Fuzzy urban sets
A conceptual framework and implementation
methodology for measuring urbanization
Eric J. Heikkila, Associate Professor
School of Policy, Planning, and Development
University of Southern California
Presentation to
The 2nd Urban Research Symposium, World Bank
Tuesday, 16 December 2003, Session C, Topic 3
MC 4-800; 11:00am – 12:45pm
Acknowledgements
This presentation is adapted from:
Eric J. Heikkila, Tiyan Shen, and Kaizhong Yang, 2003, “Fuzzy
Urban Sets: Theory and application to desakota regions in China”,
Environment and Planning B: Planning and Design, vol. 30(2),
239-254.
Partial funding support gratefully acknowledged from:
• Fulbright Program; Council for International Exchange of Scholars
• USC’s Lusk Real Estate Center
• USC’s Center for International Studies
How to measure or even define
urbanization at a global scale?
“Level of urbanization refers to the percentage of population
residing in places classified as urban. Urban and rural
settlement are defined in the national context and vary
among countries (the definitions of urban are generally
national definitions incorporated in the latest census).”
-- Habitat, Cities in a Globalizing World: Global Report on
Human Settlements, 2001
The World Bank’s approach
“Urban population is the midyear population of areas
defined as urban in each country and reported to
the United Nations … Because the estimates in the
table are based on national definitions of what
constitutes a city or metropolitan area, crosscountry comparisons should be made with
caution.”
-- World Bank, World Development Indicators, 1999
Circular reasoning
• The jurisdiction-based approach taken by Habitat and by
the World Bank does not provide insights into urbanization
as a geographic phenomenon.
• Instead, it defines urban territory (or population) as territory
(or population) found within urban jurisdictions. This
reflects the administrative orientation of those who gather
the data.
• Existing methods of measuring urbanization are inherently
limited by this administrative perspective.
Fuzzy urban sets:
a proposed alternative
• As explained in this presentation,
conceiving of urbanizing areas as fuzzy
urban sets enables one to neatly characterize
urbanization in terms of three key aspects:
– An aggregate measure of urbanization
– A measure of how “fuzzy” an area is
– An entropy measure of urbanization
Premise
1. The “fuzzy urban set” concept
is inherently suited to patterns
of human settlement
2. Measures of urbanization
derived from fuzzy set
mathematics are useful means
of characterizing the
geography of human
settlement patterns
Image from: “Fuzzy-Subsethood based Color ImageProcessing”
Mario Köppen, Ch. Nowack, G. Rösel, Fraunhofer-Institute IPK, Berlin
http://www.vision.fhg.de/ipk/publikationen/pdf/fns99slides.pdf
Peri-urbanization as
“ambiguous urbanity”
• Urbanization in Asia is
characterized by peri-urban
phenomenon termed “desakota” by Terry McGee
• The desakota regions
exemplify a kind of
ambiguous urbanity that is
well suited to a fuzzy set
framework
Remote sensing images as fuzzy sets
• Each pixel of a remote
sensing image may in
principal be interpreted as
an element whose degree of
membership ui in the fuzzy
urban set U lies somewhere
on the closed unit interval:
0 <_ ui < _1
The urban-rural dichotomy is expressed in shades of gray
Urbanization as …
2
1
Remote
Sensing Image
Summary
Statistics
3
Geographic
Phenomenon
Dynamic
Model
4
The primary focus of this paper is on [2-->3] above
Definition of fuzzy sets
• A “fuzzy set” U is one whose elements have membership
levels that may fall anywhere on the unit interval:
ui ε [0,1]
• A fuzzy set allows for degrees of membership. Note that
this is is not a question of probability or uncertainty –
rather, it is a matter of degree.
• In contrast, the membership function for a “crisp set” is
strictly binary: ui ε {0,1}
% r e f l e c t i o n
Determine each pixel’s urban membership value
We use training area method
Spectral image data
TM2
TM5 TM7
wave frequency
Pixel# TM2 TM5 TM7 m
1
2
.
.
.
j
.
0.53
0.37
.
.
.
0.82
.
.
.
N
.
.
0.77
Kosko’s fuzzy hypercube
• Following Kosko (1992),
we may define a fuzzy
hypercube of dimension N
corresponding to the
number of elements of the
fuzzy set U
I = (1,1,1)
M = (0.5,0.5,0.5)
• From this perspective, the
fuzzy set U can be
represented as a point
located somewhere within
the hypercube
U
u2
u1
O = (0,0,0)
We may characterize a peri-urban region in terms of the
location of the fuzzy set U in the context of its hypercube
Three fundamental dichotomies (1):
Urban-Rural
• Geographic
• Aggregate level of
interpretation
urbanization for the study area
• Fuzzy set
• Measure of the cardinality of
interpretation
the fuzzy set U
• Geometric
• Distance from the origin
interpretation
Three fundamental dichotomies (2):
Fuzzy-Crisp
• Geographic
• Extent of desakota
interpretation
phenomenon
• Fuzzy set
• Measure of the fuzziness of
interpretation
the set U
• Geometric
• Proximity to midpoint M
interpretation
Three fundamental dichotomies (3):
Entropy-Order
• Geographic
• Extent of spatial diffusion of
interpretation
urbanization phenomenon
• Fuzzy set
• Uniformity of membership for
interpretation
the fuzzy set U
• Geometric
• Proximity to central diagonal
interpretation
Calculate three summary measures
• Extent of urbanization
Mean U = Σi ui / N
• Level of fuzziness
F = M[U
Uc] / M[U U Uc]
• Degree of entropy
E = Σi pi log(1/pi) / log(n)
Feasibility
The feasibility of this approach is demonstrated in:
Eric J. Heikkila, Tiyan Shen, and Kaizhong Yang, 2003, “Fuzzy
Urban Sets: Theory and application to desakota regions in China”,
Environment and Planning B: Planning and Design, vol. 30(2),
239-254.
where the method is applied to remote sensing images of
Ningbo, China. I am currently working on another paper,
which uses U.S. Census data on population density to address
the question, “Is Los Angeles fuzzier than Chicago?”
Potential advantages of proposed approach
1.
The continuous membership function is inherently well suited
to classification of ambiguous peri-urban regions
2.
The three measures are derived from a somewhat abstract
mathematical formulation yet have very natural
interpretations in the context of peri-urbanizing regions
3.
The three measures are distinct and independent from one
another yet are coherently linked through their common
derivations
4.
The three measures together provide a richer descriptive basis
for peri-urbanizing regions than a single “percentage urban”
statistic can possibly do
Future prospects & considerations
• Consistent basis for measurement of peri- urbanization
over time and across locations
• A significant improvement over conventional measures of
urbanization that are tied to jurisdictional boundaries and
“crisp set” assumptions
• Summary statistics of complex phenomenon useful basis
to support new round of modeling efforts
• Remaining issues: image (study area) boundary, scale,
remote sensing interpretation
Conclusions
• The World Bank and Habitat are two organizations the
world relies on for knowledge about urbanization
• Nothing could be more fundamental than measuring or
characterizing the phenomenon – if we can’t characterize
it, how can we presume to explain it?
• Existing jurisdiction-based methods are inadequate
• The fuzzy urban set method is feasible, and provides a
ready means of supplementing existing data sources
• Data derived from this approach can help stimulate and
support “higher-order” research into urbanization