Introduction to Computational Intelligence Abraham varghese

"I am indebted to my parents for living, but to my teacher for living well." Alexander the Great

Course Objectives The primary objectives of this course are to the study of adaptive mechanisms to enable or facilitate intelligent behavior in complex and changing environments. The aim is to get machines to exhibit behaviour, which if done by humans, would be assumed to involve the use of intelligence.

Upon completion of this course, the student will be able to  Develop an understanding of fundamental computational intelligence and machine learning models.  Acquire the knowledge of Implemented neural networks, genetic algorithms, and other computational intelligence and machine learning algorithms  Apply computational intelligence and machine learning techniques to classification, prediction, pattern recognition, and optimization problems.  Design solutions for problems involving uncertain inputs or outcomes  The ability to objectively compare AI techniques and use the right ones to develop an appropriate solution

What is intelligence? Dictionaries define intelligence as the ability to comprehend, to understand and profit from experience. Definition of inelligence in biological system as the competence of system to achieve a target or sustain desired behavior under conditions of uncertainty.

Intelligent behavior can be characterized by clearly identifiable features Featurecapability AdaptabilityTo achieve specified goals or sustaining desired behavior in an environment characterized by unpredictable external changes Self - maintenance To maintain its own state of operational readiness CommunicationTo exchange information with other systems LearningBeing trained to carry out certain tasks AutonomyTo act independently from other systems including human operators Self-improvementTo improve its own future performance based on past performance combining with learning from other agents AnticipationTo predict changes in its environment which may affects its operation

Course outline

Classical Sets and Fuzzy Sets Classical Sets Operation on Classical Sets Properties of Classical (Crisp) Sets Mapping of Classical Sets to Functions Fuzzy Sets Notation Convention for Fuzzy Sets Fuzzy Set Operations Fuzzy sets

C LASSICAL S ETS AND F UZZY S ETS A classical set is defined by crisp boundaries A fuzzy set is prescribed by vague or ambiguous properties; hence its boundaries are ambiguously specified X (Universe of discourse)

C LASSICAL S ETS the universe of discourse is the universe of all available information on a given problem a universe of discourse, X, as a collection of objects all having the same characteristics The clock speeds of computer CPUs The operating currents of an electronic motor The operating temperature of a heat pump (in degrees Celsius) The Richter magnitudes of an earthquake The integers 1 to 10 The individual elements in the universe X will be denoted as x. The features of the elements in X can be discrete, countable integers or continuous valued quantities on the real line. The total number of elements in a universe X is called its cardinal number, denoted n x

C LASSICAL S ETS Collections of elements within a universe are called sets universe of discourse : The Richter magnitudes of an earthquake Set in the universe of discourse? Collections of elements within sets are called subsets The collection of all possible sets in the universe is called the whole set (power set).

C LASSICAL S ETS We have a universe comprised of three elements, X = {a, b, c} The cardinal number, n x ? The power set,P(X)? The cardinality of the power set?

O PERATION ON C LASSICAL S ETS Union A ∪ B = {x | x ∈ A or x ∈ B} The union between the two sets, denoted A ∪ B, represents all those elements in the universe that reside in (or belong to) the set A, the set B, or both sets A and B. This operation is also called the logical or Union of sets A and B (logical or) in terms of Venn diagrams

O PERATION ON C LASSICAL S ETS Intersection A ∩ B = {x | x ∈ A and x ∈ B} The intersection of the two sets, denoted A ∩ B, represents all those elements in the universe X that simultaneously reside in (or belong to) both sets A and B. This operation is also called the logical and Intersection of sets A and B.

O PERATION ON C LASSICAL S ETS The complement of a set A,is defined as the collection of all elements in the universe that do not reside in the set A. Complement of set A

O PERATION ON C LASSICAL S ETS The difference of a set A with respect to B, denoted A | B, is defined as the collection of all elements in the universe that reside in A and that do not reside in B simultaneously Difference operation A | B

P ROPERTIES OF C LASSICAL (C RISP ) S ETS Commutativity A ∪ B = B ∪ A A ∩ B = B ∩ A Associativity A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B ∩ C) = (A ∩ B) ∩ C Distributivity A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (2.7) Idempotency A ∪ A = A A ∩ A = A Identity A ∪ ∅ = A A ∩ X = A A ∩ ∅ = ∅ A ∪ X = X Transitivity If A ⊆ B and B ⊆ C, then A ⊆ C

P ROPERTIES OF C LASSICAL (C RISP ) S ETS Two special properties of set operations, The excluded middle axioms De Morgan’s principles

P ROPERTIES OF C LASSICAL (C RISP ) S ETS The excluded middle axioms not valid for both classical sets and fuzzy sets. There are two excluded middle axioms The first, called the axiom of the excluded middle, deals with the union of a set A and its complement, the second, called the axiom of contradiction, represents the intersection of a set A and its complement.

