CI ppt slide 13 Oct 2017 - myweb.cmu.ac.th

Computer Engineering, Faculty of Engineering, Biomedical Engineering Institute, Chiang Mai University

Sansanee Auephanwiriyakul, Senior Member IEEE

Sansanee Auephanwiriyakul

Computational IntelligenceComputational Intelligence

Assoc Prof. Sansanee Auephanwiriyakul, Ph.D.Assoc Prof. Sansanee Auephanwiriyakul, Ph.D.




TopicsTopics

• Introduction to Computational Intelligence• Introduction to artificial neural networks

– Neural networks foundation– Multi-layer perceptrons– Self organizing feature map

• Introduction to fuzzy system– Fuzzy set foundation– Fuzzy inference system– Fuzzy measure & fuzzy integral

• Introduction to Computational Intelligence• Introduction to artificial neural networks

– Neural networks foundation– Multi-layer perceptrons– Self organizing feature map

• Introduction to fuzzy system– Fuzzy set foundation– Fuzzy inference system– Fuzzy measure & fuzzy integral




TopicsTopics

• Introduction to evolutionary computing– Introduction to genetic algorithm– Introduction to genetic programming– Introduction to evolutionary computation– Co-evolution

• Introduction to swarm intelligence

• Introduction to evolutionary computing– Introduction to genetic algorithm– Introduction to genetic programming– Introduction to evolutionary computation– Co-evolution

• Introduction to swarm intelligence




ReferencesReferences

• [Bezdek94] J. C. Bezdek, “What is computational intelligence?” in Computational Intelligence: Imitating Life, J. M. Zurada, R. J. Marks, and C. J. Robinson (Eds.) Piscataway, NJ: IEEE Press, 1994, pp 1 – 11.

• [Bonabeau99] E. Bonabeau, M. Dorigo, and G. Theraulez, Swarm Intelligence:From Natural to Artificial Systems, Oxford University Press, Inc., New York, USA., 1999.

• [Clerc02] M. Clerc and J. Kennedy, “The particle swarm: explosion, stability, and convergence in a multi-dimensional complex space”, IEEE Transactions on Evolutionary Computation, vol 6. 2002, pp. 58 – 73.

• [Dorigo96] M. Dorigo, V. Maniezzo, and A. Colorni, “The ant system: optimization by a colony of cooperating agents”, IEEE Trans. Syst. Man Cybern. B, vol 26, 1996, pp. 29 – 41.

• [Bezdek94] J. C. Bezdek, “What is computational intelligence?” in Computational Intelligence: Imitating Life, J. M. Zurada, R. J. Marks, and C. J. Robinson (Eds.) Piscataway, NJ: IEEE Press, 1994, pp 1 – 11.

• [Bonabeau99] E. Bonabeau, M. Dorigo, and G. Theraulez, Swarm Intelligence:From Natural to Artificial Systems, Oxford University Press, Inc., New York, USA., 1999.

• [Clerc02] M. Clerc and J. Kennedy, “The particle swarm: explosion, stability, and convergence in a multi-dimensional complex space”, IEEE Transactions on Evolutionary Computation, vol 6. 2002, pp. 58 – 73.

• [Dorigo96] M. Dorigo, V. Maniezzo, and A. Colorni, “The ant system: optimization by a colony of cooperating agents”, IEEE Trans. Syst. Man Cybern. B, vol 26, 1996, pp. 29 – 41.





• [Dorigo99] M. Dorigo, Artificial Life: The swarm intelligence approach, Tutorial TD1, Congress on Evolutionary Computing, Washington, DC., 1999.

• [Dubois82] Dubois, D. and Prade, H., “A Class of Fuzzy Measures based on Triangular norms”, Internat. J. General Systems, 8, 1982, pp. 43-61.

• [Dumitrescu00] D. Dumitrescu, B. Lazzerini, L.C. Jain, and A. Dumitrescu, Evolutionary Computation, CRC Press LLC, Florida, USA., 2000.

• [Eberhart07] R. Eberhart and Y. Shi, Computational Intelligence: Concepts to Implementations, Morgan Kaufmann Publishers, USA., 2007.

• [Eng05] G. K. Eng, and A. M. Ahmad, “Malay Speech Recognition using Slef-Organizing Map and Multilayer Perceptron”, Proceedings of the Postgraduate Annual Research Seminar, 2005.

• [Dorigo99] M. Dorigo, Artificial Life: The swarm intelligence approach, Tutorial TD1, Congress on Evolutionary Computing, Washington, DC., 1999.

• [Dubois82] Dubois, D. and Prade, H., “A Class of Fuzzy Measures based on Triangular norms”, Internat. J. General Systems, 8, 1982, pp. 43-61.

• [Dumitrescu00] D. Dumitrescu, B. Lazzerini, L.C. Jain, and A. Dumitrescu, Evolutionary Computation, CRC Press LLC, Florida, USA., 2000.

• [Eberhart07] R. Eberhart and Y. Shi, Computational Intelligence: Concepts to Implementations, Morgan Kaufmann Publishers, USA., 2007.

• [Eng05] G. K. Eng, and A. M. Ahmad, “Malay Speech Recognition using Slef-Organizing Map and Multilayer Perceptron”, Proceedings of the Postgraduate Annual Research Seminar, 2005.





• [Engelbrecht02] A. P. Engelbrecht, Computational Intelligence: An Introduction, John Wiley & Sons, Ltd., West Sussex, England, 2002.


• [Fogel95] D. B. Fogel, “Review of Computational Intelligence: Imitating Life (book Review)”, Proceedings of the IEEE, Vol. 83, Issue 11, 1995, pp.1588 – 1592.

• [Fogel00a] D. B. Fogel, Evolutionary Computation: Principles and Practice for Signal Processing, SPIE-The international Society for Optical Engineering, Washington, USA., 2000.

• [Fogel00b] D.B. Fogel, Evolutionary Computation: Toward a New Philosophy of Machine Intelligence, IEEE Press, USA., 2000.



• [Fogel95] D. B. Fogel, “Review of Computational Intelligence: Imitating Life (book Review)”, Proceedings of the IEEE, Vol. 83, Issue 11, 1995, pp.1588 – 1592.

• [Fogel00a] D. B. Fogel, Evolutionary Computation: Principles and Practice for Signal Processing, SPIE-The international Society for Optical Engineering, Washington, USA., 2000.

• [Fogel00b] D.B. Fogel, Evolutionary Computation: Toward a New Philosophy of Machine Intelligence, IEEE Press, USA., 2000.





• [Goldberg05] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison Wesley Longman, Inc., USA., 2005.

• [Grabisch94] Grabisch, M. and Nicolas, J-M., “Classification by fuzzy integral: Performance and tests”, Fuzzy Set and Systems, 65, 1994, pp. 255-271.

• [Haykin94] S. Haykin, Neural Networks: A comprehensive Foundation, Macmillan Publishing Company, Inc. New Jersey, USA., 1994.

• [Haykin10] S. Haykin, Neural Networks and Learning Machines, Pearson Education, Inc., USA., 2010.

• [Karaboga05] D. Karaboga, “An Idea based on Honey Bee Swarm for Numerical Optimization”, Techincal Report –TR06, Erciyes University, Engineering Faculty, Computer Engineering Department, 2005.

• [Goldberg05] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison Wesley Longman, Inc., USA., 2005.

• [Grabisch94] Grabisch, M. and Nicolas, J-M., “Classification by fuzzy integral: Performance and tests”, Fuzzy Set and Systems, 65, 1994, pp. 255-271.

• [Haykin94] S. Haykin, Neural Networks: A comprehensive Foundation, Macmillan Publishing Company, Inc. New Jersey, USA., 1994.

• [Haykin10] S. Haykin, Neural Networks and Learning Machines, Pearson Education, Inc., USA., 2010.

• [Karaboga05] D. Karaboga, “An Idea based on Honey Bee Swarm for Numerical Optimization”, Techincal Report –TR06, Erciyes University, Engineering Faculty, Computer Engineering Department, 2005.





• [Keller94] Keller, J. M., Gader P., Tahani, H., Chiang, J-H. and Mohaned, M., “Advances in fuzzy integration for pattern recognition”, Fuzzy Sets and Systems, 65, 1994, pp.273-283.

• [Keller16] Keller, J.M., Liu, D. and Fogel, D.B., Fundamentals of Computational Intelligence:Neural Networks, Fuzzy Systems, and Evolutionary Computation, IEEE Press, Wiley, New Jersey, USA., 2016.

• [Kennedy98] J. Kennedy “The behavior of particles” in VW Porto, N. Saravanan, D. Waagen(eds), Proceedings of the 7th International Conference on Evolutionary Programming, 1998. pp. 581 – 589.

• [Klir95] G. J. Klir and B. Yuan, Fuzzy Stes and Fuzzy Logic: Theory and Applications, Prentice Hall Inc., New Jersey, USA., 1995.

• [Klir97] G. J. Klir, U. H. St. Clair, and B. Yuan, Fuzzy Set Theory: Foundations and Applications, Prentice Hall Inc., New Jersey, USA., 1997.

• [Keller94] Keller, J. M., Gader P., Tahani, H., Chiang, J-H. and Mohaned, M., “Advances in fuzzy integration for pattern recognition”, Fuzzy Sets and Systems, 65, 1994, pp.273-283.

• [Keller16] Keller, J.M., Liu, D. and Fogel, D.B., Fundamentals of Computational Intelligence:Neural Networks, Fuzzy Systems, and Evolutionary Computation, IEEE Press, Wiley, New Jersey, USA., 2016.

• [Kennedy98] J. Kennedy “The behavior of particles” in VW Porto, N. Saravanan, D. Waagen(eds), Proceedings of the 7th International Conference on Evolutionary Programming, 1998. pp. 581 – 589.

• [Klir95] G. J. Klir and B. Yuan, Fuzzy Stes and Fuzzy Logic: Theory and Applications, Prentice Hall Inc., New Jersey, USA., 1995.

• [Klir97] G. J. Klir, U. H. St. Clair, and B. Yuan, Fuzzy Set Theory: Foundations and Applications, Prentice Hall Inc., New Jersey, USA., 1997.





• [Kohavi95] R. Kohavi, “A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection”, International Joint Conference on Artificial Intelligence (IJCAI), 1995

• [Kosko87] B. Kosko, “Fuzzy associative memories”, in Fuzzy Expert Systems, A. Kandel, Ed. Reading, MA: Addison-Wesley, 1987.

• [Kosko88] B. Kosko, “Bidirectional Associative Memories”, IEEE Transactions on Systems, Man, and Cybernetics, vol 18(1), 1988, pp. 49 –60.

• [Koza92] J. R. Koza Genetic Programming: On the Programming of Computers by Means of Natural Selection, The MIT Press, England, 1992.

• [Kreyszig05] E. Kreyszig, Advanced Angineering Mathematics, John Wiley & Sons, Inc., New Yorj USA., 2005.

• [Kruse95] R. Kruse, J. Gebhardt, and F. Klawonn, Foundations of Fuzzy Systems, John Wiley & Sons Ltd., West Sussex, England, 1995.

.

• [Kohavi95] R. Kohavi, “A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection”, International Joint Conference on Artificial Intelligence (IJCAI), 1995

• [Kosko87] B. Kosko, “Fuzzy associative memories”, in Fuzzy Expert Systems, A. Kandel, Ed. Reading, MA: Addison-Wesley, 1987.

• [Kosko88] B. Kosko, “Bidirectional Associative Memories”, IEEE Transactions on Systems, Man, and Cybernetics, vol 18(1), 1988, pp. 49 –60.

• [Koza92] J. R. Koza Genetic Programming: On the Programming of Computers by Means of Natural Selection, The MIT Press, England, 1992.

• [Kreyszig05] E. Kreyszig, Advanced Angineering Mathematics, John Wiley & Sons, Inc., New Yorj USA., 2005.

• [Kruse95] R. Kruse, J. Gebhardt, and F. Klawonn, Foundations of Fuzzy Systems, John Wiley & Sons Ltd., West Sussex, England, 1995.

.




ReferencesReferences• [Mitchell98] M. Mitchell, An Introducton to Genetic Algorithms, MIT

Press, Massachusetts Institute Technology, USA., 1998• [Ohnishi90] N. Ohnishi, A. Okamoto, and N. Sugiem, “Selective

Presentation of Learning Samples for Efficient Learning in Multi-Layer Perceptron”, Proceedings of the IEEE International Joint Conference on Neural Networks, vol 1, 1990, pp. 688 – 691.

• [Rosenblatt58] F. Rosenblatt, “The Perceptron: A propabilistic model for information storage and organization in the brain”, Psychological Review, vol 65, 1958, pp. 386 – 408.

• [Ross04] T. J. Ross, Fuzzy Logic with Engineering Applications, John Wiley & Sons Ltd., West Sussex, England, 2004.

• [Suganthan99] P. N. Suganthan, “Particle swarm optimizer with neighborhood operators”, Proceedings of the IEEE Congress on Evolutionary Computation, 1999, pp. 1958 – 1961.

• [Mitchell98] M. Mitchell, An Introducton to Genetic Algorithms, MIT Press, Massachusetts Institute Technology, USA., 1998

• [Ohnishi90] N. Ohnishi, A. Okamoto, and N. Sugiem, “Selective Presentation of Learning Samples for Efficient Learning in Multi-Layer Perceptron”, Proceedings of the IEEE International Joint Conference on Neural Networks, vol 1, 1990, pp. 688 – 691.

• [Rosenblatt58] F. Rosenblatt, “The Perceptron: A propabilistic model for information storage and organization in the brain”, Psychological Review, vol 65, 1958, pp. 386 – 408.

• [Ross04] T. J. Ross, Fuzzy Logic with Engineering Applications, John Wiley & Sons Ltd., West Sussex, England, 2004.

• [Suganthan99] P. N. Suganthan, “Particle swarm optimizer with neighborhood operators”, Proceedings of the IEEE Congress on Evolutionary Computation, 1999, pp. 1958 – 1961.





• [Sugeno77] Sugeno, M., “ Fuzzy Measures and Fussy Integrals: A survey”, Fuzzy Automata and Decision Processes, Amsterdam, North Holland, 1977, pp. 89-102.

• [Tahani90] Tahani, H. and Keller, J. M., “Information Fusion in Computer Vision using the Fuzzy Integral”, IEEE Trans. on Systems, Man, and Cybernetics, 20(3), 1990, pp.773-741.

• [Theodoridis09] S. Theodoridis and K, Koutroumbas, Pattern Recognition, Academin Press, Elsevier Inc., London, UK., 2009.

• [Thodberg91] H. H. Thodberg, “Improving Generalization of Neural Networks through Prunning”, International Journal of Neural Systems, 1(4), 1991, pp. 317 – 326.

• [Turing50] A. M. Turing, Computing Machinery and Intelligence, Mind, vol 59, 1950, pp. 433 – 460.

• [Sugeno77] Sugeno, M., “ Fuzzy Measures and Fussy Integrals: A survey”, Fuzzy Automata and Decision Processes, Amsterdam, North Holland, 1977, pp. 89-102.

