1. Visualizing the Uncertainty of Urban Ontology Terms
Hyowon Ban and Ola Ahlqvist
Department of Geography, The Ohio State University
1049B Derby Hall, 154 N Oval Mall
Columbus, OH 43210, USA.
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COST C21: The 1st Towntology Workshop
Geneva, Swiss Nov. 6~7, 2006

2. Introduction Ontology in Urban Civil Engineering & Geography
For a better communication about urban environment
The differences in understanding concepts of an ontology
Ex) ontology of an urban area type: urban, suburban, exurban, and rural areas
An articulation of these differences is important
Ontology has been essential for a better communication among different groups about urban environment in Urban Civil Engineering as well as Geography

3. Exurban boundary issue
Recently exurban areas have fast growth
Separate definitions to call exurbanization but little consensus
Few research on the uncertainty of the boundaries in exurban areas
Only crisp boundaries in existing studies
- Only crisp boundaries are defined in existing studies

4. The research purposes
To compare the spatial implications of different ontological commitments from different definitions of exurban areas
To demonstrate the relevance of representing exurban areas as vague objects
Comparing the traditional crisp representation Vs. a vague, graded representation
Representing the different theoretical boundaries of exurban areas: crisp membership Vs. fuzzy membership
Visualizing them in maps: standard GIS techniques
To compare the spatial implications of different ontological commitments represented by different definitions of exurban areas
To demonstrate the relevance of representing exurban areas as vague objects by comparing the traditional crisp representation with a vague, graded representation
By representing the different theoretical boundaries of exurban areas using crisp boundaries and fuzzy membership functions
And visualizing them in maps using standard GIS techniques

5. Uncertainty in exurban boundaries
The concept of urban, suburban, exurban, and rural zones
Urban zone: within an urbanized area or an urban cluster (Fig. 1)
Suburban zone: a non-central county, metropolitan (Fig. 2)
Exurban zone: metropolitan counties outside this ring of suburban counties (Fig. 2)
Rural zone: outside of an urbanized area (Fig. 2)
No clear boundary between them
Fig. 1. Idealized spatial configuration of urban and rural area concepts
Fig. 2. Simple spatial distribution of urban, suburban, exurban, and rural areas

6. Uncertainty: error, vagueness, and ambiguity
Represented with probability
Vagueness
No unique allocation of individual objects to a class
Or, no precise spatial extent of the objects
Ambiguity
More than one definition for a term
One clearly defined object is a member of different classes
Error: can be represented with probability
Ambiguity: more than one definition for a term exist, one clearly defined object is a member of different classes under different classification schemes or interpretations

7. The idea of fuzzy membership functions
Fuzzy and rough extensions of traditional set theory
To represent semantic uncertainty of concept definitions
A rough fuzzy set
Semantic imprecision * vagueness
“closed” as a crisp and artificial definition “closed” as a continuous function (set of %) μF:U→[0, 1]
Fuzzy and rough extensions of traditional set theory (crisp membership)
Technique to represent semantic uncertainty of concept definitions
A rough fuzzy set: combinations of semantic imprecision and vagueness
Instead “closed” as a crisp and artificial definition, “closed” as a continuous function
μ(x) = Membership values indicate the degree of being a member of the fuzzy set “closed”
F = a fuzzy set of percentage values considered as “closed”
U = a universe of discourse U
[x] = An equivalence class that contains x∈U
Based on an approximation space (U,θ) defined on a universe of discourse U with an equivalence relation θ
The granularity imposed by θ on U results in a set of equivalence classes U/θ={Ei}, the quotient set.
An equivalence class that contains x∈U is denoted [x]. Now, let U be the universe of all closure percentages, and F a fuzzy set of percentage values considered as “closed”

8. Implementation of fuzzy membership functions with the exurban definition of Nelson (1992)
“Counties being those within 50 miles of the boundary of the central city of a Metropolitan Statistical Area (MSA) with a population of between 500,000 and less than 2 million, or within 70 miles of the boundary of the central city of an MSA with a population of more than 2 million”
Fig. 3. Membership function of distance in Delaware County based on Nelson’s (1992) definition

9. Implementation of fuzzy membership functions with the exurban definition of Daniels (1999)
“10 to 50 miles away from a major urban center of at least 500,000 people, or 5 to 30 miles from a city of at least 50,000 people, population density less than 500/mile2 , commute distance at 25 minutes or more”
Fig. 4. Membership function of distance (left) and population (right) in Delaware County based on Daniels’s (1999) definition

10. Conceptually synthesized definitions of exurban areas
Fig. 5. Differences between existing definitions of exurbanization of Daniels (1999) and Nelson (1992)

11. Results: definition of Nelson
Difference between crisp membership & fuzzy membership representations
Geometric + non-geometric representation
the map in the left with crisp membership represents the entire area in Delaware County as exurban.
However, the map in the right with a fuzzy membership representation shows the gradual transition of MF values of being exurban.
From the membership function based on Nelson’s definition, areas closer to the center of Columbus MSA have higher MF values of being exurban.
Fig. 6. Exurban areas based on Nelson’s (1992) definition with crisp membership (left) and fuzzy membership (right)

12. Results: definition of Daniels
The fuzzy membership: a more specific spatial pattern of sprawl than the crisp membership
The fuzzy membership is able to address a more specific spatial pattern of sprawl than the crisp membership (areal unit: block group)
Each block group is assigned to either exurban (MF value 1) or non-exurban (MF value 0).
In the middle-left map, most of the non-exurban areas are located near central urban areas such as the Columbus MSA in the lower part of the map and near the City of Delaware.
In the right map, the block groups have MF values—the degree of being exurban—ranging from 0 to 1 and the MF values are classified into 5 classes for visualization purposes.
Since this definition includes distance, the MF values generally increases with distance from the center of MSA.
The population component of this definition causes additional variation along the general distance trend. For example, the block groups in the class of values 0.6~0.8 are roughly forming a band around the 10 miles distance from the center of Columbus MSA.
Also, some small number of block groups show “leapfrog[1]” pattern in the entire study area.
In these two maps, the difference between crisp and fuzzy memberships of the definition is clearly shown.
The map with fuzzy membership is able to address a more specific spatial pattern of sprawl in the study area than the map with crisp membership.
It shows variations of degree in being exurban among the block groups within the exurban area as well as the non-exurban areas that the map with crisp membership fails to show.
[1] If there are discontinuous development patterns, this is called leapfrog or scattered development (Irwin and Bockstael 2002).
Fig. 7. Exurban areas based on Daniel’s (1999) definition with crisp membership (left) and fuzzy membership (right)

13. Concluding discussion
A clear difference between crisp membership Vs. fuzzy membership representations in defining exurban boundaries
Uncertainty reveals the heterogeneity of exurban areas in a location specific context
The crisp classification of exurban area may miss the graded phenomena
A clear difference between crisp membership and fuzzy membership representations in defining exurban boundaries using the concept of fuzzy ontology
Uncertainty in this study reveals the heterogeneity of exurban areas in a location specific context
The crisp classification of exurban area may miss the graded phenomena within such areas

14. Suggested ontology representation and prevailing approaches
The standard first-order logic representation with a fuzzy set
To explicitly recognize the vagueness of terms
To admit partial belonging to several possible categories
Comparison of different notions of exurban areas
Using standard descriptive properties (i.e. population, distance, and etc)
To compare across heterogeneous terminologies
To look for similarities and differences in a flexible manner
To generalize the standard first-order logic representation with a fuzzy set that can explicitly recognize the vagueness of terms and admit partial belonging to several possible categories
To compare different notions of exurban areas by using standard descriptive properties such as population and distance
To compare across heterogeneous terminologies and look for similarities and differences in a flexible manner

15. Future extensions
Negotiated definition with 3D geovisualization (Fig. 8)
Incorporation of the dynamic character of urbanization processes
Category descriptions with time dependent characteristics
A weighted fuzzy membership function
Comparison of the difference between definitions
MOUNTGILEAD
MANSFIELD
DELAWARE
COLUMBUS
Negitiated definition
Different definitions can be reconciled
Deviations such as a high membership in Daniels and low membership in Nelson will be in between in the negotiated value
Incorporate the dynamic character of urbanization processes
By visualizing the changing exurban boundaries through time in 3D animation of snapshot maps from different points in time
The very simplistic category descriptions with time dependent characteristics
This could help identify for example areas where an exurbanization process is just starting
Ex. fuzzy time constraints
The difference between definitions can be compared
by integrating the distribution of fuzzy set values at each location
Fig. 8. 3D visualization of the ‘negotiated’ average of fuzzy membership of Daniels and Nelson

16. Questions and answers