A Linguistic Approach to Model Urban Gro
A Linguistic Approach to Model Urban Growth
Lefteris Mantelas1, Poulicos Prastacos1 , Thomas Hatzichristos2, Kostis
Koutsopoulos2
1
Regional Analysis Division, Institute of Applied and Computational
Mathematics, Foundation for Research and Technology-Hellas,
GR 71110, Heraclion Crete, Greece
[email protected]
[email protected]
2
Department of Geography and Regional Planning, National
Technical University of Athens, I.Politechniou 9,
GR 15786, Zografou, Greece
[email protected]
[email protected]
Abstract: In this paper we present a linguistic approach for modeling urban
growth. We have developed a methodological framework which utilizes Fuzzy
Set theory to capture and describe the effect of urban features upon urban
growth and applies Cellular Automata techniques to simulate urban growth.
While there are several approaches that combine Fuzzy Logic and Cellular
Automata for urban growth modeling, we herein focus on the ability to use
partial knowledge and combine theory-driven and data driven knowledge. To
achieve this, a parallel connection between the input variables is introduced
which further allows the model to disengage from severe data limitations. In
this approach, a number of parallel fuzzy systems is used, each one of which
focuses on different types of urban growth factors, different drivers or
restrictions of development. The effects of all factors under consideration are
merged into a single internal thematic layer that maps the suitability for
urbanization for each area, providing thus an information flow familiar to the
human conceptualization of the phenomenon. Following, cellular automata
techniques are used to simulate urban growth. The proposed methodology is
applied in the Mesogeia area in the Attica basin (Athens) for the period 19902004 and provides realistic estimations for urban growth.
Keywords: Urban Growth, Fuzzy Logic, parallel conection of partial
knowledge, Cellular Automata, Mesogia - Athens
1 Introduction
The urban transition can be seen as the passage from a predominantly rural to a
predominantly urban society (Marshall, 2007) which takes place by the expansion of
existing urban areas and the development of new cities. People accumulate in urban
areas in their attempt to gain better access to goods, services, facilities and job
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
opportunities. As a result, financial, social and cultural activities flourish in large
urban areas attracting thus more people to live, work, produce and consume within the
urban environment. In 2007, 50% of the global population was living in town and
cities while it is estimated that 60 million people move to cities in a yearly basis.
What is more, these rates are expected to be preserved for the next 30 years (Marshall,
2007).
The changes in financial and social activities within the city, the settlement of new
population and the emergence of new activities lead to the reorganization of land use
and the production of buildings and services‟ networks in accordance to the
population needs (Κοηθηθός, 1986). What is more, changes are also driven by the
increasing expectations of the urban population which are reflected by peoples‟
residential choice. People‟s expectations may refer to buildings‟ attributes such as
more floor-space and better quality of construction but are also referring to locational
characteristics. People desire to live in areas that among others:
provide accessibility to high speed road networks, parking areas and public
transportation system
are in the vicinity of urban green areas and parks
provide access to goods, services and facilities
consist a healthy and safe environment
Apparently, seldom do the above criteria overlap and when they do they lead to
high real estate values. In this respect, urban growth can be described as the
spontaneous spatially referenced tradeoff between different types of human needs and
expectations. As a result, monitoring and comprehending urban growth relies heavily
on identifying the residential choice criteria and the factors that attract or repel new
settlements. For this reason, fuzzy logic has a key role to play in the challenging field
of urban modeling; a role whose importance stems from the fact that it mimics the
ways in which humans make decisions in an environment of uncertainty and
imprecision (Zadeh, 1993).
2 Challenges in Urban Modeling
The term „modeling‟ refers to creating a strictly defined analog of real world by
subtraction (Κου όπουζος, 2002). Yet there is no rigorous framework for modeling
such a spatio-temporal phenomenon as urban growth since there lies great inherent
spatial, temporal and decision-making heterogeneity (Cheng & Masser, 2003), which
results from socio-economic and ecological heterogeneity itself. Moreover there is
something special regarding the spatio-temporal nature of the urban growth. Urban
growth does not simply evolve in time; it also spreads in space and not always
continuously. This means that apart from the difficulties of studying a spatial
phenomenon, when studying urban growth we may come across first-seen qualitative
phenomena and interactions, that cannot be modeled mathematically in an easy way.
The problem seems to be that our knowledge, both theory-driven and data-driven,
is not really describing urban growth dynamics in general, but instead the part of the
urban growth dynamics that have already occurred and have been observed and
experienced. What is more, knowledge about the operational scale(s) of urban form
A Linguistic Approach to Model Urban Growth
and process, and the interaction and parallelism among different scales, is poor
(Dietzel & Clarke, 2004). We deal with a phenomenon which exists but it is also
recreated in space, extending itself both continuously and discontinuously in space
while evolving in time. Moreover, its dynamics evolve in time as well and all there is
for modeling urban growth is our experience of the phenomenon itself, which might
be inaccurate for describing its future evolution.
On top of these, for an urban model to be useful it should be able to describe the
objects, the relations and it‟s assumptions in an open, visible and comprehensible
way. This way, results and the underlying mechanisms can be challenged by experts
(Ness & Low, 2000) and get improved. At the same time, an urban model should be
able to be used both for forecasting and describing urban growth (Liu, 2009). This
allows the model not only to estimate the future evolution of an urban system but also
to unveil the underlying structure and procedures of the urban growth phenomenon.
Apparently, the usability of the estimations and the knowledge provided by the model
are as much important as the model‟s consistency to the real world.
3 Application of Fuzzy Logic for Urban Modeling
„Where is a mountain?‟ is a simple question to ask, but it is not easy to give a
consistent, precise response (Fisher & Wood, 2004). Fuzziness is an inherent attribute
of information; yet crisp GIS tend to force a binary representation of reality (Liu &
Phinn, 2001) despite the fact that issues related to vagueness, imprecision and
ambiguity can be addressed by the fuzzy set theory and fuzzy logic (Malczewski,
2004).
Fuzzy logic was originally proposed by Zadeh (1965) as a generalization of binary
logic and allows the continuous analysis between “false” and “true”. It is used to
model imprecision, vagueness and uncertainty in real world and bridges the gap
between qualitative and quantitative modeling. Fuzzy logic does not comply with the
binary property of dichotomy; hence fuzzy variables may consist of partially
overlapping fuzzy sets. The mathematical form of a fuzzy set is fully described by a
membership function which returns a membership value (η) within [0,1] for a given
object while a linguistic variable is used to describe its quality. Linguistic variables,
apart from describing primitive fuzzy sets, are also used to define new sets, based on
the primitive ones allowing thus the management of information in a way closer to
this of human conceptualization.
The knowledge base is represented as “IF…THEσ” rules, connecting hypotheses
to conclusions through a certainty factor while inference is divided (in general) in
three stages (Hatzichristos & Potamias, 2004; Kirschfink & Lieven 1999):
aggregation that returns the fulfillment of hypothesis for every rule
individually
implication that combines aggregation‟s result to the rule‟s certainty factor
(CF) resulting to the degree of fulfillment for each rule‟s conclusion
accumulation that corresponds to compromising different individual
conclusions into a final fuzzy result.
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
There is a great number of applications of fuzzy logic in spatial analysis; these
include defining and locating spatial objects (Bejaoui et al., 2007; Hatzichristos &
Potamias, 2004; Fisher et al., 2007; Fisher & Wood, 2004), mapping continuous
spatial phenomena (Fonte & Lodwick, 2005; Joss et al., 2008), defining topological
relations (Matsakis & Nikitenko, 2005; Takemura et al., 2005), satellite images
classification (Nedeljkovic, 2006), analysis of relations between geographical objects
(Guesgen, 2005, Kratochwil & Benedikt, 2005), statistical analysis of spatial data and
data mining (Kollias et al., 1999; Liu & George, 2005), development of DSS and
expert systems (Kalogirou, 2002; Stefanakis et al., 1996), spatial and temporal
interpolation (Dragicevic, 2005; Dragicevic & Marceau, 2000 & 2001).
For the case of urban modeling, the reason to apply fuzzy logic is twofold. First, it
provides improved mapping of land use and land cover (Arijit et al., 2006; Henning,
2003). Second, it provides a mechanism to represent human decision making in
linguistic terms and to approximate complex non-linear systems with simple models
(Chen & Linkens, 2004) making thus the analysis of complex systems easier (Setnes
et al., 1998).
