as theright kind of division of the fuzzy distribution are defined by trial and error methods.In the below described curve fitting method we let the method itself do this.

In the curve fitting method the values between the minimum and maximumvalues are fit for example into a third order polynomial or several values (like divisionpoints above) are calculated and a numerical polynomial is transformed from thesepoints. In this method, the estimated function’s ordinate values are divided evenlyinto the number of intervals depending on the number of labels under consideration(see Figure 4.6). The number of intervals depend on the number of labels as follows:

number of intervals = 2 × number of labels− 3 (4.7)

The number of labels can be approximated by starting from an initial number oflabels and increasing the number of labels in order to achieve the needed accuracy.This can be described procedurally as follows:

1. set initial division to2× number of initial number of labels− 3

2. divide ordinate values evenly into2× number of initial number of labels− 3 intervals

3. define areas under linear approximations of curve in all intervals (and subin-tervals) (Fig. 4.6)

4. define areas under curve in all intervals (and subintervals)

5. combine areas under subintervals together and subtract area under curve fromarea under linear approximation

6. compare difference to threshold value and decide if it is necessary to divide theinterval further into subintervals.

Naturally the definitions of the rule base determine the number of labels but inthe above mentioned method it is possible to find the needed resolution for inputvariables. In any case, membership functions can be determined as described abovefrom online data when the number of labels is defined. By designing an adaptive rulebase (to adapt in accordance with the number of labels) it is possible to estimatethe necessary resolution of labels online.

49

subint

erval

subarea

linear approximation

membership functions

Figure 4.6. Division of Fitted Curve.

4.4 Design parameters in the automatic generation of fuzzymembership functions

In this section certain design parameters for the automatic generation of fuzzymembership functions are presented. The presentation is based on the developedmodels (Papers I-VII).

The most important and significant design parameters from model developer’spoint of view are:

- period length of source data

- feature extraction methods

- division of physical domains

- shape of membership functions and

- amount of overlapping between labels.

These are considered in more detail below.

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4.4.1 Period length of source data

The period length of variable is needed to include necessary information of the mea-sured/observed variable for modelling. Such information includes, e.g., short termand long term fluctuations, dynamic scale, average level, rate of change, directionof change and existence of periodicity of the variable. Unfortunately, the periodlength of a variable used in models depends on more or less on the availability ofdata. Therefore, in these cases the length of the period is not necessarily the periodof the variable but instead the period length of the data collection time.

The source data of the Fuzzy Logic Advisory Tool (FLAT) model (Papers Iand II) for membership function generation consists of realized events of productdemand. The data collection period includes events of one year. Therefore, it coversseasonal and other short term effects, but not, for example, long term effects likerecessions. It includes 52 points for every item, because the data distribution isupdated only once a week. In order to increase the reactivity of the model thelatest events were weighted (exponentially) more compared to earlier ones.

The advantage of the developed membership function generation method forFLAT is that membership functions are updated continuously in accordance withdynamic changes of the system. It is especially important to take into considerationchanges in dependencies between different products. For example, demand mightstill be linguistically large for both master and slave products (transmission productsforms product families composed of single units connected to a main unit) in a rulebase, but now in a direct 1 to 1 relation instead of the earlier 1 to 2 relation.Furthermore, the time series scale of a product might change significantly due tomarket area enlargement and therefore oscillations of demand might always stay ina large area without the update of membership functions. Naturally the resolutionof the decision making is also higher when membership functions are updated on-line. Furthermore, the automatic generation of membership functions can operateas a starting point for more exact manual tuning.

In the mobile phone power control case (Papers III-V) the source data was origi-nated from the envelope of the fading signal. The problem was to find a suitable dataperiod for the generation of reliable and reactive membership functions in order toestimate the fading rate, change of fading rate and deepness of fades together witha compact rule base (this is especially needed for embedded systems) to control thetransmission power of a mobile. The envelope of the received signal is changed ac-cording to the movement of the receiver, movements between transmitter/receiver,weather conditions, existence of a line of sight (LOS) path of transmission andpower control of the transmission. The envelopes of the Rayleigh distributed fadingsignal (no dominant path exist), when the speed of mobile v is assumed to be 0.01m/s, 0.1 m/s and 10 m/s and carrier frequency fc is 2 GHz are presented in Figure4.7. Their Doppler spreads are about 0.13 Hz (see Figure 4.8 as an example), 1.3Hz and 133.3 Hz.

51

(a) Envelope of a Rayleigh fading signal (v=0.01 m/s).

(b) Envelope of a Rayleigh fading signal (v=0.10 m/s).

(c) Envelope of a Rayleigh fading signal (v=10.00 m/s).

Figure 4.7. Envelopes of a Rayleigh fading signal when velocity is a) 0.01 m/s,

b) 0.1 m/s and c) 10 m/s and carrier frequency is 2 GHz.

52

(a) Fourier transform of a fading signal (fc = 2 GHz,

v=0.01 m/s).

(b) Absolute value of Fourier transformed fading signal

(fc = 2 GHz, v=0.01 m/s).

Figure 4.8. a) Fourier transform and b) absolute value of Fourier transform of

a fading signal when carrier frequency fc is 2 GHz and the speed of a terminal

v is 0.01 m/s.

The envelopes of the Rician distributed fading signal (dominant path exists), whenthe speed of mobile v is assumed to be 10 m/s, K ( power in dominant path

power in scattered paths) is 10 and16 and carrier frequency fc is 2 GHz is presented in Figures 4.9. The envelopes

53

of the Rician distributed fading signal, when the speed of mobile v is assumed tobe 10 m/s, K is 5 and 10 and carrier frequency fc 900 MHz is presented in Figures4.10. The envelopes of the both Rayleigh and Rician distributed signals, when thespeed of mobile v is assumed to be 10.0 m/s and carrier frequency fc 450 MHz andK is 10 are presented in Figure 4.11.

54

(a) Envelope of a Rician fading signal with K=10 andfc = 2 GHz.

(b) Envelope of a Rician fading signal with K=16 andfc = 2 GHz.

Figure 4.9. Envelopes of a Rician fading signal with a) K=10 and b) K=16.

The carrier frequency fc is 2 GHz.

From these figures we can see, that the dynamic scales (deepness of fades and fadingrate, i.e., 2 dynamic scales on each picture) change quite a lot, when the speed of theterminal changes and/or when the dominant path appears or disappears. Thereforethere exist an infinite amount of different combinations of mobile speeds and K

55

values as well as infinite amount of different dynamic scales for the combinations.Moreover, the received power level changes due to power control (see Figure 4.12).

(a) Envelope of a Rician fading signal with K=5 andfc = 900 MHz.

(b) Envelope of a Rician fading signal with K=10 andfc = 900 MHz.

Figure 4.10. Envelopes of a Rician fading signal with a) K=5 and b) K=10.

The carrier frequency fc is 900 MHz.

56

(a) Envelope of a Rayleigh fading signal when fc = 450MHz.

(b) Envelope of a Rician fading signal with K=10 andfc = 450 MHz.

Figure 4.11. Envelopes of a a) Rayleigh and b) Rician (K=10) fading signal

when the carrier frequency fc is 450 MHz.

