What is all the Fuzz about? Engineering Fuzzy Systems CPSC 433 - - PDF document

▶
what is all the fuzz about
SMART_READER_LITE
LIVE PREVIEW

What is all the Fuzz about? Engineering Fuzzy Systems CPSC 433 - - PDF document

Aug 18, 2022 227 likes •338 views

Fuzzy Systems in Knowledge What is all the Fuzz about? Engineering Fuzzy Systems CPSC 433 Christian Jacob Dept. of Computer Science Dept. of Biochemistry & Molecular Biology University of Calgary Christian Jacob, University of Calgary

slide-1
SLIDE 1

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

CPSC 433 Christian Jacob

  • Dept. of Computer Science
  • Dept. of Biochemistry & Molecular Biology

University of Calgary

What is all the Fuzz about?

Fuzzy Systems

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Systems in Knowledge Engineering

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • 1. Motivation
  • 2. Fuzzy Sets
  • 3. Fuzzy Numbers
  • 4. Fuzzy Sets and Fuzzy Rules
  • 5. Extracting Fuzzy Models from Data
  • 6. Examples of Fuzzy Systems

Fuzzy Systems

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Fuzzy logic was introduced by Lot Zadeh

UC Berkeley in 1965.

  • Fuzzy logic is based on fuzzy set theory, an

extension of classical set theory.

  • Fuzzy logic attempts to formalize

approximate knowledge and reasoning.

  • Fuzzy logic did not attract any attention until

the 1980s fuzzy controller applications.

What does Fuzzy Logic mean?

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Humans primarily use fuzzy terms: large, sma,

fast, slow, warm, cold, ...

  • W

e say: If the weather is nice and I have a little time, I will

probably go for a hike along the Bow.

  • W

e dont say: If the temperature is above 24 degrees and the cloud

cover is less than 10, and I have 3 hours time, I will go for a hike with a probability of 0.47.

Fuzzy is Just Human

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Zadeh: Make use of the leeway of fuzziness.
  • Fuzziness as a principle of economics:
  • Precision is expensive.
  • Only apply as much precision to a problem as

necessary.

  • Example: Backing into a parking space

How long would it take if we had to park a car with a

precision of ±2 mm?

Fuzzy is Economical

slide-2
SLIDE 2

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • 1. Motivation
  • 2. Fuzzy Sets
  • 3. Fuzzy Numbers
  • 4. Fuzzy Sets and Fuzzy Rules
  • 5. Extracting Fuzzy Models from Data
  • 6. Examples of Fuzzy Systems

Fuzzy Systems

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Example: the set of young people
  • W

e can dene a characteristic function for this set:

Basics of Fuzzy Sets

young = {x ∈ P | age(x) ≤ 20}

µyoung(x) =

  • 1

: age(x) ≤ 20 : 20 < age(x)

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Fuzzy set theory oers a variable notion of

membership:

  • A person of age 25 could still belong to the set of

young people, but only to a degree of less than one, say 0.9.

  • Now the set of young contains people with ages

between 20 and 30, with a linearly decreasing degree

  • f membership.

Basics of Fuzzy Sets

µyoung(x) =    1 : age(x) ≤ 20 1− age(x)−20

10

: 20 < age(x) ≤ 30 : 30 < age(x)

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Membership Function

µyoung(x) =    1 : age(x) ≤ 20 1− age(x)−20

10

: 20 < age(x) ≤ 30 : 30 < age(x)

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Shapes for Membership Fcts.

Trapezoid: [a,b,c,d] Triangle: [a,b,c] Gaussian: [a,] Singleton: [a,m]

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Parameters of FMFs

  • Support: sA := x : Ax > 0
  • The area where the membership function is positive.
  • Core: cA := x : Ax = 1
  • The area for which elements have a maximum degree
  • f membership to the fuzzy set A.
slide-3
SLIDE 3

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Cut: A := x : Ax =
  • The cut through the membership function of A at

height a.

  • Height: hA := maxx Ax
  • The maximum value of the membership function of

A.

Parameters of FMFs

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Support: sA := x : Ax > 0

Trapezoid: [a,b,c,d] Triangle: [a,b,c] Gaussian: [a,] Singleton: [a,m]

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Core: cA := x : Ax = 1

Trapezoid: [a,b,c,d] Triangle: [a,b,c] Gaussian: [a,] Singleton: [a,m] m=1

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Cut: A := x : Ax =

Trapezoid: [a,b,c,d] Triangle: [a,b,c] Gaussian: [a,] Singleton: [a,m]

  • CPSC 433 Articial Intelligence: An Introduction

Christian Jacob, University of Calgary

Height: hA := maxx Ax

Trapezoid: [a,b,c,d] Triangle: [a,b,c] Gaussian: [a,] Singleton: [a,m] Height = m Height = 1 Height = 1 Height = 1

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Linguistic V ariables

  • The covering of a variable domain with several

fuzzy sets, together with a corresponding semantics, denes a linguistic variable.

