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A neural network approach to the analysis of city systems
Applied Geography, Vol. 18, No. I, pp. 83-96. 1998 0 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0143.6228/98 $19.00 + 0.00
Pergamon
PII: SO143-6228(97)00048-9
A neural network approach to the analysis of city systems J. Kropp Department for Integrated Systems Analysis, Potsdam Institute for Climate Impact Research, PO Box 601203, 14412 Potsdam, Germany This study describes a method for analysing systems of cities and for assessing their sensitivity to change. It is based on the premise that the macroscopic appearance of a city is a result of a larger set of underlying processes which can he indicated by useful variables. Herein, a neural approach makes use of Kohonen’s self-organizing maps (SOM) to create a phenomenological model of the (West) German city system. SOMs can display hidden patterns in input data as well as neighbourhood relations among the cities that make up the system. The 171 measurement vectors and 21 variables comprising the city system dataset can be reduced to just four dimensions that represent all relevant features of the system. The SOM technique permits classification of German cities into 24 groups that share common characteristics. By inputting a sequence of small changes to the data about a given city it is possible to observe whether and how it evolves towards the characteristics of another group. Some cities (e.g. Frankfurt, Stuttgart) are relatively insensitive to these data manipulations, whereas others respond quickly (e.g. Ntirnberg). It is believed that the former are core representatives of discrete city types. With further refinement and broader application to global datasets, this technique may be useful for identifying cities that are susceptible to perturbations of human-nature interactions, including those that involve environmental hazards and disasters. 0 1998 Elsevier Science Ltd. All
rights reserved Keywords: city systems, Kohonen urban modelling
maps, neural
networks,
systems analysis,
Despite recent progress in the field of urban analysis and modelling (Becker et al., 1994; Wegener, 1994), a comprehensive integrated model of anthropogenic structures in urban systems remains elusive. In the pursuit of one, various theoretical paradigms have been explored; most of these have been transferred from modem physics. Cellular automaton techniques and the concept of fractility have been employed to portray urban fragmentation and the evolution of land use patterns (White and Engelen, 1993; Batty and Longley, 1994; Makse et al., 1995). Bifurcation and self-organizing theory have been used to investigate suburbanization processes (Straussvogel, 1991; Becker et al., 1994; Schweitzer and Steinbrink, 1996). Sociological and economic approaches to urban analysis and modelling, partly based on dynamic decision theory, have also been common (Haag, 1989). Perhaps the most promising approach has been the application of systems theory and ecological theory to the analysis of urban evolution and flows of material 83
A neural network in the analysis of city systems: J. Kropp
84
through urban environments. For example, in the UNESCO Man and the Biosphere programme (MAB), urban agglomerations are conceived as organisms with their own circulatory systems that excrete waste and demand resources. However approaches which capture the complexity of large urban systems, and efforts to integrate the various themes are rare (see, e.g. Duvigneaud and Denayer-de Smet, 1975; Boyden et al., 1981; Giacomini, 1981). One major reason for the paucity of comprehensive integrative modelling is the overwhelming number of interdependencies between the subsystems of existing models (White, 1986; Wittig and Sukopp, 1993). Today, modelling assessments concentrate on the building of operational models, composed of up to eight sectoral subsystems (e.g. population, housing, land use, travel, etc.), either loosely coupled or tightly integrated, but without an implicit interface to city environments (Wegener, 1994). In the long term, the kinds of integrative models that are aspired to in the MAB programme are probably the only way to quantify the interactive processes of urban systems. In the short run, inductive methods can supply valuable insights of a qualitative nature, and the interrogation of existing datasets can offer limited understanding about the behaviour of urban systems (Kropp and Petschel-Held, 1996) The approach presented here uses available data about city systems and a neural network model. Assuming that data are representative of the system under investigation, complex signal processing tasks can be undertaken by an appropriate encoding of the relevant information from the underlying dataset (Ferran and Ferrara, 1992; Finch and Chater, 1992; Schweitzer, 1995; Cottrell et al., 1996). This provides the motivation for the application of neural network techniques in qualitative systems analysis. These techniques have several advantages over classical methods, such as their ability to handle noisy data and to use datasets in which the number of variables that pertain to a given unit exceed the number of measurements (observations). Neural networks also immanently discover salient features in a data distribution and categorize each pattern according to these features. However, most of the neural network models are related to or are identical to classical statistical models (e.g. primary component analysis; Joliffe, 1986). Instead, this analysis employs the well-known Kohonen algorithm, which is a form of non-linear dimensionality reduction that has no statistical analogue (Sarle, 1994). If such a method is applied carefully, additional knowledge can be extracted from data and a phenomenological model can be built, as shown by the case study of the German city system that follows. For readers who are not familiar with the methods used, two topicsthe concept of self-organizing maps and the measurement of topological ordering-are briefly reviewed (for further details see Kohonen, 1982, 1993, 1995; Ritter et al., 1990; Bauer and Pawelzik, 1992; Bauer et al., 1996b)
Methods Self-organizing
networks
This section gives a short account of Kohonen’s SOM, which is inspired by a biological example-the brains of mammals. It has been found that in areas of the neocortex the neurons are organized in ways that reflect some physical characteristics of the signals that stimulate them (Bauer et al., 1996a, b). Self-organizing networks work in the same manner. In a self-supervised (or from the external point of view, unsupervised) way, a network extracts structural information from numerical data instead of memorizing it all.