In the traditional approach the division of the dynamic scale into fuzzy membershipfunctions must be done for the worst or near to worst case. Then the resolution ofcontrol is inadequate in better conditions and causes unnecessary oscillations dueto power control. Naturally, we could also define a very large dynamic scale for

57

the deepness of fades, the fading rate and the change of the fading rate. However,we should then also define a very large rule base (increases exponentially withthe dynamic scales of variables), which unnecessarily increases the complexity ofthe model and amount of testing efforts (testing of consistency, continuity andinteraction of a set of rules). Therefore, it looks obvious to model signal behaviorby taking into consideration the dynamic scale of the signal envelope via the on-linetuning of membership functions.

(a) Envelope of a received signal, when fuzzy power con-trol is applied.

(b) Envelope of a received signal, when predefined steppower control is applied.

Figure 4.12. Envelopes of a received signal, when power control is applied: a)

fuzzy power control and b) predefined step power control.

58

The coherence times (reciprocal value of the Doppler spread) are, when the speed ofthe mobile is 0.01 m/s and 0.1 m/s and the carrier frequency fc 2 GHz, 7.5 s and 750ms, respectively. The coherence time is a measure over which a transmitted symbolwill be relatively undisturbed by channel fluctuations. Therefore the membershipfunction generation interval should be coherence time. The demand for the lengthof the membership function generation interval decreases when the Doppler spreadof the received signal increases. Because of that, we should select a long enoughtime interval to take into account the dynamic changes of the received signal if wefor some reason do not estimate Doppler spread in the receiver.

In the fuzzy synchronisation search (Papers VI) the source data has originatedfrom the defined correlation values. A very clear periodicity of 0.625 ms was found,as expected, due to WCDMA system and its slot duration. The synchronisationsymbol interval used in the research was 0.625 ms (one symbol in one slot). Themembership function generation interval for the impulse response measurement wasdefined to be the length of one frame, i.e., 10 ms, which was also the time intervalbetween impulse response measurements in the research system. All the possiblecorrelation values on the downlink (with contemporary channel conditions) werethen defined since the length of scrambling code was the length of a frame.

In the definition of membership functions from correlation values, the traditionaldivision of correlation value scale leads to problems. First of all, the searchedcorrelation value’s level changes according to the length of the spreading code.In WCDMA the length of the code in different channels might change dependingon, e.g., the data rate needed. Therefore the system should include membershipfunctions for numerous different scales. Secondly, the level of correlation valueschange according to channel conditions.

If we define membership functions by dividing the scale between zero and themaximum correlation value (or absolute value of it due to channel rounding) anduse it all the time, we can find or separate only the synchronisation value of aspecific length of spreading code in good channel conditions. A decrease in signalto noise ratio (SNR) increases the other correlation values to the biggest positivelabel area, then the system can not separate/classify searched correlation values. Anincrease in SIR (signal to inference ratio), on the other hand, decrease the searchedcorrelation value (Figure 4.13).

Furthermore, in WCDMA systems we are using a RAKE receiver, which definesthe correlation values of separate paths of transmission or separate channel correla-tion values. Therefore, if we want to use separate paths before their combinations,we can define membership functions for them separately in order to make synchroni-sation faster (provided that a strong enough path exist) or in order to find separatechannels at the same time with different channel conditions (separate fingers arededicated to separate channels).

The propagation delay when the cell radius is 10 km, is 0.03 ms and therefore theseparate paths of the signal can be assumed to arrive during the 30 µs time interval.In the impulse response measurement this means that correlation values should bedefined after the first peak value, 0.03µs

10ms × chip rate(4.096Mchip/s) ≈ 122 times inorder to guarantee that all paths are found. This information can be used in thefuzzy rule base in order to increase the reliability of the system.

59

(a) Correlation values over one frame (SNR = -5.3 dB)

(b) Correlation values over one frame (SNR = -11.3 dB)

(c) Correlation values over one frame (SNR = -17.3 dB)

Figure 4.13. Correlation values over one frame with a) SNR = -5.3 dB, b)

SNR = -11.3 dB and c) SNR = -17.3 dB.

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4.4.2 Feature extraction

In the feature extraction methods of FLAT (Papers I and II) raw data (historicalvalues of forecasts, demand and orders stored in the data base of the factory) waspreprocessed into a suitable form for constructing reliable membership functions.The preprocessing of the time series data was performed by applying upsampling,interpolating and filtering methods.

The upsampling of zeros was performed in order to achieve correct time depen-dency of data values, because the source data included only non-zero time seriesvalues (non demand periods were included into the time-series as explicit zero val-ues of demand). Interpolating the time series by spline interpolation was appliedto ”generate” data between sample points. The spline or polynomial fit is usuallyadequate as long as the interpolation is not done for too long a period.

After interpolation the extreme values, i.e., minimum and maximum values wereremoved. Extreme value filtering was used to avoid too wide a distribution of thegrade of membership. The extreme values, minimum and maximum, increase thescale of the time series and therefore the width of the so called fuzzy distributionand fuzzy membership functions, too.

The time series was then fed into a lowpass filter in order to calculate movingaverage of sequential values. It was very significant in FLAT model, where therelative effect of one value on the total behavior was notably high. On the otherhand smoothing decreases the extreme behavior of system but in the same wayfuzzified input values have on both edges of the physical domain proportionally toolarge values. However, this was compensated by calculating a so called normal valuefrom several earlier values (not to be confused with the earlier mentioned referencevalue) and fuzzifying it instead of one single event value. The side effect of this wasa weakening of reflectivity.

In principle the fuzzy distribution of an original time series describes the pos-sibility of events in magnitude more correctly than the fuzzy distribution of anupsampled time series. Therefore the time dependency of events was considered byusing probability values determined from the original time series. The probabilityvalues were defined as a number of times of realised demands divided by a numberof period length (probability determination period, sliding window, not the dataperiod) in weeks (forecasting period was one week). The probability values wereapplied to final results in order to determine a certainty level of demand. If theprobability was very low (≤0.2) the demand forecast was not considered. However,the low probability value was taken into consideration in the determination of theprobability of next week’s forecast as a realized demand value. In this way we cananticipate the demand instead of updating the probability distribution via realizeddemand afterwards.

The fuzzy power control model and fuzzy synchronisation model (Papers III-V)are examples from the signal analysis area, where traditionally lowpass filtering hasa significant role. In fuzzy power control lowpass filtering is suitable and worthwhilebut in a fuzzy synchronisation procedure it is out of the question because the rawdata consists of correlation results, where the maximum correlation should be found,since a smoothing average destroys this information.

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4.4.3 Division of the physical domain and overlapping oflabels

The right number of membership functions (labels) is essential to describe the non-linear behavior of a system accurately enough. Therefore, the models were verified,as stated earlier, from the point of view that they work better than competingdesign alternatives.

The amount of overlapping between label areas is one of the basic questions infuzzy logic, i.e., the existence and nonexistence of a phenomenon. This so called”coexistence paradox” of conflicting phenomena caused problems already to pio-neers of fuzzy logic to publish their ideas. However, our method is simple in thatrespect since we suppose that the sum of the grade of memberships is always one(1.0). Test trials did not show any significant disadvantage from this hypothesis.