  • Example: linguistic variable ag
slide-4
SLIDE 4

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Using fuzzy sets, we can incorporate the fact

that no sharp boundaries between groups, such as young, middleaged, and old, exist.

  • The corresponding membership functions
  • verlap in certain areas, forming noncrisp

fuzzy boundaries.

  • This compositional way of dening fuzzy sets
  • ver a domain of a variable is called granulation.

Granulation

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Granulation results in a grouping of objects

into imprecise clusters of fuzzy granules.

  • The objects forming a granule are drawn

together by similarity.

  • This can be seen as a form of fuzzy data

compression.

Fuzzy Granules

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • If expert knowledge on a domain is not

available, an automatic granulation is used.

  • Standard granulation using an odd number of

membership functions:

  • NL: negative large, NM: negative medium, NS:

negative small, Z: zero, ...

Finding Fuzzy Granules

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • 1. Motivation
  • 2. Fuzzy Sets
  • 3. Fuzzy Numbers
  • 4. Fuzzy Sets and Fuzzy Rules
  • 5. Extracting Fuzzy Models from Data
  • 6. Examples of Fuzzy Systems

Fuzzy Systems

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Realworld measurements are always

imprecise.

  • Usually, such imprecise measurements are

modeled through

  • a crisp number x, denoting the most typical value,
  • together with an interval, describing the amount of

imprecision.

  • In a linguistic sense: about x

Fuzzy vs. Crisp Numbers

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Fuzzy numbers are a special type of fuzzy sets

with specic membership functions:

  • A must be normalized cA ∅.
  • A must be singular. There is precisely one point

which lies inside the core, modeling the typical value = modal value of the fuzzy number.

  • A must be monotonically increasing left of the core

and monotonically decreasing on the right only one peak!.

Fuzzy Numbers as Fuzzy Sets

slide-5
SLIDE 5

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Typically, triangular membership functions are

chosen for fuzzy numbers.

Fuzzy Number Example

1 2

  • 1
  • 2

About 0 About 1 About 2 CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Number Operations

  • For practical purposes, we can calculate the

result of applying a monotonical operation ⊗

  • n fuzzy numbers as follows:
  • Subdivide Ax and Bx into monotonically

increasing and decreasing parts.

  • Perform the operation ⊗ jointly on the increasing

decreasing parts of numbers A and B.

  • Plateaus can be dealt with in a single computation

step.

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Let A and B be fuzzy numbers, and ⊗ a strongly

monotonical operation.

  • Let a1, a2 and b1, b2 be the intervals in which Ax and

Bx are monotonically increasing decreasing.

  • If there exist subintervals 1, 2 ⊆ a1, a2 and 1, 2 ⊆

b1, b2, such that

∀ xA ∈ α1, α2, ∀ xB ∈ β1, β2: μAxA = μBxB = λ

then

∀ t ∈ α1 ⊗ β1, α2 ⊗ β2: μA ⊗ Bt = λ.

Fuzzy Number Operations

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

µA + B(t) = 1.0 t = ...

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

µA + B(t) = 1.0 t = 40 + 70 = 110

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

µA + B(t) = 0.4 t [..., ...]

slide-6
SLIDE 6

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

µA + B(t) = 0.4 t [20+64, 30+64]

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

µA + B(t) = 0.4 t [20+64, 30+64] = [84, 94]

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

µA + B(t) = 0.4 t = ...

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

µA + B(t) = 0.4 t = 46 + 76 = 122

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

µA + B(t) = 0.4 t = 46 + 76 = 122

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

Support: sA + B = [..., ...]

slide-7
SLIDE 7

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

Support: sA + B = [10+60, 50+80] = [70, 130]

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Addition

… and the rest by linear interpolation: Done!