Fuzziness in urban growth originates from a series of sources which can be found
as much in a definition and representation level, as in an analysis and simulation
level. To start with, urban cover is a rather abstract concept whose definition is based
more on functional criteria rather than morphological. Additionally, there is no such
thing as sharp boundaries between urban, partly urban and rural or forest areas (Liu &
Phinn, 2001). Therefore, fuzzy classification offers a better choice in land-use
mapping (Henning, 2003). What is more, fuzzy representations can capture mixed use
and coverage (Heikkila et al., 2002).
From a technical point of view, fuzzy modeling can be interpreted as a qualitative
modeling scheme which describes system behavior using fuzzy quantities presenting
thus a hybrid quantitative-qualitative approach (Chen & Linkens, 2004). This allows
describing partial and multiple relations in a distinct way. Nevertheless, the prime
advantage off fuzzy modeling is the facility for the explicit knowledge representation
in the form of if-then rules and the mechanism of human-like reasoning in linguistic
terms (Chen & Linkens, 2004).
The application of natural language allows the direct use of empirical knowledge
for the phenomenon of urban growth which includes experts‟ opinions, theoretical
approaches and historical registrations for the development of cities. Moreover, it
allows the transfer of knowledge based on empirical similarity patterns. Nevertheless,
what is more important is the ability of fuzzy logic to mimic the human-like decision
making and describe the underlying mechanisms in terms of fuzzy multi-criteria
systems. A large part of the applications of fuzzy logic for urban growth simulation
incorporates cellular automata techniques.
4 Fuzzy Urban Cellular Automata
Cellular Automata (CA) are a computational methodology in which the system under
study is divided into a set of cells with each cell interacting with all other cells
belonging to predefined neighborhoods through a set of simple rules (Krawzyk,
A Linguistic Approach to Model Urban Growth
2003). The interactions take place in discrete time steps with each cell‟s state at any
time period estimated by considering the state of the neighboring cells. This approach
is repeated continuously in a self-reproductive mechanism with no external
interference. Growth is thus simulated through a bottom up approach and this makes
CA an appropriate technique for simulating complex phenomena that is difficult to
model with other approaches.
There is a great variety of highly sophisticated crisp approaches concerning CA
based urban growth modeling. Among them, the stochastic approach of Mulianat et al
(Mulianata & Hariadi, 2004), the object-oriented approaches of Cage (Blecic et al.,
2004) and Obeus (Benenson & Kharbash, 2006), the environment Laude that
combines CA and Genetic Algorithms (Colona et al., 1998) and some widely applied
models, such as Sleuth (Clarke et al., 1997; Dietzel & Clarke, 2004; Dietzel et al.,
2005) and Moland (Engelen et al., 2007). Crisp urban CA may be categorized in
many different ways, in general though, they might be either quantitative or
qualitative. Quantitative models express the urban dynamics in terms of numerical
equations. Such models focus on the efficiency of the estimations and can provide
rather accurate results. What such models are not capable of is to map and express the
qualitative characteristics of urban growth phenomenon, which are a result of the
socio-economic decision making of the urban population.
Qualitative models on the other hand are rule-based and are capable of such
mapping and expression since they focus on the quality of causes and effects. They
fail though to compete with numerical models in following the exact numerical path
of urban growth. What is more, in the binary world of crisp rule-based systems,
qualities, objects and relations are strictly defined and either exist or not. There is no
such thing as partial, uncertain or imprecise fact, membership or relation and this is
not the way reality works.
The combination of CA and fuzzy logic under the term Fuzzy Cellular Automata
presents a hybrid approach that bridges qualitative and quantitative CA. In this
approach quantitative information can be converted to qualitative and vice versa
through the implementation of membership functions. Fuzzy Cellular Automata are a
rather new development; yet they have been implemented in a series of spatial
applications including fire propagation models (Mraz & Zimic, 1999) forest insect
infestation (Bone et al., 2006), and spatial load of electricity demand (Miranda &
Monteiro, 1999).
On the field of urban modeling, fuzzy cellular automata have been used both in
theoretical approaches (Dragicevic, 2004; Vancheri et al., 2004) and real world
applications. The first application by Wu (1996 & 1998) applies a fuzzy system to
govern the system‟s evolution using a max-min inference and crisp input and output.
This means that for each area only the maximum effect of the parameters that favor
urban growth is applied. In a similar fashion, only the minimum effect of the
parameters that restrict urban growth is applied. What is more, despite the fact that
information is processed within a fuzzy system, there is no gradation in the results and
each area is either considered to be urban or non-urban. In Wu‟s model fuzzy sets and
rules are defined empirically while fuzzy hedges may also be used. The model was
applied on real data and appears to simulate urban growth in a rather accurate way,
yet no evaluation index was used.
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
Liu and Phinn (2001, 2003 & 2005) proposed an approach with significant
differences. To start with, both input and output are represented as fuzzy sets which
are empirically designed. Additionally, population density was used to initialize the
fuzzy urban status of each area and define its propensity to urbanization. Finally, in
this approach, rules express logistic patterns of evolution while the input parameters
determine the speed of growth. The model was applied in Sydney for the period 19711996 during which urban cover grew by 28%. For this case study, the Kappa1 index of
agreement calculated in 3X3 neighborhoods was 90%. Despite the high levels of
accuracy, the model appears to focus in the partly urban to urban transition but it is
not equally effective when it comes to rural to urban transition. Our approach
attempts to combine the advantages of these two models and reduce their
disadvantages in an effective modeling environment that can use both theory-driven
and data-driven knowledge while retains a knowledge base form that is adjusted to the
available data.
5 The Modeling Framework
While most contemporary urban models go through an in depth analysis of selected
parameters, we propose a modeling structure that allows the parallel “in-width”
analysis of any given spatial or spatially referenced variable. We have developed a
modeling structure (figure 1) based on our previous work (Mantelas et al., 2007 &
2008) that attempts to describe a work flow familiar to the human perception of the
urban growth process. A structure based upon the relations between facts and
procedures so that even partial results may be helpful to the better understanding of
the process.
Apparently the model‟s structure and the systems‟ connection are tightly related to
the model‟s efficiency, the information‟s flow and the interpretability of the extracted
knowledge. Moreover, a generic form of the model is sustained which allows to avoid
severe data limitation (more data provide better results, less data still provide results
though); hence the model can be transferred to both data-rich and data-poor
applications.
Our approach uses spatially variable rules that may be either data-driven or theorydriven; hence they may fit better to reality allowing the user to overcome the
limitations of the available data by using exogenous knowledge adapted to the model
by empirical similarity patterns. Empirical rules are enabled by the fact that the
knowledge base is expressed in common language, which makes this model friendly
and usable, especially to the non-expert users.
We propose a structure of fuzzy systems each one of which receive different input
and conclude over certain aspects of the overall dynamics while the system that
simulates the evolution of the urban growth further incorporates CA techniques. An
1
The kappa coefficient of agreement expresses the agreement between two categorical datasets,
corrected for the agreement as can be expected by chance, which depends on the distribution of
class sizes in both datasets (Bishop et al., 1975). Kappa is well suited to compare a pair of land
use maps with its values ranging from 1, indicating a perfect agreement, to -1 indicating no
agreement (Jasper, 2009).
A Linguistic Approach to Model Urban Growth
important feature of the model is that the Growth Simulator does not access the input
data directlyν instead it operates upon a single intermediate layer called „suitability for
urbanization‟ that is the output of the Suitability Calculator.
Fig. 1. The structure of the modeling framework
Though the main idea of combining CA and fuzzy logic exists in previous
approaches, the herein presented approach presents some advantages; more
specifically:
it allows the combination of theory-driven and data-driven knowledge
it supports a reducible/extensible form of knowledge base which means that:
o it does not require specific data apply
o variables and rules can me added or removed without altering the
rest of the knowledge base
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
spatial parameters may be taken into consideration within each rule’s
certainty factor.
advanced fuzzy hedges are introduced
simulation of spontaneous growth is supported
The model can be used to reproduce the spatial patterns of urban growth but
requires the amount of urban cover to be allocated in a certain period to be given as an
input parameter. This allows us to measure real time as a function of the allocated
growth while at the same time functions as the exit criterion.