4.5 Illustrative example of the modelling problem - powercontrol of mobile terminal

Power control in the DS/WCDMA is needed to compensate for the fluctuation of amobile’s transmitting power received at a base station (to equalize the signal powerreceived at the base station of all mobile subscribers in a specified cell coveragearea) and to increase the capacity of mobile communication systems. Fluctuation ofpower level is mainly due to both fast varying short-term fading and slowly varyinglong-term fading of the transmitted signal and co-channel interference phenomenon.The developed fuzzy proportional-integral (PI) control has been used to stabilizethe power level and to decrease the overshoot and the rise time of it. The inputvariables of the fuzzy controller are received power error and the change of receivedpower error. The output of the fuzzy controller is the power step size. Membershipfunctions are estimated automatically during operation. The determination of fuzzyrules has been done by analysing the behavior of the power control system andproperties of the fading signal. Achieved linguistic fuzzy rules are presented inmatrix form.

The user transmitted signal power ptk(dB) (more correctly SIR, signal to

interference ratio, that is the ratio of narrowband power to wideband power) isupdated by a step ∆p(dB) every Tp seconds according to the following PI typecontrol rules:

Rulei: if e is Ai and ∆e is Bi then ∆p is Ci,

where Ai, Bi, Ci are linguistic terms, Tp is the power control sampling period, e isthe power error and ∆e is the power error change. The power error is simply theset point level minus received signal power level at the base station:

62

ei = prsp − pr

k. (4.8)

As an example, let prsp = 0dB, pr

k = 10.96dB and ei = −12.04dB (see Table 4.1,where integer numbers -2, -1, 0, 1 and 2 denote labels negative big, negative small,zero, positive small and positive big, respectively).

Table 4.1. Inputs and fired labels of the 15 first power control steps.

errordB

change oferror dB

firederror

label 1

firederror

label 2

firedchangeof errorlabel 1

firedchangeof errorlabel 2

-12.04 -12.04 -2 -1 -2 0

-8.48 3.56 -2 -1 2 -

-4.97 3.51 - -1 2 -

-4.70 0.27 - -1 - 1

-3.19 1.50 -2 - 0 1

-0.87 2.33 - -1 -2 -

-2.17 -1.30 -1 0 2 -

-2.41 -0.24 - -1 -2 0

-2.41 -0.24 - -1 - 0

-1.91 0.49 - -1 0 1

-1.38 0.54 - -1 0 1

-0.82 0.56 -1 0 0 1

-0.02 -0.80 - 0 - 1

-0.10 -0.08 - 0 - 0

-0.18 -0.08 - 0 - 0

-0.26 -0.08 - 0 - 0

The desired mobile unit is initially placed in a position, which causes a -23 dB pathloss. The error change is the difference between the current received error and lastreceived error:

∆ei = ei − ei−1. (4.9)

Let ei−1 = 0dB and ∆ei = −12.04dB (see Table 4.1). The e and ∆e are nowfuzzified for the fuzzy reasoning (see Table 4.2).

However, the fuzzy membership functions are updated before fuzzification ac-cording to the fading rate and deepness of the fades of the signal in the transmissionmedia. The period length of the data (time series of the received signal power levels)

63

required for the update was considered in the section 4.4.1. The fuzzy membershipfunctions are generated from the time series for the fuzzification phase using themethods considered in the sections 4.2 and 4.2. For example, suppose that thetime series is 750 ms long and includes 22500 symbols, i.e., power levels and theaverage based division of the fuzzy distribution is used. Then the fuzzy distributionis composed from the time series values. If we suppose that the signal dynamic isbetween -35dB and +10 dB, the width of the fuzzy distribution is then 45 dB (seeFigure 4.14). The fuzzy distribution is divided into fuzzy membership functions ineither way described earlier in the section 4.2.

Table 4.2. Grades of membership of the 15 first power control steps.

grade ofmembership

for errorlabel 1

grade ofmembership

for errorlabel 2

grade ofmembershipfor changeof errorlabel 1

grade ofmembershipfor changeof errorlabel 2

0.41 0.59 1.00 0.00

0.34 0.66 1.00 -

- 1.00 1.00 -

0.00 1.00 0.00 1.00

1.00 - 0.28 0.72

- 1.00 1.00 -

0.32 0.68 1.00 -

- 1.00 1.00 -

- 1.00 - 1.00

- 1.00 0.15 0.85

- 1.00 0.23 0.77

0.19 0.81 0.02 0.98

- 1.00 - 1.00

- 1.00 - 1.00

- 1.00 - 1.00

- 1.00 - 1.00

After fuzzification, in the reasoning phase, the user transmitted power ptk is

updated according to the rule base. The inferred result is defuzzified in order toget a crisp value. The transmitted power pt

k is the last transmitted power plus theincrement of power step:

ptk = pt

k−1 + ∆p. (4.10)

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Hence, the received power prk of the kth user at the base station is:

prk = pt

k + gk + nIk (4.11)

where ptk is the transmitted power by the mobile of the kth user, gk is the channel

link gain to the kth user due to path loss, multipath fading and shadowing and nIk

is the interference noise from the other users to the kth user. Upper indices r, t andI denote receiver, transmitter and interference, respectively. After control actionsthe power update command is sent to the mobile over the return channel.

Figure 4.14. Fuzzy distribution and related membership functions when the

dynamic range is 45 dB.

Figure 4.15. Fuzzy distribution and related membership functions when the

dynamic range is 32 dB.

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Suppose now, that on the second iteration prsp = 0dB, pr

k = 9.95dB and ei =−8.47dB. Furthermore, suppose that ei−1 = −12.04dB and ∆ei = 3.56dB. Valuese and ∆e are now fuzzified (see Table 4.2), as mentioned earlier, for the fuzzyreasoning. However, the fuzzy membership functions are updated again beforefuzzification according to the fading rate and deepness of the fades of the signalin the transmission media. Furthermore, suppose that the period length of thedata (time series of the received signal power levels) required for the update isnow only 500 ms and includes 15000 symbols. From the time series the fuzzymembership functions are generated for the fuzzification phase using the methodsconsidered earlier, but now only the newest 15000 power level values are taken intoconsideration. Then the fuzzy distribution is composed from these time series valuesin the above described way and differs from the earlier defined fuzzy distributioneven if the period length is same or if the deepness of the fades or the fadingrate differs. If we suppose that the signal dynamic is now between -25dB and+7 dB, the width of the fuzzy distribution is then only 32 dB (see Figure 4.15).Moreover, for example the label zero is narrower than earlier. If we now suppose that

width of the current label zerowidth of the current distribution is smaller than the width of the earlier label zero

width of the earlier distribution thensmaller amount of the values are near the reference power level (this might due to,for example, the change of fading rate or the disappearance of line of sight signalcomponent on the received signal) and the model has taken it to consideration.

Table 4.3. Results of 15 power control steps.

reasonedlabel 1

reasonedlabel 2

grade ofmembershipfor label 1

grade ofmembershipfor label 2 output

2 1 0.70 0.79 4.57

2 1 0.66 0.83 4.53

-1 1 0.49 1.00 1.29

0 1 0.49 1.00 2.52

2 0 0.86 0.00 3.34

-1 1 0.65 0.84 -0.28

-1 1 0.49 1.00 0.77

1 0 1.00 0.49 1.51

0 1 0.41 0.92 1.55

0 1 0.38 0.88 1.58

0 1 0.58 0.89 0.89

0 0 1.00 0.00 0.00

0 0 1.00 0.00 0.00

0 0 1.00 0.00 0.00

0 0 1.00 0.00 0.00

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The fuzzy distribution is divided into fuzzy membership functions in either waydescribed earlier for the fuzzy distribution, which naturally differ from the earlierdefined membership functions. Continuing in the same way we update the fuzzymembership functions continuously before the fuzzification phase according to thesignal power level. Tables 4.1, 4.2 and 4.3 present 15 power control steps withinput data (error, change of error, theirs fired labels and grades of memberships) andoutput (transmission power level) with reasoned labels and corresponding grade ofmemberships, respectively. The speed of the terminal was 10 m/s. For illustration,unnormalised values are shown and quantization is not used.

4.6 Evaluation of the developed methods

The advantages of the presented methods are in application areas where other kindsof methods are difficult to implement, i.e., in an environment with fast dynamicchanges or a large number of different physical domains with continuous changes ora high dimensionality of input data. Therefore, the presented methods have to beconsidered primarily from the implementational point of view in order to see theseadvantages. In all the attached applications (described in Papers I-VII) other typesof fuzzy modelling methods were considered to be either impossible or too laboriousto implement.

For example, in the forecasting case (Papers I-II), it is important to take intoconsideration changes in dependencies between different products. Demand mightstill be linguistically large for both master and slave products (transmission productsform product families composed of single units connected to a main unit) in therule base but now in a direct 1 to 1 relation instead of the earlier 1 to 2 relation.Furthermore, a product time series scale might change significantly due to marketarea enlargement and, therefore oscillations of the demand might always stay in thelarge area without the update of membership functions. Naturally the resolution ofthe decision making is also higher when membership functions are updated on-line.Moreover, it is very laborious to define membership functions for an input set of an8-dimensional vector for every product (at least when the number of products is ashigh as 1000).

In the fuzzy code tracking model (Paper VI) the source data originated fromdefined correlation values over a 10 ms time period (length of one frame). The defi-nition of membership functions from correlation values in the traditional way leadsto problems. First of all, the searched correlation value’s level changes accordingto the length of the spreading code. In WCDMA systems the length of the code indifferent channels might change depending on, e.g., the data rate needed. There-fore, the system should include membership functions for numerous different scales.Secondly, the level of correlation values changes according to the channel condi-tions. If we define membership functions by dividing the scale between zero and themaximum correlation value (or absolute value of it because of channel rounding)and use it all the time, we can find or separate only the synchronization value of aspecific length of spreading code in good channel conditions. A decrease in signal

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to noise relation (SNR) increases other correlation values to the biggest positivelabel area, the system the can not separate/classify searched correlation values. Anincrease in SIR (signal to inference ratio), on the other hand, decreases also thesearched correlation value.

Paper IV presented a comparison of adaptive (using the automatic generationand tuning of membership functions) and semi-adaptive (the dynamic changes of thephysical domain were predicted by dividing it into different sub-ares, i.e., the modeluses different sets of membership functions depending the level of state variables,thus imitating the automatic generation of membership functions) fuzzy power con-troller performance evaluation. In this application area static fuzzy controllers aretheoretically applicable, and were rigorously studied also in this research. However,the implementation of them demanded a very large rule base and consumed a lot ofvery limited memory and processing time of the embedded signal processor of themobile terminal. Therefore, they were considered impossible to implement and forthese reasons those performances are not evaluated here.

In Figures 4.16, 4.17, 4.18 and 4.19 the waveforms of the fixed-step powercontrolled signal, semi-adaptive fuzzy PI controlled signal and adaptive fuzzy PIcontrolled signal are presented, respectively, when the velocities are 3 m/s, 10 m/s,20 m/s and 25 m/s and the bit error rate (BER) is 5 % (forward error correctioncodes and interleaving can not be used in fast closed loop power control). Thefixed-step power control waveforms (see more details from the Papers III-V) arepresented in order to improve the comparison. The desired mobile unit is initiallyplaced in a position which causes -23 dB path loss.

In this comparison one can notice that the rise time and the variance (see Table4.4) of the semi-adaptive fuzzy and adaptive fuzzy PI controllers are considerablysmaller than in predefined step control systems. Furthermore, one can see that theadaptive fuzzy controller causes smaller variations and smaller overshoot than asemi-adaptive fuzzy controller. However, we want to note that the enhancement ofpredefined step controller to the variable size step controller with some kind of non-fuzzy decision making logic could improve also its results. The increased variation ofthe adaptive controller with small velocities (3 m/s in Table 4.4) is due to the factthat on smaller velocities near the set point value the adaptive controller changesto a normal fuzzy controller (as well as an semi-adaptive controller), which explainsthe same size of variations in both cases.

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Table 4.4. Variances of power control methods with different velocities.

Velocity

m/sPredefined steppower control

Fuzzypower control

Adaptive fuzzypower control

3 17.25 2.07 2.01

10 12.99 3.90 1.23

20 13.46 3.76 1.55

25 16.51 5.81 2.00

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(a) Predefined step controlled signal, v=3 m/s.

(b) Fuzzy controlled signal, v=3 m/s.

(c) Adaptive fuzzy controlled signal, v=3 m/s.

Figure 4.16. Typical waveforms of a) predefined step, b) fuzzy and c) adaptive

fuzzy power controlled signal with a velocity v of 3 m/s.

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Figure 4.17. Typical waveforms of a) fixed-step, b) fuzzy and c) adaptive fuzzy

power controlled signal with a velocity of 10 m/s.

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72






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4.7 Adaptation of the membership functions

In this section the adaptivity of the average based division and cumulative distribu-tion based division of the fuzzy distribution into membership function generationin a changing environment are demonstrated. These divisions were selected for il-lustration because they turn out to be best and easiest methods to implement inpractice. The curve fitting method, described earlier, is aimed more at the eval-uation of the required number of linguistic labels. It is quite cumbersome and ifthe threshold value is not appropriately selected the iteration time of the algorithmmight be very long.

In the evaluation we used a second order process. The response Y(s) in thes-plane is

Y (s) =R(s)K

(1 + τ1s)(1 + τ2s)(4.12)

where R(s) denotes input, K describes amplification and τ1 and τ2 are the first andsecond time constants, respectively. Ramp signal t ( 1

s2 in the s-plane), whose slopewas 0.1, was used as an input signal. This could describe for example a slow growthof demand or a fast amplification of a signal depending on the time constants. Theinitial values used were τ1 = 1, τ2 = 0.5, K=1 and y(0)=0. In the t-plane theequation is

y(t) = t+ 2 × e−t − 12× e−2t. (4.13)

Furthermore, the output is summed with random noise weighted with the multiplier0.1. The example response for the 50 first time step is presented in Figure 4.20.The upper curve presents the response with random noise whereas the lower onepresents the response without noise.

The response of the system is described with a fuzzy variable. It can take fivelinguistic values: negative big, negative small, zero, positive small and positive big.Each of the values are described with trapezoidal membership functions. Therefore,the 8 corner points are required to define membership functions as shown in Figures4.1 and 4.2. The development of corner points during the first 5 time steps areshown for the illustration in the Tables 4.5 and 4.6 for average based divisionof the fuzzy distribution and cumulative distribution based division, respectively.The membership functions were generated in discrete time steps by using the N=50earlier response values from the above mentioned model. The development of thepoints as a function of time is presented in Figures 4.21 and 4.22.