P ROPERTIES OF C LASSICAL (C RISP ) S ETS De Morgan’s principles information about the complement of a set (or event), or the complement of combinations of sets (or events), rather than information about the sets themselves

P ROPERTIES OF C LASSICAL (C RISP ) S ETS De Morgan’s principles Example: A shallow arch consists of two slender members as shown in Fig. If either member fails, then the arch will collapse. E1 = survival of member 1 and E2 = survival of member 2, Survival of the arch = ? Collapse of the arch = ? Logically, collapse of the arch will occur if either of the members fails

P ROPERTIES OF C LASSICAL (C RISP ) S ETS De Morgan’s principles Example 1: A shallow arch consists of two slender members as shown in Fig. If either member fails, then the arch will collapse. E1 = survival of member 1 and E2 = survival of member 2, Survival of the arch = E1 ∩ E2 Collapse of the arch = Collapse of the arch will occur if either of the members fails: Illustration of De Morgan’s principle:

M APPING OF C LASSICAL S ETS TO F UNCTIONS Mapping is an important concept in relating set-theoretic forms to function- theoretic representations of information. In its most general form it can be used to map elements or subsets on one universe of discourse to elements or sets in another universe. If an element x is contained in X and corresponds to an element y contained in Y, it is generally termed a mapping from X to Y, f : X →Y

M APPING OF C LASSICAL S ETS TO F UNCTIONS The characteristic (indicator) function χ A is defined by Membership function is a mapping for crisp set A.

M APPING OF C LASSICAL S ETS TO F UNCTIONS Example : a universe with three elements, X = {a, b, c}, we desire to map the elements of the power set of X, i.e., P(X), to a universe, Y, consisting of only two elements (the characteristic function), Y = {0, 1} the elements of the power set? the elements in the value set V(P(X))?

M APPING OF C LASSICAL S ETS TO F UNCTIONS Example : a universe with three elements, X = {a, b, c}, we desire to map the elements of the power set of X, i.e., P(X), to a universe, Y, consisting of only two elements (the characteristic function), Y = {0, 1} the elements of the power set P(X) = { ∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} the elements in the value set V(P(X)) V{P(X)} = {{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 1, 0}, {0, 1, 1}, {1, 0, 1}, {1, 1, 1}}

M APPING OF C LASSICAL S ETS TO F UNCTIONS The union of these two sets in terms of function-theoretic terms is given as follows (the symbol ∨ is the maximum): Union A ∪ B −→ χA ∪ B(x) = χA(x) ∨ χB(x) = max(χA(x), χ B(x)) The intersection of these two sets in function-theoretic terms is given by (the symbol ∧ is the minimum operator): Intersection A ∩ B −→ χA∩B(x) = χA(x) ∧ χB(x) = min(χA(x), χB(x)) The complement of a single set on universe X, say A, is given by Example

F UZZY S ETS The boundaries of the fuzzy sets are vague and ambiguous. Hence, membership of an element from the universe in this set is measured by a function that attempts to describe vagueness and ambiguity

F UZZY S ETS the set of tall people

F UZZY S ETS

Elements of a fuzzy set are mapped to a universe of membership values using a function-theoretic form. fuzzy sets are denoted by a set symbol with a tilde understrike; A ∼ would be the fuzzy set A. This function maps elements of a fuzzy set A ∼ to a real numbered value on the interval 0 to 1. If an element in the universe, say x, is a member of fuzzy set A ∼, then this mapping is given by

N OTATION C ONVENTION FOR F UZZY S ETS When the universe of discourse, X, is discrete and finite, is as follows for a fuzzy setA ∼ : When the universe, X, is continuous and infinite, the fuzzy setA ∼ Membership function for fuzzy set A ∼ Example

F UZZY S ET O PERATIONS Three fuzzy sets A, B, and C on the universe X For a given element x of the universe, the following function-theoretic operations for the set-theoretic operations of union, intersection, and complement are defined for aA, B, and C on X Standard fuzzy operations

F UZZY S ET O PERATIONS Union of fuzzy sets A ∼ and B ∼ Intersection of fuzzy sets A ∼ and B ∼ Complement of fuzzy sets A ∼ and B ∼ Example

F UZZY S ET O PERATIONS Proof, classical and fuzzy sets? All other operations on classical sets also hold for fuzzy sets, except for the excluded middle axioms

