introduction to fuzzy logic and applications in...

Fuzzy Logic and Applications in GIS

Wolfgang Kainz 1

Introduction to Fuzzy Logic and Applications in GISWolfgang KainzCartography and GeoinformationDepartment of Geography and Regional ResearchUniversity of Vienna, Universittsstrae 7, A-1010 Vienna, AustriaTel.: +43 (1) 4277 48640 Fax: +43 (1) 4277 9486e-mail: [email protected]://homepage.univie.ac.at/wolfgang.kainz

Wolfgang Kainz Introduction to Fuzzy Logic and Applications in GIS 2

Schedule

Examples: fuzzy site analysis, fuzzy spatial reasoning

03.30pm - 05.00pmBreak03.00pm - 03.30pmFuzzy reasoning, software tools01.30pm - 03.00pmLunch12.00pm - 01.30pm

Fuzzy relations, linguistic variables and hedges, fuzzy boundaries

10.30am - 12.00pmBreak10.00am - 10.30am

Introduction, fuzzy sets, membership functions, operations, -cuts

08.30am - 10.00am

Introduction

Crisp exampleFuzziness and probabilityFuzzy sets


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Example: problem statementGiven a topographic data set, find all areaswith

flat slope,favorable aspect, andmoderate elevation; that areclose to a lake or reservoir,not near a major road, and arenot located in a park or military installation.


CriteriaFlat slope is less than 10 degrees.Aspect is favorable when the terrain is flat or oriented towards SE, S, or SW, i.e., aspect is -1 or between 135 and 225.Elevation is moderate when it is between 1,350 and 2,150 meters.Close to a lake or reservoir means within a buffer of 1,000 meters.Not near a major road means not within a buffer of 300 meters.


FuzzinessFuzziness refers to vagueness and uncertainty, in particular to the vagueness related to human language and thinking.

the set of tall people. all people living close to my home. all areas that are very suitable for growing corn.


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Fuzziness vs. probabilityProbability gives us an indication about the likelihood an event will occur. Whether it is going to happen or not, is not known. Fuzziness is an indication to what degreesomething belongs to a class. We know that it exists. What we do not know, however, is its extent, i.e., to which degree members of a given universe belong to the class.


Terminology is member of set , belongs to is not a member of , does not belong to is subset of , is included in united with intersected with

for all

x Ax A

x Ax A

x Ax AA B

A BA B

A B A BA B A B


Characteristic function of crisp sets

=AxAx

xX AA iff 0 iff 1

)(where}1,0{:

Let A be a subset of the universe X. Then the characteristic function A is defined as:

Where iff means if and only if.


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Characteristic function: example

Let X be the set of all persons attending the ESRI user conference.Let A X be the set of all persons who attend this seminar.The characteristic function of all xthat are member of A is 1.


Fuzzy sets

A fuzzy (sub-)set A of a universe X is defined by a membership function A.

. in of the is)( where]1,0[:

Axvalue membershipxX AA

The universe is never fuzzy!


Crisp sets versus fuzzy setsCharacteristic function Membership function

=

AxAx

x

X

A

A

iff 0 iff 1

)(

where}1,0{:

. in ofvalue membership

the is)( where]1,0[:

Ax

xX

A

A


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ExampleHeight of three persons: A is 185cm (6 0.8), B is 165cm (5 5.0), and C is 186cm (6 1.2)

short average tall

0

1

165 185

Height

short average tall

0

1

165 185

HeightAB C CAB

short average tallA 0 1 0B 1 0 0C 0 0 1

Characteristic Functions Membership Functions

short average tallA 0 0.60 0.50B 0.50 0.60 0C 0 0.56 0.53


Notation of fuzzy sets:discrete expressionWhen the universe X = {x1, x2, , xn} is finite a fuzzy set A on X can be expressed as

ii

n

iAnnAA xxxxxxA /)(/)(/)(

111

=

=++=

The symbol / is called separator. The symbols and + function as aggregation and connection of terms.


