Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ

Slides:

Advertisements

Similar presentations

Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ

Advertisements

Clustering Validity Adriano Joaquim de O Cruz ©2006 NCE/UFRJ

Fuzzy Control - Example. Adriano Joaquim de Oliveira Cruz NCE e IM, UFRJ ©2002.

Learning from Examples Adriano Cruz ©2004 NCE/UFRJ e IM/UFRJ.

Clustering: Introduction Adriano Joaquim de O Cruz ©2002 NCE/UFRJ

Chapter 6 FUZZY FUNCTION Chi-Yuan Yeh. Kinds of fuzzy function Crisp function with fuzzy constraint Fuzzy extension function which propagates the fuzziness.

Automatic Histogram Threshold Using Fuzzy Measures 呂惠琪.

ผศ. ดร. สุพจน์ นิตย์ สุวัฒน์ ตอนที่ interval, 2. the fundamental concept of fuzzy number, 3. operation of fuzzy numbers. 4. special kind of fuzzy.

Fuzzy Sets - Introduction If you only have a hammer, everything looks like a nail. Adriano Joaquim de Oliveira Cruz – NCE e IM, UFRJ

Economics 214 Lecture 6 Properties of Functions. Increasing & Decreasing Functions.

Geometry of Fuzzy Sets.

Fuzzy Systems Adriano Cruz NCE e IM/UFRJ

Fuzzy Expert System.

Extension Principle — Concepts

Economics 2301 Lecture 5 Introduction to Functions Cont.

12 April 2013Lecture 2: Intervals, Interval Arithmetic and Interval Functions1 Intervals, Interval Arithmetic and Interval Functions Jorge Cruz DI/FCT/UNL.

Neuro-Fuzzy Control Adriano Joaquim de Oliveira Cruz NCE/UFRJ

Linguistic Descriptions Adriano Joaquim de Oliveira Cruz NCE e IM/UFRJ © 2003.

Economics 214 Lecture 31 Univariate Optimization.

Operations on Fuzzy Sets One should not increase, beyond what is necessary, the number of entities required to explain anything. Occam's Razor Adriano.

August 12, 2003II. BASICS: Math Clinic Fall II. BASICS – Lecture 2 OBJECTIVES 1. To define the basic ideas and entities in fuzzy set theory 2. To.

WELCOME TO THE WORLD OF FUZZY SYSTEMS. DEFINITION Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept.

Fussy Set Theory Definition A fuzzy subset A of a universe of discourse U is characterized by a membership function which associate with each element.

Math – Getting Information from the Graph of a Function 1.

CHAPTER Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h  0 ( f(a + h) – f(a))

CSCI 115 Chapter 5 Functions. CSCI 115 §5.1 Functions.

3. Rough set extensions  In the rough set literature, several extensions have been developed that attempt to handle better the uncertainty present in.

6.2 Polynomials and Linear Functions

Discrete Mathematics and Its Applications Sixth Edition By Kenneth Rosen Copyright  The McGraw-Hill Companies, Inc. Permission required for reproduction.

Image Processing Fundamental II Operation & Transform.

Functions Section 1.4. Relation The value of one variable is related to the value of a second variable A correspondence between two sets If x and y are.

Practical Considerations When t , the supp, then there will be a value of t when supp folds, it becomes multi-w-to-one-x mapping.

Unit 1 Review Standards 1-8. Standard 1: Describe subsets of real numbers.

Rosen 1.6, 1.7. Basic Definitions Set - Collection of objects, usually denoted by capital letter Member, element - Object in a set, usually denoted by.

3.1 Derivative of a Function Objectives Students will be able to: 1)Calculate slopes and derivatives using the definition of the derivative 2)Graph f’

Fuzzy Expert System n Introduction n Fuzzy sets n Linguistic variables and hedges n Operations of fuzzy sets n Fuzzy rules n Summary.

3.2 Properties of Functions. If c is in the domain of a function y=f(x), the average rate of change of f from c to x is defined as This expression is.

Increasing & Decreasing Functions A function f is increasing on an interval if, for any x 1 and x 2, in the interval, x 1 < x 2 implies f(x 1 ) < f(x 2.

FUNCTIONS COSC-1321 Discrete Structures 1. Function. Definition Let X and Y be sets. A function f from X to Y is a relation from X to Y with the property.

1.1 - Functions. Ex. 1 Describe the sets of numbers using set- builder notation. a. {8,9,10,11,…}

Ch.3 Fuzzy Rules and Fuzzy Reasoning

Fuzzy Relations( 關係 ), Fuzzy Graphs( 圖形 ), and Fuzzy Arithmetic( 運算 ) Chapter 4.

Symmetry and Coordinate Graphs Section 3.1. Symmetry with Respect to the Origin Symmetric with the origin if and only if the following statement is true:

Chapter 6 Fuzzy Function. 6.1 Kinds of Fuzzy Function 1. Crisp function with fuzzy constraint. 2. Crisp function which propagates the fuzziness of independent.

Chapter 3 FUZZY RELATION AND COMPOSITION

W-up 3-27 When is the function increasing?

Learning Target: I will determine if a function is increasing or decreasing and find extrema using the first derivative. Section 3: Increasing & Decreasing.

3.3: Increasing/Decreasing Functions and the First Derivative Test

6.1 One-to-One Functions; Inverse Function

3.1 Extrema on an Interval Define extrema of a function on an interval. Define relative extrema of a function on an open interval. Find extrema on a closed.