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0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time

resp

onse

example response of the second order system

Figure 4.20. The response of the second order system to the ramp signal plus

random noise.

Table 4.5. Development of corner points during the first 5 time steps when averagebased division of the fuzzy distribution is used.

point 1 point 2 point 3 point 4 point 5 point 6 point 7 point 8

5.095 5.587 6.249 6.860 8.156 8.762 9.419 9.968

5.153 6.216 6.599 7.793 8.917 9.212 9.670 10.091

5.209 6.358 6.714 7.988 9.064 9.329 9.769 10.148

5.319 6.469 6.819 8.107 9.175 9.433 9.864 10.222

5.466 6.581 6.926 8.217 9.283 9.539 9.965 10.395

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Table 4.6. Development of corner points during the first 5 time steps when cumu-lative distribution based division of the fuzzy distribution is used.

point 1 point 2 point 3 point 4 point 5 point 6 point 7 point 8

5.095 5.779 6.355 6.930 7.998 8.491 9.230 9.968

5.153 6.494 6.772 7.189 8.103 8.848 9.345 10.091

5.209 6.628 7.032 7.301 8.194 8.927 9.415 10.148

5.319 6.737 7.004 7.405 8.289 9.014 9.497 10.222

5.466 6.846 7.111 7.509 8.405 9.151 9.649 10.395

The sensitivity of the membership function adaptation can be regulated by chang-ing the value N. Figures 4.21 and 4.22 also clarifies that adaptation of the mem-bership functions affects the operating range of them. The smaller the operatingrange is, the more sensitive the membership functions are to changes. Here, weassumed constant number of membership functions in order to keep the rule baseunchanged.

0 5 10 15 20 25 30 35 40 45 502.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

time

corn

erpo

ints

cornerpoints of the membership functions

Figure 4.21. The development of the fuzzy membership functions with the

function of time, when average based division of the fuzzy distribution method

were used.

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0 5 10 15 20 25 30 35 40 45 505

6

7

8

9

10

11

12

13

14

15

time

corn

erpo

ints

cornerpoints of the membership functions

Figure 4.22. The development of the fuzzy membership functions with the

function of time, when cumulative distribution based division of the fuzzy

distribution method were used.

5 Compact rule base

In the ”traditional” approach the division of the system’s dynamic scale into fuzzymembership functions must be done for the worst or near the worst case. Thereforea very large dynamic scale for the variables of the model must be defined. However,we should also define a very large rule base which increases exponentially with thedynamic scales of variables in order to achieve high enough resolution. This, onthe other hand, increases the complexity of the model and amount of testing efforts(testing of consistency, continuity and interaction of a set of rules). The requirementof high resolution was primary reasons for developing fuzzy membership functionsfrom online data in the reported cases (Papers I-VII). Therefore, it looks obvious tomodel changes in dynamic scales of system (for example dynamic of signal behaviorin Papers III-VII) via the on-line tuning of membership functions.

For example, fuzzy power control (Papers III-V and Paper VII) and code tracking(Papers VI-VII) of a mobile transmitter in a cellular network needs adaptive estima-tion and tuning of membership functions in order to avoid a rule base with overhead(in the reported cases there exist an infinite amount of different dynamic scales, seesection 4.4.1 for more details). The behavior of the signal envelope is ”self-similar”(the shape of the signal envelope remains the same, but the scale changes depend-ing on the transmission power level, interference, noise, path loss, shadowing andfading). Therefore, the increase of variables’ resolution with the number of labels(dividing the dynamic scale by the number of labels) instead of membership func-tions adaptation, increases the size of the rule base with the increased number oflabels.

6 Introduction to the papers

In the following sections, the contents of each enclosed paper is briefly described.The papers are applications oriented, however in every paper the automatic fuzzymembership function generation and on-line tuning of membership functions arestudied, developed and applied to case studies.

6.1 Paper I, An adaptive, hierarchical fuzzy logic advisorytool (FLAT) for anticipating the demand of transmission

products

An adaptive, hierarchical Fuzzy Logic Advisory Tool (FLAT), presented in Paper I,was developed for material purchasing. The justification to build FLAT grew fromthe need to have a forecasting system for component purchasing, because even asuboptimal system would bring in huge advantages. Fuzzy modelling was the finalchosen method because other more traditional forecasting methods were inadequatefor this application domain Frantti & Mahonen (2001).

FLAT can forecast demand and therefore control a very complex and highly non-linear materials purchasing system. The membership functions of expert system’sfuzzy rules are generated through online data. The expert system is divided intoseveral hierarchical subsystems. The rule and knowledge bases are presented bylinguistic relations. The inference function in FLAT is performed by hierarchicalreasoning in which linguistic relations in each subsystem are expressed by equations.The potential benefits of fuzzy logic with linguistic relations are better and moreaccurate decision-making than expert assisted customer forecasts due to the model-based approach and systematic data management for materials purchasing.

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6.2 Paper II, Fuzzy logic based forecasting model

In this paper a fuzzy logic based software tool, the Fuzzy Logic Advisory Tool(FLAT), for demand forecasting of signal transmission products is presented. FLATwas developed for the prediction of demand of about 1000 different products inorder to aid the materials purchasing process of about 14000 different componentsin the electronics manufacturing processes of Nokia Network Systems’s Haukipudasfactory.

The prediction values of different products are inferred by starting from a set ofeight input values. Each input value is fuzzified by FLAT. Thereafter, fuzzy resultsare inferred in three sequential phases. In each phase the number of variables is splitdue to the hierarchical structure of the inference module. A data base and a rulebase are divided accordingly into three hierarchical levels. Rules are represented bylinguistic relations changed into their matrix equations form in order to apply thelinguistic equations framework technique Juuso (1992).

Fuzzy membership functions for input values are determined on-line from earlierinput values of the products. Fuzzy rules were inferred by analyzing the behaviorof the products together with market experts and product experts of the company.

The model is able to produce more accurate decision-making support than moretraditional approaches, which was verified via parallel use of different approaches.This is probably due to the model-based approach and systematic data management.

6.3 Paper III, Fuzzy power control for mobile radio systems

Fuzzy logic based power control for a direct sequence, wideband code-divisionmultiple-access (DS/WCDMA) cellular mobile phone radio system is introducedin Paper III. Power control in DS/WCDMA is needed to compensate for the fluc-tuation of a mobile’s transmitting power received in a base station, to fight againstthe far-near problem and co-channel inference and to increase the capacity of mo-bile communication systems. Fluctuation of power level is mainly due to both fastvarying short-term fading and slowly varying long-term fading of the transmittedsignal and co-channel interference. The developed fuzzy proportional-integral (PI)control has been used to stabilize the power level and to decrease the overshootand rise time of it. The input variables of the fuzzy controller are received powererror and change of received power error. The output of the fuzzy controller isa power step size. The membership functions are estimated automatically duringthe operation. The determination of fuzzy rules has been done by analysing thebehavior of the power control system and properties of the fading signal. Achievedlinguistic fuzzy rules are represented in matrix form. Simulation results show thatfuzzy power control offers a considerably shorter rise time and smaller standarddeviation of received power at a given base station compared to the predefined steppower control model.