Notation of fuzzy sets:continuous expression

When the universe X = {x1, x2, } is infinite a fuzzy set A on X can be expressed as

xxAX A

/)(= The symbols / and function as separator and aggregation.NOTE: The symbol has nothing to do with the integral!


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Empty setThe empty set is defined as

0)( , = xXx

For every element of the universe we have trivially

1)( , = xx X

Membership Functions

Linear membership functionsSinusoidal membership functionsSemantic import approach


Membership functionThe membership function must be a real valued function whose values are between 0 and 1.The membership values should be 1 at the center of the set, i.e., for those members that definitely belong to the set.


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Membership functionThe membership function should fall off in an appropriate way from the center through the boundary.The points with membership value 0.5 (crossover point) should be at the boundary of the crisp set, i.e., if we would apply a crisp classification, the class boundary should be represented by the crossover points.


Choice of membership function

The membership function depends on the application.

Example: moderate elevation may be defined differently in the Netherlands than in Tibet.


Choice of membership function

Classification based on attributesSemantic Import Approach (using a priorimembership functions)Fuzzy k-means or c-means (data driven, not discussed here)


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Types of membership functions

Linear membership functionsSinusoidal membership functionsGaussian membership function


Linear membership function

a

b c

d

0 20 40 60 80 100U

0.10.20.30.40.50.60.70.80.9

1Membership Value

>

+


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trapezoidala

b c

dtriangulara d

b=c

S-shapeda

b=c=d

L-shaped

a=b=c

d

Shapes of membership functions


2

2( )

2( )x c

A x e

=

-20 -10 0 10 20

0.10.20.30.40.50.60.70.80.9

1Membership Value

Uc

2

Gaussian membership function


Assignment 1List five phenomena in your work environment that can better be described as fuzzy sets than with a crisp classification.


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Operations on Fuzzy Sets

Support, HeightEquality, InclusionUnion, IntersectionComplement


Support of a fuzzy setThe support of a fuzzy set is the set of all elements of the universe that have a membership degree greater than 0.

supp( ) { | ( ) 0}AA x X x= >


Core of a fuzzy setThe core of a fuzzy set is the set of all elements of the universe that have a membership degree equal to 1.

core( ) { | ( ) 1}AA x X x= =


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Height of a fuzzy setThe height of a fuzzy set A, hgt(A), is the largest membership degree in A.

If the height is 1 then the fuzzy set is called normal.


Equality of fuzzy setsTwo fuzzy sets A and B are said to be equal (written as A = B) iff

).()(, xxXx BA =


Inclusion of fuzzy setsThe inclusion of fuzzy set A in B is defined as )()( iff , xxBAXx BA

20 10 0 10 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Membership Value

A

B


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Fuzzy set theoretic operations: UnionThe union of two fuzzy sets is defined as one of the following operators:

, ( ) max( ( ), ( )), ( ) ( ) ( ) ( ) ( ), ( ) min(1, ( ) ( ))

A B A B

A B A B A B

A B A B

x X x x xx X x x x x xx X x x x

= = + = +

where AB is the membership function of AB.

(2)(3)

(1)


120 140 160 180 200 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Type 1

120 140 160 180 200 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Type 2

120 140 160 180 200 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Type 3

short average


Fuzzy set theoretic operations: IntersectionThe intersection of two fuzzy sets is defined as one of the following operators:

, ( ) min( ( ), ( )), ( ) ( ) ( ), ( ) max(0, ( ) ( ) 1)

A B A B

A B A B

A B A B

x X x x xx X x x xx X x x x

= = = +

where AB is the membership function of AB.

(2)(3)

(1)


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120 140 160 180 200 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Type 1

120 140 160 180 200 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Type 2

120 140 160 180 200 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Type 3

short average


Fuzzy set theoretic operations: ComplementThe complement of a fuzzy set A is defined as

)(1)(, xxXx AA =

where A is the membership function of A.