6 The Suitability Calculator
The suitability calculator receives input data and concludes over the suitability for
urbanization for each cell. The role of the suitability is twofold; on the one hand
defines the maximum urban cover that can be allocated to a non-urban cell and on the
other determines the maximum densification speed of an already urban cell. Initially a
number of fuzzy systems is used to process the input data and estimate the thematic
suitability for urbanization for distinct groups of input variables. The themes that can
be used depend on data availability with physical/natural (slope, land use) and
accessibility (road networks) being the cardinal themes.
Data-driven rules in these systems have a plain hypothesis premise – which results
to trivial aggregation – and depict the average distribution of urban cover appearance
for each fuzzy set of the variable. Theory-driven rules can be added without affecting
the data-driven ones and may have a complex hypothesis while the aggregation
operators used relying on the syntax of the premise. Both types of rules though, apply
the product operator for implication and conclude over thematic suitability indexes
that consist of a single fuzzy set (high).
In order to extract the rules for these systems we calculate the fitting between each
fuzzy set of each input variable and the urban fuzzy set. This way, each fuzzy set of
each input variable forms the rule‟s hypothesis while as the correspondent certainty
factor is taken the value of the fitting without any further calibration being required at
this stageν i.e. the sets “low distance from secondary road network” and “high urban
cover” fit each other by 56%, hence the rule becomesμ
IF “distance from secondary road network is Low”, THEN “suitability is High”
CF=0.56
|
What is more, the thematic suitability systems apply accumulation using the
algebraic sum operator – also known as the „probabilistic τR‟ – in the paradigm of
the Dempster-Shafer theory of evidence (Ahmazadeh & Petrou, 2001). The central
notion is that the more rules that lead to the same conclusion, the less likely it is for
this conclusion to be false. Its advantage is that it takes into account not only the
strength of each result but also the number of rules that conclude to this specific
resultν it‟s disadvantage on the other hand is that it tends to return higher values than
normal as the number of rules whose hypotheses are relatively dependent increase.
The certainty factors of the thematic suitability rules systems may be spatially
variable, which means that the same rule performs differently in different locations.
A Linguistic Approach to Model Urban Growth
This allows us in example to assign different suitability to urbanization for a certain
land use in the North-east and South-west parts of the area under study. Spatial
variability is gained by expressing rules‟ certainty factors as a function in terms of a
spatial 2-D fuzzy variable with 9 fuzzy sets which expresses the relative location of a
cell within the study area (figure 2).
The thematic indexes are considered to be either constant or variable in time. Static
indexes are calculated only once while variable ones may change in consecutive steps.
A suitability index may be treated as variable for three reasons:
input variables change in time i.e. road network or subway lines
input variables do not change in time, but their effect on urban growth is, i.e.
slope or distance from the sea
the combination of the above.
It is evident that in order to introduce variable suitability indexes additional
information is required. For the first case, insight to the planned projects and
developments is needed while the second case requires a time-series of data or
historical knowledge of them. Obviously, the third case needs both of them.
Once the thematic suitability indexes are calculated, they are merged into a single
overall suitability index which in the same fashion consists of a single fuzzy set
(high). This is accomplished by a separate fuzzy system that applies exclusively the
Algebraic Sum as an aggregation operation upon the following rule:
IF “Index A, … Index N are High ”, THEN “overall suitability is High” | CF=1.
Fig. 2. graphs of the fuzzy sets of the 2-D spatial variable, the horizontal level is a square reparameterization of the area under study while the vertical axis is the membership values for
each spatial fuzzy set: center, N, S, E, W (top) and center, NE, NW, SE, SW (bottom)
The suitability systems, apart from providing useful intermediate results, are used
for three purposes. To start with, they result into decreasing the number of variables
used in the following systems, making thus easier the further analysis and allowing
the user to use more simple and hence more comprehensible rules; for modeling,
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
simplification is both necessary and useful (Ness & Low, 2000). What is next, each
thematic system focuses on both the relations and the diversifications of the input
variables (i.e. if density of road network is positive, then obviously the distance from
road network is zero and vice versa) and as a result the outputs – the thematic indexes
– tend to be less correlated to each other than the input variables. Finally, it is much
easier and less risky to update the (variable) thematic indexes rather than the initial
variables.
7 The Urban Growth Simulator
The area under study is divided into three groups – static non urban, dynamic non
urban and urban. The static and dynamic non-urban groups are described by a crisp
set (membership is 0/1) while urban is a fuzzy set. Static areas are assumed not to
change and in this respect are excluded by the model. The exclusion criterion is a full
or high membership in the “not high overall suitability” fuzzy set. Additional
empirical criteria may be added to exclude cells that are be covered by forests, have
very high slopes or similar attributes.
Dynamic non urban areas are then processed by a hybrid fuzzy system that
incorporates CA techniques in order to simulate the transition from non urban to
urban. This system operates in two distinct levels, edge expansion and spontaneous
growth, each one of which is applied by one rule that uses the overall suitability as a
roof for the conclusions‟ valueμ
Level 1 - edge expansion: IF “Cell Suitability is High” AND “Neighborhood is
Urban ”, THEN “Cell is Urban” | CF=cf1.
Level 2 - spontaneous growth : IF “Cell Suitability is EXTREMELY High” AND
“Neighborhood is NOT non-Urban ”, THEN “Cell is Urban” | CF=cf2.
Level 2 applies the fuzzy hedge „extremely‟ that is designed to increase membership
values that are greater than 80% while decreases smoothly but very fast as initial
membership values decrease (figure 3). The numerical formula of „extremely‟ is
given by the following equations:
out in 2(0.8
in ) / 2
if
,
out in 2 4*( 0.8
in
)
, if
in 0.8
in 0.8
Cells that are urbanized to any extend are accessed by a third level, the
intensification module that applies an exponential implication operator. The syntax of
the rule is essential the same as in Level 1:
Level 3 - intensification: IF “Cell Suitability is High” AND “Neighborhood is
Urban”, THEN “Cell is Urban” | CF=cf3.
A Linguistic Approach to Model Urban Growth
Fig. 3. Visualization of the membership of each cell in the „overall suitability high‟ set (left)
and the effect of the fuzzy hedge „extremely‟ on the same set (right)
The computational difference though is that the new operator raises the membership
value of the conclusion premise in the power of the hypothesis complement. Given
the fact that membership values, certainty factors and aggregation results are bounded
in [0,1], if the hypothesis of the rule is not met at all, such a rule results to no change
in the fuzzy set of the conclusion; it returns the initial membership value. On the other
hand if the hypothesis is fully met and the rule is deterministic, it results to a certain
conclusion – a membership value 1. In any other case it returns a membership value
within (m,1) where m is the initial membership value.
In the hybrid fuzzy systems we may additionally use simple empirical or common
sense rules while all rules are subjected to a calibration process to determine the
certainty factors which may vary in space. What is more, spatial variation allows us to
potentially use neighborhoods of different radius in different areas. Nevertheless, due
to the (in general) irreversible nature of CA, data for two time-points may depict the
result of the CA process but not the dynamic or the specific form of its transition
engine; in such a system, simulation is the only way to predict outcomes (Clarke et
al., 1997). As a result, at this stage, calibration takes place manually through trial and
error. Having though reduced the number of variables that are used, it is relatively
easy - and even kind of fun in a more literal sense of Urban Gaming Simulations
(Cecchini & Rizzi, 2001) to experiment with each rule‟s syntax and its parameters
fine tuning.
8 Application of the Model
The model was applied in the Mesogia area (figure 4) at the eastern part of the Attica
basin (Athens), an area of 632 square km that is located within 25km from the
historical center of Athens, yet it used to be agricultural until 15-20 years ago when it
started to develop rapidly; more specifically, from 1990 till today urban cover in
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
Mesogia has been doubled. Recently the new international airport, the extension of
the subway and train lines and the construction of a new highway in Mesogia have
influenced urban growth not only in terms of the growth rate but also in terms of the
location of the growth. What is more, Mesogia can be considered relatively
autonomous when studying urban growth since it is physically separated by the mount
of Hemmetus in the West and Aegean Sea in the East, while neither Northern, nor
Southern areas are significantly urbanized.
In our study the available data for Mesogia include the Corine Land Cover
database (100X100m spatial resolution) for 1990 which are available from the
European Environmental Agency and a Corine based classification for 2004 produced
by GeoInformation S.A2. Most of the area is classified as Agricultural or Forest/Seminatural while the vast majority of Artificial Surface is mainly Urban Fabric with the
exception of the airport. The road network was provided by Infocharta Ltd. 3 for the
year 2004 while an estimation of the road network for 1990 was produced based on a
satellite image of the area for 1990. Road network data were classified in primary and
secondary and layers of distance and density were derived. What is more, a 90m
resolution DEM of the area was acquired from the SRTM webpage 4.
Fig. 4. Mesogia are in east part of Athens - the wider and specific area under study
2
http://www.g-i.gr/
http://www.infocharta.gr/
4 http://www2.jpl.nasa.gov/srtm/
3
A Linguistic Approach to Model Urban Growth
Fig. 5. The singleton representation of land use classes‟ suitability for urbanization (left) and
the membership functions of the fuzzy sets (low, average and high) for the slope variable
(right)
Fig. 6. The membership functions of the fuzzy sets (low, average and high) for primary road
density (top left), primary road distance (top right), secondary road density (bottom left) and
secondary road distance (bottom right)
For the application of the model, land cover data were represented as singletons
(figure 5a) while for the quantitative variables of slope (figure 5b) and
distance/density of primary and secondary road network (figure 6) three fuzzy sets
(low, average and high) were empirically defined. In the thematic suitability systems,
each singleton and each fuzzy forms a separate high suitability rule whose certainty
factor equals the fitting degree of each singleton/set with the urban cover fuzzy set.
This procedure took place for data referring to 1990 and produced the rules initial
rules for suitability calculation.
These rules were separated in three groups, land use, slope and road network and
produced three correspondent thematic suitability indexes which were merged into the
overall suitability using the probabilistic sum operator. At this point the urban growth
simulator was applied using Moore neighborhoods of radius 1 while the certainty
factors were initially given the value 1. What is next, manual calibration of the model
through trial and error shaped the final rules of the suitability calculator (table 1) and
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
urban growth simulator (table 2) which in some cases were determined to vary in
space.
Table 1: Rules for high thematic suitabilty
IF
{HYPOTHESIS}
Corine is 111 or 112
Corine is 221or 222
Corine is 223 or 243
Corine is 242
Corine is 321 or 323 or 333
Slope is low
Slope is average
Slope is high
Distance from primary road is low
Distance from primary road is average
Distance from primary road is high
Density of primary road is low
Density of primary road is average
Density of primary road is high
Distance from secondary road is low
Distance from secondary road is average
Distance from secondary road is high
Density of secondary road is low
Density of secondary road is average
Density of secondary road is high
THEN
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
{CONCLUSION}
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
CF
1.
0.25+0.4SE
0.33
0.45
0.1
0.6
0.4
0.1
0.8
0.6
0.2
0.2
0.5
0.6+0.3E
0.9
0.6
0.1
0.5
0.7
0.9
1 is
THEN
=>
{CONCLUSION}
cell is urban
CF
0.8.
2 is
=>
cell is urban
0.9
and
THEN
=>
{CONCLUSION}
cell is urban
CF
0.7
1 is
THEN
=>
{CONCLUSION}
cell is urban
CF
0.8.
2 is
=>
cell is urban
Table 2: Rules applied in the growth simulator
LEVEL 1 – edge expansion
IF {HYPOTHESIS}
Cell Suitability is high and Moore
urban and location is not SE
Cell Suitability is high and Moore
urban and location is SE
LEVEL 2 – spontaneous growth
IF {HYPOTHESIS}
Cell Suitability is extremely high
Moore 3 is not non-urban
LEVEL 3 - intensification
IF {HYPOTHESIS}
Cell Suitability is high and Moore
urban and location is not NW
Cell Suitability is high and Moore
urban and location is NW
1.
The initial configuration appeared to underestimate the urban density of the Northwest part of the area and the spatial extend of urban cover in the South-east part. For
this reason the suitability of Corine classes 221 and 222 was increased in the South-
A Linguistic Approach to Model Urban Growth
east areas while the suitability of „high primary road density‟ was increased for the
whole east area. Additionally, while the growth simulator uses Moore neighborhoods
of radius 1 at all levels for the general case, level 1 and level 3 used Moore
neighborhoods of radius 2 for the South-east and the North-west areas
correspondingly. Figure 7 shows the final thematic and overall suitability indexes.
The final rules were applied upon the 1990 data in order to estimate the 2004 urban
cover (figure 8) while to evaluate the results four indicators were calculated (table 3):
the map error which the percentage of the misallocated cells to the whole
area
the model error which is the percentage of the misallocated cells to the total
number of cells allocated by the model
the Kappa index of agreement calculated for the whole area
the Lee-Shale index of agreement calculated for the whole area.
The error indexes suggest that the model can provide useful estimations of future
urban growth. Nevertheless, a more thorough view of the error indexes and
specifically their evolution over the iterations of the algorithm suggests that the
optimum – in terms of error indexes – is reached a few steps before the algorithm
exits and hence error is slightly increased during the last few steps. This is partially
due to the exit criterion which introduces dependencies between the two types of
possible error - either a rural cell is considered urban or vice versa. That means that,
more or less, for each mistakenly considered urban cell there is another cell,
mistakenly considered rural and after some steps more cells are allocated inaccurately
rather than accurately.
On top of the numerical indexes, the results of the models appear to visually fit in
an excellent way upon actual urban cover. Yet, to some extent, this is due to the low
spatial resolution of the Corine database which cannot capture the detailed urban
cover but rather provides a homogenous shape with high autocorrelation which is
easier to simulate. Even so, a significant part of the error cannot be avoided for two
reasons. First, the case study uses only a very small amount of data that cannot
diversify easily which cells present higher suitability than others. Second, statistical
analysis of the growth occurred during a period can elucidate the average way in
which people make their residential choice but cannot describe relatively rare
processes that are responsible for a significant part of the growth i.e. constructing the
airport or a sport facility.
Table 3: Error indexes for cases studied
Period
1990 - 2004
Growth
occurred
Map error
Model Error
Kappa
Lee-Shale
6%
21%
0.75
0.81
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
Fig. 7. Thematic suitability indexes based on land use (top left), slope (middle left) and road
network (bottom left) are merged into overall suitability for urbanization (right)
Fig. 8. Actual urban cover for 1990 and 2004 (left) and the fuzzy estimation of the 2004 urban
cover based on 1990 (right)
A Linguistic Approach to Model Urban Growth
9 Conclusions and Future Work
We developed and presented an urban growth model that applies a parallel connection
between input variables and can easily be transferred to both data rich and data poor
cases. The model was calibrated and applied in Mesogia area in east Attica (Athens)
and is capable of providing satisfactory estimations of the future urban growth
patterns at least in the short-term. The fitting indicators suggest that the model
simulates efficiently the qualitative patterns of the urban expansion in the study area
which is further certified by visual comparison.
This is partially due to the CA techniques incorporated, that are proved very
efficient in simulating the spread of existing urban cover. What they lack, and this is
because of the CA nature, is the ability to capture the urbanization of detached areas.
Nevertheless, in our case study, the low spatial resolution of the used data and the
homogenous spatial patterns of urban cover in the area conceal this inability. For this
reason, the spontaneous growth mechanism (level 3) does not improve significantly
the accuracy of the model.
Fuzzy logic is an advisable way to deal with vague data and stochastic relations
since it enhances the potential qualitative resolution of the model and provides the
proper tools for uncertainty management. It allows the process of information in both
a qualitative and a quantitative way and expresses the dynamic of the phenomenon in
common linguistic terms. The rules syntax is kept simple and hence comprehensible;
as a result the user can easily calibrate the model empirically and introduce changes
such as spatial variability or fuzzy hedges to improve the performance of the model.
What is more, theory driven rules can be added and applied along with the rules that
are extracted by data analysis.
Acquiring a rich data set to experiment on temporal rules is one possible direction
towards which future work could be directed. More challenging though is to test this
model in data of higher spatial resolution that are less auto-correlated and present
more complex patterns of growth. Finally it would be very interesting to investigate
the potentials of the modeling framework to study population and employment change
along with urban growth.
Acknowledgements. The research leading to these results has received funding from
the European Community's Seventh Framework Programme FP7/2007-2013 under
grant agreement n° 212034
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Mantelas, L., Hatzichristos, Th., & Prastacos P. (2008). Modeling Urban Growth using Fuzzy
Cellular Automata . Paper presented at the 11th AGIL
Lefteris Mantelas1, Poulicos Prastacos1 , Thomas Hatzichristos2, Kostis
Koutsopoulos2
1
Regional Analysis Division, Institute of Applied and Computational
Mathematics, Foundation for Research and Technology-Hellas,
GR 71110, Heraclion Crete, Greece
[email protected]
[email protected]
2
Department of Geography and Regional Planning, National
Technical University of Athens, I.Politechniou 9,
GR 15786, Zografou, Greece
[email protected]
[email protected]
Abstract: In this paper we present a linguistic approach for modeling urban
growth. We have developed a methodological framework which utilizes Fuzzy
Set theory to capture and describe the effect of urban features upon urban
growth and applies Cellular Automata techniques to simulate urban growth.
While there are several approaches that combine Fuzzy Logic and Cellular
Automata for urban growth modeling, we herein focus on the ability to use
partial knowledge and combine theory-driven and data driven knowledge. To
achieve this, a parallel connection between the input variables is introduced
which further allows the model to disengage from severe data limitations. In
this approach, a number of parallel fuzzy systems is used, each one of which
focuses on different types of urban growth factors, different drivers or
restrictions of development. The effects of all factors under consideration are
merged into a single internal thematic layer that maps the suitability for
urbanization for each area, providing thus an information flow familiar to the
human conceptualization of the phenomenon. Following, cellular automata
techniques are used to simulate urban growth. The proposed methodology is
applied in the Mesogeia area in the Attica basin (Athens) for the period 19902004 and provides realistic estimations for urban growth.
Keywords: Urban Growth, Fuzzy Logic, parallel conection of partial
knowledge, Cellular Automata, Mesogia - Athens
1 Introduction
The urban transition can be seen as the passage from a predominantly rural to a
predominantly urban society (Marshall, 2007) which takes place by the expansion of
existing urban areas and the development of new cities. People accumulate in urban
areas in their attempt to gain better access to goods, services, facilities and job
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
opportunities. As a result, financial, social and cultural activities flourish in large
urban areas attracting thus more people to live, work, produce and consume within the
urban environment. In 2007, 50% of the global population was living in town and
cities while it is estimated that 60 million people move to cities in a yearly basis.
What is more, these rates are expected to be preserved for the next 30 years (Marshall,
2007).
The changes in financial and social activities within the city, the settlement of new
population and the emergence of new activities lead to the reorganization of land use
and the production of buildings and services‟ networks in accordance to the
population needs (Κοηθηθός, 1986). What is more, changes are also driven by the
increasing expectations of the urban population which are reflected by peoples‟
residential choice. People‟s expectations may refer to buildings‟ attributes such as
more floor-space and better quality of construction but are also referring to locational
characteristics. People desire to live in areas that among others:
provide accessibility to high speed road networks, parking areas and public
transportation system
are in the vicinity of urban green areas and parks
provide access to goods, services and facilities
consist a healthy and safe environment
Apparently, seldom do the above criteria overlap and when they do they lead to
high real estate values. In this respect, urban growth can be described as the
spontaneous spatially referenced tradeoff between different types of human needs and
expectations. As a result, monitoring and comprehending urban growth relies heavily
on identifying the residential choice criteria and the factors that attract or repel new
settlements. For this reason, fuzzy logic has a key role to play in the challenging field
of urban modeling; a role whose importance stems from the fact that it mimics the
ways in which humans make decisions in an environment of uncertainty and
imprecision (Zadeh, 1993).
2 Challenges in Urban Modeling
The term „modeling‟ refers to creating a strictly defined analog of real world by
subtraction (Κου όπουζος, 2002). Yet there is no rigorous framework for modeling
such a spatio-temporal phenomenon as urban growth since there lies great inherent
spatial, temporal and decision-making heterogeneity (Cheng & Masser, 2003), which
results from socio-economic and ecological heterogeneity itself. Moreover there is
something special regarding the spatio-temporal nature of the urban growth. Urban
growth does not simply evolve in time; it also spreads in space and not always
continuously. This means that apart from the difficulties of studying a spatial
phenomenon, when studying urban growth we may come across first-seen qualitative
phenomena and interactions, that cannot be modeled mathematically in an easy way.
The problem seems to be that our knowledge, both theory-driven and data-driven,
is not really describing urban growth dynamics in general, but instead the part of the
urban growth dynamics that have already occurred and have been observed and
experienced. What is more, knowledge about the operational scale(s) of urban form
A Linguistic Approach to Model Urban Growth
and process, and the interaction and parallelism among different scales, is poor
(Dietzel & Clarke, 2004). We deal with a phenomenon which exists but it is also
recreated in space, extending itself both continuously and discontinuously in space
while evolving in time. Moreover, its dynamics evolve in time as well and all there is
for modeling urban growth is our experience of the phenomenon itself, which might
be inaccurate for describing its future evolution.
On top of these, for an urban model to be useful it should be able to describe the
objects, the relations and it‟s assumptions in an open, visible and comprehensible
way. This way, results and the underlying mechanisms can be challenged by experts
(Ness & Low, 2000) and get improved. At the same time, an urban model should be
able to be used both for forecasting and describing urban growth (Liu, 2009). This
allows the model not only to estimate the future evolution of an urban system but also
to unveil the underlying structure and procedures of the urban growth phenomenon.
Apparently, the usability of the estimations and the knowledge provided by the model
are as much important as the model‟s consistency to the real world.
3 Application of Fuzzy Logic for Urban Modeling
„Where is a mountain?‟ is a simple question to ask, but it is not easy to give a
consistent, precise response (Fisher & Wood, 2004). Fuzziness is an inherent attribute
of information; yet crisp GIS tend to force a binary representation of reality (Liu &
Phinn, 2001) despite the fact that issues related to vagueness, imprecision and
ambiguity can be addressed by the fuzzy set theory and fuzzy logic (Malczewski,
2004).
Fuzzy logic was originally proposed by Zadeh (1965) as a generalization of binary
logic and allows the continuous analysis between “false” and “true”. It is used to
model imprecision, vagueness and uncertainty in real world and bridges the gap
between qualitative and quantitative modeling. Fuzzy logic does not comply with the
binary property of dichotomy; hence fuzzy variables may consist of partially
overlapping fuzzy sets. The mathematical form of a fuzzy set is fully described by a
membership function which returns a membership value (η) within [0,1] for a given
object while a linguistic variable is used to describe its quality. Linguistic variables,
apart from describing primitive fuzzy sets, are also used to define new sets, based on
the primitive ones allowing thus the management of information in a way closer to
this of human conceptualization.
The knowledge base is represented as “IF…THEσ” rules, connecting hypotheses
to conclusions through a certainty factor while inference is divided (in general) in
three stages (Hatzichristos & Potamias, 2004; Kirschfink & Lieven 1999):
aggregation that returns the fulfillment of hypothesis for every rule
individually
implication that combines aggregation‟s result to the rule‟s certainty factor
(CF) resulting to the degree of fulfillment for each rule‟s conclusion
accumulation that corresponds to compromising different individual
conclusions into a final fuzzy result.
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
There is a great number of applications of fuzzy logic in spatial analysis; these
include defining and locating spatial objects (Bejaoui et al., 2007; Hatzichristos &
Potamias, 2004; Fisher et al., 2007; Fisher & Wood, 2004), mapping continuous
spatial phenomena (Fonte & Lodwick, 2005; Joss et al., 2008), defining topological
relations (Matsakis & Nikitenko, 2005; Takemura et al., 2005), satellite images
classification (Nedeljkovic, 2006), analysis of relations between geographical objects
(Guesgen, 2005, Kratochwil & Benedikt, 2005), statistical analysis of spatial data and
data mining (Kollias et al., 1999; Liu & George, 2005), development of DSS and
expert systems (Kalogirou, 2002; Stefanakis et al., 1996), spatial and temporal
interpolation (Dragicevic, 2005; Dragicevic & Marceau, 2000 & 2001).
For the case of urban modeling, the reason to apply fuzzy logic is twofold. First, it
provides improved mapping of land use and land cover (Arijit et al., 2006; Henning,
2003). Second, it provides a mechanism to represent human decision making in
linguistic terms and to approximate complex non-linear systems with simple models
(Chen & Linkens, 2004) making thus the analysis of complex systems easier (Setnes
et al., 1998).
Fuzziness in urban growth originates from a series of sources which can be found
as much in a definition and representation level, as in an analysis and simulation
level. To start with, urban cover is a rather abstract concept whose definition is based
more on functional criteria rather than morphological. Additionally, there is no such
thing as sharp boundaries between urban, partly urban and rural or forest areas (Liu &
Phinn, 2001). Therefore, fuzzy classification offers a better choice in land-use
mapping (Henning, 2003). What is more, fuzzy representations can capture mixed use
and coverage (Heikkila et al., 2002).
From a technical point of view, fuzzy modeling can be interpreted as a qualitative
modeling scheme which describes system behavior using fuzzy quantities presenting
thus a hybrid quantitative-qualitative approach (Chen & Linkens, 2004). This allows
describing partial and multiple relations in a distinct way. Nevertheless, the prime
advantage off fuzzy modeling is the facility for the explicit knowledge representation
in the form of if-then rules and the mechanism of human-like reasoning in linguistic
terms (Chen & Linkens, 2004).
The application of natural language allows the direct use of empirical knowledge
for the phenomenon of urban growth which includes experts‟ opinions, theoretical
approaches and historical registrations for the development of cities. Moreover, it
allows the transfer of knowledge based on empirical similarity patterns. Nevertheless,
what is more important is the ability of fuzzy logic to mimic the human-like decision
making and describe the underlying mechanisms in terms of fuzzy multi-criteria
systems. A large part of the applications of fuzzy logic for urban growth simulation
incorporates cellular automata techniques.
4 Fuzzy Urban Cellular Automata
Cellular Automata (CA) are a computational methodology in which the system under
study is divided into a set of cells with each cell interacting with all other cells
belonging to predefined neighborhoods through a set of simple rules (Krawzyk,
A Linguistic Approach to Model Urban Growth
2003). The interactions take place in discrete time steps with each cell‟s state at any
time period estimated by considering the state of the neighboring cells. This approach
is repeated continuously in a self-reproductive mechanism with no external
interference. Growth is thus simulated through a bottom up approach and this makes
CA an appropriate technique for simulating complex phenomena that is difficult to
model with other approaches.
There is a great variety of highly sophisticated crisp approaches concerning CA
based urban growth modeling. Among them, the stochastic approach of Mulianat et al
(Mulianata & Hariadi, 2004), the object-oriented approaches of Cage (Blecic et al.,
2004) and Obeus (Benenson & Kharbash, 2006), the environment Laude that
combines CA and Genetic Algorithms (Colona et al., 1998) and some widely applied
models, such as Sleuth (Clarke et al., 1997; Dietzel & Clarke, 2004; Dietzel et al.,
2005) and Moland (Engelen et al., 2007). Crisp urban CA may be categorized in
many different ways, in general though, they might be either quantitative or
qualitative. Quantitative models express the urban dynamics in terms of numerical
equations. Such models focus on the efficiency of the estimations and can provide
rather accurate results. What such models are not capable of is to map and express the
qualitative characteristics of urban growth phenomenon, which are a result of the
socio-economic decision making of the urban population.
Qualitative models on the other hand are rule-based and are capable of such
mapping and expression since they focus on the quality of causes and effects. They
fail though to compete with numerical models in following the exact numerical path
of urban growth. What is more, in the binary world of crisp rule-based systems,
qualities, objects and relations are strictly defined and either exist or not. There is no
such thing as partial, uncertain or imprecise fact, membership or relation and this is
not the way reality works.
The combination of CA and fuzzy logic under the term Fuzzy Cellular Automata
presents a hybrid approach that bridges qualitative and quantitative CA. In this
approach quantitative information can be converted to qualitative and vice versa
through the implementation of membership functions. Fuzzy Cellular Automata are a
rather new development; yet they have been implemented in a series of spatial
applications including fire propagation models (Mraz & Zimic, 1999) forest insect
infestation (Bone et al., 2006), and spatial load of electricity demand (Miranda &
Monteiro, 1999).
On the field of urban modeling, fuzzy cellular automata have been used both in
theoretical approaches (Dragicevic, 2004; Vancheri et al., 2004) and real world
applications. The first application by Wu (1996 & 1998) applies a fuzzy system to
govern the system‟s evolution using a max-min inference and crisp input and output.
This means that for each area only the maximum effect of the parameters that favor
urban growth is applied. In a similar fashion, only the minimum effect of the
parameters that restrict urban growth is applied. What is more, despite the fact that
information is processed within a fuzzy system, there is no gradation in the results and
each area is either considered to be urban or non-urban. In Wu‟s model fuzzy sets and
rules are defined empirically while fuzzy hedges may also be used. The model was
applied on real data and appears to simulate urban growth in a rather accurate way,
yet no evaluation index was used.
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
Liu and Phinn (2001, 2003 & 2005) proposed an approach with significant
differences. To start with, both input and output are represented as fuzzy sets which
are empirically designed. Additionally, population density was used to initialize the
fuzzy urban status of each area and define its propensity to urbanization. Finally, in
this approach, rules express logistic patterns of evolution while the input parameters
determine the speed of growth. The model was applied in Sydney for the period 19711996 during which urban cover grew by 28%. For this case study, the Kappa1 index of
agreement calculated in 3X3 neighborhoods was 90%. Despite the high levels of
accuracy, the model appears to focus in the partly urban to urban transition but it is
not equally effective when it comes to rural to urban transition. Our approach
attempts to combine the advantages of these two models and reduce their
disadvantages in an effective modeling environment that can use both theory-driven
and data-driven knowledge while retains a knowledge base form that is adjusted to the
available data.
5 The Modeling Framework
While most contemporary urban models go through an in depth analysis of selected
parameters, we propose a modeling structure that allows the parallel “in-width”
analysis of any given spatial or spatially referenced variable. We have developed a
modeling structure (figure 1) based on our previous work (Mantelas et al., 2007 &
2008) that attempts to describe a work flow familiar to the human perception of the
urban growth process. A structure based upon the relations between facts and
procedures so that even partial results may be helpful to the better understanding of
the process.
Apparently the model‟s structure and the systems‟ connection are tightly related to
the model‟s efficiency, the information‟s flow and the interpretability of the extracted
knowledge. Moreover, a generic form of the model is sustained which allows to avoid
severe data limitation (more data provide better results, less data still provide results
though); hence the model can be transferred to both data-rich and data-poor
applications.
Our approach uses spatially variable rules that may be either data-driven or theorydriven; hence they may fit better to reality allowing the user to overcome the
limitations of the available data by using exogenous knowledge adapted to the model
by empirical similarity patterns. Empirical rules are enabled by the fact that the
knowledge base is expressed in common language, which makes this model friendly
and usable, especially to the non-expert users.
We propose a structure of fuzzy systems each one of which receive different input
and conclude over certain aspects of the overall dynamics while the system that
simulates the evolution of the urban growth further incorporates CA techniques. An
1
The kappa coefficient of agreement expresses the agreement between two categorical datasets,
corrected for the agreement as can be expected by chance, which depends on the distribution of
class sizes in both datasets (Bishop et al., 1975). Kappa is well suited to compare a pair of land
use maps with its values ranging from 1, indicating a perfect agreement, to -1 indicating no
agreement (Jasper, 2009).
A Linguistic Approach to Model Urban Growth
important feature of the model is that the Growth Simulator does not access the input
data directlyν instead it operates upon a single intermediate layer called „suitability for
urbanization‟ that is the output of the Suitability Calculator.
Fig. 1. The structure of the modeling framework
Though the main idea of combining CA and fuzzy logic exists in previous
approaches, the herein presented approach presents some advantages; more
specifically:
it allows the combination of theory-driven and data-driven knowledge
it supports a reducible/extensible form of knowledge base which means that:
o it does not require specific data apply
o variables and rules can me added or removed without altering the
rest of the knowledge base
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
spatial parameters may be taken into consideration within each rule’s
certainty factor.
advanced fuzzy hedges are introduced
simulation of spontaneous growth is supported
The model can be used to reproduce the spatial patterns of urban growth but
requires the amount of urban cover to be allocated in a certain period to be given as an
input parameter. This allows us to measure real time as a function of the allocated
growth while at the same time functions as the exit criterion.
6 The Suitability Calculator
The suitability calculator receives input data and concludes over the suitability for
urbanization for each cell. The role of the suitability is twofold; on the one hand
defines the maximum urban cover that can be allocated to a non-urban cell and on the
other determines the maximum densification speed of an already urban cell. Initially a
number of fuzzy systems is used to process the input data and estimate the thematic
suitability for urbanization for distinct groups of input variables. The themes that can
be used depend on data availability with physical/natural (slope, land use) and
accessibility (road networks) being the cardinal themes.
Data-driven rules in these systems have a plain hypothesis premise – which results
to trivial aggregation – and depict the average distribution of urban cover appearance
for each fuzzy set of the variable. Theory-driven rules can be added without affecting
the data-driven ones and may have a complex hypothesis while the aggregation
operators used relying on the syntax of the premise. Both types of rules though, apply
the product operator for implication and conclude over thematic suitability indexes
that consist of a single fuzzy set (high).
In order to extract the rules for these systems we calculate the fitting between each
fuzzy set of each input variable and the urban fuzzy set. This way, each fuzzy set of
each input variable forms the rule‟s hypothesis while as the correspondent certainty
factor is taken the value of the fitting without any further calibration being required at
this stageν i.e. the sets “low distance from secondary road network” and “high urban
cover” fit each other by 56%, hence the rule becomesμ
IF “distance from secondary road network is Low”, THEN “suitability is High”
CF=0.56
|
What is more, the thematic suitability systems apply accumulation using the
algebraic sum operator – also known as the „probabilistic τR‟ – in the paradigm of
the Dempster-Shafer theory of evidence (Ahmazadeh & Petrou, 2001). The central
notion is that the more rules that lead to the same conclusion, the less likely it is for
this conclusion to be false. Its advantage is that it takes into account not only the
strength of each result but also the number of rules that conclude to this specific
resultν it‟s disadvantage on the other hand is that it tends to return higher values than
normal as the number of rules whose hypotheses are relatively dependent increase.
The certainty factors of the thematic suitability rules systems may be spatially
variable, which means that the same rule performs differently in different locations.
A Linguistic Approach to Model Urban Growth
This allows us in example to assign different suitability to urbanization for a certain
land use in the North-east and South-west parts of the area under study. Spatial
variability is gained by expressing rules‟ certainty factors as a function in terms of a
spatial 2-D fuzzy variable with 9 fuzzy sets which expresses the relative location of a
cell within the study area (figure 2).
The thematic indexes are considered to be either constant or variable in time. Static
indexes are calculated only once while variable ones may change in consecutive steps.
A suitability index may be treated as variable for three reasons:
input variables change in time i.e. road network or subway lines
input variables do not change in time, but their effect on urban growth is, i.e.
slope or distance from the sea
the combination of the above.
It is evident that in order to introduce variable suitability indexes additional
information is required. For the first case, insight to the planned projects and
developments is needed while the second case requires a time-series of data or
historical knowledge of them. Obviously, the third case needs both of them.
Once the thematic suitability indexes are calculated, they are merged into a single
overall suitability index which in the same fashion consists of a single fuzzy set
(high). This is accomplished by a separate fuzzy system that applies exclusively the
Algebraic Sum as an aggregation operation upon the following rule:
IF “Index A, … Index N are High ”, THEN “overall suitability is High” | CF=1.
Fig. 2. graphs of the fuzzy sets of the 2-D spatial variable, the horizontal level is a square reparameterization of the area under study while the vertical axis is the membership values for
each spatial fuzzy set: center, N, S, E, W (top) and center, NE, NW, SE, SW (bottom)
The suitability systems, apart from providing useful intermediate results, are used
for three purposes. To start with, they result into decreasing the number of variables
used in the following systems, making thus easier the further analysis and allowing
the user to use more simple and hence more comprehensible rules; for modeling,
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
simplification is both necessary and useful (Ness & Low, 2000). What is next, each
thematic system focuses on both the relations and the diversifications of the input
variables (i.e. if density of road network is positive, then obviously the distance from
road network is zero and vice versa) and as a result the outputs – the thematic indexes
– tend to be less correlated to each other than the input variables. Finally, it is much
easier and less risky to update the (variable) thematic indexes rather than the initial
variables.
7 The Urban Growth Simulator
The area under study is divided into three groups – static non urban, dynamic non
urban and urban. The static and dynamic non-urban groups are described by a crisp
set (membership is 0/1) while urban is a fuzzy set. Static areas are assumed not to
change and in this respect are excluded by the model. The exclusion criterion is a full
or high membership in the “not high overall suitability” fuzzy set. Additional
empirical criteria may be added to exclude cells that are be covered by forests, have
very high slopes or similar attributes.
Dynamic non urban areas are then processed by a hybrid fuzzy system that
incorporates CA techniques in order to simulate the transition from non urban to
urban. This system operates in two distinct levels, edge expansion and spontaneous
growth, each one of which is applied by one rule that uses the overall suitability as a
roof for the conclusions‟ valueμ
Level 1 - edge expansion: IF “Cell Suitability is High” AND “Neighborhood is
Urban ”, THEN “Cell is Urban” | CF=cf1.
Level 2 - spontaneous growth : IF “Cell Suitability is EXTREMELY High” AND
“Neighborhood is NOT non-Urban ”, THEN “Cell is Urban” | CF=cf2.
Level 2 applies the fuzzy hedge „extremely‟ that is designed to increase membership
values that are greater than 80% while decreases smoothly but very fast as initial
membership values decrease (figure 3). The numerical formula of „extremely‟ is
given by the following equations:
out in 2(0.8
in ) / 2
if
,
out in 2 4*( 0.8
in
)
, if
in 0.8
in 0.8
Cells that are urbanized to any extend are accessed by a third level, the
intensification module that applies an exponential implication operator. The syntax of
the rule is essential the same as in Level 1:
Level 3 - intensification: IF “Cell Suitability is High” AND “Neighborhood is
Urban”, THEN “Cell is Urban” | CF=cf3.
A Linguistic Approach to Model Urban Growth
Fig. 3. Visualization of the membership of each cell in the „overall suitability high‟ set (left)
and the effect of the fuzzy hedge „extremely‟ on the same set (right)
The computational difference though is that the new operator raises the membership
value of the conclusion premise in the power of the hypothesis complement. Given
the fact that membership values, certainty factors and aggregation results are bounded
in [0,1], if the hypothesis of the rule is not met at all, such a rule results to no change
in the fuzzy set of the conclusion; it returns the initial membership value. On the other
hand if the hypothesis is fully met and the rule is deterministic, it results to a certain
conclusion – a membership value 1. In any other case it returns a membership value
within (m,1) where m is the initial membership value.
In the hybrid fuzzy systems we may additionally use simple empirical or common
sense rules while all rules are subjected to a calibration process to determine the
certainty factors which may vary in space. What is more, spatial variation allows us to
potentially use neighborhoods of different radius in different areas. Nevertheless, due
to the (in general) irreversible nature of CA, data for two time-points may depict the
result of the CA process but not the dynamic or the specific form of its transition
engine; in such a system, simulation is the only way to predict outcomes (Clarke et
al., 1997). As a result, at this stage, calibration takes place manually through trial and
error. Having though reduced the number of variables that are used, it is relatively
easy - and even kind of fun in a more literal sense of Urban Gaming Simulations
(Cecchini & Rizzi, 2001) to experiment with each rule‟s syntax and its parameters
fine tuning.
8 Application of the Model
The model was applied in the Mesogia area (figure 4) at the eastern part of the Attica
basin (Athens), an area of 632 square km that is located within 25km from the
historical center of Athens, yet it used to be agricultural until 15-20 years ago when it
started to develop rapidly; more specifically, from 1990 till today urban cover in
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
Mesogia has been doubled. Recently the new international airport, the extension of
the subway and train lines and the construction of a new highway in Mesogia have
influenced urban growth not only in terms of the growth rate but also in terms of the
location of the growth. What is more, Mesogia can be considered relatively
autonomous when studying urban growth since it is physically separated by the mount
of Hemmetus in the West and Aegean Sea in the East, while neither Northern, nor
Southern areas are significantly urbanized.