• [Tahani90] Tahani, H. and Keller, J. M., “Information Fusion in Computer Vision using the Fuzzy Integral”, IEEE Trans. on Systems, Man, and Cybernetics, 20(3), 1990, pp.773-741.

• [Theodoridis09] S. Theodoridis and K, Koutroumbas, Pattern Recognition, Academin Press, Elsevier Inc., London, UK., 2009.

• [Thodberg91] H. H. Thodberg, “Improving Generalization of Neural Networks through Prunning”, International Journal of Neural Systems, 1(4), 1991, pp. 317 – 326.

• [Turing50] A. M. Turing, Computing Machinery and Intelligence, Mind, vol 59, 1950, pp. 433 – 460.





• [Wang92] Wang, Z. and Klir, G., “ Fuzzy Measure Theory”, New York, Plenum Press, 1992.

• [Zadeh95] L. A. Zadeh, “Fuzzy Sets”, Information and Control, 8(3), 1965,pp 338 -353.

• [Wang92] Wang, Z. and Klir, G., “ Fuzzy Measure Theory”, New York, Plenum Press, 1992.

• [Zadeh95] L. A. Zadeh, “Fuzzy Sets”, Information and Control, 8(3), 1965,pp 338 -353.




Introduction to Computational IntelligenceIntroduction to Computational Intelligence




HistoryHistory

• Aristotle (384-322 BC) explain and codify styles of deductive reasoningsyllogisms

• Ramon Llull (1235-1316) Ars Magna machine consisting of a set of wheels supposed to be able to answer all questions.

• Gottfried Leibniz (1646 – 1716) calculus philosophicus a universal algebra that can be used to represent all knowledge in a deductive system

• George Boole (1854) developed the foundations of propositional logic part of AI tool

• Gottlieb Ferge (1879) developed the foundations of predicate calculus part of AI tool

• Alan Turing(1950) definition of AIstudy how machinary could be used to mimic processes of the human brain Book called “The chemical Basis of Morphogenesis artificial life

• Aristotle (384-322 BC) explain and codify styles of deductive reasoningsyllogisms

• Ramon Llull (1235-1316) Ars Magna machine consisting of a set of wheels supposed to be able to answer all questions.

• Gottfried Leibniz (1646 – 1716) calculus philosophicus a universal algebra that can be used to represent all knowledge in a deductive system

• George Boole (1854) developed the foundations of propositional logic part of AI tool

• Gottlieb Ferge (1879) developed the foundations of predicate calculus part of AI tool

• Alan Turing(1950) definition of AIstudy how machinary could be used to mimic processes of the human brain Book called “The chemical Basis of Morphogenesis artificial life




HistoryHistory

• Artificial intelligence first coined at the Dartmoutn conference organized by John Maccarthy (1956 )father of AI

• 1956 – 1969– Biological neurons

• Rosenblatt perceptrons• Widrow and Hoff adaline• Minsky and Papert (1969) cause a major set back to artificial neural networks conclude that the extension of simple perceptrons to multilayer perceptrons “is sterile”

• NN hibernation until the mid 1980s• Grossberg, Carpenter, Amari, Kohonen and Fukushimacontinue researching in

NN • Hopfield, Hinton and Rumelhart and McLelland (early and mid 1980s)

resurrection of NN research

• Artificial intelligence first coined at the Dartmoutn conference organized by John Maccarthy (1956 )father of AI

• 1956 – 1969– Biological neurons

• Rosenblatt perceptrons• Widrow and Hoff adaline• Minsky and Papert (1969) cause a major set back to artificial neural networks conclude that the extension of simple perceptrons to multilayer perceptrons “is sterile”

• NN hibernation until the mid 1980s• Grossberg, Carpenter, Amari, Kohonen and Fukushimacontinue researching in

NN • Hopfield, Hinton and Rumelhart and McLelland (early and mid 1980s)

resurrection of NN research




HistoryHistory

• Fraser, Bremermann and Reed (1950s) Genetic algorithm• John Holland father of Evolutionary Computation (EC)genetic

algorithms• Rechenberg (1960s) evolutionary strategies (ES)• Lawrence Fogel evolutionary programmingevolved behavior models• De Jong, Schaffer, Goldberge, Koza, Schwefel, Storn, and Price , important contricutors in EC field

• Gautama Buddha (563 BC) described things in shade of gray starting point of fuzzy logic

• Aristotle2-valued logic the birth of fuzzy logic• Lukasiewicz (1920)3-valued logic• Max Blackquasi-fuzzy sets

• Fraser, Bremermann and Reed (1950s) Genetic algorithm• John Holland father of Evolutionary Computation (EC)genetic

algorithms• Rechenberg (1960s) evolutionary strategies (ES)• Lawrence Fogel evolutionary programmingevolved behavior models• De Jong, Schaffer, Goldberge, Koza, Schwefel, Storn, and Price , important contricutors in EC field

• Gautama Buddha (563 BC) described things in shade of gray starting point of fuzzy logic

• Aristotle2-valued logic the birth of fuzzy logic• Lukasiewicz (1920)3-valued logic• Max Blackquasi-fuzzy sets




HistoryHistory

• Lotfi Zadehcontribute most in the field of fuzzy logicfather of Fuzzy Set

• Mamdani, Sugeno, Takagi and Bezdek famous in the fuzzy field• 1980 dark age for fuzzy research revive again in the late 1980• Pawlak (1991)rough set theory• Eugene N Marais (1871 – 1936) poetintroduce swarm intelligence

Books called “The Soul of the White Ant” and “The Soul of the Ape”• Eberhart and Kennedy (1995) particle swarm optimization• Marco Dorigo (1992)ant colonies

• ETC.• A lot of people are working in the CI field

• Lotfi Zadehcontribute most in the field of fuzzy logicfather of Fuzzy Set

• Mamdani, Sugeno, Takagi and Bezdek famous in the fuzzy field• 1980 dark age for fuzzy research revive again in the late 1980• Pawlak (1991)rough set theory• Eugene N Marais (1871 – 1936) poetintroduce swarm intelligence

Books called “The Soul of the White Ant” and “The Soul of the Ape”• Eberhart and Kennedy (1995) particle swarm optimization• Marco Dorigo (1992)ant colonies

• ETC.• A lot of people are working in the CI field




Connection in CIConnection in CI

CI Study of adaptive mechanisms to enable or facilitateintelligent behavior in complex and changing environments

Artificial immune systems

Neural Networks

Evolutionary Computation

Fuzzy Systems

Swarm Intelligence

Probabilistic Techniques




Neural Networks (NN)Neural Networks (NN)Nucleus

Dendrite

Cell Body

Axon

synapse

Biological Neural




NNNN

f(net)

output

input1

input2

inputp

weight1

weight2

weightp

Artificial neuron




NNNN

input layer

hidden layer

output layer

xp

x2

x1

p-m-2 :p input nodes, m hidden nodes, 2 output nodesNumber of hidden nodes or hidden layers will be shown here p-m1-m2-2 means that there are

2 hidden layers: one with m1 nodes and the second one is with m2 nodesNumber of output nodes normally corresponds to the number of classes

p-4-2

input layer

1st hiddenlayer

output layer

xp

x2

x1

2nd hiddenlayer

p-4-3-2




Fuzzy SystemsFuzzy Systems

• Not only 0 or 1, there is a gray area• Unsharp boundary • Cope with uncertainty• Approximate reasoning

• Not only 0 or 1, there is a gray area• Unsharp boundary • Cope with uncertainty• Approximate reasoning




Evolutionary ComputationEvolutionary Computation• Individualchromosomeinherit characteristic• Population of chromosome• Each characteristic --<gene– allel(gene value)• Each generationindividual compete to reproduce offspring• Best survival capability have the best chance to reproduce• Crossovercombine part of parents to generate offspring• Mutationalter some of allele of the chromosome• Survuval strength measured using fitness function objective function• Each generation individual culling or survive• Behaviorphenotype influence evolutionary process in genetic change and evolve

separately

• Individualchromosomeinherit characteristic• Population of chromosome• Each characteristic --<gene– allel(gene value)• Each generationindividual compete to reproduce offspring• Best survival capability have the best chance to reproduce• Crossovercombine part of parents to generate offspring• Mutationalter some of allele of the chromosome• Survuval strength measured using fitness function objective function• Each generation individual culling or survive• Behaviorphenotype influence evolutionary process in genetic change and evolve

separately




Evolutionary ComputationEvolutionary Computation• EC algo

– Genrtic algorithmmodel genetic evolutiona– Genetic programmingsimilar to GA but individual are program– Evolutionary programmingderived from the simulation of adaptive behavior in

evolution (phenotype evolution)– Evolution strategiesmodel strategic parameters that control variation in

evolution– Differential evolutionsimilar to GA except reproduction mechanism used– Cultural evolution model the evolution of culture of a population and how the

culture influences the genetic and phenotypic evolution of individual– Coevolution “dump” individual evolve through cooperation or in competition

with one another, acquiring the necessary characteristics to survive

• EC algo– Genrtic algorithmmodel genetic evolutiona– Genetic programmingsimilar to GA but individual are program– Evolutionary programmingderived from the simulation of adaptive behavior in

evolution (phenotype evolution)– Evolution strategiesmodel strategic parameters that control variation in

evolution– Differential evolutionsimilar to GA except reproduction mechanism used– Cultural evolution model the evolution of culture of a population and how the

culture influences the genetic and phenotypic evolution of individual– Coevolution “dump” individual evolve through cooperation or in competition

with one another, acquiring the necessary characteristics to survive




Swarm IntelligenceSwarm Intelligence• Study of colonies or swarm of social organisma• Study of social behavior of organisms (individuals) in swarm prompted the design f

very efficient optimization and clustering algorithm• Particle swarm optimization (PSO) global optimization approach modeled on social

behavior of bird flocks– A population based search procedure where the individuals (particles) are grouped

into a swarm– Particleindividualcandidate solution to the optimization problem

• Flown through the multidimensional search spaceadjusting its position in search space according to its own experience and that of neighboring particles

• Best position encounter by itself and that of its neighborposition itself to the global minimum• Performancemeasured according to a predefined fitness function which related to the problem

being solved

• Study of ant coloniesmodeling of pheromone depositing by ants in their search for the shortest paths to food sources resulted in the development of shortest path optimization algorithm

• Study of colonies or swarm of social organisma• Study of social behavior of organisms (individuals) in swarm prompted the design f

very efficient optimization and clustering algorithm• Particle swarm optimization (PSO) global optimization approach modeled on social

behavior of bird flocks– A population based search procedure where the individuals (particles) are grouped

into a swarm– Particleindividualcandidate solution to the optimization problem

• Flown through the multidimensional search spaceadjusting its position in search space according to its own experience and that of neighboring particles

• Best position encounter by itself and that of its neighborposition itself to the global minimum• Performancemeasured according to a predefined fitness function which related to the problem

being solved

• Study of ant coloniesmodeling of pheromone depositing by ants in their search for the shortest paths to food sources resulted in the development of shortest path optimization algorithm




Introduction to Artificial Neural NetworksIntroduction to Artificial Neural Networks




Neural Networks (NN)Neural Networks (NN)• Mapping function form I input space to K output space• Mapping function form I input space to K output space

: I KNNf

f(net)

o

x1

x2

xI

w1

w2

wI

orf(net)

o

x1

x2

xI

w1

w2

wI

1

I

i ii

net x w

or and called bias or threshold1

iI

wi

i

net x




NNNN

• Activation function or and

– linear functionwhere is a constant

• Activation function or and

– linear functionwhere is a constant

0f 1f 1f

f v v

v

f(v)




NNNN

• Step function or unit step function

normally 11 and 20 or 1

• Step function or unit step function

normally 11 and 20 or 1 1

2

if

if

vf v

v

v

f(v)

1

2




NNNN• Ramp function• Ramp function

if

if

if

v

f v v v

v

v

f(v)




NNNN

• Sigmoid function – Logistic function

– Hyperbolic tangent function

• Sigmoid function – Logistic function

– Hyperbolic tangent function

1

1 expf v

av

0

0

0.2

0.4

0.6

0.8

1

v

f(v)

1 exp 2tanh 1

2 1 exp 1 exp

v vf v

v v

0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

v

f(v)




NNNN

• Gaussian function

where mean and standard deviation

• Gaussian function

where mean and standard deviation 22exp

vf v

00

v

f(v)




Multi-layer perceptronsMulti-layer perceptrons• Example: AND Logic input vector x1,x2t with output y =0 are in the same class (1st class), one

with output 1 are in another class (2nd class) use unit step function with 11, 20 and 0as activation function

• Example: AND Logic input vector x1,x2t with output y =0 are in the same class (1st class), one with output 1 are in another class (2nd class) use unit step function with 11, 20 and 0as activation function

x1 x2 Y(desired output)

0 0 00 1 01 0 01 1 1

o

x1

x2

1

1

1

1.5

1 2 1.5v x x

x1

x2

1

1

0 1.5

1.5

x1x21.50

Decision boundary

Gray area 2nd class

White area 1st class

x1 x2 v o (program output)

0 0 1.5 00 1 0.5 01 0 0.5 01 1 0.5 1




Multi-layer perceptronsMulti-layer perceptrons• Example : XOR Logic input vector x1,x2t with output y =0 are in the same class (1st class),

one with output 1 are in another class (2nd class) use unit step function with 11, 20 and 0 as activation function

• Example : XOR Logic input vector x1,x2t with output y =0 are in the same class (1st class), one with output 1 are in another class (2nd class) use unit step function with 11, 20 and 0 as activation function

x1 x2 Y(desired output)

0 0 00 1 11 0 11 1 0

x1 x2 v1o1 v2 o2 o (program

output)0 0 1.5 0 0.5 0 0.50 1 0.5 0 0.5 1 0.51 0 0.5 0 0.5 1 0.51 1 0.5 1 1.5 1 1.5

o1

1

o

x1

x2

1

1

1

1.5

1

1

1

0.5

0.5

2

1

1

2

3

o2




x1

x2

1

1

0 1.5

1.5

x1x21.50Output = 0

Output = 1

x1

x2

1

1

0 0.5

0.5

x1x20.50

Output = 1

Output = 0

x1

x2

1

1

0 0.5

0.5

Output = 0

1.5

1.5

Output = 1

Output = 0




x1

x2

1

2

3

4

5

x3

g1 g2

g3 g4

One hidden node create one decision boundary 4hidden nodes creates 4 decision boundary theseboundary is combined at the output node to createdecision for the decision making




w0w(t)

Ew

w(t1)

Ew

Ew

Gradient descent learning ruleDirection of gradient vector (Ew is in the increasingdirectionif we invert the direction we will have

New weight that is in the decreasing direction of the error space• if is small, the transient response of the algorithm will be

overdamped• If is large , the transient response of the algorithm will be

underdamped• If is larger than the critical value, the algorithm will be

unstable and may be diverged

( 1) ( )w t w t E E

1t t w w w E




Backpropagation 2 passes;Forward pass and Backward pass

21

2j

j C

t e t

E

1

1 N

avn

nN

E E




Multilayer PerceptronsMultilayer Perceptrons

• Initialization• Forward Computation

– Output from neuron j in layer l

wherewhere is the output signal of neuron i in the previous layer l1and is the synaptic weight of neuron j in layer l that is fed from neuron i in layer l1if l = 1, and if l = L,compute the error: ej(n) = dj(n) oj(n)

• Initialization• Forward Computation

– Output from neuron j in layer l

wherewhere is the output signal of neuron i in the previous layer l1and is the synaptic weight of neuron j in layer l that is fed from neuron i in layer l1if l = 1, and if l = L,compute the error: ej(n) = dj(n) oj(n)

( 1) ( )liy n

( ) ( )ljiw n

1( ) ( ) ( 1)

0( ) ( ) ( )

lml l lj ji i

iv n w n y n

( ) ( )l

j j jy n v n

(0) ( )j jy n x n (0) ( )j jy n o n




• Backward computation– Local gradients

where denotes differentiation with respect to the argument – Update weights

• Iteration: until the stopping criterion is met

• Backward computation– Local gradients

where denotes differentiation with respect to the argument – Update weights

• Iteration: until the stopping criterion is met

1 1

for neuron in output layer

for neuron in output layer

L Lj j j

lj l ll

j j k kjk

e n v n j Ln

v n n w n j l

j

11 1l l l l lji ji ji j jw n w n w n n y n

11 1l l l l l lji ji ji ji j jw n w n w n w n n y n




• For

output layer

hidden layer

• For

output layer

hidden layer

( ) ( ) ( )( )

1

1 exp

l l lj j j l

j

y t v tv t

( )( )( ) ( )

( ) 2( )

exp

1 exp

lljj l l

j jl lj j

v ty tv t

v t v t

( ) ( )( ) ( )

1 11

1 exp 1 exp

l lj j l l

j j

v tv t v t

( ) ( ) ( ) ( )1l l l lj j j jv t y t y t

( ) ( ) 1L Lj j j jt e t o t o t

( ) ( ) ( ) ( 1) ( 1)1l l l l lj j j k kj

k

t y t y t t w t




• For

output layer

hidden layer

• For

output layer

hidden layer

( )( ) ( ) ( )

( )

( ) 2tanh 1

2 1 exp

ljl l l

j j j lj

v ty t v t

v t

( )( )( ) ( )

( ) 2( )

2exp

1 exp

lljj l l

j jl lj j

v ty tv t

v t v t

( ) ( )( ) ( )

2 11

1 exp 1 exp

l lj j l l

j j

v tv t v t

( ) ( ) ( ) ( )2 1l l l lj j j jv t y t y t

( ) ( ) 2 1L Lj j j jt e t o t o t

( ) ( ) ( ) ( 1) ( 1)2 1l l l l lj j j k kj

k

t y t y t t w t




• Example: suppose at iteration t and and 0.2, 0.1. Find

• Example: suppose at iteration t and and 0.2, 0.1. Find

1

1t

x 0

1t

d (1)

32 1w t

x1

x2

1

2

3

1

2

-1.2

0.9

-0.6

-0.5

0.8

0.3-0.8

-2.1

-1.6

0.5-0.6

0.2-0.3

-0.20.5

-1.1

-0.7

l0 l1 l2

x1

x2

1

2

3

1

2

-1.0

-0.2

-0.2

-0.1

0.2

0.2-0.3

0.1

0.01

0.2-0.54

-1.10.2

-0.10.4

-0.5

1.2

l0 l1 l2

tt 1




2

(1) (1)1 1

0

1.2 (0.3)(1) ( 0.6)(1) 1.5i ii

v t w x

(1)1 (1)

1

1 1( ) 0.1824

1 exp(1.5)1 expy t

v t

2

(1) (1)2 2

0

0.9 ( 0.8)(1) (0.5)(1) 0.6i ii

v t w x

(1)2

1( ) 0.6457

1 exp( 0.6)y t

2

(1) (1)3 3

0

0.6 (0.5)(1) ( 1.6)(1) 1.7i ii

v t w x

(1)3

1( ) 0.1545

1 exp(1.7)y t

3

(2) (2) (1)1 1

0

0.5 (0.2)(0.1824) ( 0.2)(0.6457) ( 1.1)(0.1545) 0.7626i ii

v t w y

(2)1

1( ) 0.3181

1 exp(0.7626)y t

(2)1 ( ) 0 0.3181 0.3181e t

3

(2) (2) (1)2 1

0

0.8 ( 0.3)(0.1824) (0.5)(0.6457) ( 0.7)(0.1545) 0.96i ii

v t w y




(2)2

1( ) 0.7231

1 exp( 0.96)y t

(2)2 ( ) 1 0.7231 0.2769e t

(2) (2)1 1 1 11 ( 0.3181) 0.3181 1 0.3181 0.0690t e t o t o t

(2) (2)2 2 2 21 (0.2769) 0.7231(1 0.7231) 0.0554t e t o t o t

2

(1) (1) (1) (2) (2)3 3 3

1

1 k kjk

t y t y t t w t

(1)3 0.1545 1 0.1545 0.0690 1.1 0.0554 0.7 0.0048t

(2) (2) (2) (1)12 12 1 21w t w t t y t

(2)12 0.1 0.2 0.1 0.2 0.0690 0.6457 0.0189w t

(2) (2) (2)12 12 121 0.2 0.0189 0.2189w t w t w t

(1) (1) (1)32 32 3 21w t w t t x t

(1)32 0.1 1.6 0.01 0.2 0.0048 1 0.16w t

(1) (1) (1)32 32 321 1.6 0.16 1.76w t w t w t




• Generalization– Overtrainlook up tabledo not want– Properly fitwant– Factor

• Size of training data set• Architecture of NN• Physical complexity of the problem at hand

• Performance factor– Data preparation

• Missing valuediscard if have a lot of train data or replace the missing value with the average of that feature or with the most frequent

• Coding input value• Outlier discard that outlier if known and have a lot of train data if not,

adjust the objective function so that it can cope with the outlierrobustness

• Generalization– Overtrainlook up tabledo not want– Properly fitwant– Factor

• Size of training data set• Architecture of NN• Physical complexity of the problem at hand

• Performance factor– Data preparation

• Missing valuediscard if have a lot of train data or replace the missing value with the average of that feature or with the most frequent

• Coding input value• Outlier discard that outlier if known and have a lot of train data if not,

adjust the objective function so that it can cope with the outlierrobustness




• Scaling and normalization if some features dominate other featuresnormalize each feature with its own mean and standard deviation so that every features are in the same range

k feature of sample i where

and

– output value should be (0.1 or 0.9) or (-0.9 or 0.9) because if we use logistic function or hyperbolic tangent the computed valued will never goes to 0 or 1 (or -1 or 1) suppose there are 3 class: sample in class 1 will have the desire output be 0.9 0.1 0.1, that in class 2 will be 0.1 0.9 0.1 and that in class 3 will be 0.1 0.1 0.9 if use tanh, it should be 0.9 -0.9 0.9 (class 1), -0.9 0.9 -0.9 (class 2) and -0.9 -0.9 0.9 (class 3)

• Or – Noise injectionaround decision boundary– Training set manipulation selective presentationtypical pattern and

confusing pattern need priori information

• Scaling and normalization if some features dominate other featuresnormalize each feature with its own mean and standard deviation so that every features are in the same range

k feature of sample i where

and

– output value should be (0.1 or 0.9) or (-0.9 or 0.9) because if we use logistic function or hyperbolic tangent the computed valued will never goes to 0 or 1 (or -1 or 1) suppose there are 3 class: sample in class 1 will have the desire output be 0.9 0.1 0.1, that in class 2 will be 0.1 0.9 0.1 and that in class 3 will be 0.1 0.1 0.9 if use tanh, it should be 0.9 -0.9 0.9 (class 1), -0.9 0.9 -0.9 (class 2) and -0.9 -0.9 0.9 (class 3)

• Or – Noise injectionaround decision boundary– Training set manipulation selective presentationtypical pattern and

confusing pattern need priori information

ˆ ik kik

k

x xx

1

1 for 1,2,...,

N

k iki

x x k pN

22

1

1

1

N

k ik ki

x xN




– Initialize weight with

– Learning rate and momentum rate

– Initialize weight with

– Learning rate and momentum rate

1 1,

fanin fanin




Self organizing feature map (SOFM)Self organizing feature map (SOFM)

input (x)

winning neuron




• Competitive

Input neuron j withThere are l neurons

Winning neuron

• Cooperation

– hj,i(x) : symmetric about the maximum point defined by dji=0 (at winning neuron)– Amplitude of hj,i(x) decrease monotonically with increasing lateral distance dji , decaying

to 0 for dji=necessary condition for convergence

• Competitive

Input neuron j withThere are l neurons

Winning neuron

• Cooperation

– hj,i(x) : symmetric about the maximum point defined by dji=0 (at winning neuron)– Amplitude of hj,i(x) decrease monotonically with increasing lateral distance dji , decaying

to 0 for dji=necessary condition for convergence

1 2, , ,t

mx x xx 1 2, , ,t

j j j jmw w w w

argmin for jj

i x j x w A

1

hj,i(x)

i(x)

•hj,i(x

)

1-d lattice2-d lattice

22,j i j id r r




Where between neuron j and winning neuron i in case of 1-d lattice and for 2-d lattice between ri and rj are

vector of neuron i and j

2,

, 2=exp for 2

j ij i

dh j

x A

00

1

dj,i

hj,i

2

01

exp for 0,1,2, ,n

n n

2,

, 2=exp for 0,1,2, ,2

j ij i

dh n n

n

x

,j id j i 22

,j i j id r r




• Synaptic adaptationsince update in 1 directionmight saturate, hence,include forgetting term (g(yj))

and

If neighborhood function is rectangle function

If neighborhood function is guassian

• Synaptic adaptationsince update in 1 directionmight saturate, hence,include forgetting term (g(yj))

and

If neighborhood function is rectangle function

If neighborhood function is guassian

for j j j jy g y j w x w A

This image cannot currently be displayed.

,j j iy h x

, for j j i jh j xw x w A

if is inside neighborhood function for

0 if is outside neighborhood function

jj

jj

j

x ww A

This image cannot currently be displayed.

if j is inside the neighborhood function1

if j is not inside the neighborhood function

j jj

j

t t t tt

t

w x ww

w

02

exp for 0,1,2, ,n

n n




• 2 phase– Self-organizing or ordering phase

• might start from 0.1and decrease till the value is around 0.01 never goes to 0• Neighborhood function start from all neuron centered on the winning neuron

and shrink slowly with time– Convergence phase

• maintained at a small value on the order of 0.01 do not decrease to 0• Neighborhood function contain only the nearest neighbor of a winning neuron

eventually reduce to 1 or 0 neighbor

• 2 phase– Self-organizing or ordering phase

• might start from 0.1and decrease till the value is around 0.01 never goes to 0• Neighborhood function start from all neuron centered on the winning neuron

and shrink slowly with time– Convergence phase

• maintained at a small value on the order of 0.01 do not decrease to 0• Neighborhood function contain only the nearest neighbor of a winning neuron

eventually reduce to 1 or 0 neighbor




• Example

Input learning rate 0.1 use and hji(x) has the radius of 1

d(1) = 1.5033, d(2) =1.3038, d(3) = 0.1414, d(4) = 2.0616, d(5) = 3.4713Winning neuron is neuron 3 (i(x) = 3) , neighbor with radius of 1 will be node 2 and 4 only, hence, update weight at nodes 2, 3,and 4

• Example

Input learning rate 0.1 use and hji(x) has the radius of 1

d(1) = 1.5033, d(2) =1.3038, d(3) = 0.1414, d(4) = 2.0616, d(5) = 3.4713Winning neuron is neuron 3 (i(x) = 3) , neighbor with radius of 1 will be node 2 and 4 only, hence, update weight at nodes 2, 3,and 4

1 2 3 4 5

x

1

0.5

0.9

w 2

0.3

0.1

w 3

0.9

0.9

w4

0.3

0.6

w 5

1.3

1.6

w

1

1

x 2 21 1 2 2d x w x w

2

0.3 1 0.3 0.370.1

0.1 1 0.1 0.01

w 3

0.9 1 0.9 0.910.1

0.9 1 0.9 0.91

w

4

0.3 1 0.3 0.170.1

0.6 1 0.6 0.44

w 1

0.5

0.9

w5

1.3

1.6

w

1 2 3 4 5

x

hj,i(x)winning neuron




• SOFM properties– Approximation of the input space– Topological ordering– Density matching– Feature selection

• SOFM properties– Approximation of the input space– Topological ordering– Density matching– Feature selection

x (input vector from linear predict code)

hidden layer

output layer




Introduction to Fuzzy SystemsIntroduction to Fuzzy Systems




Fuzzy systemsFuzzy systems

• Uncertainty and complexity– Driving in the unfamiliar condition

• Natural language– Cold: people in the north and in the south know what cold is but when it

happens people in these two area will say differently people in the north might say that 10 degree celsius is cold but people in the south might say that 20 degree celsius is cold

• Heap paradox

• Uncertainty and complexity– Driving in the unfamiliar condition

• Natural language– Cold: people in the north and in the south know what cold is but when it

happens people in these two area will say differently people in the north might say that 10 degree celsius is cold but people in the south might say that 20 degree celsius is cold

• Heap paradox




Fuzzy Set FoundationFuzzy Set Foundation

• Characteristic functioncrisp set

• Membership function fuzzy setA: X 0,1 or A: X 0,1

• Characteristic functioncrisp set

• Membership function fuzzy setA: X 0,1 or A: X 0,1

ana1 a2

1

a-1 a0 an+10

A(x)

X

X

x1x2

xn

0 1

A




0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

age

mem

bers

hip

valu

eMembership function of“old”

height

weight

0.3

0.5

0.9

Membership function of“big”: Countour diagram




No. Education level0 No education

1 Elementary school2 High school

3 2-year college4 Bachelor’s degree

5 Master’s degree6 Doctoral degree

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

education level

mem

bers

hip

valu

e

little education highly education

very highly education

Universal set 3 fuzzy sets (“little”, “high” and “very high” education)




name (symbol) Membership value in fuzzy set A

Carry (x1) 0.8Bill (x2) 0.3J-H (x3) 0.5

Wabei (x4) 0.9

0.8 0.3 0.5 0.9

Carry Bill J-H Wabei A

( )x

x

AA

( )

x

x

x A

A

How to write membership function as an equation

discrete continuous

How to write membership function




Geometric representation

x1

x2

(1,0)

(1,1)(0,1)

(0,0)

A

0.5

0.3

universal set

1 2

0.5 0.3

x xA

Every point in this area are fuzzy setincluding 4corner pointsFuzzy empty set




Analytic representation : function

A

5 5 6

( ) 7 6 7

0 else

x x

x x x

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

numberm

embe

rshi

p va

lue

example

Analytic representation




a

b

s s

A1

( )

0 else

x ab a s x a s

x s

a b dc

e

A( )

0 else

a x ea x b

a be b x c

xd x e

c x dd c

b

c

a

A

2

( )

x a

bx ce

Triangular shape

Trapezoidal shape

Bell-shape




a

0.5

b c

1

2

2

0 0

1

2

11

2

x a

x ax a x b

b a

x cb x c

c b

S

a

0.5

b c

1

d e f

-function is from S-function + reflected of S-function

S-function




Interval-valued fuzzy set

A : ([0,1])X

0,1 denote the family of all closed interval of real number in [0,1] (0,1 P0,1)

1

a

2

A(x)

A(a) = [1, 2]




Type-2 fuzzy set

denote the set of all ordinary fuzzy sets that can be defined within the universal [0,1]

: ([0,1])X A P

([0,1])P

1a

2

A(x)

3

4

1Ia(y)

y

b1

2

3

4

y

Ib(y)1

A(a) = fuzzy set (1, 2, 3, 4) A(b) = fuzzy set (1, 2, 3, 4)




Level 2 fuzzy set

denote fuzzy power set of X( )XP : ([0,1])X A P

‘todd

ler’

‘young’‘c

hild

’

‘teen

age

’

‘child’

0 5 10

15




Standard fuzzy complement

1 x x xA A X

Ax express degree to which x belong to fuzzy set A

express degree to which x does not belong to fuzzy set A xA

0 50 100 15010 20 30 40 60 70 80 90 110 120 130 1401400

0.2

0.4

0.6

0.8

1

0.1

0.3

0.5

0.7

0.90.9

credit hours

mem

bers

hip

valu

e

Inexperience

Experience




Standard fuzzy union

ABx maxAx, Bx

patient high blood pressure(A)

High fever (B) High blood pressure or high

fever (AB)1 1 1 12 0.5 0.6 0.63 1 0.1 1

n 0.1 0.7 0.7

Do not satisfy the law of excluded middle

will not equal to universal setxA A




Standard fuzzy intersection

ABx minAx, Bx

river Long river (A) Nevigable (B) Long and nevigable (AB)

Amazon 1 0.8 0.8Nile 0.9 0.7 0.7

Yang-Tsi 0.8 0.8 0.8Danube 0.5 0.6 0.5Rhine 0.4 0.3 0.3

Do not satisfy the law of contradiction

will not equal to fuzzy empty setxA A




Inclusion: AB if AxBxxX

Equality: AB if Ax BxxX

Unary operation: Aax Axaif a>1more specific, if a <1less specific

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

age

mem

bers

hip

valu

e

Old

Very Old

Somewhat Old

Very old are from old2

somewhat old are from old0.5




Support of fuzzy set A (supp(A) ):

supp(A) xX Ax>0

Height of fuzzy set A (h(A)) :

Largest value of membership value obtained by any element in that set that is >0

If hA1 normal fuzzy setIf hA<1 supnormal fuzzy set

If hA>1 supernormal fuzzy set

supx

h xA A

Core of fuzzy set A (core(A)) :

coreAxX AxhA since height is h(A) then we can havecoreAxX Ax = hA




-cut of fuzzy set A (A ): A xX Ax

Strong -cut of fuzzy set A (+A ): +A xX Ax>

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

mem

bers

hip

valu

e

0E 0,100 or 0.2E 30,100 or 1E 70,100 (core(E))

0+E 20,100 (supp(E)) or 0.2+E 30,100 or 1+E

If 1<2 then 1A2Aand 1A2A2A , 1A2A1A

If 1<2 then 1+A2+Aand 1+A2+A2+A , 1+A2+A1+A




Level set of fuzzy set A (LA or A ): LA A Ax=; xX

Special fuzzy set from -cut (A)

Ax Ax xX

ExampleA 0.2/x1 + 0.4/x2 + 0.6/x3 + 0.8/x4 + 1/x5

We have LA 0.2, 0.4, 0.6, 0.8, 1

-cut of A will be0.2A x1, x2, x3, x4, x5 1/ x1 + 1/ x2 + 1/ x3 +1/ x4 +1/ x50.4A 0/ x1 + 1/ x2 + 1/ x3 +1/ x4 +1/ x50.6A 0/ x1 + 0/ x2 + 1/ x3 +1/ x4 +1/ x50.8A 0/ x1 + 0/ x2 + 0/ x3 +1/ x4 +1/ x51A 0/ x1 + 0/ x2 + 0/ x3 +0/ x4 +1/ x5

Special fuzzy set will be0.2A 0.2/ x1 + 0.2/ x2 + 0.2/ x3 +0.2/ x4 +0.2/ x5

0.4A 0/ x1 + 0.4/ x2 + 0.4/ x3 +0.4/ x4 +0.4/ x5

0.6A 0/ x1 + 0/ x2 + 0.6/ x3 +0.6/ x4 +0.6/ x5

0.8A 0/ x1 + 0/ x2 + 0/ x3 +0.8/ x4 +0.8/ x5

1A 0/ x1 + 0/ x2 + 0/ x3 +0/ x4 +1/ x5




Decomposition theorem

From special fuzzy set in the previous example0.2A 0.2/ x1 + 0.2/ x2 + 0.2/ x3 +0.2/ x4 +0.2/ x5

0.4A 0/ x1 + 0.4/ x2 + 0.4/ x3 +0.4/ x4 +0.4/ x5

0.6A 0/ x1 + 0/ x2 + 0.6/ x3 +0.6/ x4 +0.6/ x5

0.8A 0/ x1 + 0/ x2 + 0/ x3 +0.8/ x4 +0.8/ x5

1A 0/ x1 + 0/ x2 + 0/ x3 +0/ x4 +1/ x5

We can see that (0.2A0.4A0.6A0.8A1A) A




First theorem:

For and A ,

where Ax Ax , xX and is a standard fuzzy union

xP

0,1

A A

A

A

A

The standard fuzzy union of 3 special fuzzy sets A




Second theorem:

For and A ,

where +Ax +Ax , xX and is a standard fuzzyunion

xP

0,1

A A

Third theorem:

For and A ,

where Ax Ax , xX and is a standard fuzzy union

xP

A

A A




A 0, 0.3, 0.6, 1 and 0A

therefore A is a union of 0.3A0.6A and 1A

This example shows that fordiscrete fuzzy set only 1st and 3rd

theorem will work fine




Convex fuzzy set

A is a convex fuzzy set if

Meaning that membership values of all the points on the lineconnecting and is bigger than or equal to the minimummembership values of and

Or

If A is a convex fuzzy set then (1) -cut for all are notdisconnected, and (2)each -cut is convex in crisp sense

1 min , where 0 1r s r s A A A

r

rs

s




height

weight

0.90.5

0.3

r

s

All the points of the dashedline have a membershipvalues bigger than or equal tothe minimum membershipvalue of vector r and vector s

r

s

height

weight

Not convex fuzzy set

Convex fuzzy set




Fuzzy Inference SystemFuzzy Inference System

• Hedge– Dilation: HAx = Ax1/2more or less of A– Concentration: HAx = Ax2 very A– Plus: HAx = Ax1.25

– Intensification:

slightly

– Vague

seldom

• Hedge– Dilation: HAx = Ax1/2more or less of A– Concentration: HAx = Ax2 very A– Plus: HAx = Ax1.25

– Intensification:

slightly

– Vague

seldom

2

2

12 if 0,

2

1 2 1 otherwise

x xx

x

A AHA

A

5.0)( if 2

)(1 1

5.0)( if 2

)(

)(xx

xx

xAA

AA

HA




• Generalized modus ponenimplication: If is A, then Y is Bpremise: is Aconclusion: Y is B

• Generalized modus ponenimplication: If is A, then Y is Bpremise: is Aconclusion: Y is B

x X

y

Y yf(x)

AX

B

Y yf(x)

x we get Yyfx is in A we get Y in

B yY yfx, xA




u X

B

Y

R

AX

B

Y

R

supmin , ,B A Rx X

y x x y

X X X

Relation R on XY if u and R we get YB where ByY<x,y>R

Relation R on XY if A we get YBwhereB yY <x,y>R, xA




X

B

YR

11 A

If R is a fuzzy relation on XY where Aand A are fuzzy sets on X, B and B arefuzzy sets on Y

If R (relation between A and B) and A aregiven, we can find B using

' supmin , ,x X

y x x y

B A R

Or BAR operator is a composition operatorHence, this is called compositional rule of inference




• Exampleimplication: if x and y are approximately equalpremise: x is littleconclusion: ?

Suppose little A 1,1, 2,0.6, 3,0.2, 4,0Suppose

if x and y are approximately equal

From B AR

• Exampleimplication: if x and y are approximately equalpremise: x is littleconclusion: ?

Suppose little A 1,1, 2,0.6, 3,0.2, 4,0Suppose

if x and y are approximately equal

From B AR

1.0 0.5 0.0 0.0

0.5 1.0 0.5 0.0

0.0 0.5 1.0 0.5

0.0 0.0 0.5 1.0

R

1.0 0.5 0.0 0.0

0.5 1.0 0.5 0.01.0 0.6 0.2 0.0

0.0 0.5 1.0 0.5

0.0 0.0 0.5 1.0

B B 1.0 0.6 0.5 0.2Max-min composition




• Several ways to create relation from implication:

– If is A, then Y is B for xX and yY , relation will beRx,y IAx, By

Where I is fuzzy implication• Lukasiewicz: IAx,By = min1, 1AxBy• Zadeh: IAx,By max1Ax, minAx,By• Kleen-Dienes: IAx,By max1Ax,By• Mamdani (correlation-min): IAx,By minAx,By• Correlation-product: IAx,By AxBy• Goedel:

• Several ways to create relation from implication:

– If is A, then Y is B for xX and yY , relation will beRx,y IAx, By

Where I is fuzzy implication• Lukasiewicz: IAx,By = min1, 1AxBy• Zadeh: IAx,By max1Ax, minAx,By• Kleen-Dienes: IAx,By max1Ax,By• Mamdani (correlation-min): IAx,By minAx,By• Correlation-product: IAx,By AxBy• Goedel:

1 if,

if

x yI x y

y x y

A BA B

B A B




– If is A, then Y is BIf is A, then Y is B else V is UNKNOWN

where , Y and V are variables in X, Y and V and A, B and UNKNOWN

are fuzzy sets on X, Y and V

R AB UNKNOWN minimum, maximum

– If is A, then Y is B else Z is C

R AB C

– If is A, then Y is BIf is A, then Y is B else V is UNKNOWN

where , Y and V are variables in X, Y and V and A, B and UNKNOWN

are fuzzy sets on X, Y and V

R AB UNKNOWN minimum, maximum

– If is A, then Y is B else Z is C

R AB C

A

A




• ExampleA 1/1 0.4/2 , B 0.4/2 1/3 and C 1/1 0.6/2

If is A, then Y is B else Z is C

• ExampleA 1/1 0.4/2 , B 0.4/2 1/3 and C 1/1 0.6/2

If is A, then Y is B else Z is C

1.0 0.0

0.4 0.0 0.4 1.0 0.6 1.0 0.6 0.0

0.0 1.0

R

0.0 0.4 1.0 0.0 0.0 0.0

0.0 0.4 0.4 0.6 0.6 0.0

0.0 0.0 0.0 1.0 0.6 0.0

R

0.0 0.4 1.0

0.6 0.6 0.4

1.0 0.6 0.0

R




If is A, then Y is B If is A, then Y is B

1.0 0.0

0.4 0.0 0.4 1.0 0.6 1.0 1.0 1.0

0.0 1.0

R

0.0 0.4 1.0 0.0 0.0 0.0

0.0 0.4 0.4 0.6 0.6 0.6

0.0 0.0 0.0 1.0 1.0 1.0

R

0.0 0.4 1.0

0.6 0.6 0.6

1.0 1.0 1.0

R




Fuzzy Control: Simple Diagram




• Example: Inverted pendulum

input: angle and angular velocity output: current vt

membership functions same name for , and vt

negative large (NL), negative medium (NM), negative small (NS), zero (ZE), positive small (PS), positive medium (PM), and positive large (PL)

rule j: if is Aj and is Bj then vt is Cj

e.g. if is NL and is NL then vt is PL (NL,NL;PL)or if is ZE and is ZE then vt is ZE (ZE,ZE;ZE)

• Example: Inverted pendulum

input: angle and angular velocity output: current vt

membership functions same name for , and vt

negative large (NL), negative medium (NM), negative small (NS), zero (ZE), positive small (PS), positive medium (PM), and positive large (PL)

rule j: if is Aj and is Bj then vt is Cj

e.g. if is NL and is NL then vt is PL (NL,NL;PL)or if is ZE and is ZE then vt is ZE (ZE,ZE;ZE)

motor




NL NM NS ZE PS PM PL

NL PL

NM PM

NS PS

ZE PL PM PS ZE NS NM NL

PS NS

PM NM

PL NL

Total number of rules 7 7 7 =343 but some of rules do not make sense.Hence, the number of usable rules will be less than that




• Fuzzy associative memory (FAM)rule 1: If is A1, then Y is B1 (A1,B1)R1

rule 2: If is A2, then Y is B2 (A2,B2)R2

rule n: If is An, then Y is Bn (An,Bn)Rn

premise: is Aconclusion: Y is B

B w1B1 w2B2 ... wnBn where Bi ARi

• Fuzzy associative memory (FAM)rule 1: If is A1, then Y is B1 (A1,B1)R1

rule 2: If is A2, then Y is B2 (A2,B2)R2

rule n: If is An, then Y is Bn (An,Bn)Rn

premise: is Aconclusion: Y is B

B w1B1 w2B2 ... wnBn where Bi ARi




Supposed: Ai a1/x1 a2/x2 ... am/xm Ai a1, a2,..., am

Correlation-min:

OR

Supposed: Ai a1/x1 a2/x2 ... am/xm Ai a1, a2,..., am

Correlation-min:

OR

Ti i i R A B

1

2

i

ii

m i

a

a

a

B

BR

B

T T T1 2i i i p ib b b R A A A




Correlation-product :

OR

Correlation-product :

OR

Ti i i R A B

1

2

i

ii

m i

a

a

a

B

BR

B

T T T1 2i i i p ib b b R A A A




Conclude B AR A

A

Conclude B AR A

A

n

jj R

supn

jj

R

supn

jj

A R

Tsupn

j jj

A A B

Tsupn

j jj

A A B

T T T T1 1 2 2sup , ,..., ,...,i i n n

B A A B A A B A A B A A B




Example: A 0/x1 0/x2 ... 1/xi 0/xi+1 ... 0/xm

Correlation-min AAjT Bj minAixi, Bj

Correlation-product AAjT Bj AixiBj

Example: A 0/x1 0/x2 ... 1/xi 0/xi+1 ... 0/xm

Correlation-min AAjT Bj minAixi, Bj

Correlation-product AAjT Bj AixiBj

YX

Mem

bers

hip

Mem

bers

hip

A1B1

xi

1

ai1

min(ai1,B1)

1

Mem

bers

hip

Aj

xi

ai

Mem

bers

hip

Bj

aiBj




• Example: A 0/x1 0/x2 ... 1/xi 0/xi+1 ... 0/xm

Supposed that there are 2 rules (A1;B1) and (A2; B2)

B sup AA1T B1, AA2

T B2

where AA1T ai1 and AA2

T ai2

• Example: A 0/x1 0/x2 ... 1/xi 0/xi+1 ... 0/xm

Supposed that there are 2 rules (A1;B1) and (A2; B2)

B sup AA1T B1, AA2

T B2

where AA1T ai1 and AA2

T ai2




• Example

A is a fuzzy set not crisp set

Supposed that there are 2 rules(A1;B1) and (A2; B2)

B sup AA1T B1, AA2

T B2

where

• Example

A is a fuzzy set not crisp set

Supposed that there are 2 rules(A1;B1) and (A2; B2)

B sup AA1T B1, AA2

T B2

where supmin ,Tj j j

x Xr x x

A A A A A




YX

Mem

bers

hip

Mem

bers

hip

A1B1

xi

1

r1(A)

min(r1(A),B1))

YX

Mem

bers

hip

Mem

bers

hip

1

xi

A2B2

min(r2(A),B2)r2(A)

Y

Mem

bers

hip

B

A

A




• If there are more than 1 input A,B;CA a1/x1 a2/x2 ... am/xm and B b1/y1 b2/y2 ... bm/ym

split into A;C and B;Cand

fact: A Ixi 0/x1 0/x2 ... 1/xi 0/xi+1 ... 0/xm and

B Iyj 0/y1 0/y2 ... 1/yj 0/ji+1 ... 0/ym

compute: FA, B C AMACBMBC where AMAC Ix

iMAC

aiC

• If there are more than 1 input A,B;CA a1/x1 a2/x2 ... am/xm and B b1/y1 b2/y2 ... bm/ym

split into A;C and B;Cand

fact: A Ixi 0/x1 0/x2 ... 1/xi 0/xi+1 ... 0/xm and

B Iyj 0/y1 0/y2 ... 1/yj 0/ji+1 ... 0/ym

compute: FA, B C AMACBMBC where AMAC Ix

iMAC

aiC

TACM A C TBCM B C

1

2Iix

m

a

a

a

C

C

C




and BMBC IyjMBC

bjC

Hence, the conclusion will be C aiC bjC minai,bjC

If use correlation-product C aiCbjC minai,bjC

and BMBC IyjMBC

bjC

Hence, the conclusion will be C aiC bjC minai,bjC

If use correlation-product C aiCbjC minai,bjC

1

2=Ijy

m

b

b

b

C

C

C




• Mamdani modelRule: If 1 is A(1), and 2 is A(2) and ... and n is A(n) then is B

where A(1), A(2 and ... and A(n) are linguistic terms in Txi for 1in and

B is linguistic term in Ty andsuppose there are more than 1 ruleLet Tx1A1

(1), A2(1),... ,AN1

(1)

Tx2 A1(2), A2

(2),... ,AN2(2)

Txn A1(n), A2

(n),... ,ANn(n)

and Ty B1, B2,... ,BN0

rule j: If 1 is Ai1,j(1), and 2 is Ai2,j

(2) and ... and n is Ain,j(n) then is Bi,j

where i11,2,...,N1, i21,2,...,N2,..., in1,2,...,Nn and

i1,2,...,N0

• Mamdani modelRule: If 1 is A(1), and 2 is A(2) and ... and n is A(n) then is B

where A(1), A(2 and ... and A(n) are linguistic terms in Txi for 1in and

B is linguistic term in Ty andsuppose there are more than 1 ruleLet Tx1A1

(1), A2(1),... ,AN1

(1)

Tx2 A1(2), A2

(2),... ,AN2(2)

Txn A1(n), A2

(n),... ,ANn(n)

and Ty B1, B2,... ,BN0


(2) and ... and n is Ain,j(n) then is Bi,j

where i11,2,...,N1, i21,2,...,N2,..., in1,2,...,Nn and

i1,2,...,N0




Firing strength of rule j will be j minAi1,j

(1)(x1),Ai2,j(2)(x2),...,Ain,j

(n)(xn)Output of rule j will be

Overall output

Firing strength of rule j will be j minAi1,j

(1)(x1),Ai2,j(2)(x2),...,Ain,j

(n)(xn)Output of rule j will be

Overall output

1 2

1 21, 1 2, 2 , ,, ,..., min , ,..., ,

n

j ni j i j in j n i jx x x y x x x y OUT A A A B

1 2

1 2, ,..., 1, 1 2, 2 , ,

1,2,...,max min , ,..., ,

n

nx x x i j i j in j n i j

j ky x x x y

OUT A A A B




• Defuzzification

– Max membership select de_y where Bde_yBde_y for all

de_yYif Myy1, y2 ByhB where hB is a height of B

– Center of area (centroid)

• Defuzzification

– Max membership select de_y where Bde_yBde_y for all

de_yYif Myy1, y2 ByhB where hB is a height of B

– Center of area (centroid)

_ k

ky M

y

de yM

_

k kk

kk

y y

de yy

B

B




– Simplified centroid

where cj is firing strength ( aij or rjA ) and y0Bj is the mode

of output of rule j

– Simplified centroid

where cj is firing strength ( aij or rjA ) and y0Bj is the mode

of output of rule j

N

N

0

_j

n

n

jj

jj

c y

de yc

B




• Examplerule 1: If x1 is L1 และ x2 is H2, then y is Lrule 2: If x1 is M1 และ x2 is M2, then y is H

input (-4,3)

1 minL1(-4),H2(3) min (0.67,0.5) 0.52 minM1(-4),M2(3) min (0.33,0.5) 0.5

• Examplerule 1: If x1 is L1 และ x2 is H2, then y is Lrule 2: If x1 is M1 และ x2 is M2, then y is H

input (-4,3)

1 minL1(-4),H2(3) min (0.67,0.5) 0.52 minM1(-4),M2(3) min (0.33,0.5) 0.5




H

6 42 0 2 4 6X1

0.2

0.40.6

0.8

1.0L1 M1 H1

6 42 0 2 4 6X2

0.2

0.40.6

0.8

1.0L2 M2 H2

6 42 0 2 4 6Y

0.2

0.40.6

0.8

1.0L M

6 42 0 2 4 6X1

0.2

0.40.6

0.8

1.0L1 M1 H1

6 42 0 2 4 6X2

0.2

0.40.6

0.8

1.0L2 M2 H2

6 42 0 2 4 6Y

0.2

0.40.6

0.8

1.0L M H

6 42 0 2 4 6Y

0.2

0.40.6

0.8

1.0

Defuzzify the output fuzzy setto get the final output




• Takagi-Sugeno modelRule: If 1 is A(1), and 2 is A(2) and ... and n is A(n) then is B

where A(1), A(2 and ... and A(n) are linguistic terms in Txi for 1in and B is linguistic term in Ty and suppose there are more than 1 ruleLet Tx1A1

(1), A2(1),... ,AN1

(1)

Tx2 A1(2), A2

(2),... ,AN2(2)

Txn A1(n), A2

(n),... ,ANn(n)


(2) and ... and n is Ain,j(n) then

j fj1, 2,..., nwhere i11,2,...,N1, i21,2,...,N2,..., in1,2,...,Nn and

fjx1, x2,..., xn a1jx1 a2

jx2 ... anjxn a0

j

• Takagi-Sugeno modelRule: If 1 is A(1), and 2 is A(2) and ... and n is A(n) then is B

where A(1), A(2 and ... and A(n) are linguistic terms in Txi for 1in and B is linguistic term in Ty and suppose there are more than 1 ruleLet Tx1A1

(1), A2(1),... ,AN1

(1)

Tx2 A1(2), A2

(2),... ,AN2(2)

Txn A1(n), A2

(n),... ,ANn(n)


(2) and ... and n is Ain,j(n) then

j fj1, 2,..., nwhere i11,2,...,N1, i21,2,...,N2,..., in1,2,...,Nn and

fjx1, x2,..., xn a1jx1 a2

jx2 ... anjxn a0

j




overall output

where j is the firing degree of rule j

No need for defuzzification

overall output

where j is the firing degree of rule j

No need for defuzzification

1 21

1

, ,...k

j j nj

k

jj

f x x x




• Example turning steering wheel control systeminput: 1, 2, 3 and 4

universal set: X10, 150cm, X20, 150 cm, X390, 90 degree and X40, 150 cm

output: rotation speed () of the steering wheel

Rule j: If 1 is Ai1,j(1), และ 2 is Ai2,j

(2) และ 3 is Aj3,j(3) และ 4 is Aj4,j then p0 p11

p22 p33 p44

• Example turning steering wheel control systeminput: 1, 2, 3 and 4

universal set: X10, 150cm, X20, 150 cm, X390, 90 degree and X40, 150 cm

output: rotation speed () of the steering wheel

Rule j: If 1 is Ai1,j(1), และ 2 is Ai2,j

(2) และ 3 is Aj3,j(3) และ 4 is Aj4,j then p0 p11

p22 p33 p44

1

24

3




rule 1 2 3 4 p0 p1 p2 p3 p41 outwar

dssmall

3.000

0.000

0.000

-0.045

-0.004

2 forward

small

3.000

0.000

0.000

-0.030

-0.090

3 small small outwards

3.000

-0.041

0.004

0.000

0.000

4 small small forward

0.303

-0.026

0.061

-0.050

0.000

5 small small inwards

0.000

-0.025

0.070

-0.075

0.000

6 small big outwards

3.000

-0.066

0.000

-0.034

0.000

7 small big forward

2.990

-0.017

0.000

-0.021

0.000

8 small big inwards

1.500

0.025

0.000

-0.050

0.000

9 medium

small outwards

3.000

-0.017

0.005

-0.036

0.000

10 medium

small forward

0.053

-0.038

0.080

-0.034

0.000

rule

1 2 3 4 p0 p1 p2 p3 p4

11 medium

small

inwards

-1.220

-0.016

0.047

-0.018

0.000

12 medium

big outwards

3.000

-0.027

0.000

-0.044

0.000

13 medium

big forward

7.000

-0.049

0.000

-0.041

0.000

14 medium

big inwards

4.000

-0.025

0.000

-0.100

0.000

15 big small

outwards

0.370

0.000

0.000

-0.007

0.000

16 big small

forward

-0.900

0.000

0.034

-0.030

0.000

17 big small

inwards

-1.500

0.000

0.005

-0.100

0.000

18 big big outwards

1.000

0.000

0.000

-0.013

0.000

19 big big forward

0.000

0.000

0.000

-0.006

0.000

20 big big inwards

0.000

0.000

0.000

-0.010

0.000




Input1 10 cm, 2 30 cm, 3 0 degree (straight forward) and 4 50 cm

Only rule 4 and 7 fire4 0.25 and 4 0.3030.026100.061300.05000.00050 1.873

7 0.167 and 7 2.9900.017100.000300.02100.00050 2.820

Output of the system will be

Input1 10 cm, 2 30 cm, 3 0 degree (straight forward) and 4 50 cm

Only rule 4 and 7 fire4 0.25 and 4 0.3030.026100.061300.05000.00050 1.873

7 0.167 and 7 2.9900.017100.000300.02100.00050 2.820

Output of the system will be

0.25 1.873 0.167 2.8202.252

0.25 0.167




Fuzzy measure & fuzzy integralFuzzy measure & fuzzy integral

• Fuzzy measure Address the ambiguity axis of uncertainty how likely can the answer to a question be found in various subsets of the sources of informationX = x1, x2, … xn, g: 2X [0,1]Properties1. g() = 02. g(A) g(B) if AB

gi=gxi fuzzy density function importance of the single information source xi in determine the answer to a particular question

• Fuzzy measure Address the ambiguity axis of uncertainty how likely can the answer to a question be found in various subsets of the sources of informationX = x1, x2, … xn, g: 2X [0,1]Properties1. g() = 02. g(A) g(B) if AB

gi=gxi fuzzy density function importance of the single information source xi in determine the answer to a particular question




• Sugeno Lambda measuresatisfy 2 properties with additional one3. For all A,B X with AB =

From

• Sugeno Lambda measuresatisfy 2 properties with additional one3. For all A,B X with AB =

From 1 and 1

XgxXn

ii

n

i

ig1

11

nnn

j

n

k

n

i

ikjn

j

n

jk

kjn

j

j ggggggggg ......1 212

1

1

1 1

31

1 1

2

1

1 BgAgBgAgBAg




• ExampleX=x1, x2, x3 fuzzy densities g1=0.2, g2=0.3, g3=0.1

find using

Hence =3.2g(x1, x2) = 0.2+0.3+3.2(0.2)(0.3) =

0.69g(x1, x3) = 0.2+0.1+3.2(0.2)(0.1) =

0.36g(x2, x3) = 0.3+0.1+3.2(0.3)(0.1) =

0.5g(x1, x2 ,x3) = 0.69+0.1+3.2(0.69)(0.1)

= 1.011

• ExampleX=x1, x2, x3 fuzzy densities g1=0.2, g2=0.3, g3=0.1

find using

Hence =3.2g(x1, x2) = 0.2+0.3+3.2(0.2)(0.3) =

0.69g(x1, x3) = 0.2+0.1+3.2(0.2)(0.1) =

0.36g(x2, x3) = 0.3+0.1+3.2(0.3)(0.1) =

0.5g(x1, x2 ,x3) = 0.69+0.1+3.2(0.69)(0.1)

= 1.011

Subset g 0 from

propertyx1 0.2

x2 0.3

x3 0.1

x1, x2 0.69

x1, x3 0.36

x2, x3 0.5

x1, x2, x3 1 from property

1.013.012.011 20.006 0.11 0.4 0

20.11 0.11 4(0.006)( 0.4)3.2, 21.44

2(0.006)




• Fuzzy integral– Sugeno fuzzy integralContiuous case:

Finite case: X=x1, x2,…, xn reorder

Hence, sugeno fuzzy integral

• Fuzzy integral– Sugeno fuzzy integralContiuous case:


Hence, sugeno fuzzy integral

0,1

supmin , where h x g g A A xh x

1 2 1 2, ,... so that ...n nX x x x h x h x h x

1 21maxmin , where , ,...

n

g i i i iiS h h x g A A x x x




• Example

X=x1, x2, x3 h(x1) = 0.7, h(x2) = 0.9, h(x3) = 0.2

and assume that g are the same as previous example

Reorder h(x2) > h(x1) > h(x3)

Sg=(0.9g (x2 )) (0.7 g ( x1, x2 )) (0.2 g ( x1, x2, x3 )) =0.69

• Example

X=x1, x2, x3 h(x1) = 0.7, h(x2) = 0.9, h(x3) = 0.2

and assume that g are the same as previous example


Sg=(0.9g (x2 )) (0.7 g ( x1, x2 )) (0.2 g ( x1, x2, x3 )) =0.69




– Choquet fuzzy integralContiuous case:


Hence, choquet fuzzy integral

– Choquet fuzzy integralContiuous case:


Hence, choquet fuzzy integral

1 2 1 2, ,... so that ...n nX x x x h x h x h x

1 1 2 11

where , ,... and 0n

g i i i i i ni

C h h x h x g A A x x x h x

1

0

where X

h x g g A d A x h x




Let i(g) = g(Ai) - g(Ai-1)

Hence

Let i(g) = g(Ai) - g(Ai-1)

Hence

11

1 2 1 2 3 2 1

1 1 2 2 1 1 1

...

...

n

g i i ii

n n n

n n n n n

C h h x h x g A

h x h x g A h x h x g A h x h x g A

h x g A h x g A g A h x g A g A h x g A

11

where n

g i ii i ii

C h g h x g g A g A




We can have linear order statistic

where w1, w2,…, wn satisfy wi [0,1] and

• ExampleX=x1, x2, x3 h(x1) = 0.7, h(x2) = 0.9, h(x3) = 0.2 and assume that g are the same as previous example


Cg=(0.9-0.7)(0.3)+(0.7-0.2)(0.69+(0.2-0.1)(1) = 0.605

We can have linear order statistic

where w1, w2,…, wn satisfy wi [0,1] and

• ExampleX=x1, x2, x3 h(x1) = 0.7, h(x2) = 0.9, h(x3) = 0.2 and assume that g are the same as previous example


Cg=(0.9-0.7)(0.3)+(0.7-0.2)(0.69+(0.2-0.1)(1) = 0.605

11

n

kkw

k

k k

w k kx w

LOS h x w h x




• Optimization trainingLet training set T=(oj, j)|j=1,2,…,m where oj=jth object and j= desired output of jth object, and there are m training samplesSuppose there are n sensors/algorithmsOutput from choquet integral of jth object will be

Want to find fuzzy measure

• Optimization trainingLet training set T=(oj, j)|j=1,2,…,m where oj=jth object and j= desired output of jth object, and there are m training samplesSuppose there are n sensors/algorithmsOutput from choquet integral of jth object will be

Want to find fuzzy measure

;;)(1

1

n

iiijijjg AgxohxohohC No h(x1) h(x2) … h(xn)

12…m

Training set with msamples each has desiredoutput (). There are nsensors/algorithms

X

n

g

ggg

gg

g

...

...

13

12

2

1




such that Cg(h(oj)) is close to j

Let

We have

Then

such that Cg(h(oj)) is close to j

Let

We have

Then

else 0

if ;; 1,

iAAxohxohv iijij

Aj

0

0

;;

;;0

0

1

21

njnj

jj

j

xohxoh

xohxoh

v

;)( njT

jjg xohgvohC

m

jjjg ohCE

1

22

2 2 with only 1 possible nonzeronjv R n




Objective function

Hence

Objective function

Hence

m

jjjg ohCE

1

22

21

m

jjnj

m

j

Tjjnj

m

j

Tjj

T

m

jjnj

Tjjnj

Tjj

T

m

jjnj

Tj

T

jnjT

j

xohgvxohgvvg

xohgvxohgvvg

xohgvxohgv

1

2

11

1

2

1

;21 ;

21

; ;221

; ;21

22

1 1

1; and ; 2

m mT

j j j j jn nj j

v h o x v h o x

2 2

1

1 where ,2

mT T T

j jj

E g g v g v v

D D




Similar toSimilar to1minimize 2

subject to and 0 1

T T

gg g v g

g b g

D

A




Fromg1 g120g1 g130…g1 g12…n0… andg123…n-1 g123…n0…g23…n g123…n0

Fromg1 g120g1 g130…g1 g12…n0… andg123…n-1 g123…n0…g23…n g123…n0

1 0 0 0 0 0 0

0 0 1 0 0 0 0

0 0 1 0 0 0 10 0 0 1 0 0 1

A

1

10

0

b




• Example

Let find such

that E2 is minimized

• Example

Let find such

that E2 is minimized

No h(x1) h(x2) h(x3)

1 0.68 0.53 0.81 0.743

2 0.74 0.99 0.86 0.926

3 0.45 0.07 0.08 0.301

23

13

12

3

2

1

gggggg

g




O1: Cg(O1) = (0.81 – 0.68) g3+ (0.68 – 0.53) g13+ (0.53) = 0.13g3+ 0.15 g13+ 0.53

O2: Cg(O2) = (0.99 – 0.86) g2+ (0.86 – 0.74) g23+ (0.74) = 0.13g2+ 0.12 g23+ 0.74

O3: Cg(O3) = (0.45 – 0.08) g1+ (0.08 – 0.07) g13+ (0.07) = 0.37g3+ 0.01 g13+ 0.07

O1: Cg(O1) = (0.81 – 0.68) g3+ (0.68 – 0.53) g13+ (0.53) = 0.13g3+ 0.15 g13+ 0.53

O2: Cg(O2) = (0.99 – 0.86) g2+ (0.86 – 0.74) g23+ (0.74) = 0.13g2+ 0.12 g23+ 0.74

O3: Cg(O3) = (0.45 – 0.08) g1+ (0.08 – 0.07) g13+ (0.07) = 0.37g3+ 0.01 g13+ 0.07




Hence

And

Hence

And

001.0

000

37.0

,

12.0000

13.00

,

015.0

013.0

00

321 vvv

3

1

0.0860.0240.028

; 0

0.0340.022

Tj j jn

j

v h o x v

3

1

0.14 0 0 0 0.004 00 0.017 0 0 0 0.0160 0 0.017 0 0.02 00 0 0 0 0 0

0.004 0 0.02 0 0.023 00 0.016 0 0 0 0.014

Tj j

j

v v

D




100000010000001000100100

010100100010

001010010001001001

A

111000000

b

Now we can compute1minimize 2

subject to and 0 1

T T

gg g v g

g b g

D

A




Introduction to Evolutionary ComputationIntroduction to Evolutionary Computation




Evolutionary computingEvolutionary computing

Without mutationpopulation tends to converge to a homogeneous state where individuals vary only slightly from each other

Without mutationpopulation tends to converge to a homogeneous state where individuals vary only slightly from each other

Initialize the population

Create offspring through random

variation

Evaluate the

“fitness” of each candidate solution

Apply selection Terminate ?

Yes

No

Component1. Chromosome encoding2. Fitness value or survival strength of

individual3. Initial population4. Selection operator5. Reproduction operator




Introduction to Genetic AlgorithmIntroduction to Genetic Algorithm

• Genetic AlgorithmAlgo

1. Set t=02. Initial population P(t)3. Calculate each individual fitness value4. While not converge do

1. Select individual to P1 (intermediate population)Mating Pool (MP)2. Select from MP to mate P2 mutate chromosome in P2 P3

3. Select chromosome in P3 and P(t) for replacement P(t+1)4. Set t= t+1

5. End while

If (P3)=N then P3 become P(t+1) else if (P3)<N select q missing chromosome from P(t) or P1

• Genetic AlgorithmAlgo

1. Set t=02. Initial population P(t)3. Calculate each individual fitness value4. While not converge do

1. Select individual to P1 (intermediate population)Mating Pool (MP)2. Select from MP to mate P2 mutate chromosome in P2 P3

3. Select chromosome in P3 and P(t) for replacement P(t+1)4. Set t= t+1

5. End while

If (P3)=N then P3 become P(t+1) else if (P3)<N select q missing chromosome from P(t) or P1




• ExampleChromosome A, B, C, D each with 8 genesFitness function: number of 1 in the string

• ExampleChromosome A, B, C, D each with 8 genesFitness function: number of 1 in the string

chromosome label

chromosome string

fitness value

A 00000110 2B 11101110 6C 00100000 1D 00110100 3

Average fitness value 12/4

CrossoverB and Dat crossing site: 1get E and FAnd B and C arecopied

chromosome label

chromosome string

fitness value

B 11101110 6C 00100000 1E 10110100 4F 01101110 5

B is mutated at 1st bitE is mutated at 6th bit

chromosome label

chromosome string

fitness value

B 01101110 5

C 00100000 1E 10110000 3

F 01101110 5Average fitness value 14/4

P(t)

P(t+1)




• Initial population– random gene values of each gene in each chromosomeuniform

representation of the entire search space– Small population small part of search space, time complexity is low,

need more generations to converge EA force to explore a larger search space by increasing the rate of mutation

– Large population large area of search space, less generation to converge, time complexity is increased

• Selection operation– Selection techniques exist

• Explicit fitness remappingfitness values is mapped into a new range, e.g., normalization to [0,1]

• Implicit fitness remappinguse actual fitness values for selection several selection operators

• Initial population– random gene values of each gene in each chromosomeuniform

representation of the entire search space– Small population small part of search space, time complexity is low,

need more generations to converge EA force to explore a larger search space by increasing the rate of mutation

– Large population large area of search space, less generation to converge, time complexity is increased

• Selection operation– Selection techniques exist

• Explicit fitness remappingfitness values is mapped into a new range, e.g., normalization to [0,1]

• Implicit fitness remappinguse actual fitness values for selection several selection operators




– Random selectionrandom no reference to fitness each individual has an equal

chance of being selected– Proportional selection

total fitness

where f(xi)fitness

value of chromosome xiselection probability of xi

Expected number of copied of xi

Or

– Random selectionrandom no reference to fitness each individual has an equal

chance of being selected– Proportional selection

total fitness

where f(xi)fitness

value of chromosome xiselection probability of xi

Expected number of copied of xi

Or

1 2, ,..., NP t x x x

1

Ni

i

F f x

for 1,2,...,

i

i

f xp i N

F

for 1,2,...,

i i

i

f x f xn N i N

F f

for 1,2,...,i in Np i N




– AlgorithmFor i=1 to N

Calculate

End forFor i=1 to N

random 0,1uniform distributionIf 0q1, chromosome x1 is selectedIf qi1qi for i1,2,…N then chromosome xi is selected

End for

– AlgorithmFor i=1 to N

Calculate

End forFor i=1 to N

random 0,1uniform distributionIf 0q1, chromosome x1 is selectedIf qi1qi for i1,2,…N then chromosome xi is selected

End for

1

for 1,2,...,i

i kk

q p i N




• Example with x is varied between 0-31,

assume random 4 times with the value of 0.8, 0.5, 0.1 and 0.6

• Example with x is varied between 0-31,

assume random 4 times with the value of 0.8, 0.5, 0.1 and 0.6

No Chromosome

x (in real number)

f(x) pi qi Ni Number of

copies1 01101 13 169 0.14 0.14 0.58 12 11000 24 576 0.49 0.63 1.97 23 01000 8 64 0.06 0.69 0.22 04 10011 19 361 0.31 1.00 1.23 1

Total 1170 1.00

Average 293 0.25

maximum 576 0.49

2max maxx x

f x x




Crossover x1and x2 at crossover site 4x3and x4 at crossover site 2

Mutation probability is 0.001 there are 20 bits expected number of bit undergoes mutation is 20(0.001) = 0.02 bits suppose no mutationHence, New population will be

Crossover x1and x2 at crossover site 4x3and x4 at crossover site 2

Mutation probability is 0.001 there are 20 bits expected number of bit undergoes mutation is 20(0.001) = 0.02 bits suppose no mutationHence, New population will be

No Chromosome (x)

mate Crossoversite

New population

X (in real number)

f(x)

1 01101 2 4 01100 12 1442 11000 1 4 11001 25 6253 11000 4 2 11011 27 7294 10011 3 2 10000 16 256 Total 1754 Average 439

maximum 729




• Disadvantage– Premature convergence– Slow convergence

• Disadvantage– Premature convergence– Slow convergence




Introduction to Genetic AlgorithmIntroduction to Genetic Algorithm

• Selection Based on Scaling and Ranking Mechanisms– Static Scaling Mechanisms

• linear scaling

• Power law scaling

• Selection Based on Scaling and Ranking Mechanisms– Static Scaling Mechanisms

• linear scaling

• Power law scaling

S x ax b

tf x S f x

Evaluation function: eval(x) tf(x) S(f(x)) af(x) b

ueval x tf x S f x f x




• Logarithmic scaling

T is controlling the selective pressure >decrease during search process

Selection pressuredegree to which highly fit individuals are allowed to produce offspring (copies) in the next generationLow values of selection pressure provide each individual in the population with a reasonable selection prob.High values of selection pressurestrongly favor the best individuals in the population population diversity decrease quickly

• Logarithmic scaling

T is controlling the selective pressure >decrease during search process

Selection pressuredegree to which highly fit individuals are allowed to produce offspring (copies) in the next generationLow values of selection pressure provide each individual in the population with a reasonable selection prob.High values of selection pressurestrongly favor the best individuals in the population population diversity decrease quickly

f x

Ttf x S f x e




• Dynamic Scaling mechanism– Sigma truncation

OR

• Dynamic Scaling mechanism– Sigma truncation

OR

if 0

else

x xT x

o

t tS x T x m c

t teval x tf x S f x T f x m c

if

0 if

t t t t

t t

f x m c f x m ceval x

f x m c

1+ if 0

0 if 0

tt

t

t

f x m

eval x




– Window scaling

OR

where W is the set of all individuals contained in the last ggeneration, g1

– Window scaling

OR

where W is the set of all individuals contained in the last ggeneration, g1

miny P t

eval x f x f y

miny W

eval x f x f y




• Fitness remapping for minimization problem

where M is the upperbound of the objective function

where x(t)P(t) and Mt is max value of function f found so far or max value of f found within generation P(t)

• Fitness remapping for minimization problem

where M is the upperbound of the objective function

where x(t)P(t) and Mt is max value of function f found so far or max value of f found within generation P(t)

tf x M f x

teval x t tf x t M f x t




OR

where K is positive constant and fmin(t) is the minimum observed value of function f up to time t

OR

OR

where K is positive constant and fmin(t) is the minimum observed value of function f up to time t

OR

min

1eval x t

K f x t f t

1eval x

K f x




• Rank-based Selectionlet number of individuals in each generation N (P(t)N)– Linear Ranking selection

selection probability of xi

where ri rank of individual xi (ranked in increasing order)expected value of offspring of xi

And Then

• Rank-based Selectionlet number of individuals in each generation N (P(t)N)– Linear Ranking selection

selection probability of xi

where ri rank of individual xi (ranked in increasing order)expected value of offspring of xi

And Then

i in Np

1 1

1i

ir

p Min Max MinN N

1 1

N N

i ii i

n Np N

1

N

ii

Np N

1 1 1

1 1 1

1 1

N N Ni i

i i i

r rN Min Max Min Min Max Min N

N N N

1

11

1

N

ii

NMin Max Min r NN




But, since

Hence,

Min 2 Max OR Max Min 2

Since Min 0 and Min Max Max 2 and Max 1 1 Max 2

Hence and

Alternatively,

But, since

Hence,

Min 2 Max OR Max Min 2

Since Min 0 and Min Max Max 2 and Max 1 1 Max 2

Hence and

Alternatively,

1

1 1 0

11 1

2

N N N

ii i i

N Nr i i

NMax

pN

1

2 Max Minp

N N

1

12

Min Max Min

1 1

2 1 11

ii

rp Max Max

N N




• nonlinear ranking selectionSelection probability will be

• SUS Algorithm

guarantee that xi will reproduce at least but not more than

• nonlinear ranking selectionSelection probability will be

• SUS Algorithm

guarantee that xi will reproduce at least but not more than

111 ir

ip ec

1. ptr Rand(); /*random within [0,1] (uniformdistribution)*/

2. For (sum=i=0; i<N;i++)1. For (sum +=ni; sum>ptr; ptr++)2. Select(i);3. End For

3. End For

in in




• Example

Suppose ptr is 0.8 the selection will be x1, x3 and x3 , respectively

• Example

Suppose ptr is 0.8 the selection will be x1, x3 and x3 , respectively

Chomosome f(x)x2 r1 pi ni

x110 100 1 3(0.27)0.81

x215 225 2 3(0.33)0.99

x320 400 3 0.4 3(0.4)=1.2

1 2 1

0.8 1.2 0.8 0.333 3 1

2 1.20.27

3




• Tournament selectionSimilar to rank selection in terms of selection pressure but more efficient – Binary tournament (deterministic)