120 140 160 180 200 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Average

120 140 160 180 200 220

0.10.2

0.30.4

0.5

0.60.7

0.80.9

1Complement of Average


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Properties valid for both fuzzy and crisp sets

,A A A A A A = =Idempotent law

,A B B A A B B A = = Commutative law( ) ( )( ) ( )

A B C A B CA B C A B C = =

Associative law

( ) ( ) ( )( ) ( ) ( )

A B C A B A CA B C A B A C = =

Distributive law

A A=Double negation

A B A BA B A B =

= De Morgan's law


Properties in general valid only for crisp sets

Law of the excluded middle A A X =

Law of contradiction A A =


220120 140 160 180 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

120 140 160 180 200 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1A A A A

A = average


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Assignment 2Can you give an explanation why the Min / Max operators for intersection / union are called non-interactive, whereas the alternative operators using the product and sum are called interactive?

-Cuts

-level sets


-Cuts (or -level sets)

The (weak) -cut A (with (0,1]) of a fuzzy set A is defined as

})(|{ = xXxA A

A strong -cut is defined as

})(|{ >= xXxA A


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5 7 9 11 13 15 17 19 21 23 25U

0.2

0.4

0.6

0.8

1Membership Grade

Example

A0.6


-cuts (or -level sets)An -cut is the set of all elements of the universe that typically belong to a fuzzy set.With -cuts we can decompose a membership function into an infinite number of rectangular membership functions (decomposition principle).

Fuzzy Relations

Binary fuzzy relations


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Binary fuzzy relations:Continuous expression

A binary fuzzy relation R between sets X and Y is defined as

= YX R yxyxR ),/(),(where R is the membership function of R as

]1,0[: YXR


Binary fuzzy relations:Discrete expression

A binary fuzzy relation R between sets X = {x1,,xn} and Y = {y1,,ym} is denoted as a fuzzy matrix

=

),(),(

),(),(

1

111

mnRnR

mRR

yxyx

yxyxR


Binary Fuzzy Relations: Example

X = {Vienna, Graz, Salzburg}Y = {Bratislava, Budapest, Ljubljana}Fuzzy relation: close

Bratislava Budapest LjubljanaVienna 0.9 0.6 0.5Graz 0.7 0.5 0.6

Salzburg 0.5 0.4 0.5


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Binary Fuzzy Relations: Example

Vienna

Graz

SalzburgBratislava

Budapest

Ljubljana

00.20.40.60.81

Grade

Linguistic Variables and Hedges

OperatorsHedges


Linguistic variableA linguistic variable is a variable that assumes linguistic values (linguisticterms).

ExampleVariable: heightValues: short, average, tall


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HedgesA hedge h functions as modifier of a meaning of a term x, thus resulting in a composite term hx, e.g., very steep.Examples for hedges are very, sort of, slightly, etc.They are implemented with operators on fuzzy sets.


Operators as a basis for hedges

=

=

=

=

otherwise))(1(21]5.0,0[)( for)(2)(:ationintensificcontrast

)()(:dilation

)()(:ionconcentrat

)(hgt)()(:ionnormalizat

2

2

)(int

)(dil

2)(con

)(norm

xxxx

xx

xx

xx

A

AAA

AA

AA

A

AA


Models of hedgesvery A = con(A)

more or less A(fairly A)

= dil(A)

plus A = A 1.25

slightly A = int[norm(plus A andnot (very A))]


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Fuzzy Boundaries

Map unit approachIndividual approach


Fuzzy polygon boundariesMap unit approach

All boundaries in the data set are assumed to be equally fuzzy

Individual boundary approachFuzziness determined for each feature class


Fuzzy polygon boundaries (in raster representation)

Extract the boundaries (e.g., with an edge filter)Use a spread function to compute the zones around the boundariesThe membership function has the cross over point at the original boundary


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Fuzzy boundariesBoundary

(membership degree = 0.5)Boundary width

Inside(membership between 1 and 0.5)Outside

(membership between 0.5 and 0)

Well inside

Well outside


Polygon with fuzzy boundary

1.0 0.5

0.0


Assignment 3How would you measure (determine) the width of a boundary in practice? Give examples for different phenomena (e.g., parcels, land use units, soil types,)?


Wolfgang Kainz 22

Fuzzy Reasoning

Direct methodSimplified method


Fuzzy Reasoning: Rules of inference

In binary logic reasoning is based on Deduction (modus ponens)

Premise 1: If x is A then y is B Premise 2: x is A Conclusion: y is B

Induction (moduls tollens) Premise 1: If x is A then y is B Premise 2: y is not B Conclusion: x is not A


Fuzzy Reasoning: Rules of inference

ExampleDeduction (modus ponens)

Premise 1: If it rains then I get wet Premise 2: It rains Conclusion: I get wet

Induction (moduls tollens) Premise 1: If it rains then I get wet Premise 2: I do not get wet Conclusion: It does not rain


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Fuzzy Reasoning: generalized modus ponens

Premise 1: If x is A then y is BPremise 2: x is A'Conclusion: y is B'

A, B, A', and B' are fuzzy sets where A'and B' are not exactly the same as Aand B.


Fuzzy Reasoning: generalized modus ponens (example)Premise 1: If temperature is low then set the

heater to highPremise 2: Temperature is very lowConclusion: Set the heater to very high


Fuzzy ReasoningDirect methods

Mamdanis Direct MethodTagaki & Sugenos Fuzzy ModelingSimplified Method

Indirect MethodNot discussed here


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Mamdanis Direct Method:Inference Rule

If x is A and y is B then z is C

where A, B, and C are fuzzy sets,x and y are premise variables,z is the consequence variable

premise consequence

fuzzy set


Mamdanis Direct Method:Inference Rule Example

set the air conditioner setting to highthen

humidity is fairly highand

room temperature is highIf

where the room temperature is measured in degreesCentigrade, the humidity in percent, and the air conditioner setting range from 1 to 10


Tagaki & Sugenos Fuzzy Modeling:Inference Rule

If x is A and y is B then z = ax+by+c

where A and B are fuzzy sets, x and y are premise variables, z = ax+by+c is the consequence part linear equation with the consequence part parameters a, b and c

premise consequence

linear function


Wolfgang Kainz 25


Tagaki & Sugenos Fuzzy Modeling: Inference Rule Example

set the air conditioner setting to room temperature x 0.2 + humidity x 0.05then





Simplified Method:Inference Rule

If x is A and y is B then z = c

where A and B are fuzzy sets, x and y are premise variables, c is the consequence, a real value (fuzzy singleton)

premise consequence

fuzzy singleton


Simplified Method :Inference Rule Example

set the air conditioner setting to 9then





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Generalized modus ponens with two premise variables

1 1 1

2 2 2

1

1

If is and is then is If is and is then is

:

If is then is then is : is , is : z is

n n n

x A y B z Cx A y B z C

p q

x A y B z Cp x A y Bq C


1. Apply the input values to the premise variables for every rule andcompute the minimum of 0( )iA x and 0( )iB y :

1 1

2 2

1 1 0 0

2 2 0 0

n 0 0

Rule : min( ( ), ( ))Rule : min( ( ), ( ))

Rule : min( ( ), ( ))n n

A B

A B

n A B

m x ym x y

m x y

=

=

=

2. Cut the membership function of the consequence ( )iC

z at im :

1 1

2 2

1 1 1

2 2 2

n

Conclusion of rule : ( ) min( , ( ))Conclusion of rule : ( ) min( , ( ))

Conclusion of rule : ( ) min( , ( ))n n

C C

C C

C n C n

z m z z Cz m z z C

z m z z C

=

=

=

3. Compute the final conclusion by determining the union of allindividual conclusions from step 2:

1 2( ) max( ( ), ( ), , ( ))

nC C C Cz z z z =

Mam

dani

Mam

dani

ssD

irec

t M

etho

dD

irec

t M

etho

d


To derive a single value from the fuzzy set of the conclusion we defuzzify it by the center of area method.