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6.4 Paper IV, Adaptive fuzzy power control for WCDMAmobile radio systems

Fuzzy logic based power control for a DS/WCDMA cellular mobile phone radiosystem is introduced in Paper IV . Power control in DS/WCDMA is needed tocompensate for the fluctuation of a mobile’s transmitting power received in a basestation. The developed fuzzy proportional-integral (FPI) control has been usedto solve stability problems and decrease the overshoot and rise time. The inputvariables of the fuzzy controller are received power error and change of receivedpower error. The output of the fuzzy controller is power control step size. Themembership functions are generated automatically online for the FPI controller.

In the paper a short introduction to DS/WCDMA is given and the backgroundfor fuzzy power control research need is discussed. Thereafter conventional powercontrol for a CDMA system (IS95) is described and fuzzy power control is depictedwith automatic membership function generation. Finally the simulation model isdescribed and the results of power control simulations are given.

6.5 Paper V, Fading rate based adaptive fuzzy power controlfor WCDMA cellular systems

This paper introduces an adaptive fading rate based fuzzy power control algorithmfor a DS/WCDMA cellular phone system. A power control procedure is needed tocompensate the fluctuation of mobile’s transmitting power received in a base stationand to increase the capacity of mobile communication systems. The input variablesof the controller are the power error and the change of the error. The output isthe power step size. The fading rate is used to control the parameter value of theaveraging operator in fuzzy reasoning. Simulation results show that the developedadaptive fading rate based fuzzy proportional-integral (PI) control stabilizes thepower level and decreases overshoot and rise time of it.

6.6 Paper VI, Fuzzy decision making in synchronisationprocedure of DS/WCDMA based cellular mobile phone

radio system

In this paper an initial fuzzy channel synchronisation and handover procedure forDS/WCDMA cellular mobile phone system is presented and discussed. A synchroni-sation procedure can be performed by searching the maximum complex correlationpoint of the received searching code. Correlations are calculated over several radioframes slots and the highest complex correlation point is detected afterwards fromthe stored values. Combining the fuzzy decision procedure with a synchronisationsearch, the synchronisation point can be detected during the correlation calcula-

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tion. Furthermore, the fuzzy decision procedure guarantees that the selected pointis really a synchronisation point.

The input variables of the fuzzy model are a ratio (complex cross-correlationvalue) / (theoretical maximum cross-correlation value) and determined ’level’ ofcorrelation values. The output value of the fuzzy decision-maker is the linguisticdescription of the correlation value defuzzified to the synchronisation time. Themembership functions are estimated automatically from on-line data during thesynchronisation procedure. The determination of fuzzy rules has been done byanalysing the behaviour of the system and are presented in matrix form.

In the structure of the paper an introduction to WCDMA system and spreadingcode generation with correct properties is first presented. The initial synchronisa-tion procedure is depicted secondly. Thirdly, the fuzzy initial synchronisation searchis discussed as well as the reasons for the selection of fuzzy logic. In the same phasethe automatic membership function generation used in the model is also shown.Finally, the simulation model with achieved results is presented.

Fuzzy decision making guarantees, that the point found is a real synchronisationpoint. It is easy and fast to apply this method to practical systems. It effectivelyalso limits the size of the rule base, which would be dramatically increased by us-ing fixed membership functions to take into consideration dynamical variations ofthe system. Simulation results show a significantly shorter synchronisation time.The method will also notice, when there is no synchronisation point. Moreover,the power consumption of the mobile phone can be decreased because of fastersynchronisation time. The detection of unsynchronisation also decreases the inter-ference level of the nearby area. The algorithm can be straightforwardly applied tohandover situations with the same benefits.

6.7 Paper VII, Fuzzy reasoning in WCDMA radio resourcefunctions

This paper presents the detailed descriptions of wide-band code-division multipleaccess (WCDMA) radio resource functions, i.e., power control, random access, ini-tial cell search, and handover procedures for the universal mobile telecommunica-tions system (UMTS). The paper gives first a presentation of the physical layer ofthe UTRA (UMTS Terrestrial Radio Access) followed by a description of the ra-dio resource functions. Then the paper presents the enhancement of radio resourcefunctions with fuzzy reasoning. Finally the paper describes, as a comparative exam-ple, self-organising map (SOM) based estimation procedure for the radio resourcefunctions. The WCDMA system was selected as an example system for fuzzy rea-soning illustrations in cellular systems mainly because of its important role as athird generation cellular system.

7 Discussion

7.1 Why the generation of fuzzy membership functions?

Concerning the generation of fuzzy membership functions for industrial systemsor applications, many non-trivial questions have to be answered. It is necessaryto understand the actual application area. Developed methods and systems mustbe integrated into existing systems and environments with their expectations andlimitations. They have to survive with a finite amount of data and their computa-tional complexity is often limited. These aspects are not usually considered in puretheoretical research.

The primary target of the work was to automate the generation and tuningmethod of membership functions for the nonlinear fuzzy models fast adaptationinto new circumstances dynamically. In the connection of hypothesis formulationwe also stated that the application area of the fuzzy model have to be understoodand taken into consideration.

Here, the selected approach connects the preprocessing and analysis of data andgeneration and tuning of fuzzy membership functions phases as a part of the moregeneral approach of the development of fuzzy systems presented by Isomursu Iso-mursu (1995). The chapter 3 considers mostly the static fuzzy modelling frame,which enlargement for the dynamical systems is presented in this thesis. The chap-ter 4 presented the developed methods for the feature extractions and the on-linegeneration and tuning of fuzzy membership functions as a part of the fuzzy mod-elling process. The numerical examples are presented in the related papers and inthe sections 4.5, 4.6 and 4.7. The model development is out of the scope of thisthesis. The models are given by the domain expert, which in these cases has beenthe same person (the author). The primary reasons for developing fuzzy member-ship functions from online data in these reported cases are due to the nature of theapplications areas

I number of state variables and/or dimensionality of data was high

II data changed rapidly

III high resolution was needed with a compact rule base

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IV generation time was very limited

V behavior of the system was difficult to understand and

VI nonexistence of expert knowledge.

Considering the generation of membership functions against the nature of the ap-plication areas (see Section 4.4.1), it is clear that the automatic generation of mem-bership function was a necessary requirement to flexibly adapt models into variablesituations.

The high number of state variables, for example in the case of FLAT with 8input variables for every 1000 fuzzified items, complicates system design if thefuzzy membership functions for each item are different. In these cases, automationor mechanization is needed to prevent manual work and to unify the handling ofinputs.

The physical domains of input and output variables might change rapidly andfrequently making earlier created membership functions out of date and making thecontinuous generation of membership functions necessary in order to preserve highresolution without an extended rule base. In some applications the time availablefor membership function generation is also very limited after a change of physicaldomain, therefore applications need a very fast estimation of functions.

The very complex behavior of a system or nonexistence of expert knowledge alsomake the generation of membership functions more difficult and too slow. Thisgeneration must then be based on the analysis of data.

In other words, we can conclude the question of this sections by answering withthe research assumption: ”Fast adaptation and tuning of fuzzy models in an envi-ronment with fast dynamic changes or a large number of different physical domainswith continuous changes or a high dimensionality of input data needs a specificautomatic generation algorithm for fuzzy membership functions as a part of fuzzymodel development”.