E XAMPLES OF F UZZY S ET O PERATIONS Problem 1 : We want to compare two sensors based upon their detection levels and gain settings. The following table of gain settings and sensor detection levels with a standard item being monitored provides typical membership values to represents the detection levels for each of the sensors. Gain setting Sensor 1 detection levels Sensor 2 detection levels The universe of discourse is X = {0, 20, 40, 60, 80, 100}. Find the membership function for the two sensors: Find the following membership functions using standard set operations: 1) 2) 3) 4) 5) 6) 7) 8)

Solution

E XAMPLES OF F UZZY S ET O PERATIONS Example: We are asked to select an implementation technology for a numerical processor. Computation throughput is directly related to clock speed. We are considering whether the design should be implemented using medium-scale integration (MSI) with discrete parts, field-programmable array parts (FPGA), or multichip modules (MCM). Define the universe of potential clock speeds as MHz; and define MSI, FPGA, and MCM as fuzzy sets of clock frequencies that should be implemented in each of these technologies. The following table defines the membership values for each of the three fuzzy sets. Representing the three sets as MSI = M ∼, FPGA = F ∼, and MCM= C ∼, find the following:

E XAMPLES OF F UZZY S ET O PERATIONS Example: Samples of a new microprocessor IC chip are to be sent to several customers for beta testing. The chips are sorted to meet certain maximum electrical characteristics, say frequency and temperature rating, so that the ‘‘best’’ chips are distributed to preferred customer 1. Suppose that each sample chip is screened and all chips are found to have a maximum operating frequency in the range 7–15 MHz at 20◦C. Also, the maximum operating temperature range (20◦C ±T ) at 8 MHz is determined. Suppose there are eight sample chips with the following electrical characteristics: The following fuzzy sets are defined: ?

feb CLASSICAL RELATIONS AND FUZZY RELATIONS

April RELATIONS  Relations represent mappings between sets and connectives in logic.  A classical binary relation represents the presence or absence of a connection or interaction or association between the elements of two sets.  Fuzzy binary relations are a generalization of crisp binary relations, and they allow various degrees of relationship (association) between elements.

April CRISP CARTESIAN PRODUCT

April CRISP RELATIONS

April CRISP RELATIONS

CRISP BINARY RELATIONS Examples of binary relations April

April OPERATIONS ON CRISP RELATIONS

April PROPERTIES OF CRISP RELATIONS The properties of crisp sets (given below) hold good for crisp relations as well.  Commutativity,  Associativity,  Distributivity,  Involution,  Idempotency,  DeMorgan’s Law,  Excluded Middle Laws.

April COMPOSITION ON CRISP RELATIONS

April FUZZY CARTESIAN PRODUCT Let R be a fuzzy subset of M and S be a fuzzy subset of N. Then the Cartesian product R  S is a fuzzy subset of N  M such that Example: Let R be a fuzzy subset of {a, b, c} such that R = 1/a + 0.8/b + 0.2/c and S be a fuzzy subset of {1, 2, 3} such that S = 1/ /2+ 0.9/3. Then R x S is given by

April FUZZY RELATION

April FUZZY RELATION

April OPERATIONS ON FUZZY RELATION The basic operation on fuzzy sets also apply on fuzzy relations.

OPERATIONS ON FUZZY RELATION April

April PROPERTIES OF FUZZY RELATIONS The properties of fuzzy sets (given below) hold good for fuzzy relations as well.  Commutativity,  Associativity,  Distributivity,  Involution,  Idempotency,  DeMorgan’s Law,  Excluded Middle Laws.

April COMPOSITION OF FUZZY RELATIONS

April COMPOSITION OF FUZZY RELATIONS

April COMPOSITION OF FUZZY RELATIONS

April COMPOSITION OF FUZZY RELATIONS

April CLASSICAL EQUIVALENCE RELATION

April CLASSICAL TOLERANCE RELATION

April FUZZY EQUIVALENCE RELATION

April FUZZY TOLERANCE RELATION A binary fuzzy relation that possesses the properties of reflexivity and symmetry is called fuzzy tolerance relation or resemblance relation. The equivalence relations are a special case of the tolerance relation. The fuzzy tolerance relation can be reformed into fuzzy equivalence relation in the same way as a crisp tolerance relation is reformed into crisp equivalence relation, i.e., where ‘n’ is the cardinality of the set that defines R 1.

April SUMMARY The chapter provides the definition, properties, and operations of the following:  Classical sets,  Fuzzy sets,  Classical relations,  Fuzzy relations. The chapter also discusses crisp and fuzzy equivalence and tolerance relations.