Fuzzy Relation &

Neural Network

CRISP AND FUZZY RELATIONS• crisp relation represents the presence or absence of

association, interaction or interconnectedness between the elements of two or more sets. Thia concept can be generalized to allow for various degrees or strengths of association or interaction between elements.

• Degrees of association can be represented by membership grades in a fuzzy relation in the same way as degrees of set membership are represented in the fuzzy set. In fact, just as the crisp set can be viewed as a restricted case of the more general fuzzy set concept, the crisp relation can be considered to be a restricted case of the fuzzy relation.

• A relation among crisp sets Xi, X2, . Xn is a subset of the Cartesian product. It is denoted either by R (X1,X2… . Xn)

• Each crisp relation R can be defined by a characteristic function which assigns a value of 1 to every tit& of the universal set belonging to the relation and a 0 to every tuple not belonging to it. Denoting -a relation and its characteristic function by the same symbol R, we have

• Relationships among objects are fundamental to decision making and dynamic systems applications. A classical (crisp) relation represents the presence or absence of a connection or association among elements of two or more sets. Fuzzy relations, on the other hand, allow degrees of strength to such connections and associations.

• Example:The elements in two sets A and B are given as A = {0, 1} and B = {a, b, c}.Various Cartesian products of these two sets can be written as shown:

Classical Relations• A binary relation associates elements of only two sets

to each other. A relation R from set A to set B is a subset of the Cartesian product of the two:

• Example• Let A = {desk, bag, book}, and B = {wood, plastic,

paper}.• The Cartesian product of these two sets leads to• A × B = {(wood,desk), (wood,bag), (wood,book),

(plastic,desk), (plastic,bag), (plastic,book), (paper,desk), (paper,bag), (paper,book)}.

• From this set one may select a subset such that• R = {(wood,desk), (plastic,bag), ((paper,bag),

(paper,book)}

The coordinate diagram of a relation.

a)Matrix formb) Mapping of relation

Fuzzy Relation• Fuzzy relation generalizes classical relation into

one that allows partial membership and describes a relationship that holds between two or more objects. Example: a fuzzy relation “Friend” describe the degree of friendship between two persons (in contrast to either being friend or not being friend in classical relation!)

Fuzzy Advice

• Beware of the math!

Fuzzy Cartesian Product

Neural Network• It is massively parallel distributed processor made

up of simple processing units which has anatural propensity for storing experimental knowledgeand making it avail.

• It resembles the brain in two aspects:i) knowledge is acquired by network from its

environment through a learning processIi) interneuron connection strengths, known as

synaptic weights azre used to store the acquired knowledge.

Benefits of neural network• Nonlinearity-to process nonlinear inputs.• Input-output mapping-learning with a teacher,

network modifies weights to minimize differences between actual output & desired response of system.

• Adaptively-buit-in capability to adapt their synaptic weights to changes in surrounding environment

• Fault Tolerance: capable of robust computation.

Biological Neuron

Natural and artificial neural networks***where is the caption??***

Schematic representation of mathematical neuron

a) Single layer b) Multi-layer Neural Network

Artificial Neural Network

• The aim of neural network is to mimic the human ability to adapt to changing circumstances & current environment.

• Ex: decisions made by experience & trainee doctor

Historical development of NN

• 1943-McCulloch-pitts:start of modern era of nn

• 1949:Hebb:organization of behavour• 1958:Rosenblatts:introduces perceptron• 1960:Widrow &Hoff : introduce adaline• 1982:John Hopfield network• 1990: Vapnik: support vector machine

Terminologies

• NN consists of large number of processing elements called neurons. Neurons are connected to each other by directed communication links which are associated with weights.

• “weight is an information used by neural net to solve a problem.”

Simple neural net

w1

w2

x1

x2

y

Terminologies Contd….

• In figure of simple neural net the weights carry information are denoted by w1 & w2, they may be fixed or random values.

• Net is summation of products of the weights & input signal

• Net= x1w1+x1w2• Net input = ∑ xiwi

Activation Functions• AF are used to calculate response of neuron.

The sum of the weighted input signal is applied with an activation to obtain the response.

• Linear/Nonlinear

Characteristics ANN Biological Neural Network

speed Faster; cycle time of execution one step of program-few nanoseconds

Slow; milliseconds

Processing Sequential mode Parallel

storage Memory, replaceable In strengths of interconnections

Control unit present Absent

Size & complexity Not involve much computational neurons

10^11 neuron & 10^15 interconnections

Building Blocks of ANN

• Network Architecture• Setting the Weights• Activation Function

ANN Connection Structures

Learning paradigms• There are three major learning paradigms, each

corresponding to a particular abstract learning task. These are supervised learning, unsupervised learning and reinforcement learning.

• Feed forward: data flow in forward direction• Competitive net: negative connections• Recurrent net: all units are interconnected