Smart Systems and Computational Intelligence - Lecture 8

Smart Systems and Computational Intelligence

• Lecture 8 is part of a course on smart systems and computational intelligence
• Topics covered include clustering methods, fuzzy logic, and fuzzy inference systems.

Types of Clustering Methods

• Hard clustering: Elements belong to exactly one cluster; clusters do not overlap
• Soft clustering (Fuzzy): Elements can belong to multiple clusters; clusters may overlap

Hard Clustering

• Every node is assigned to only one cluster.
• Example data shows the assignment of nodes (1, 2, 3, 4, 5, 6, 7) in different clusters (Cluster 1, Cluster 2, Cluster 3) in binary format.

Soft Clustering

• Every node can belong to multiple clusters with a fractional degree of membership in each.
• Example data illustrates the assignment of nodes (1, 2, 3, 4, 5, 6, 7) in different clusters (Cluster 1, Cluster 2, Cluster 3) using binary fractional values.

Fuzzy Logic and Fuzzy Inference System

• Fuzzy logic is used to compute based on degrees of truth rather than Boolean logic

Fuzzy Logic

• Introduced by Lotfi Zadeh in the 1960s to model uncertainties in natural language
• Utilizes fuzzy sets, simulating human reasoning, and handling imprecise decisions.
• Fuzzy logic variables have various degrees of membership, ranging from 0 to 1.

Fuzzy Logic - Definitions

• Dictionary definition of "fuzzy" is "not clear, distinct, precise, blurred"
• The world is imprecise, not clear, blurred
• Fuzzy logic adapts to cases that may not fit predefined criteria, using contextual information.

Traditional Representation of Logic

• Traditional logic uses binary values (0 or 1) to represent concepts like "slow" (0) and "fast" (1) in terms of speed

Fuzzy Logic - How Fast is Fast?

• The definition of fast/slow is subjective and context-dependent.
• The goal is to define the semantics for various linguistic terms (e.g., very slow, slow, fast, very fast).

Fuzzy Logic - Classical View

• Establishing clear ranges for linguistic variables like Very slow, Slow, Fast, Very fast.
• Examples of intervals are Very slow [0–0.25], Slow [0.25–0.5], Fast [0.5–0.75], Very fast [0.75–1]

Fuzzy Logic - Fuzzy View

• Defining a degree of membership for each observation belonging to a linguistic term.
• Defining membership functions to map input space (observations) to values between 0 and 1 indicating their degree of membership to a linguistic term

Membership Functions

• Show the semantics of how observations are related to linguistic terms
• Graph illustrations show different functions (triangular, trapezoidal, Gaussian)

Foundation of Fuzzy Logic

• Key components include fuzzy sets, membership functions, logical operations, and if-then rules.

Fuzzy Sets

• Fuzzy sets assign degrees of membership 0–1 for each element in the universe of discourse
• Illustrates a contrast to crisp sets which only assign elements to be either members or non-members

Crisp Sets vs. Fuzzy Sets

• Crisp sets sort elements into members and non-members in a universe of discourse while fuzzy sets define membership grades.
• Example: a crisp set might define "tall" people as those above a certain height, while a fuzzy set might assign a degree of membership (0–1) indicating how tall someone is relative to the definition of "tall".

Fuzzy Logic Example: Tallness

• A classic example showing how fuzzy sets assign degrees of membership to heights that suggest being "tall" according to thresholds defined.
• This illustrates the assignment of crisp and fuzzy membership ratings for individuals.

Crisp and Fuzzy Sets

• Graphs illustrating crisp and fuzzy membership functions for height ranges, indicating varying degrees of membership associated with "short", "average", and "tall" height classifications.

Fuzzy Logic Example: Tallness

• A table showing data values (height) corresponding to names and different degrees of membership in a crisp and fuzzy set.

Membership Functions

• Membership functions map input values to membership degrees using curves or shapes.
• Illustrations demonstrate different shapes like triangular, trapezoidal, Gaussian.
• The shape can depend on context.

Membership Function for Fuzzy Set "Speed"

• Graph illustrates membership degrees for linguistic speed terms (slow, medium, fast) varying with change in speed.

Fuzzy Inference System (FIS)

• Maps inputs to outputs via fuzzy logic by using membership functions and rules.

Fuzzy Inference System (FIS)

• Rules in a FIS can take the form of "if (antecedent)-then (consequent)", where those involved are fuzzy statements
• For example "If x is low and y is high, then z is medium".

Fuzzy Inference System (FIS)

• Steps involved include Fuzzification, Fuzzy Inference (or Rules Evaluation), Aggregation, and Defuzzification.
• FIS may allow more than one conclusion for a given rule.

Structure of a Fuzzy Expert System

• Diagram illustrating the order of steps leading from inputs to outputs
• This order is determined according to a set of rules embedded in a fuzzy expert system.

Fuzzy Inferencing Process

• Illustration showing the sequence of steps involved in inferencing from fuzzy inputs to outputs.

Fuzzification

• Converting crisp inputs into fuzzy values.
• Requires membership functions.
• Steps that help resolve truth for a fuzzy statement.
• Helps determine degree of fit for specified rules.

Fuzzification

• In this stage, membership functions are used to apply to input values
• This stage helps determine the membership degree for each rule premise, ranging from 0 to 1.

Fuzzy Inferencing or Rule Evaluation

• Evaluating rules based on fuzzy inputs.
• Also called fuzzy inference.

Fuzzy Inferencing

• Finding truth values for rule premises and using these values for further rule conclusion.
• Results are generated as fuzzy sets for specific output variables.

Aggregation of all outputs

• Combining fuzzy outputs from multiple rules.
• Used to generate a single fuzzy set for each output variable.
• Output is generated from the aggregation method, which in turn is defined by predefined operations.

Aggregation of all outputs

• Combining individual fuzzy outputs to form a single fuzzy output
• Common operators such as Maximum, Sum, Probabilistic Sum are used in aggregation.

Defuzzification

• Transforming the fuzzy set into a crisp value.
• The result of defuzzification process returns a definite crisp value.
• Methods include centroid and maximum methods.

Fuzzy Rules

• If-then statements describing outcomes for various fuzzy inputs.
• Based on common sense behavior of the system using linguistic labels of membership functions.

Fuzzy Rules

• Example Rules for a Tipping Scenario are provided.

Example: Tipping Problem

• Describing an example scenario involving calculating appropriate tips based on quality of food and service using fuzzy rules.

Rules for Tipping

• Specific rules outlined, explaining how quality of service and food rating may produce various tips amounts.

The Reasoning Process for FIS (the tipping example)

• Diagram illustrating the reasoning process involved in the tipping application example.

Antecedent for each rule

• Steps to find the degree of support for a rule when determining the output of each rule

Rule's Conclusion

• Illustration of how the antecedent and consequent are linked by implication operations in rule conclusion process.

Aggregate Conclusions

• Diagram illustrating how to aggregate the conclusion of each rule.

Fuzzy Applications

• Numerous applications from manufacturing to business.
• Illustrates the variety and versatility of FIS in many sectors.

Advantages of Fuzzy Logic

• Natural conceptualization
• Easy understanding
• Flexibility and forgiving behavior
• System development time is reduced, maintaining the system is easier.
• Uses simpler hardware.
• Handles problems not defined by mathematics.

Explore the concepts from Lecture 8 of the Smart Systems and Computational Intelligence course. This quiz covers clustering methods such as hard and soft clustering, as well as fuzzy logic and fuzzy inference systems. Test your understanding of these advanced topics and their applications.