Data Engineering

Fuzzy Mathematics in

System Theory and Data Analysis

Olaf Wolkenhauer

Control Systems Centre

UMIST

o.wolkenhauer@umist.ac.uk

www.csc.umist.ac.uk/people/wolkenhauer.htm

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Introduction

General Issues:

✘ Teaching

✘ Lecture Notes [1]

✘ Recommended Literature

✘ Internet Resources

✘ Examination

✘ Questions

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Motivation

We see an ever-increasing move toward inter andtrans- disciplinary attacks upon problems in thereal world.

The system scientist has a central role to play inthis new order, and that role is to first of all un-derstand ways and means of how to encode thenatural world into “good” formal structures.

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Data Engineering

.. and the rest of the world:

� Statistics

� System Theory

� Pattern Recognition

� Data Mining

� Soft Computing

Why...?

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What has changed..

✘ Data:� Imprecise, fuzzy, random.� Sparse, and large data sets.

✘ Systems:� Nonlinearity.� Complexity.

✘ Methodologies:� Learning.� Strategy.

✘ Technology:� Computer Power.� Database- and Web-Technology.� Imaging.

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Data Engineering

.. is reasoning about data:

1. Decision Making... a unified framework forprediction, control, prioritisation, classification.

2. System Theory... the conceptual framework: systems as graphs.

3. Learning from Data... matching data with models.

4. Uncertainty Techniques... being precise about uncertainty.

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Prediction Control Prioritisation Classification

DECISION MAKING

MODELING

...quantitative ...qualitative

Multivariate Analysis Rule-Based Reasoning

Data Information

SYSTEM, PROCESS

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Data Engineering: Decision Making

future state x(k + 1)

X

current state x(k)

X

class C

P(E)

event e

E

control action u(k + 1)

U

error e(k)

E

priorities, schedule

P

alternatives

A

a1 a2 a3

1 2 3

Forecasting Fault-Detectionand Diagnosis

Feeback- andAnticipatory Systems Maintenance

PREDICTION CLASSIFICATION CONTROL PRIORITISATION

DECISION MAKING

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Data Engineering: System Theory

System Analysis

Fact Explanation:

{theory, data} → singular factual statement

Descriptive Data Analysis

Law Explanation:

{theory, subsidiary assumption(s), data} → law

Inferential Formal Modelling

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Data Engineering: The Modelling Relation

Phenomenal World Mathematical World

causal entailment inferential entailment

ambience the self

NATURALSYSTEM

FORMALSYSTEM

decoding

encoding

Natural Law

propositions, axioms,production rules,

algorithms.

components, functionphenomena,organisation.

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Data Engineering: Learning from Data

DECISION MAKING

FORMAL MODELM

EXPERIMENTATION

SYSTEM, PROCESSS IDENTIFICATION

enco

ding

y = f (·)

control

prioritisation

forecasting,classification

E[·]

observation ξ(·)

parametersθ

data M

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Course Outline

Data, Systems, and Uncertainty:

� System Theory

� Uncertainty Techniques

� Learning from Data

� Clustering, Classification

� Fuzzy Systems Identification

� Fuzzy Mathematics

� Fuzzy Systems

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System Theory:

� The Modelling Relation.

� Observables.

� Representation of (dynamic) systems bymappings (as sets - graphs).

� Classical Modelling: differential equations,state-space modelling.

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Uncertainty Techniques:

� The Expectation Operator.

� Descriptive Statistics.

� The Least-Squares Criterion.

� Linear Regression.

� Maximum Likelihood Estimation.

� Stochastic Processes, Kalman-Bucy Filtering.

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Learning from Data:

� System Identification.

� The Probabilistic Perspective.

� Basis Function Approximation.

� Kernel Density Estimation.

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Clustering:

� Pattern Recognition.

� Hard c-Means Algorithm.

� Fuzzy c-Means Algorithm.

� Gustavson-Kessel Algorithm.

..with application to

� Classification.

� System Identification.

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Fuzzy Systems Identification:

� Fuzzy Systems Model Structures.

� Parameter Identification.

� Takagi-Sugeno Modelling.

� Switching Regression Models.

� Forecasting.

� Control.

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Fuzzy Mathematics:

� Fuzzy Sets.

� Fuzzy Logic.

� Fuzzy Relations: Similarity Relations.

� Possibility Theory.

� Approximate Reasoning.

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Fuzzy Systems:

� Fuzzy Inference Engines.

� Fuzzy Classification.

� Fuzzy Control.

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Internet Resources

1. Control Systems Centre : http://www.csc.umist.ac.uk/

2. Principia Cybernetica : http://pespmc1.vub.ac.be/

3. Pattern Recognition Information :http://www.ph.tn.tudelft.nl/PRInfo/

4. Support Vector Machines : http://svm.first.gmd.de/

5. NFS Group Magdeburg : http://fuzzy.cs.uni-magdeburg.de/

6. BISC - The Berkeley Initiative in Soft Computing :http://www.cs.berkeley.edu/Research/Projects/Bisc/

...

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Internet Resources

1. Virtual Laboratories in Probability and Statistics :http://www.math.uah.edu/stat/

2. Probability Net : http://www.probability.net/

3. WWW Virtual Libraries :

• Systems and Control :http://www-control.eng.cam.ac.uk/

• Mathematics :http://euclid.math.fsu.edu/Science/math.html

4. The MathWorks : http://www.mathworks.com/

5. Wolfram Research : http://www.wri.com/

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Recommended Literature

Fuzzy Mathematics:

1. Babuska, R. : Fuzzy Modelling for Control. Kluwer, 1998.See http://lcewww.et.tudelft.nl/.

2. Dubois, D. and Prade,H. : Fuzzy Sets and Systems. AcademicPress, 1980.

3. Kruse, et.al. : Foundations of Fuzzy Systems. Wiley, 1994.

4. Nguyen, H.T. and Walker, E.A. : A First Course in Fuzzy Logic.CRC Press, 1997.

5. Pedrycz, W. : Fuzzy Control and Fuzzy Systems. RSP, 1993.

6. Wang, L.-X. : A Course in Fuzzy Systems and Control.Prentice Hall, 1997.

7. Zimmermann, H.-J. : Fuzzy Set Theory. Kluwer, 1996.

...

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Recommended Literature

Probability Theory, Statistical Inference:

1. Freedman, D. and Pisani, R. and Purves, R. : Statistics.Norton, 1997.

2. Larsen, H. : Introduction to Probability Theory and StatisticalInference. Wiley.

3. Papoulis, A. : Probability, Random Variables, and StochasticProcesses. McGraw Hill.

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References

[1] Wolkenhauer, O. : Data Engineering.http://www.csc.umist.ac.uk/people/wolkenhauer.htm. 2

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