7.2 Effects on the application model

In the connection of hypothesis formulation we also stated that the developmentprocess, methods, environment and algorithm development related matters have tobe understood and taken into consideration.

The development process and methods of the fuzzy model were considered bytaking into account:

- Preprocessing and Analysis of Data (data for membership functions generationand tuning)

- Fuzzification (type of membership functions)

- Inference (number of labels, size of rule base)

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- Defuzzification (type of membership functions)

The modelling environment was considered by taking into account the type of thegenerated software, i.e., embedded software or distributed software.Algorithm related matters, which were considered, were the amount of data needed

for membership functions generation and the suitability of the algorithm for variousmembership function types.

The application area were considered in the section 7.1. Combining the restric-tions set by the development process, development methods, modelling environmentand algorithm development related matters with it framework for automatic gener-ation and tuning procedures of fuzzy membership functions is achieved.

7.3 Results achieved in real-world applications

In Papers I and II was focused on decision making for materials purchasing processof about 1000 products (about 14000 different components) in an uncertain envi-ronment with the aid of fuzzy set theory. The processing of eight input parametersfor every single product in order to achieve reasonable conclusions for the mate-rials purchasing process was possible with fuzzy set theory, hierarchical reasoning,linguistic equation approach and on-line generation and tuning of membership func-tions. Automatisation of data preprocessing and on-line generation and tuning ofmembership functions made it possible to continuously process and update a largeamount of data in a robust way. The developed system proved to be successful, andis now in production use.

The used approach is very promising and useful also in several other applicationdomains, e.g., in manufacturing. The method can predict non-linear behavior betterthan many traditional time-series methods that are based, for example, on polyno-mial approximations. The methods based on characteristics such as AMI (AverageMutual Information), Lyapunov polynom, Nearest Neighborhood approximation orHurst-parameter approximation were providing interesting insights into the under-lying process. Although these methods were proving the inherent non-linearity of aprocess, the traditional time-series prediction methods could not compete againstthe FLAT approach. The maximum χ2-error between estimates from FLAT andnon-linear approximators was especially considerable. This is probably due to theadaptivity of the fuzzy logic algorithm and increased prediction ability due to therulebase and expert information of the input-set.

A comparison of actual demand, the demand forecasted by a customer and anexpert and that forecasted by FLAT shows that the FLAT curves (Papers I and II)follow the actual demand more precisely than does the forecasted curve of expertand customer, and the squared error of FLAT is less than that of the forecasted.Over the first period the customer forecast was better in 12% of cases whereas theFLAT forecast was better in 60% of cases, results being the same in 28% of cases,while the corresponding figures in the second period were 19% of cases, 56% of cases

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and 25% of cases. The square root of squared errors of the customer forecasts were505.4 and 1836.1, while those of the FLAT forecasts were 271.8 and 356.2.

In Papers III-VII the behavior of the fading signal was analysed and the fuzzypower level control studied. The fuzzy power control has a faster rise time andsmaller overshoot of power level and decreases the standard deviation of the mo-bile’s transmitting power received in a base station compared to the predefined stepcontrol. Hence we were able to increase the capacity of the mobile communica-tion systems. The fuzzy PI -controller was developed combining tradional fuzzy settheory and on-line generation and tuning methods of membership functions. Withthe aid of fuzzy set theory conclusions from input variables were made accordingto the linguistic control policy of a rule base. The power control model adaptedto quickly changing dynamics via the automation of data preprocessing and on-linegeneration and tuning of membership functions. The automatic fuzzy membershipfunction generation and tuning decreases the variance of the power level, when, forexample, the velocity of a terminal increases. The small rule base with fast mem-bership function estimation algorithm is easy and straightforward to implement.Furthermore, the decrease of overshoot and fluctuation also decreases the powerconsumption of mobiles and therefore increases speech and standby times.

In Paper VI the synchronisation procedure of a DS/WCDMA cellular mobilephone radio system with fuzzy decision was presented and described. The pro-cedure is suitable for initial and channel synchronisation including Rake receiverfinger allocations as well as handover situations. Combining the fuzzy decisionmaking procedure with a synchronisation search, a high enough correlation point,i.e. the synchronisation time, was found when it appeared. Furthermore, fuzzy de-cision making guaranteed that the noticed point was really a synchronisation point.The fuzzy membership functions were generated in real time from correlation dis-tributions to take a quickly changing dynamic environment into consideration. Theapplied method is easy and fast to apply in practice and effectively limits the size ofthe rule base, which would dramatically increase by using fixed membership func-tions. The fuzzy rules were represented in matrix form. Simulation results showsa significantly shorter synchronisation time and the detection of the lack of a syn-chronisation point. Moreover, the power consumption of the mobile phone can bedecreased because in the case of the lack of a synchronisation point, the synchro-nisation procedure and/or phone connection procedure can be stopped instead ofcontinuing it unnecessarily, which finally also decreases the inference level of thenearby area.

Instead of the fuzzy set theory based method, the SOM based method was alsoconsidered in Paper VII. However, the SOM method is more uncertain and compli-cated than the fuzzy set theory method. Its implementation is also more compli-cated.

Automatic generation and tuning of membership functions for input and/or out-put variables is useful at least in the following ways:

I the development of the fuzzy model becomes faster

II the development process becomes more structured and easier to control andmanage

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III the development process is easier to understand

IV the system under consideration becomes analysed in a robust way

V the modification of membership functions is easier and faster

VI a system can adapt into new circumstances with a compact rule base.

The development time for the fuzzy model becomes faster when the developmentprocess and analysis of data is well structured in all modelling phases. Automaticgeneration of membership functions can also operate as a starting point for moreexact manual tuning.

The more structured development process for membership function generationreduces costs, development time and increases the quality of the approach. Thecontrol of the development process is also easier because of the structured method,which makes the managing of the process more stable and understandable.

Using the above presented automatic methods for membership function genera-tion, the physical domains of input variables as well as output variables are analysedfor every variable in the same way. This kind of robustness also helps to under-stand the testing and tuning phases of the model. The modification of membershipfunctions is also reasonable because it forces developers to analyse the possible rea-sons for doing so. The presented automatic generation methods of membershipfunctions make fuzzy modelling easier. Moreover, a fuzzy model can adapt to newcircumstances with the same resolution and compact rule base via the modificationof membership functions.

8 Directions for further research

The work presented in this thesis has already initiated further research on a projectsat Technical Research Centre of Finland. The common interest of the projects, onthe fuzzy side point of view, is fast adaptation and intelligent decision making inthe wireless networks.

The possible further research targets of the work presented in this thesis is con-sidered to be:

1. the definition of optimal number of labels

2. the definition of adaptive rule base

3. the definition of very simple and extremely fast membership function adapta-tion algorithm for the software radios.

The definition of optimal number of labels and adaptive rule base are closely con-nected to each other. In this thesis we performed adaptation via membership func-tions in order to keep a rule base compact. Therefore, we could not change numberof labels adaptively. However, in order to increase accuracy it might be neces-sary especially in the applications of telecommunication software. The drawbackof the rule base adaptivity is more complex system model and increased amount ofprocessing time. The definition of very simple and fast membership function adap-tation algorithm for the software radios seems challenging research topic. One veryinteresting application could be, for example, fuzzy amplitude modulation, wherelinguistic areas of voltage are used instead of the numerical voltage quantizationlevels. This kind of algorithms would make possible the use of totally different kindof algorithms development for the terminal devices.