II. BASICS: Math Clinic Fall 2003

Do your homework meticulously!!!

Absolute or Global Maximum Absolute or Global Minimum

Let’s Review Functions

A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2). A function f is decreasing.

-20 is an absolute minimum 6 is an absolute minimum

Warm-up: Determine which of the following are functions. A. B.

Lesson 5.8 Graphing Polynomials.

4.1 One-to-One Functions; Inverse Function

I. Finite Field Algebra.

6.1 One-to-One Functions; Inverse Function

Combinations of Functions

4.1 – Graphs of the Sine and Cosine Functions

2.3 Properties of Functions

f(x) g(x) x x (-8,5) (8,4) (8,3) (3,0) (-4,-1) (-7,-1) (3,-2) (0,-3)

Let’s Review Functions

Let’s Review Functions

FUZZY SETS AND CRISP SETS PPTS

Let’s Review Functions

Presentation transcript:

Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ

@2002 Adriano Cruz NCE e IM - UFRJNo. 2 Fuzzy Numbers n A fuzzy number is fuzzy subset of the universe of a numerical number. –A fuzzy real number is a fuzzy subset of the domain of real numbers. –A fuzzy integer number is a fuzzy subset of the domain of integers.

@2002 Adriano Cruz NCE e IM - UFRJNo. 3 Fuzzy Numbers - Example u(x) x51015 Fuzzy real number 10 u(x) x51015 Fuzzy integer number 10

@2002 Adriano Cruz NCE e IM - UFRJNo. 4 Functions with Fuzzy Arguments n A crisp function maps its crisp input argument to its image. n A fuzzy arguments have membership degrees. n When computing a fuzzy mapping it is necessary to compute the image and its membership value.

@2002 Adriano Cruz NCE e IM - UFRJNo. 5 Functions applied to intervals n Compute the image of the interval. n An interval is a crisp set. x y I y=f(I)

@2002 Adriano Cruz NCE e IM - UFRJNo. 6 Monotonic Continuous Functions n For each point in the interval –Compute the image of the interval. –The membership degrees are carried through. I

@2002 Adriano Cruz NCE e IM - UFRJNo. 7 Monotonic Continuous Functions x y x y u(x) u(y)

@2002 Adriano Cruz NCE e IM - UFRJNo. 8 Monotonic Continuous Ex. n Function: y=f(x)=0.6*x+4 n Input: Fuzzy number - around-5 –Around-5 = 0.3/ / /7 n f(around-5) = 0.3/f(3) + 1/f(5) + 0.3/f(7) –f(around-5) = 0.3/ / /8.2 I

@2002 Adriano Cruz NCE e IM - UFRJNo. 9 Monotonic Continuous Ex. f(x) x 510 u(x) x

@2002 Adriano Cruz NCE e IM - UFRJNo. 10 Nonmonotonic Continuous Functions n For each point in the interval –Compute the image of the interval. –The membership degrees are carried through. –When different inputs map to the same value, combine the membership degrees.

@2002 Adriano Cruz NCE e IM - UFRJNo. 11 Nonmonotonic Continuous Functions x y x y u(x) u(y)

@2002 Adriano Cruz NCE e IM - UFRJNo. 12 Nonmonotonic Continuous Ex. n Function: y=f(x)=x 2 -6x+11 n Input: Fuzzy number - around-4 Around-4 = 0.3/2+0.6/3+1/4+0.6/5+0.3/6 y = 0.3/f(2)+0.6/f(3)+1/f(4)+0.6/f(5)+0.3/f(6) y = 0.3/3+0.6/2+1/3+0.6/6+0.3/11 y = 0.6/2+(0.3 v 1)/3+0.6/6+0.3/11 y = 0.6/2 + 1/ / /11 I

@2002 Adriano Cruz NCE e IM - UFRJNo. 13 Nonmonotonic Continuous Functions x y x y u(x) u(y)

@2002 Adriano Cruz NCE e IM - UFRJNo. 14 Extension Principle n Let f be a function with n arguments that maps a point in X 1 xX 2 x...xX n to a point in Y such that y=f(x 1,…,x n ). n Let A 1 x…xA n be fuzzy subsets of X 1 xX 2 x...xX n n The image of A under f is a subset of V defined by

@2002 Adriano Cruz NCE e IM - UFRJNo. 15 Arithmetic Operations n Applying the extension principle to arithmetic operations it is possible to define fuzzy arithmetic operations n Let x and y be the operands, z the result. n Let A and B denote the fuzzy sets that represent the operands x and y respectively.

@2002 Adriano Cruz NCE e IM - UFRJNo. 16 Fuzzy addition n Using the extension principle fuzzy addition is defined as

@2002 Adriano Cruz NCE e IM - UFRJNo. 17 Fuzzy addition - Examples n A = 0.3/ /2 +1/ / /5 n B = 0.5/10 + 1/ /12 n Getting the minimum of the membership values n A+B=0.3/ / / / / / / / / / / / / / /17 n Getting the maximum of the duplicates n A+B= 0.3/ / /13 + 1/ / / /17

@2002 Adriano Cruz NCE e IM - UFRJNo. 18 Fuzzy addition A, x=3 B, y= C, x=14

@2002 Adriano Cruz NCE e IM - UFRJNo. 19 Fuzzy Arithmetic n Using the extension principle the remaining fuzzy arithmetic fuzzy operations are defined as: