Case Study 4.1 (M)

LECTURE 4. Fuzzy Sets & Fuzzy

Geographical Objects.

David O’Sullivan, David Unwin “Geographic Information

Analysis”, chapter 11, pp. 316-356 Peter Burrough, Rachael

McDonnell “Principles of GIS”, chapter 11, pp. 265 - 297

Fuzzy Sets & Fuzzy Geographical Objects

 Lecture outline:  Imprecision as a way of thought.  Fuzzy objects.  Operations on several fuzzy sets  Combining fuzzy

boundaries & fuzzy attributes  Fuzzy k-means  Advantages & Disadvantages  Application of fuzzy classification

I’m fuzzy…..

BASICS

 To model geographical phenomena - first necessary to divide the world either into CRISP ENTITIES or into CONTONUOUS FIELDS.

 These fundamental spatial entities  - points, lines, polygons, pixels - are described by their:  location,  attributes,  topology.  All these statements in conventional logic can have only 2

values: TRUE & FALSE: 0 & 1.  The principles of 2-valued logic.  Lies at the heart of most of mathematics & computer

science.

BASICS

 Many geographical phenomena are not simple clear-cut entities.  The patterns vary over many spatial and temporal scales.

Defined by many interacting attributes.  Until recently we had no means in GIS, apart from statistics,

for dealing with entities that are not crispy defined.  By limiting the rules of logic to binary decisions we limit the

retrieval and classification of data to situations in which ONLY A COMPLETE MATCH is possible.

 In real life we make compromises based on the DEGREE with which an object meets our specifications.

 E.g. House. Rock types, soil/vegetation classes, socio-economic groupings, decisions in law courts (guilty, not, or not proven), nationality, even the borders of the nation state



Imprecision

 The Law of the EXCLUDED middle and its role in mathematical proof - of paramount importance in scientific & philosophical development.

 The rules of logic used in PC query languages are based on EXACT ideas of truth or falsehood.

 In environmental data this is not necessarily so…  “He says that he always lies” Neither true nor false. It is a

paradox.  Many users of Geo Info have a clear notion of what they need.

Land users & evaluators.  Imprecisely formulated requests (which areas are under the threat

of flooding?).  Must be translated in terms of the basic units of information

available. Not all information is EXACT !!!

Geographical phenomena & Imprecision

 Geo phenomena are more complicated.  We must consider grouping both in ATTRIBUTE (whether

all entities are of the same kind) & GEOGRAPHIC space(whether entities of the same kind occupy a region).

 Very often we ‘ve concentrated only on building class definitions from attributes with assumption that similar entities will cluster together…

 This may be the case, BUT…  How to deal with imprecision in overlapping attribute

classes ?????



Fuzzy Sets & Fuzzy objects

 Conventional or crisp sets allow ONLY binary membership functions (T or F). CRISP BOOLEAN SETS.

 An individual IS or IS NOT a member of any set. All members match the class concept, the class boundaries are SHARP.

 Fuzzy sets admits the possibility of partial membership.  The class boundaries are NOT or CANNOT be SHARPLY defined.

• Boolean CRISP Set. FUZZY Set.

Fuzzy Sets & Fuzzy objects

 In Fuzzy sets the grade of membership is expressed in terms of a scale that can vary CONTINUOUSLY between 0 & 1.

 Individuals to different degrees can be members of more than one set…

 THE BOUNDARY VALUES based on attributes.  In CRISP sets: - on the basis of expert knowledge; - using

methods of numerical taxonomy.  Both options are possible with FUZZY sets.  1. Uses a priori membership function with which individuals

can be assigned a membership grade.  The Semantic Import Approach or Model SI  2. The value of the membership function is a function of the

classifier used. METHOD of FUZZY K-MEANS

Membership Functions

 1. THE SEMANTIC IMPORT APPROACH - SI  Useful in situations where users have a very good,

QUALITATIVE IDEA of how to group data,  but for various reasons - difficulties with exactness.  The membership function should ensure that the GRADE of

membership is 1.00 at the center of the set, that it falls off through the fuzzy boundaries to the regions outside the set, where it takes the value 0.

 Boolean Fuzzy

A

1.0 1.0

0.0 0.0

ANot A

Not A

Not A

Not A

Membership Functions

 The SI approach to polygon boundaries.  You can incorporate information about the nature

of the boundaries and also to calculate sensible area measures.

 2 separate approaches: the map-unit, and the individual boundary approach.

 The SI approach can be used to add information about the abruptness of boundaries to a polygon database.

 Picture of spatial variation across and along boundaries.

Membership Functions

 2. FUZZY K-MEANS.  Very often users may not know which classification is

useful and appropriate.  Continuous classification.  Soil science, geohydrology, vegetation mapping.  Translating a multiple attribute description of an

object into k membership values to k classes or clusters.

 Rainforest types, heavy metal pollution.

Advantages & disadvantages

 SI approach to exactly delineated polygons can improve their information content, providing information about the nature of the sharpness or diffuseness of the identified boundaries.

 Results are more congruent with reality.  Membership values can be easily interpolated

over space.  THE GREATEST DIFFICULTIES come with

choosing the values of the control parameters to obtain the best results: the kind of membership functions, boundary values, transition widths, etc.

Applications

 In situations where a well-defined and functional scheme - SI APPROACH.

 SI continuous classes are more robust and less prone to errors and extremes than simple Boolean classes that use the same attribute boundaries.

 Fuzzy k-means approach is appropriate when information about the number and definition of classes is lacking.

 Fuzzy k-means methods yield sets of optimal, overlapping classes that can be also mapped in data space and in geographical space.