The more application oriented further research targets where developed methodsprobably are useful are:

1. definition of fuzzy circuits for different proprietary purposes in digital com-munications

2. definition of intelligent buffering technology into data link control protocol’s

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3. definition of routing servers with intelligent decision making in the networkingtechnology

4. optimisation of power control and handover decisions.

The definition of fuzzy circuits of the described methods for proprietary purposesis already ongoing. However, in the digital communication there is a lot of otherapplication possibilities, for example in the interleaving of symbols and modulationof carriers. The intelligent buffering research for the various link layer protocols aswell as transmission control protocols is ongoing, too. The routing of the networktraffic in the combination of different networks (in other words in the Internet)seems to be very promising application area for the developed methods. The veryinteresting application area is also optimisation of combined decision making in theCDMA based cellular networks power control and handover. The combined decisionmaking includes numerous amount of parameters. Therefore the optimisation ofthe decisions in varying conditions is very difficult task. The human like reasoningcombined with the adaptive algorithms in order to adapt varying situations mightbe in a key role in the listed examples.

9 Conclusions

The primary target of the work was to automate the generation and tuning methodof membership functions for the nonlinear fuzzy models fast adaptation into newcircumstances. In the beginning we assumed that fast adaptation and tuning offuzzy models in an environment with fast dynamic changes or a large number ofdifferent physical domains with continuous changes or a high dimensionality of inputdata needs an automatic generation algorithm for fuzzy membership functions.

We considered that appropriate division of fuzzy modelling clarifies the designprocesses of fuzzy membership function generation and tuning. In this work thegeneral development model Isomursu (1995) was modified to include the prepro-cessing and analysis of data and membership function generation phases as theirown separate phases. Therefore we consider here that the fuzzy model developmentprocess includes the following phases:

I Preprocessing and Analysis of Data

II Normalisation Phase

III Membership Function Generation Phase

IV Fuzzification Phase

V Reasoning Phase

VI Defuzzification Phase

VII Denormalisation Phase

VIII Representation of Results Phase

The focus of the research were in the fast adaptation of nonlinear fuzzy models intonew circumstances via generation and tuning of membership functions. Thereforethree kinds of automatic generation methods for fuzzy membership functions weredeveloped:

1. average based division of fuzzy distribution

95

2. cumulative distribution based division of fuzzy distribution and

3. curve fitting method.

The developed methods generate membership functions from online time series data.For that reason they are straightly usable for tuning purposes, too. The developedmethods and algorithms are applicable in embedded systems and computer con-trolled environments or processes. Especially, they are applicable and practical fortime critical applications where online adaptation is needed, such as terminal powercontrol in cellular mobile phone systems (Papers III-V), synchronisation search ofcellular phone systems (Paper VI) and radio resource functions parameters estima-tion of cellular phone systems (Paper VII). Another successful application areas aredecision support systems with a large number of input variables (Papers I and II)and different multidimensional classification problems Mahonen & Frantti (2000).

We considered here only fuzzy systems combined with numerical methods. How-ever, this does not mean that developed algorithms are not suitable, for instance,in neuro-fuzzy applications or other kind of hybrid methods with fuzzy logic.

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Appendix 1/1

The fuzzy set theory

The membership function µF (u) of a fuzzy set F is mostly defined in the form

µF (u) : U → L (A1.1)

where L denotes any set at least partially ordered (L -fuzzy set, L stands for lattice).Usually L is the interval of real numbers from 0 to 1 ([0, 1]). The membershipfunction assigns to each u ∈ U a value from the unit interval [0, 1] instead of thedual value set {0,1}. The sets defined with generalised membership functions arecalled fuzzy sets. F is completely defined by the set of tuples Zadeh (1965)

F = {(u, µF (u))|u ∈ U}. (A1.2)

The value of µF (u) can be a single number or an interval of real numbers. Fuzzysets of this type are interval valued fuzzy sets and formally defined in the formZadeh (1965)

µF (u) : U → P (L) (A1.3)

where P(L) denotes the set of all fuzzy subsets of L.The fuzzy set F is a type 1 fuzzy set or ordinary fuzzy set. Type 2 fuzzy sets have

membership grades that are themselves type 1 fuzzy sets defined on some universalset Y. In the same way, higher types of fuzzy sets can be defined. Zadeh (1975)

The support of a fuzzy set F in the universal set U contains all the elements ofU that have nonzero membership grade in F, that is

APPENDIX 1/2

supp(F ) = {u ∈ U |µF (u) > 0}. (A1.4)

Sometimes the term width is mistakenly used instead of support. Width can bedefined by Klir & Folger (1988)

width(F ) = supremum(supp(F ))− infimum(supp(F )). (A1.5)

In the literature the terms left width and right width are also used quite often.An empty fuzzy set is defined by Zadeh (1965)

µF (u) = 0, ∀u. (A1.6)

An empty fuzzy set has an empty support. The height of a fuzzy set is the largestmembership grade attached to any element in that set. A fuzzy set is called nor-malised, when at least one of its elements attains the grade of membership 1.0. Thenucleus of a fuzzy set F is Klir & Folger (1988)

nucleus(F ) = {u ∈ U |µF = 1}. (A1.7)

An α -cut of a fuzzy set F contains those elements of the universal set U that havea membership grade greater than or equal to α. This can be represented as followsZadeh (1965)

Fα = {u ∈ U |µF ≥ α}. (A1.8)

The set of all levels α ∈ [L] or [0, 1] (in the following treatment, we refer to thenormalised interval [0, 1] instead of L) that represents distinct α -cuts of a fuzzyset F is called a level set of F. A fuzzy set is called convex if each of its α -cuts isa convex set. That is Klir & Folger (1988)

µF (λ× r + (1 − λ) × s) ≥ min[µF (r), µF (s)] (A1.9)

for all r, s ∈ Rn and all λ ∈ [0, 1].Fuzzy sets F1, F2 are equal if Zadeh (1965)

∀u ∈ U : µF1(u) = µF2(u). (A1.10)

F1 is a subset of F2 if Zadeh (1965)

APPENDIX 1/3

∀u ∈ U : µF1(u) ≤ µF2(u). (A1.11)

In the same way the concepts strict subset, superset and strict superset are defined.A fuzzy number is defined as a convex and normalised fuzzy set defined on R

whose membership function is piecewise continuous, i.e., it contains the real num-bers within some interval to varying degrees.

The standard operations of fuzzy set theory are complement, union (more gener-ally triangular co-norms) and intersection (more generally triangular norms) Klir& Folger (1988):

µF (u) = 1 − µF (u) (A1.12)

µF1∪F2 = T �(µF1(u), µF2(u)) = ∪[µF1 , µF2 ] (A1.13)

µF1∩F2 = S�(µF1(u), µF2(u)) = ∩[µF1 , µF2 ] (A1.14)

where ∪ denotes union and ∩ intersection. The operator used to define union andintersection sets depends on the task. Zadeh defined in his seminal paper 1965union as a maximum and intersection as minimum.

Timing of fuzzy membership functions from datajultika.oulu.fi/files/isbn9514264355.pdf · Frantti, Tapio, Timing of fuzzy membership functions from data Department of Process and

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