Fuzzy Logic Lecture

[Clide Birkner]

EE 576 - Neural Networks and Fuzzy Logic(3 units)

Prerequisites: Open to Electrical Engineering MS and Engineering MS students only.

Principles and application of artificial neural networks and fuzzy logic. Mechanisms of supervised and unsupervised neural networks. Fuzzy control systems. Applications in signal processing, communications, control, and other areas. Additional projects required for EE 576.

Letter grade only (A-F). (Lecture-problems 3 hours). Double Numbered with: E E 476

Basic concepts and methods of artificial intelligence; Heuristic search procedures for general graphs; game playing strategies; resolution and rule based deduction systems; knowledge representation; reasoning with uncertainty.

Basic concepts in neural computing; functional equivalence and convergence properties of neural network models; associative memory models; associative, competitive and adaptive resonance models of adaptation and learning; selective applications of neural networks to vision, speech, motor control and planning; neural network modeling environments.

Introduces the basics of fuzzy logic and its role in developing intelligent systems; topics include fuzzy set theory, fuzzy rule inference, fuzzy logic in control, fuzzy pattern recognition, neural fuzzy systems and fuzzy model identification using genetic algorithms.

The architecture of the mammalian cerebral cortex; its modular organization and its network for distributed and parallel processing; cortical networks in perception and memory; neuronal microstructure and dynamical simulation of cortical networks; the cortical network as a proven paradigm for the design of cognitive machines.

James K. Peckol received his BS in engineering from Case Institute of Technology in 1966 and his MS and Ph.D. degrees in Electrical & Computer Engineering from the University of Washington 1975 and 1985 respectively.

Peckol has spent over 45 years in industry and in universities developing embedded systems, conducting research and teaching. He is currently a Principal Lecturer on real time embedded system design, computer architecture, digital logic design and elementary circuit design. He has served as Professor (1996) and Maitre de Conferences (1993) at the University of Nantes in France while working as an invited lecturer in fuzzy logic and researcher in hardware/software co-design and computer performance monitoring. He has held the position of Senior Lecturer (Associate Professor) at the University of Aberdeen in Scotland (1987) while conducting research in artificial intelligence and knowledge management. Working as an invited visiting Professor in the Advanced Program at Danang Technical University in Vietnam (summers 2008, 2009, 2011, 2013 and 2015), he evaluated the program, advised and taught in the areas of computer architecture, digital / embedded systems and C programming. One team of students from the 2011 embedded systems class won a national competition in Vietnam for a project they developed based upon what they had learned in class.

In standard logic, every statement must have an absolute value: true or false. In fuzzy logic, truth values are replaced by degrees of "membership" from 0 to 1, where 1 is absolutely true and 0 is absolutely false.

Data mining is the process of identifying significant relationships in large sets of data, a field that overlaps with statistics, machine learning, and computer science. Fuzzy logic is a set of rules that can be used to reach logical conclusions from fuzzy sets of data. Since data mining is often applied to imprecise measurements, fuzzy logic is a useful way of determining relevant relationships from this kind of data.

Fuzzy logic is often grouped together with machine learning, but they are not the same thing. Machine learning refers to computational systems that mimic human cognition, by iteratively adapting algorithms to solve complex problems. Fuzzy logic is a set of rules and functions that can operate on imprecise data sets, but the algorithms still need to be coded by humans. Both areas have applications in artificial intelligence and complex problem-solving.

An artificial neural network is a computational system designed to imitate the problem-solving procedures of a human-like nervous system. This is distinct from fuzzy logic, a set of rules designed to reach conclusions from imprecise data. Both have applications in computer science, but they are distinct fields.

The point of fuzzy logic is to map an input space to an output space. The primary mechanism for doing this is a list of if-then statements called rules. All rules are evaluated in parallel and the order of the rules is unimportant. The rules themselves are useful because they refer to variables and the adjectives that describe those variables. Before you can build a system that interprets rules, you must define all the terms you plan on using and the adjectives that describe them. To say that the water is hot, you need to define the range within which the water temperature can be expected to vary as well as what you mean by the word hot.

To understand what a fuzzy set is, first consider the definition of a classical set. A classical set is a container that wholly includes or wholly excludes any given element. For example, the set of days of the week unquestionably includes Monday, Thursday, and Saturday. It just as unquestionably excludes butter, liberty, dorsal fins, and so on.

Individual perceptions and cultural background must be taken into account when you define what constitutes the weekend. Even the dictionary is imprecise, defining the weekend as the period from Friday night or Saturday to Monday morning.

Reasoning in fuzzy logic is a matter of generalizing the familiar yes-no (Boolean) logic. If you give true the numerical value of 1 and false the numerical value of 0, fuzzy logic then permits in-between values like 0.2 and 0.7453. For instance:

Q: Is Saturday a weekend day? A: 1 (yes, or true) Q: Is Tuesday a weekend day? A: 0 (no, or false) Q: Is Friday a weekend day? A: 0.8 (for the most part yes, but not completely) Q: Is Sunday a weekend day? A: 0.95 (yes, but not quite as much as Saturday).

The plot on the left shows the truth values for weekend-ness if you are forced to respond with an absolute yes or no response. On the right is a plot that shows the truth value for weekend-ness if you are allowed to respond with fuzzy in-between values.

Technically, the representation on the right is from the domain of multivalued logic (or multivalent logic). If you ask the question "Is X a member of set A?" the answer might be yes, no, or any one of a thousand intermediate values in between. Thus, X might have partial membership in A. Multivalued logic stands in direct contrast to the more familiar concept of two-valued (or bivalent yes-no) logic.

By making the plot continuous, you define the degree to which any given instant belongs in the weekend rather than an entire day. In the plot on the left, at midnight on Friday, the weekend-ness truth value jumps discontinuously from 0 to 1.

The plot on the right shows a smoothly varying curve that accounts for the fact that all of Friday, and, to a small degree, parts of Thursday, partake of the quality of weekend-ness and thus deserve partial membership in the fuzzy set of weekend moments. The curve that defines the weekend-ness of any instant in time is a function that maps the input space (time of the week) to the output space (weekend-ness). Such a function is a membership function.

As another example of fuzzy sets, consider the question of seasons. What season is it right now? In the northern hemisphere, summer officially begins at the exact moment in the earth's orbit when the North Pole is pointed most directly toward the sun. It occurs exactly once a year, in late June. Using the astronomical definitions for the season, you get sharp boundaries as shown on the left in the figure that follows. But what you experience as the seasons vary continuously as shown on the right in the following figure (in temperate northern hemisphere climates).

A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. The input space is often referred to as the universe of discourse.

One of the most commonly used examples of a fuzzy set is the set of tall people. In this case, the universe of discourse is all potential heights, say from three feet to nine feet. The word tall corresponds to a curve that defines the degree to which any person is tall. If the set of tall people is given the well-defined (crisp) boundary of a classical set, you might say all people taller than six feet are officially considered tall. However, it is unreasonable to call one person short and another one tall when they differ in height by an inch.

If the kind of distinction shown previously is unworkable, then what is the right way to define the set of tall people? Much as with the plot of weekend days, the following figure shows a smoothly varying curve that passes from not-tall to tall. The output axis is a number known as the membership value between 0 and 1. The curve is known as a membership function and is often given the designation of . For example, the following figure shows both crisp and smooth tall membership functions. In the top plot, the two people are classified as either entirely tall or entirely not-tall. In the bottom plot, the smooth transition allows for different degrees of tallness. Both people are tall to some degree, but one is significantly less tall than the other. The taller person, with a tallness membership of 0.95 is definitely a tall person, but the person with a tallness membership of 0.3 is not very tall.

Subjective interpretations and appropriate units are built into fuzzy sets. If you say "She's tall," then the tall membership function should already take into account whether you are referring to a six-year-old or a grown woman. Similarly, the units are included in the curve since it makes no sense to say "Is she tall in inches or in meters?"

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