In our study the available data for Mesogia include the Corine Land Cover
database (100X100m spatial resolution) for 1990 which are available from the
European Environmental Agency and a Corine based classification for 2004 produced
by GeoInformation S.A2. Most of the area is classified as Agricultural or Forest/Seminatural while the vast majority of Artificial Surface is mainly Urban Fabric with the
exception of the airport. The road network was provided by Infocharta Ltd. 3 for the
year 2004 while an estimation of the road network for 1990 was produced based on a
satellite image of the area for 1990. Road network data were classified in primary and
secondary and layers of distance and density were derived. What is more, a 90m
resolution DEM of the area was acquired from the SRTM webpage 4.
Fig. 4. Mesogia are in east part of Athens - the wider and specific area under study
2
http://www.g-i.gr/
http://www.infocharta.gr/
4 http://www2.jpl.nasa.gov/srtm/
3
A Linguistic Approach to Model Urban Growth
Fig. 5. The singleton representation of land use classes‟ suitability for urbanization (left) and
the membership functions of the fuzzy sets (low, average and high) for the slope variable
(right)
Fig. 6. The membership functions of the fuzzy sets (low, average and high) for primary road
density (top left), primary road distance (top right), secondary road density (bottom left) and
secondary road distance (bottom right)
For the application of the model, land cover data were represented as singletons
(figure 5a) while for the quantitative variables of slope (figure 5b) and
distance/density of primary and secondary road network (figure 6) three fuzzy sets
(low, average and high) were empirically defined. In the thematic suitability systems,
each singleton and each fuzzy forms a separate high suitability rule whose certainty
factor equals the fitting degree of each singleton/set with the urban cover fuzzy set.
This procedure took place for data referring to 1990 and produced the rules initial
rules for suitability calculation.
These rules were separated in three groups, land use, slope and road network and
produced three correspondent thematic suitability indexes which were merged into the
overall suitability using the probabilistic sum operator. At this point the urban growth
simulator was applied using Moore neighborhoods of radius 1 while the certainty
factors were initially given the value 1. What is next, manual calibration of the model
through trial and error shaped the final rules of the suitability calculator (table 1) and
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
urban growth simulator (table 2) which in some cases were determined to vary in
space.
Table 1: Rules for high thematic suitabilty
IF
{HYPOTHESIS}
Corine is 111 or 112
Corine is 221or 222
Corine is 223 or 243
Corine is 242
Corine is 321 or 323 or 333
Slope is low
Slope is average
Slope is high
Distance from primary road is low
Distance from primary road is average
Distance from primary road is high
Density of primary road is low
Density of primary road is average
Density of primary road is high
Distance from secondary road is low
Distance from secondary road is average
Distance from secondary road is high
Density of secondary road is low
Density of secondary road is average
Density of secondary road is high
THEN
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
{CONCLUSION}
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
suitability is high
CF
1.
0.25+0.4SE
0.33
0.45
0.1
0.6
0.4
0.1
0.8
0.6
0.2
0.2
0.5
0.6+0.3E
0.9
0.6
0.1
0.5
0.7
0.9
1 is
THEN
=>
{CONCLUSION}
cell is urban
CF
0.8.
2 is
=>
cell is urban
0.9
and
THEN
=>
{CONCLUSION}
cell is urban
CF
0.7
1 is
THEN
=>
{CONCLUSION}
cell is urban
CF
0.8.
2 is
=>
cell is urban
Table 2: Rules applied in the growth simulator
LEVEL 1 – edge expansion
IF {HYPOTHESIS}
Cell Suitability is high and Moore
urban and location is not SE
Cell Suitability is high and Moore
urban and location is SE
LEVEL 2 – spontaneous growth
IF {HYPOTHESIS}
Cell Suitability is extremely high
Moore 3 is not non-urban
LEVEL 3 - intensification
IF {HYPOTHESIS}
Cell Suitability is high and Moore
urban and location is not NW
Cell Suitability is high and Moore
urban and location is NW
1.
The initial configuration appeared to underestimate the urban density of the Northwest part of the area and the spatial extend of urban cover in the South-east part. For
this reason the suitability of Corine classes 221 and 222 was increased in the South-
A Linguistic Approach to Model Urban Growth
east areas while the suitability of „high primary road density‟ was increased for the
whole east area. Additionally, while the growth simulator uses Moore neighborhoods
of radius 1 at all levels for the general case, level 1 and level 3 used Moore
neighborhoods of radius 2 for the South-east and the North-west areas
correspondingly. Figure 7 shows the final thematic and overall suitability indexes.
The final rules were applied upon the 1990 data in order to estimate the 2004 urban
cover (figure 8) while to evaluate the results four indicators were calculated (table 3):
the map error which the percentage of the misallocated cells to the whole
area
the model error which is the percentage of the misallocated cells to the total
number of cells allocated by the model
the Kappa index of agreement calculated for the whole area
the Lee-Shale index of agreement calculated for the whole area.
The error indexes suggest that the model can provide useful estimations of future
urban growth. Nevertheless, a more thorough view of the error indexes and
specifically their evolution over the iterations of the algorithm suggests that the
optimum – in terms of error indexes – is reached a few steps before the algorithm
exits and hence error is slightly increased during the last few steps. This is partially
due to the exit criterion which introduces dependencies between the two types of
possible error - either a rural cell is considered urban or vice versa. That means that,
more or less, for each mistakenly considered urban cell there is another cell,
mistakenly considered rural and after some steps more cells are allocated inaccurately
rather than accurately.
On top of the numerical indexes, the results of the models appear to visually fit in
an excellent way upon actual urban cover. Yet, to some extent, this is due to the low
spatial resolution of the Corine database which cannot capture the detailed urban
cover but rather provides a homogenous shape with high autocorrelation which is
easier to simulate. Even so, a significant part of the error cannot be avoided for two
reasons. First, the case study uses only a very small amount of data that cannot
diversify easily which cells present higher suitability than others. Second, statistical
analysis of the growth occurred during a period can elucidate the average way in
which people make their residential choice but cannot describe relatively rare
processes that are responsible for a significant part of the growth i.e. constructing the
airport or a sport facility.
Table 3: Error indexes for cases studied
Period
1990 - 2004
Growth
occurred
Map error
Model Error
Kappa
Lee-Shale
6%
21%
0.75
0.81
L. Mantelas, P. Prastacos, Th. Hatzichristos, K. Koutsopoulos
Fig. 7. Thematic suitability indexes based on land use (top left), slope (middle left) and road
network (bottom left) are merged into overall suitability for urbanization (right)
Fig. 8. Actual urban cover for 1990 and 2004 (left) and the fuzzy estimation of the 2004 urban
cover based on 1990 (right)
A Linguistic Approach to Model Urban Growth
9 Conclusions and Future Work
We developed and presented an urban growth model that applies a parallel connection
between input variables and can easily be transferred to both data rich and data poor
cases. The model was calibrated and applied in Mesogia area in east Attica (Athens)
and is capable of providing satisfactory estimations of the future urban growth
patterns at least in the short-term. The fitting indicators suggest that the model
simulates efficiently the qualitative patterns of the urban expansion in the study area
which is further certified by visual comparison.
This is partially due to the CA techniques incorporated, that are proved very
efficient in simulating the spread of existing urban cover. What they lack, and this is
because of the CA nature, is the ability to capture the urbanization of detached areas.
Nevertheless, in our case study, the low spatial resolution of the used data and the
homogenous spatial patterns of urban cover in the area conceal this inability. For this
reason, the spontaneous growth mechanism (level 3) does not improve significantly
the accuracy of the model.
Fuzzy logic is an advisable way to deal with vague data and stochastic relations
since it enhances the potential qualitative resolution of the model and provides the
proper tools for uncertainty management. It allows the process of information in both
a qualitative and a quantitative way and expresses the dynamic of the phenomenon in
common linguistic terms. The rules syntax is kept simple and hence comprehensible;
as a result the user can easily calibrate the model empirically and introduce changes
such as spatial variability or fuzzy hedges to improve the performance of the model.
What is more, theory driven rules can be added and applied along with the rules that
are extracted by data analysis.
Acquiring a rich data set to experiment on temporal rules is one possible direction
towards which future work could be directed. More challenging though is to test this
model in data of higher spatial resolution that are less auto-correlated and present
more complex patterns of growth. Finally it would be very interesting to investigate
the potentials of the modeling framework to study population and employment change
along with urban growth.
Acknowledgements. The research leading to these results has received funding from
the European Community's Seventh Framework Programme FP7/2007-2013 under
grant agreement n° 212034
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