85
A neural network in the analysis of city systems: J. Kropp
During this learning process information from an m-dimensional continuous input space V is mapped (a) onto an n-dimensional discrete space A, representing the structural information of the input data, and where normally n < m. The discrete space is defined by the network, whose geometry can be either rectangular or hexagonal. Kohonen’s algorithm can be formulated in a shortened form as follows (Figure 1): (1) Initialize the synaptic weights (reference vectors) w at random on the unity sphere. (2) At each step t a stochastically chosen sensory input v is presented to the network, where all nodes compete to represent the input pattern. (2.1) Compute the Euclidean distance to all nodes (neurons): IIv - wi(( 5 min+J(v
- will,
and select as output node i that which is most similar to v (best matching unit; BMU), whereby similarity means that the vectors are adjacent in the spatial sense. (2.2) Modify the weights of the winner and of its neighbours according to the update rule: w;(t + 1) = w,(t) + e(t).hij.[v(t)
- w,(t)]
where e(f) is a decreasing to zero learning parameter and hi,, a time-dependent neighbourhood function (Gaussian) which defines the vicinity of the mesh in which other nodes learn from the same input stimulus. (2.3) Calculate the average change rate of the map or the mapping error, and if
Figure 1 Schematic representation
of the learning process of a self-organizing feature map (after Ritter et al., 1990). The grey ‘towel’ on the left denotes the m-dimensional input space; the ndimensional grid on the right represents the neural network. Both are connected via a set of socalled reference vectors. The dark-grey shaded circles label the neighbourhood of the neuron activated by an input stimulus. These nodes are also affected by the following learning step
A
86
neural
network in the analysis of city systems: J. Kropp
it is greater than a defined threshold training process.
go to step (2); otherwise
terminate
the
Artificial neural networks are adaptive models. They generalize what they have learned from input data and can process non-linear relations of multidimensional data. This technique offers a convenient way of reducing the amount of information in a complex environment (e.g. megacity) and provides an implicit model, without the necessity of creating a traditional physical model of the underlying problem. Topological
ordering
The aggregation of similar objects onto a neuron, during the training process, is a topology-preserving representation of the input data. Mapping divides the input space in a kind of Voronoi segmentation (Voronoi, 1908). Each node represents one of the gravity centres of the partially segmented input space V. If the dimensionality of the input data and the network differ, it is impossible to describe the data in a simple way and with optimal quality (Li et al., 1993). In addition, if the training parameters are inappropriate the topological map may be distorted (Figure 2). Therefore it is necessary to include a measure that estimates the quality of the mapping. This can be done by calculating the topographic product P (Bauer and Pawelzik, 1992). It measures the preservation of the neighbourhood between the neural units i in A and their weight vectors wi lying on V. If the preservation of the neighbourhood relations is achieved, P = 0; P > 0 indicates an embedding dimension that is too large, and P < 0 a dimension that is too small. Adjustment of the learning parameters is an additional precondition necessary to minimize lP(. Only in this case the error caused by the stochastic training process can also be minimized and thereby the network converges to a topographic map (Ritter and Schulten, 1988). Computational
methods
The dutuset. The dataset used here is for cities in the former Federal Republic of Germany (West Germany) that have populations between 45,000 and 2.2 million. It includes 171 measurement vectors of 21 variables (components; Table I). For some vectors (37 items), certain variables are not determined. Assuming that the data are representative
Figure 2 Example of two different topologies: a well-adapted (left) and a distorted (twisted) network topology due to an unsuitable training process (right)
A neural network in the analysis of city systems: J. Kropp Table 1
The 21 variables
in the dataset
1 2 3 4 5 6 7 8 9 10
Non-German residents Total city area Built-up area Number of motorcycles Total power consumption Total gas consumption Total water consumption Gas consumption by households Gas consumption by authorities Water consumption by households 11Number of flats
12 13 14 15 16 17 18 19 20 21
Single-room flats Double-room flats Triple-room flats Hats with 4 rooms Flats with 5 rooms Flats with 6 rooms Flats with > 6 rooms Net tax yield Trade tax yield Social expenditure
of the German city system, a subset of 134 measurements (including all items with a complete component vector) is used for the training process. Some of the variables involve physical data (e.g. city area; area occupied by buildings); others reflect lifestyles (e.g. total number of motor vehicles; size of apartments); a few variables represent social characteristics (e.g. number of non-German residents; social expenditure); and the remainder pertain to the economy of cities (e.g. net tax yield; trade tax yield). All variables are normalized with respect to city population sizes. Data preprocessing.