0

( )( )

C

C

z zz

z

=

Mam

dani

Mam

dani

ssD

irec

t M

etho

dD

irec

t M

etho

d


Wolfgang Kainz 27


Example

If distance between cars is long and speed is high then maintain speed

Rule 4

If distance between cars is long and speed is low then increase speed

Rule 3

If distance between cars is short and speed is high then reduce speed

Rule 2

If distance between cars is short and speed is low then maintain speed

Rule 1


Distance between cars

0 10 20 30 40m

0.2

0.4

0.6

0.8

1Membership


Speed

0 20 40 60 80 100kmh

0.2

0.4

0.6

0.8

1Membership


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Acceleration

- 20 - 10 0 10 20kmh2

0.2

0.4

0.6

0.8

1Membership


Step 1: distance = 15, speed = 60

0.250.750.254

0.250.250.253

0.750.750.752

0.250.250.751

MinHighLowLongShortRule


- 10 0 10 20U

- 20

0.2

0.4

0.6

0.8

1Rule 1 Rule 2

- 20 - 10 0 10 20U

0.2

0.4

0.6

0.8

1

Rule 3

- 20 - 10 0 10 20U

0.2

0.4

0.6

0.8

1Rule 4

- 10 0 10 20U

- 20

0.2

0.4

0.6

0.8

1



Wolfgang Kainz 29


- 10 0 10 20U

- 20

0.2

0.4

0.6

0.8

1Rule 1 Rule 2

- 20 - 10 0 10 20U

0.2

0.4

0.6

0.8

1

Rule 3

- 20 - 10 0 10 20U

0.2

0.4

0.6

0.8

1Rule 4

- 10 0 10 20U

- 20

0.2

0.4

0.6

0.8

1



- 10 0 10 20U

- 20

0.2

0.4

0.6

0.8

1


20 10 0 10 20U

0.2

0.4

0.6

0.8

1


Defuzzification

Reduce speed a little

20 10 0 10 20U

0.2

0.4

0.6

0.8

1Membership Grade

Center of area is -5.45833


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1. Apply the input values to the premise variables for every rule and compute the minimum of 0( )iA x and 0( )iB y :

1 1

2 2

1 1 0 0

2 2 0 0

n 0 0

Rule : min( ( ), ( ))Rule : min( ( ), ( ))

Rule : min( ( ), ( ))n n

A B

A B

n A B

m x ym x y

m x y

=

=

=

2. Compute the conclusion value per rule as: 1 1 1 1

2 2 2 2

n

Conclusion of rule :Conclusion of rule :

Conclusion of rule : n n n

c m cc m c

c m c

= =

=

3. Compute the final conclusion as:

1

1

nii

nii

cc

m=

=

=

Sim

plif

ied

Met

hod

Sim

plif

ied

Met

hod


Example

If slope is steep and aspect is unfavorable then risk is 4

Rule 4

If slope is flat and aspect is unfavorable then risk is 1

Rule 3

If slope is steep and aspect is favorable then risk is 2

Rule 2

If slope is flat and aspect is favorable then risk is 1

Rule 1


Example: slope

0 10 20 30 40Percent

0.2

0.4

0.6

0.8

1Membership

flat

steep


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0 50 100 150 200 250 300 350Aspect

0.2

0.4

0.6

0.8

1Membership

favorable

unfavorable

Example: aspect


Example:slope = 10, aspect = 180

0000.2Rule4

0000.5Rule3

0.40.210.2Rule2

0.50.510.5Rule1

ConclusionMin(s,a)Aspect (a)

Slope (s)


Example: final conclusion

0.5 0.4 0 0 1.290.5 0.2 0 0

c + + + = =+ + +


Wolfgang Kainz 32

Fuzzy Example

Fuzzy site analysisFuzzy reasoning


Fuzzy Site AnalysisFind all areas with flat slope, favorable aspect and moderate elevation that are close to a water body, not near a major road and are not located in a park or military installation.


Fuzzy Risk AnalysisDerive a risk map from the slope and aspect according to the following rules:

If slope is steep and aspect is unfavorable then risk is 4

Rule 4

If slope is flat and aspect is unfavorable then risk is 1

Rule 3

If slope is steep and aspect is favorable then risk is 2

Rule 2

If slope is flat and aspect is favorable then risk is 1

Rule 1

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