IS THERE ANY A NOVEL BEST THEORY FOR UNCERTAINTY?

Andino Maseleno

andinomaseleno@yahoo.com

DEPARTMENT OF COMPUTER SCIENCE

FACULTY OF SCIENCE

UNIVERSITI BRUNEI DARUSSALAM

WEDNESDAY, SEPTEMBER 14, 2011

In the early 1960s, L.A. Zadeh, a professor at the University of California at Berkeley well

respected for his contributions to the development of system theories, began to feel that

traditional systems analysis techniques were too precise for many complex real-world problems.

The idea of grade of membership, which is the concept that became the backbone of fuzzy set

theory, occurred to him in 1964, which lead to the publication of his seminal paper on fuzzy sets

in 1965 and the birth of fuzzy logic technology. The concept of fuzzy sets and fuzzy logic

encountered sharp criticism from the academic community; however, scholars and scientists

around the world—ranging from psychology, sociology, philosophy and economics to natural

sciences and engineering—became Zadeh’s followers.

FUZZY LOGICSo far as the laws of mathematics refer to reality, they are not certain.

And so far as they are certain, they do not refer to reality.

TRADITIONAL REPRESENTATION OF LOGIC

Slow Fast

Speed = 0 Speed = 1

bool speed; get the speed if ( speed == 0)

// speed is slow else

// speed is fast

FUZZY LOGIC REPRESENTATION

For every problem must represent in terms of fuzzy sets.

What are fuzzy sets?

Slowest

Fastest

Slow

Fast

[ 0.0 – 0.25 ]

[ 0.25 – 0.50 ]

[ 0.50 – 0.75 ]

[ 0.75 – 1.00 ]

FUZZY LOGIC REPRESENTATION CONT.

Slowest Fastestfloat speed; get the speed if ((speed >= 0.0)&&(speed < 0.25))

// speed is slowest else if ((speed >= 0.25)&&(speed < 0.5))

// speed is slowelse if ((speed >= 0.5)&&(speed < 0.75))

// speed is fastelse // speed >= 0.75 && speed < 1.0

// speed is fastest

Slow Fast

65° F Cool = 0.4, Warm= 0.7

25% Cover Sunny = 0.8, Cloudy = 0.2

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1

40 60 80 100200

Cloud Cover (%)

OvercastPartly CloudySunny

0

1

FUZZY LOGICMeasurement devices in technical systems provide crisp measurements, like 65° F or 25% Cover. At first, these crisp values must be transformed into linguistic terms (fuzzy sets) . This is called fuzzification.

50 75 100250

Speed (mph)

Slow Fast

0

1

Speed is 20% Slow and 70% Fast

Find centroids: Location where membership is 100%

7

Defuzzification is the process of producing a quantifiable result in fuzzy logic, given fuzzy sets and corresponding membership degrees.

FUZZY LOGIC

DEMPSTER-SHAFER THEORY

The Dempster-Shafer theory was first introduced by Dempster in the 1968 and then

extended by shafer in the 1976, but the kind of reasoning the theory uses can be found as

far back as the seventeenth century. This theory is actually an extension to classic

probabilistic uncertainty modeling. Whereas the Bayesian theory requires probabilities for

each question of interest, belief functions allow us to base degrees of belief for on question

on probabilities for a related question. These degrees of belief may or may not have the

mathematical properties of probabilities will depend on how closely the two questions are

related. In terms of previous work using Dempster-Shafer theory, most prior research with

this system has been theoretical, for example, in pursuing the use of belief functions for

propagating uncertainty in AI/expert systems in addition or instead of using probabilities.

Dempster Shafer theory of evidence has attracted considerable attention within the AI

community as a promising method of dealing with uncertainty in expert systems as Zadeh

said in his paper in the 1986.