• 2 individuals directly compete for selection• With or without reinsertion of the competing individual into the original

population• based on direct pairwise comparison of individuals

– 2 chromosomes are chosen at random from the population– The fitness of the chosen chromosomes is calculated– The most successful chromosome (the winner) is selected to intermediate

population

– Probability tournament– Step 3 random R 0,1 if R<p then the fitter of the two is selected;

otherwise the less fit is selectedwhere p is preselected or p varied with time

or where p00,1 and c is positive constant

• Tournament selectionSimilar to rank selection in terms of selection pressure but more efficient – Binary tournament (deterministic)

• 2 individuals directly compete for selection• With or without reinsertion of the competing individual into the original

population• based on direct pairwise comparison of individuals

– 2 chromosomes are chosen at random from the population– The fitness of the chosen chromosomes is calculated– The most successful chromosome (the winner) is selected to intermediate

population

– Probability tournament– Step 3 random R 0,1 if R<p then the fitter of the two is selected;

otherwise the less fit is selectedwhere p is preselected or p varied with time

or where p00,1 and c is positive constant

0ctp t p e

01 ln

cp t p

t




– Boltzmann tournamentx current solution, y alternative solution binary tournament

between x and y with

Where T is the temperature reduced in a predefined manner in successive iterationThe winner of the trial will be the current solution in the next trial

• Elitism selection of a set of individual from the current generation to survive to the next generation

The fraction of new individual at each generation has been called generation gap

– Boltzmann tournamentx current solution, y alternative solution binary tournament

between x and y with

Where T is the temperature reduced in a predefined manner in successive iterationThe winner of the trial will be the current solution in the next trial

• Elitism selection of a set of individual from the current generation to survive to the next generation

The fraction of new individual at each generation has been called generation gap

1

1

f x f y

T

p x

e




• Recombination operatorcross over: C: XX XX Cx,yx,y

– One-point crossoverx x1 x2 x3… xL and y y1 y2 y3… yL after cross over x x1 x2 x3… xk1 xk yk1… yL and y y1 y2 y3… yk1 yk xk1… xL

• Recombination operatorcross over: C: XX XX Cx,yx,y

– One-point crossoverx x1 x2 x3… xL and y y1 y2 y3… yL after cross over x x1 x2 x3… xk1 xk yk1… yL and y y1 y2 y3… yk1 yk xk1… xL

Crossover site

1st parent

2nd parent

Crossover site

1st child

2nd child




– Two-point cross over

– N-point cross over

– Two-point cross over

– N-point cross over

1st Crossover

site

2nd Crossover site

chromosome x

chromosome

y

1st Crossover

site

2nd Crossover site

chromosome

x

chromosome

y

2nd Crossover site

chromosome

x

chromosome

y

3rdCrossover site

1st Crossover site

1st

Crossover site

2nd

Crossover site

Chromosome x

chromosome

y

3rd Crossover site




– N-point cross over algorithmP1: for each chromosome from intermediate population (P1)

P1.1: random q 0,1 uniform distributionP1.2: if q < pc (where pc is mating probability) then respective

chromosome is considered mating go to mating pool else not matingP2: let m*number of chromosome selected at P1

if m* is even then select pair chromosome (random)if m* is odd then discard or select chromosome from P1

P3: Cross over is applied on pairs in P2P3.1: for each pair, generate k (kL)

and random crossing site k1, k2,…,kN for (N2) P3.2: descendants obtained at P3.1 become potential members of P(t+1) candidate for mutation

P3.3: parents of newly generated chromosome are discarded from P1

P3.4: chromosomes left in P1 are added to P(t+1)

– N-point cross over algorithmP1: for each chromosome from intermediate population (P1)

P1.1: random q 0,1 uniform distributionP1.2: if q < pc (where pc is mating probability) then respective

chromosome is considered mating go to mating pool else not matingP2: let m*number of chromosome selected at P1

if m* is even then select pair chromosome (random)if m* is odd then discard or select chromosome from P1

P3: Cross over is applied on pairs in P2P3.1: for each pair, generate k (kL)

and random crossing site k1, k2,…,kN for (N2) P3.2: descendants obtained at P3.1 become potential members of P(t+1) candidate for mutation

P3.3: parents of newly generated chromosome are discarded from P1

P3.4: chromosomes left in P1 are added to P(t+1)




– Punctuated crossover• Record the crossover points in the chromosome• If a certain crossover point has generated a very poor offspring this

crossover point will not be considered further• If the fitness of the descendant is good, the crossover point is maintained as

active• Cross-over point self adjust according to the previous dynamics of the

search processC x1 x1… xLk1 k1... kL when ki 1 active crossover point, ki

0 not a crossover point

During initialization probability of generating ki 1 is smallSet of crossover point used in the recombination of 2 chromosomes is the union of the parents

– Punctuated crossover• Record the crossover points in the chromosome• If a certain crossover point has generated a very poor offspring this

crossover point will not be considered further• If the fitness of the descendant is good, the crossover point is maintained as

active• Cross-over point self adjust according to the previous dynamics of the

search processC x1 x1… xLk1 k1... kL when ki 1 active crossover point, ki

0 not a crossover point

During initialization probability of generating ki 1 is smallSet of crossover point used in the recombination of 2 chromosomes is the union of the parents




– Uniform crossover– Uniform crossover

1. Set mi 0 for i 1,2,…,L2. For each i

1. random U0,12. If pat_i then mi 1 where pat_i

is the crossover probability at ith

position x

y

1 0 1 0 0 1 0 0 0 1

m

x

y




• MutationAverage number of position that will undergo a mutation B NLpmwhere pmmutation probability (pm 0.001,0.01 ), N number of chromosomes, Llength of chromosome

Algorithm

Weak mutationstep 2 changed to if q <pm then 1 of 0 or 1 is chosen at random

• MutationAverage number of position that will undergo a mutation B NLpmwhere pmmutation probability (pm 0.001,0.01 ), N number of chromosomes, Llength of chromosome

Algorithm

Weak mutationstep 2 changed to if q <pm then 1 of 0 or 1 is chosen at random

For each chromosome และ For each position of thechromosome

1. Generate a random number q from U(0,1)2. If q <pm invert that position

If qpm do not mutate




– Non-uniform mutation

where 1

OR

where ,,Real-valued positive parameter

– Non-uniform mutation

where 1

OR

where ,,Real-valued positive parameter

tm mp t p e

12

21

2

t

me

p t

L N




OR

where 0tT pm(0) 0.5 and

OR

OR

where 0tT pm(0) 0.5 and

OR

12

2mp t

Lt

T 1

mp TL

1

2 1mp x

f x L




• Inversion operatorActs on a single chromosome. It inverts the values between 2 position of a chromosome. The 2 positions are chosen at random

• Inversion operatorActs on a single chromosome. It inverts the values between 2 position of a chromosome. The 2 positions are chosen at random




• Simple genetic algorithm• Simple genetic algorithm

1. Set t 0Initialize (P(0)) random a selection mechanism is chosen and if necessary a scaling mechanism

2. The chromosome of P(t) are evaluated using the fitness function. The most successfulindividual from P(t)is kept.3. Selection operator is applied N times

Selected chromosome make up an intermediate population P1 (having N members as well)representing candidate from the mating pool (MP). Some chromosomes from P(t) mayhave more copies in P1 while some others may have none

4. Crossover operator is applied to the chromosomes from MPNewly generated chromosomes forท P2

Parents of the chromosomes obtained through crossover are discarded from P1

Chromosomes that remained in P1 become member of P2

5. Mutation operator is applied to P2 outcome is the new generation(P(t1))6. set t t1

if t < T (T is the maximum number of generation) then go to step 2 otherwise stop




• Solution best member of the last population or best individual in all generation

• Can involve inversion or other types of operators

• Solution best member of the last population or best individual in all generation

• Can involve inversion or other types of operators




• Examplewhere

• Examplewhere

1 2 2 21 2

1,

1f x x

x x

chromosome

x1 x2 Fitnessvalue

C0 1 2 0.167C1 2 3 0.007C2 1.5 0.0 0.31C3 0.5 1.0 0.44

C3, C3, C2 and C0 areselected to P1

C3, C2 and C3, C0

chromosome

x1 x2 Fitnessvalue

C0 0.6 0 0.8C1 1.6 1.0 0.24C2 0.6 2.0 0.19C3 1.0 1.0 0.33

1 2

1 2,

max ,x x

f x x




Introduction to Genetic ProgrammingIntroduction to Genetic Programming

• Each individual/chromosome 1 computer program represented using a tree structure

• Terminal set specifies all variables and constant elements of the terminal set form the leaf modes

• Function set all functions that can be applied to the elements of the terminal set may be mathematical arithmetic and/or Boolean function or if-then-elseform the non-leaf nodes.

• Each individual/chromosome 1 computer program represented using a tree structure

• Terminal set specifies all variables and constant elements of the terminal set form the leaf modes

• Function set all functions that can be applied to the elements of the terminal set may be mathematical arithmetic and/or Boolean function or if-then-elseform the non-leaf nodes.




• Examplefunction AND, OR, NOT and terminal set x1, x2 where

x1,x20,1

• Examplefunction AND, OR, NOT and terminal set x1, x2 where

x1,x20,1

x1

OR

AND AND

NOT

x2

NOT

x1

x2

XOR x1 AND NOT x2 OR NOT x1 AND x2




• ExampleFunction ,,, and terminal set a,x,z,3.4 where a,x,z • ExampleFunction ,,, and terminal set a,x,z,3.4 where a,x,z

x

+

ln

a

3.4

y

sin exp

z

x

ln( ) sin( ) / exp( ) 3.4y x a z x




exp

a

3.4 x

sin

3.4 a

exp

a

3.4 x

z sin

3.4

+

Crossover 2 parents generating 1 child




Crossover 2 parents generating 2 children

exp

a

3.4 x

sin

3.4 a

+




• Mutationchoose a random point in a tree and replacing the subtree beneath that point by a randomly generated subtree

• Calculate the fitness of each program by running it on a set of “fitness case”

• Mutationchoose a random point in a tree and replacing the subtree beneath that point by a randomly generated subtree

• Calculate the fitness of each program by running it on a set of “fitness case”




Evolutionary programmingEvolutionary programming• Algorithm• Algorithm

1. Let current generation be t0

2. Initialize all parameters3. Initialize population C(0) with ns individual4. For each individual xi(t) C(t)

calculate fitness value f(xi(t))End

5. While no convergence• For each individual xi(t) C(t)

mutate xi(t) to produce offspring xi(t)

calculate fitness value (f(xi(t)))

add xi(t) in offspring set C(t)

End• Select new population C(t1) from C(t)C(t) using selection operator• t t1End




CoevolutionCoevolution

• Predator-pray competitive coevolution– Insects that eat plant

• To survive plant needs to evolve mechanism to defend itself from the insects, and• The insects need the plants as food source to survive.

– Inverse fitness interaction between 2 specieswin for 1 species failure for the other

• Symbiosis– Different species cooperate instead of competesuccess in 1 species improve

the survival strength of the other speciespositive feedback among the species that take part in this cooperating process

• Predator-pray competitive coevolution– Insects that eat plant

• To survive plant needs to evolve mechanism to defend itself from the insects, and• The insects need the plants as food source to survive.

– Inverse fitness interaction between 2 specieswin for 1 species failure for the other

• Symbiosis– Different species cooperate instead of competesuccess in 1 species improve

the survival strength of the other speciespositive feedback among the species that take part in this cooperating process




• Coevolutonary algorithm (CoEA)– Competitive coevolution

• Competition• Amensalism

– Cooperative coevolution• Mutualism• Commensalism• Parasitism

• Competitive coevolution (CCE)– Evolve 2 population simultaneously

• individual in 1 population solution (evolve to solve as many test cases as possible) fitness the number of test cases solved by the solution

• individual in another population test case (evolve to present an incrementally increasing level of difficult to the solution individuals)fitness is inversely proportional to the number of strategies that solve it

• Coevolutonary algorithm (CoEA)– Competitive coevolution

• Competition• Amensalism

– Cooperative coevolution• Mutualism• Commensalism• Parasitism

• Competitive coevolution (CCE)– Evolve 2 population simultaneously

• individual in 1 population solution (evolve to solve as many test cases as possible) fitness the number of test cases solved by the solution

• individual in another population test case (evolve to present an incrementally increasing level of difficult to the solution individuals)fitness is inversely proportional to the number of strategies that solve it




– Relative fitness functionexpress the performance of individual in one population is comparison with individuals in another population

• Which individual from the competing population is used• Exactly how these competing individuals are used to compute the relative

fitness– Sampling scheme

• All versub all sampling• Random sampling• Tournament sampling• All versus best sampling• Shared sampling

– Relative fitness function f(C1.xi)• Simple function2 population C1 and C2aim to calculate the relative fitness of each individual C1.xi of C1

Sample of individuals is taken from C2 S

sum C1.xi = count the number of individuals in S that C1.xi winf(C1.xi) = sum C1.xi

– Relative fitness functionexpress the performance of individual in one population is comparison with individuals in another population

• Which individual from the competing population is used• Exactly how these competing individuals are used to compute the relative

fitness– Sampling scheme

• All versub all sampling• Random sampling• Tournament sampling• All versus best sampling• Shared sampling

– Relative fitness function f(C1.xi)• Simple function2 population C1 and C2aim to calculate the relative fitness of each individual C1.xi of C1

Sample of individuals is taken from C2 S

sum C1.xi = count the number of individuals in S that C1.xi winf(C1.xi) = sum C1.xi




• Fitness sharingsim C1.xi = similarities of C1.xi with all the other individuals in that population e.g., the number of individuals that also beats individual from C2 samples

• Competitive fitness sharing

Where C2.x1, C2.x2,…, C2.xC2.ns, form C2 and C1.nl = total number of individuals in C1 that defeat individual C2.xl

• Tournament fitness– Binary tournament determine a relative fitness ranking– Tournament fitness results in a tournament tree with root element as the best

individual• Each level 2 opponent are randomly selected from that level and the best of the 2

advances to next level– If odd number a single individual from the level moves to the next level– After tournament use any selection operator

• Fitness sharingsim C1.xi = similarities of C1.xi with all the other individuals in that population e.g., the number of individuals that also beats individual from C2 samples

• Competitive fitness sharing

Where C2.x1, C2.x2,…, C2.xC2.ns, form C2 and C1.nl = total number of individuals in C1 that defeat individual C2.xl

• Tournament fitness– Binary tournament determine a relative fitness ranking– Tournament fitness results in a tournament tree with root element as the best

individual• Each level 2 opponent are randomly selected from that level and the best of the 2

advances to next level– If odd number a single individual from the level moves to the next level– After tournament use any selection operator

11

1

...

ii

i

sumC xf C xsimC x

2 .

