Soft Computing
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Transcript of Soft Computing
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SOFT COMPUTING FUNDAMENTS
á
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OUTLINE
• Introduction to Soft Computing
• Probabilistic Computing (PC)
• Fuzzy computing (FC)
▫ Fuzzy Theory
▫ Fuzzy Systems
• Neural computing (NC) ▫ Artificial Neural Networks
• Evolutionary Computing (EC)
▫ Genetic Algorithms
• Summary
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INTRUDUCTION
The idea of soft computing started with Lofti A. Zadeh in 1981. Soft Computing is the fusion of methodologies designed to model and enable solutions to real world problems. * Basically, soft computing is not a homogeneous body of concepts and techniques. *The role model for soft computing is the human mind
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“The dominant aim of soft computing is to exploit the tolerance for imprecision, uncertainty and partial truth to achieve tractability, robustness, low solution-cost, and better raptor with reality .” (Zadeh,1981)
Rather than a precise definition for soft computing, it is instead defined by extension.
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AI
SOFT COMPUTING
Machine Learning
Symbolic manipulation
Automatic improvement with experience
Cognitive Psychology
Study of the mind
Statistics
Probability (not possibility)
Uncertainty and Imprecision
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Probabilistic Computing (PC)
• Supervised learning
▫ Classification problems
▫ Regression problems
• Unsupervised learning
▫ Clustering
▫ Dimensionality reduction.
*Bayes Theory
*Dampster-Shafer Theory
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Supervised Learning
• Inputs • Outputs
Training Set
• Linear regression, polynomial regression, logistic regression, etc.
• Optimization functions
Learning algorithm
• Obtain probable solution.
Hypothesis
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Input
Output
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Generate a hypothesis that classifies the data.
A decision boundary can be established using linear regression.
Optimization Algorithms:
• Gradient descendent.
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X1
X2
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Unsupervised Learning
• Input • Not Feedback
DATA
• Generate a probabilistic model.
• Bayes Networks…
Learning algorithm • Obtain
probable solution.
Model
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Input
Output
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There is not correct answer.
There are not parameters to feet.
The only data available is the input
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X1
X2
In general, the true distribution of the data is unknown
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Bayes Decision Theory
Belief Theory:
• 𝑃 𝑦 𝑥 =𝑃 𝑥 𝑦 𝑃(𝑦)
𝑃(𝑥) Classification
• From the Bayes rule is assumed to have a models m where
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Ethem Alpaydin, 2004
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Bayesian Network
• Belief networks or probabilistic networks
• Used as graphical models to represent the interaction between variables.
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Taken from Wikipedia
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Fuzzy computing (FC)
• Introduced by Lofti Zadeh in 1965, as the fuzzy set theory.
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Human Think
• Vague • Inexact
Nature
• Uncertain • Imprecise
Linguistic Knowledge
• Not quantifiable
Ambiguous
Probabilistic Probable truth
Imprecisely defined
REAL WORLD
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Fuzzy theory
Imprecise definition
Membership functions
• Created by experts
Rules • Qualificative
Probable truth
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Try to reproduce the reasoning to make a decision.
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Fuzzy set is an extension of classical set theory where elements have degrees of membership.
Classical theory
𝜇𝐴 = 1 𝑥𝜖𝑋0 𝑒𝑙𝑠𝑒
Fuzzy theory
𝜇𝐴 = 𝜇𝐴 𝑥 𝑥𝜖𝑋0 𝑒𝑙𝑠𝑒
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Fuzzy Systems
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• Fuzzification:
With an input variable – find the numeric values of the membership function.
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Fuzzy inference: 1. Using linguistic rules, determine with the inputs which rules apply. 2. Determining conclusions.
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• Desfuzzification:
Operation to obtain the most certain output.
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NEURAL COMPUTING
• Artificial Neural Networks (ANN):
Non-linear approximation
Emulate a simplified biological neuron.
Mathematical or computational model.
Advantage:
Size is not directly related with the number of features.
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Biological Neuron
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INPUTS
OUTPUT
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McCulloch-Pitts Neuron
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𝑦𝑖
𝑢𝑖 = 𝑤𝑖𝑗𝑥𝑗
𝑗
𝑦𝑖 = 𝑓(𝑢𝑖) INPUTS
OUTPUT 𝑥1
𝑥𝑛
𝑥𝑗
……
…
…
𝑤1
𝑤𝑗
Synaptic Weights
𝑤𝑛
Activation function
ɸ
Threshold
)()( i
j
iijii xwfty
𝜃𝑖
Bias
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• Weight factors, Wj
Determine the strain of the input vector.
• Activation Function:
Describes the relationship of input-output.
Could be a threshold function, sigmoid function(smooth), piecewise linear function, tangent hyperbolic function…
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Perceptron: example AND
I1
I2
W1
W2
2
1i
iiIWx O O = x
1
+
F(x) = x
0 0
0 1
1 1
1 0
I1 I2
0
0
1
0
I1 AND I2
-1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
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Single layer feed-forward network
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Multi Layer Feed-forward netowrk
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LEARNING METHODS IN ANN
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ANN Learning algorithms
Supervised Learning
Stochastic Gradient descent
EMS Back
propagation
Reinforced Learning
Unsupervised Learning
Hebbian Competitive
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Back propagation algorithm
• The artificial neurons are organized in layers, and send their signals “forward”, and then the errors are propagated backwards.`
• The idea of the backpropagation algorithm is to reduce the error in the output.
• The backpropagation algorithm calculates how the error depends on the output, inputs, and weights
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Classification of ANN
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Optimization
Optimization Methods
Linear Programming
Non-Linear Programming
Classical Methods
Newton, Fibonacci
Enumerative Methods
Min-max algorithms
Stochastic Methods
EA, GA,SA
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Evolutionary Computing
• Evolutionary algorithms (EA) are population-based metaheuristics optimization algorithms.
• Use biology-inspired mechanisms like mutation, crossover, natural selection, and survival of the fittest.
• Is more robust than other techniques in cases of noise, or input changes.
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Basic cycle of evolutionary computing
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Weise Thomas (2009)
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A basic evolutionary algorithm
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GENETIC ALGORITHMS
Is a subclass of Evolutionary algorithms.
• Mimic natural process as: selection, crosses over, mutation and accepting
• Random search
• Adaptive heuristic search.
• Survival of the fittest
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Basic process
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GENETIC OPERATORS
ADJUSTMENT AND EVALUATION
SELECTION
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Basic genetic algorithm
• 1. Generate random population
• 2.Evaluate fitness f(x) for each chromosome x
• 3. New Population.
▫ A. Selection select parents – best fitness
▫ B. Crossover crossover the parents –new offspring child
▫ C. Mutation using mutation probability – mutate new offspring
• 4. Raplace
• 5. Test
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Summary
• Soft computing techniques deal with the uncertainty and imprecision of the real world, using algorithms that in contrast to hard computing can treat this problems with robustness and generate low cost solutions.
• Systems based in probabilistic computing and fuzzy logic are approximation to the reasoning, so that it can be obtained a similar model to the real one but not exactly the same.
• Neural networks and evolutionary algorithms create functional approximations, for unknown systems, or very complex systems that can’t be mathematically described.
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References
• J.-S.R. Jang, K. Goebel, Neuro-Fuzzy and Soft Computing.
• Ethem Alpaydin (2004), Introduction to Machine Learning, MIT Press, Cambridge, Massachustts.
• http://www.uncertainty-in-engineering.net/uncertainty_models/fuzziness
• Weise Thomas (2009), Global Optimization Algorithms, Theory and Application, http://www.it-weise.de/
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