DEMPSTER-SHAFER THEORY

The Dempster-Shafer theory could be considered as a generalization of

probability theory. A mapping a set of values: ᴦ: x → PΩ, where PΩ is the set of all

non fuzzy subsets from Ω. Assume a probability measure ρ over x; now, what can

be said about a probability measure over Ω is induced by ᴦ? This is a basic

question, where Dempster showed that for every B Ω, P(B) belongs to the

following intervals:

where Aj PΩ is any nonempty member of ᴦ and

Shafer introduced his evidence theory and defined bel and pls functions. Consider a referential

set Ω = w1, w2,…wn; a body of evidence is defined as follows:

A1, A2,…, Al m1, m2,…,ml

in which each Aj is a focal element, and mj is the corresponding mass value. Evidence theory could be

considered as a direct generalization of Bayesian statistics. One may think of mass value as probability

density values; but in evidence theory, mass values are assigned to the subsets of Ω instead of the

elements of Ω; so, it conveys a higher level of uncertainty and is capable of modeling both ignorance

and indeterminism. Shafer defined the concepts of belief and plausibility as two measures over the

subsets of in axiomatic manner and then he showed that bel and pls with the following definitions were

belief and plausibility functions.

DEMPSTER-SHAFER THEORY

Trend that contributed to research in fuzzy model identification is the increasing visibility of

neural network research in the late 1980s. Because of certain similarities between neural

networks and fuzzy logic, researchers began to investigate ways to combine the two

technologies. The most important outcome of this trend is the development of various

techniques for identifying the parameters in a fuzzy system using neural network learning

techniques. A system built this way is called a neuro-fuzzy system.

NEURAL NETWORK

NEURAL NETWORK

How do they work?

The network is trained with a set of known facts that cover the

solution space

During the training the weights in the network are adjusted

until the correct answer is given for all the facts in the training

set

After training, the weights are fixed and the network answers

questions not in the training data.

These “answers” are consistent with the training data

INSPIRATION FROM NEUROBIOLOGY

A neuron: many-inputs / one-output unit

output can be excited or not excited

incoming signals from other neurons determine if the neuron shall excite ("fire")

Output subject to attenuation in the synapses, which are junction parts of the neuron

SYNAPSE CONCEPT

The synapse resistance to the incoming signal can be changed during a "learning" process [1949]

Hebb’s Rule: If an input of a neuron is repeatedly and persistently

causing the neuron to fire, a metabolic change happens in the synapse of that particular input to

reduce its resistance

MATHEMATICAL REPRESENTATION

The neuron calculates a weighted sum of inputs and compares it to a threshold. If the sum is higher than the threshold, the output is set to 1, otherwise to -1.

Non-linearity

The 1990s is an era of new computational paradigms. In addition to fuzzy logic and neural

networks, a third nonconventional computational paradigm also became popular —evolutionary

computing, which includes genetic algorithms, evolutionary strategies, and evolutionary

programming. Genetic algorithms (GA) and evolutionary strategies are optimization techniques

that attempt to avoid being easily trapped in local minima by simultaneously exploring multiple

points in the search space and by generating new points based on the Darwinian theory of

evolution— survival of the fittest. The popularity of GA in the 1990s inspired the use of GA for

optimizing parameters in fuzzy systems. Various synergistic combinations of neural networks,

genetic algorithms, and fuzzy logic help people to view them as complementary. To

distinguish these paradigms from the conventional methodologies based on precise

formulations, Zadeh introduced the term soft computing in the early 1990s.

GENETIC ALGORITHM

Based on Darwinian Paradigm

Intrinsically a robust search and optimization mechanism

Reproduction Competition

SelectionSurvive

GENETIC ALGORITHM

GENETIC ALGORITHM

Inspired by Darwinian Paradigm or natural evolution Population of individuals

Individual is feasible solution to problem Each individual is characterized by a Fitness function

Higher fitness is better solution Based on their fitness, parents are selected to reproduce

offspring for a new generation Fitter individuals have more chance to reproduce New generation has same size as old generation; old

generation dies Offspring has combination of properties of two parents If well designed, population will converge to optimal

solution

GENETIC ALGORITHM

SO FAR…

Avian Influenza warning system with Dempster Shafer Theory and Web Map

Shortest Path Routing with Hopfield Neural Network and Dempster Shafer Theory

Dempster Shafer Theory with Fuzzy Logic Membership Function to Determine Student Research Interest

Thank You

I wish I could finish all the things I started