11 1

1..

sC n

il l

f C xC n




• Hall of frame– At each generation the best individual of a population is stored in that

population’s hall of frames– May have limited size– New individual inserted in hall of frame will replace the worst or the

oldest individual– Individual from 1 population complete against a sample of the

current opponent population and its hall of frame– Prevents overspecialization

• Hall of frame– At each generation the best individual of a population is stored in that

population’s hall of frames– May have limited size– New individual inserted in hall of frame will replace the worst or the

oldest individual– Individual from 1 population complete against a sample of the

current opponent population and its hall of frame– Prevents overspecialization




• Competitive coevolution with 2 population Algo– Initialize C1 (solution), C2 (test) population– While stopping condition(s) not true do

• For each C1.xi for i=1,…, C1.ns do– Select a sample of opponents from C2

– Evaluate the relative fitness of C1.xi w.r.t. this sample• End• For each C2.xi for i=1,…, C2.ns do

– Select a sample of opponents from C1

– Evaluate the relative fitness of C2.xi w.r.t. this sample• End• Evolve population C1 for 1 generation• Evolve population C2 for 1 generation

– end– Select the best individual from C1 as solution

• Competitive coevolution with 2 population Algo– Initialize C1 (solution), C2 (test) population– While stopping condition(s) not true do

• For each C1.xi for i=1,…, C1.ns do– Select a sample of opponents from C2

– Evaluate the relative fitness of C1.xi w.r.t. this sample• End• For each C2.xi for i=1,…, C2.ns do

– Select a sample of opponents from C1

– Evaluate the relative fitness of C2.xi w.r.t. this sample• End• Evolve population C1 for 1 generation• Evolve population C2 for 1 generation

– end– Select the best individual from C1 as solution




• Competitive coevolution with 1 population Algo– Initialize 1 population C– While stopping condition(s) not true do

• For each C.xi for i=1,…, C.ns do– Select a sample of opponents from C (exclude C.xi )– Evaluate the relative fitness of C.xi w.r.t. the sample

• End• Evolve population C for 1 generation

– end– Select the best individual from C as solution

• Competitive coevolution with 1 population Algo– Initialize 1 population C– While stopping condition(s) not true do

• For each C.xi for i=1,…, C.ns do– Select a sample of opponents from C (exclude C.xi )– Evaluate the relative fitness of C.xi w.r.t. the sample

• End• Evolve population C for 1 generation

– end– Select the best individual from C as solution




• Cooperative coevolution– Similar to divide and conquer– Only mutualism here– Individual from different species or subpopulation have to cooperate in

some way to solve a global task– Fitness depends on that individual’s ability to collaborate with

individual from other species– Ex. For an nx dimensional problemnx subpopulation ( 1 for each)

• each subpopulation responsible for optimizing one of the parameters of problem and no subpopulation can form a complete solution by itself

• Merge representative from each subpopulation• Consider j subpopulation (Cj) each Cj.xi perform a single collaboration

with the best individual from each other subpopulation complete solution

• Credit assign to Cj.xi fitness of the complete solution• Evolve independently from one another separate population is used to

evolve each subcomponent using some EA

• Cooperative coevolution– Similar to divide and conquer– Only mutualism here– Individual from different species or subpopulation have to cooperate in

some way to solve a global task– Fitness depends on that individual’s ability to collaborate with

individual from other species– Ex. For an nx dimensional problemnx subpopulation ( 1 for each)

• each subpopulation responsible for optimizing one of the parameters of problem and no subpopulation can form a complete solution by itself

• Merge representative from each subpopulation• Consider j subpopulation (Cj) each Cj.xi perform a single collaboration

with the best individual from each other subpopulation complete solution

• Credit assign to Cj.xi fitness of the complete solution• Evolve independently from one another separate population is used to

evolve each subcomponent using some EA




Introduction to Swarm IntelligenceIntroduction to Swarm Intelligence




Swarm Intelligence (SI)Swarm Intelligence (SI)

• Structured collection of interacting organisms (or agents)• Individual organism

– Simple in structure but their collective behavior can become complex• Tight coupling between individual behavior and the behabior of the entire

swarm• Social interactiondirect (through visual audio, chemical context), or

indirect (occur when one changes the environment and the other individuals respond to the new environment)

• 2 concepts– Self-organization

• Positive feedback

• Structured collection of interacting organisms (or agents)• Individual organism

– Simple in structure but their collective behavior can become complex• Tight coupling between individual behavior and the behabior of the entire

swarm• Social interactiondirect (through visual audio, chemical context), or

indirect (occur when one changes the environment and the other individuals respond to the new environment)


• Positive feedback




Swarm IntelligenceSwarm Intelligence


• Positive feedback• Negative feedback• Fluctuation• Multiple interaction

– Division of labor• SIno one is in charge, no one is giving orders, which the swarm

individuals should obey or follow• Social networkstructureof swarmform an integral part of the existence

of that swarmprovide the communication channel, able to self-organize to form the optimal nest structures


• Positive feedback• Negative feedback• Fluctuation• Multiple interaction

– Division of labor• SIno one is in charge, no one is giving orders, which the swarm

individuals should obey or follow• Social networkstructureof swarmform an integral part of the existence

of that swarmprovide the communication channel, able to self-organize to form the optimal nest structures




• Partical swarm optimization (PSO)– Simulation of social behavior of birds within flock– Particles are “flown” through hyperdimensional search space– Neighborhood topology

• Star topologyfirst version PSO gbest

• Partical swarm optimization (PSO)– Simulation of social behavior of birds within flock– Particles are “flown” through hyperdimensional search space– Neighborhood topology

• Star topologyfirst version PSO gbest




• Ring topology

n=2

• Wheel topology

• Ring topology

n=2

• Wheel topologyFocal particle




• Individual best (pbest)– Initialize the swarm P(t) at t0 of particles such that the position (xi(t))

of ith particle (PiP(t)) is random within the hyperspace– Evaluate the performance F of each particle using xi(t)– Compare the performance of each individual to its best performance

• If F(xi(t)) < pbestipbesti F(xi(t))

• End– Change the velocity vector of each particle

where is positive random number

– Move each particle to a new position

– Set tt1– Repeat until converge (go to step 2)

• Individual best (pbest)– Initialize the swarm P(t) at t0 of particles such that the position (xi(t))

of ith particle (PiP(t)) is random within the hyperspace– Evaluate the performance F of each particle using xi(t)– Compare the performance of each individual to its best performance


• End– Change the velocity vector of each particle

where is positive random number



ipbest it tx x

1ii i pbest it t tv v x x

1i i it t tx x v




• Global best (gbest)– Initialize the swarm P(t) at t0 of particles such that the position (xi(t)) of ith

particle (PiP(t)) is random within the hyperspace– Evaluate the performance F of each particle using xi(t)– Compare the performance of each individual to its best performance


• End– Compare the performance of each individual to its best performance

• If F(xi(t)) < gbestgbest F(xi(t))

• Global best (gbest)– Initialize the swarm P(t) at t0 of particles such that the position (xi(t)) of ith




• If F(xi(t)) < gbestgbest F(xi(t))

ipbest it tx x

Where 1 and 2 are random number 1 = r1 c1 and 2 = r2 c2 with r1

and r2 U0,1, c1 c2 4

gbest it tx x




– Change the velocity vector of each particle






1i i it t tx x v

1 21ii i pbest i gbest it t t tv v x x x x




• Local best (lbest)– Initialize the swarm P(t) at t0 of particles such that the position (xi(t)) of ith




• If F(xi(t)) < lbestlbest F(xi(t))

• Local best (lbest)– Initialize the swarm P(t) at t0 of particles such that the position (xi(t)) of ith




• If F(xi(t)) < lbestlbest F(xi(t))

ipbest it tx x

lbest it tx x

Where 1 and 2 are random number 1 = r1 c1 and 2 = r2 c2 with r1

and r2 U0,1, c1 c2 4










1i i it t tx x v

1 21ii i pbest i lbest it t t tv v x x x x




– Fitness calculationcan be objective function same idea as in GA– Convergenceexecuted for a fixed number of iteration or velocity

changes are close to 0 for all particles– PSO system parameter

• Dimension of problem• Number of individual• Upper limit of • Upper limit on the maximum velocity (vmax )

If vij(t) vmax then vij(t) vmax and If vij(t) vmax then vij(t) vmax

where vij(t) is a velocity of ith particle at jth dimension at time tOR if do not want to use vmax

Where

and 12 4.0

– Fitness calculationcan be objective function same idea as in GA– Convergenceexecuted for a fixed number of iteration or velocity

changes are close to 0 for all particles– PSO system parameter

• Dimension of problem• Number of individual• Upper limit of • Upper limit on the maximum velocity (vmax )

If vij(t) vmax then vij(t) vmax and If vij(t) vmax then vij(t) vmax

where vij(t) is a velocity of ith particle at jth dimension at time tOR if do not want to use vmax

Where

and 12 4.0


2 411

2




• Neighborhood size– gbestsimply lbest with the entire swarm as the neighborhood– gbestmore susceptible to local minima

• Inertia weight

Where is inertia weight (1) and

– Modified PSO• Using selection perform selection before velocity updateAlgo

– for each particle in the swarm, score the performance of that particle w.r.t a random selected group of k particles

– Rank the particles according to these performance scores– Select the top half of the particle and copy their current position onto that of the

bottom half of the swarm without changing the personal best values of the bottom half of the swarm

• Neighborhood size– gbestsimply lbest with the entire swarm as the neighborhood– gbestmore susceptible to local minima

• Inertia weight

Where is inertia weight (1) and

– Modified PSO• Using selection perform selection before velocity updateAlgo

– for each particle in the swarm, score the performance of that particle w.r.t a random selected group of k particles

– Rank the particles according to these performance scores– Select the top half of the particle and copy their current position onto that of the

bottom half of the swarm without changing the personal best values of the bottom half of the swarm


1 21

12

c c




• Breeding PSOAlgorithm

– Calculate the particle velocities and new positions– For each particle assign a breeding probability pb

– Select 2 particles assume Pa and Pb and produce 2 offsprings using

Where r1, r2U0,1– Set personal best position of each particle involved in the breeding process to its

current position

• Breeding PSOAlgorithm

– Calculate the particle velocities and new positions– For each particle assign a breeding probability pb

– Select 2 particles assume Pa and Pb and produce 2 offsprings using

Where r1, r2U0,1– Set personal best position of each particle involved in the breeding process to its

current position

1 11 1a a bt r t r tx x x

2 21 1b b at r t r tx x x

1 a b

a aa b

t tt t

t t

v vv v

v v

1 a b

b ba b

t tt t

t t

v vv v

v v




• Neighborhood topology– Use of indices– Use spatial neighbor

Pb is said to be in the neighborhood of Pa if

where dmax is the largest distance between any 2 particles and

• Neighborhood topology– Use of indices– Use spatial neighbor

Pb is said to be in the neighborhood of Pa if

where dmax is the largest distance between any 2 particles and

max

a b

d

x x

max

max

3 0.6t t

t




• Ant Colony Optimization– tasks

• Reproduction• Defense• Foor collection• Brood care• Nest brooming• Nest building

– Natural stigmergy• Lack of central coordination• Communication and coordination among individuals in a colony are based

on local modifications of the environment• Positive feedback

• Ant Colony Optimization– tasks

• Reproduction• Defense• Foor collection• Brood care• Nest brooming• Nest building

– Natural stigmergy• Lack of central coordination• Communication and coordination among individuals in a colony are based

on local modifications of the environment• Positive feedback




– Shortest path problem traveling salesman problem (TSP)– Virtual pheromone– Negative feedbackavoid premature feedback– Time scalenot too large and not too short– Ant System (AS)

• Find a closed tour of minimal length connected n given cities

distance between city i and city j

• Tabu list and set of all cities that ant k is has to visit when it is on city i

– Shortest path problem traveling salesman problem (TSP)– Virtual pheromone– Negative feedbackavoid premature feedback– Time scalenot too large and not too short– Ant System (AS)

• Find a closed tour of minimal length connected n given cities

distance between city i and city j

• Tabu list and set of all cities that ant k is has to visit when it is on city i

2 2

ij i j i jd x x y y

1

2

3

4Graph N,E With 4 cities and they are all connected

kiJ




– AS algorithm– AS algorithm

For every edge (i,j)ij(0) 0 #initialize pheromone

End ForFor k 1 to m #for each ant k

Place ant k on a randomly chosen cityEnd ForSet T and LFor t 1 to tmax

For k 1to mbuild tour Tkt by applying n1 times the following steps: choose the next city j when the current city

is city i using the transition rule

Where and are adjustable and visibility

End For

if

0 else

ki

ij ij ki

kil ilij

l J

tj J

tp t 1

ijijd




For k 1 to mcompute length Lkt of tour Tkt produced by ant k

End ForIf an improved tour is found then

update T and L

End If

For every edge (i,j)

where

End ForEnd For

1 1 eij ij ij ijt t t e t

if ,

0 if ,

eij

Qi j T

Lt

i j T

1

mk

ij ijk

t t

if ,

0 if ,

kkk

ijk

Qi j T t

L tt

i j T t




– Ant colony system (ACS)• Different transition rule• Different pheromone trail update rule• Use of local updates of pheromone trail to favor exploration• Use of candidate list cl closest cities ordered by increasing distance ant

first restricts the choice of the next city to those in the list, and consider other cities if all in the list have been visited

– Ant colony system (ACS)• Different transition rule• Different pheromone trail update rule• Use of local updates of pheromone trail to favor exploration• Use of candidate list cl closest cities ordered by increasing distance ant

first restricts the choice of the next city to those in the list, and consider other cities if all in the list have been visited




For every edge (i,j)

ij(0) 0 #initialize pheromoneEnd For

For k 1 to m #for each ant kPlace ant k on a randomly chosen city

End ForSet T and L

For t 1 to tmax

For k 1to mbuild tour Tkt by applying n1 times the following steps:if exits at least one city jcandidate list then choose the next cityt j among cl citiy in the candicate

list as

where q U0,1 , 0 q0 1

where J is chosen according to

else choose the closest city j

end ifafter each transition ant k applies the local update rule where

End For

kiJ

0

0

argmax if

if

ki

iu iuu J

t q qj

J q q

kiJ

ki

iJ iJkiJ

il ill J

tp t

t

kiJ

01ij ijt t 10 nnnL




For k 1 to mcompute length Lkt of tour Tkt produced by ant k

End ForIf an improved tour is found then

update T and L

End IfFor every edge (i,j)

where

End ForEnd For

1ij ij ijt t t

1

ij tL

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