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Monotonic Function

µf(A)(y) = max{µA(x) | ∀x : f(x) = y}

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • 1. Motivation
  • 2. Fuzzy Sets
  • 3. Fuzzy Numbers
  • 4. Fuzzy Sets and Fuzzy Rules
  • 5. Extracting Fuzzy Models from Data
  • 6. Examples of Fuzzy Systems

Fuzzy Systems

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Let A and B be fuzzy sets.
  • Intersection conjunction:

A∩Bx = min μAx, μBx

  • Union disjunction:

A∪Bx = max μAx, μBx

  • Complement:

¬Ax = 1 μAx

Operations on Fuzzy Sets

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Union & Intersection

A∩Bx = min μAx, μBx A∪Bx = max μAx, μBx

slide-8
SLIDE 8

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Union & Intersection

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Union & Intersection: Alternatives

µA∩B(x) = min{µA(x),µB(x)} µA∪B(x) = max{µA(x),µB(x)} µA∩B(x) = µA(x)·µB(x) µA∪B(x) = min{µA(x)+µB(x),1}

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Fuzzy Rules

IF temperature = low THEN cooling valve = half open. IF temperature = medium THEN cooling valve = almost

  • pen.

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

MaxMin Inference

Max-Min Inference

Approximate Reasoning

IF temperature = low THEN cooling valve = half open IF temperature = medium THEN cooling valve = almost open

temperature = medium temperature = low valve = almost open valve = half open in

  • ut

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

MaxMin Inference

Max-Min Inference

Approximate Reasoning

IF temperature = low THEN cooling valve = half open IF temperature = medium THEN cooling valve = almost open

temperature = medium temperature = low valve = almost open valve = half open in

  • ut
  • 1. Input of crisp

value

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

MaxMin Inference

Max-Min Inference

Approximate Reasoning

IF temperature = low THEN cooling valve = half open IF temperature = medium THEN cooling valve = almost open

temperature = medium temperature = low valve = almost open valve = half open in

  • ut
  • 1. Input of crisp

value

  • 2. Fuzzification
slide-9
SLIDE 9

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

MaxMin Inference

Max-Min Inference

Approximate Reasoning

IF temperature = low THEN cooling valve = half open IF temperature = medium THEN cooling valve = almost open

temperature = medium temperature = low valve = almost open valve = half open in

  • ut
  • 1. Input of crisp

value

  • 2. Fuzzification
  • 3. Inference

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

MaxMin Inference

Max-Min Inference

Approximate Reasoning

IF temperature = low THEN cooling valve = half open IF temperature = medium THEN cooling valve = almost open

temperature = medium temperature = low valve = almost open valve = half open in

  • ut
  • 1. Input of crisp

value

  • 2. Fuzzification
  • 3. Inference
  • 4. Output set

Fuzzy OR

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

MaxMin Inference

Max-Min Inference

Approximate Reasoning

IF temperature = low THEN cooling valve = half open IF temperature = medium THEN cooling valve = almost open

temperature = medium temperature = low valve = almost open valve = half open in

  • ut
  • 1. Input of crisp

value

  • 2. Fuzzification
  • 3. Inference
  • 4. Output set

Fuzzy OR

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

MaxMin Inference

Max-Min Inference

Approximate Reasoning

IF temperature = low THEN cooling valve = half open IF temperature = medium THEN cooling valve = almost open

temperature = medium temperature = low valve = almost open valve = half open in

  • ut
  • 1. Input of crisp

value

  • 2. Fuzzification
  • 3. Inference
  • 4. Output set
  • 5. Defuzzification

Center of gravity CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • 1. Motivation
  • 2. Fuzzy Sets
  • 3. Fuzzy Numbers
  • 4. Fuzzy Sets and Fuzzy Rules
  • 5. Extracting Fuzzy Models from Data
  • 6. Examples of Fuzzy Systems

Fuzzy Systems

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Approximate representation of functions,

contours, and relations

Extracting Fuzzy Models: Graphs

slide-10
SLIDE 10

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

Global Granulation of Input Space

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

FixedGrid Rules Extraction

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

AdaptiveGrid Rules Extraction

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • 1. Motivation
  • 2. Fuzzy Sets
  • 3. Fuzzy Numbers
  • 4. Fuzzy Sets and Fuzzy Rules
  • 5. Extracting Fuzzy Models from Data
  • 6. Examples of Fuzzy Systems

Fuzzy Systems

CPSC 433 Articial Intelligence: An Introduction Christian Jacob, University of Calgary

  • Berthold, M., and Hand, D. J. 2003. Inteigent Data
  • Analysis. Berlin, Springer.
  • Bothe, H. 1995. Fuzzy Logic Einfhrung in die Theori

und Anwendungen. Berlin, Springer.

  • Kasabov, N. 1996. Foundations of Neural Networks, Fuzzy

Systems, amd Knowledge Engineering. Cambridge, MA, MIT Press.

References