When the data are encoded at different scales it is useful to transform them into comparable sets. However, data standardization presents problems (Kohonen, 1995). During the learning process the map orientates itself in the direction of the components with the largest variance. It is easy to see that this can depend on the normalization method. Here, the clear-cut separation of feature domains should be the major goal, and because the similarity measure loses identity of component differences via summation, the components must contribute as much as possible to the similarity measure. Consequently the data were scaled to the unit interval by using the minimum/maximum values of each component of the measurement vector. Network simulations.
The source code of the network simulator, including the algorithm for the calculation of the topographical product, was developed in the programming language C. The programs used for the analysis of the trained networks and the graphical presentations were generated in PV-WAVE. Numerical calculations were carried out on an IBM RS60001410 workstation. Because the learning is a Markovian process, five runs each were performed for the different network geometries and chosen embedding space dimensions.
Results and discussion The results of simulating the German city system by this technique three headings: reduction of information; classification of cities; change. Dimensionality
are discussed under and sensitivities to
reduction
In order to quantify the quality of the mapping, the topographical product was averaged for each geometry and for numbers of network nodes (Table 2). The simulations show
A neural network in the analysis of city systems: J. Kropp
88 Table 2 metries
Topographic
product for the map of the 21.dimensional
P
Dimension
input space onto different network geo-
Geometry
4d 3d 2d
0.0 176900 + 0.003448 -0.016001 k 0.002371 -0.020414 + 0.001561
2X2X2X2 4x2x2 4x4
Id
-0.090070 k 0.004376
16
4d 3d 2d Id
-0.021756 + 0.002762 -0.031520 + 0.001240 -0.089628 f 0.009103
5X2X2 5x4 20
4d 3d 2d Id
-0.002568 -0.018259 -0.028588 -0.092010
3x2~2~2 4x3x2 6x4 24
Nodes
16
_
+ 0.000374 rt0.001540 f 0.001724 + 0.005483
20
24
that the 21-dimensional data manifold can be mapped onto a 4-dimensional network representing the relevant features of the underlying dataset. This subspace is defined by a 3 x 2 x 2 x 2 geometry. A network of this dimensionality is used for further analysis. After the termination of the training process, each of the nodes represents a phenomenological city type, which can be characterized by an associated feature spectra. In addition, the method yields an upper limit of the degrees of freedom of the dynamic system under investigation (Liebert et al., 1991). This number is also equivalent to the minimal number of equations needed for modelling the underlying systems dynamic. Extraction
of phenomenological
city types
Trained city distribution. The classified city distribution obtained from the training process makes intuitive sense (Figure 3). Cities with a sound economic basis, mainly dominated by services, are grouped together (node 21: Munich, Frankfurt, Dtisseldorf). Cities containing large universities also show up on the same node (node 6: Heidelberg, Freiburg, Tubingen). Cities that are major tourist centres cluster together (node 11: Baden Baden, Cuxhaven, Hameln), as do cities in the Ruhr that have undergone recent economic restructuring (node 15: Essen, Dortmund, Duisburg). It is possible to obtain similar groupings by using clustering algorithms but the network methodology applied here allows us a more holistic view on city systems, and to observe how the system might change. ClassiJcation of incomplete information. Normally measurement vectors containing unknown components cannot be subjected to statistical examination because a network stores information learned in a content-addressable way. It is possible to forecast a missing variable by seeking the appropriate BMU. Predicted values are fetched from the associated reference vector. (In this case the network behaves as an adaptive reference table for a qualitative system decryption.) In order to estimate the quality of such forecasts we have computed the extent to which training measurements can be reconstructed from the information stored within the network (Figure 4). This is done by successively excluding single components from the dataset. Then the BMU for each measurement is
A neural network in the analysis of city systems: J. Kropp
A neural network in the analysis of city systems: J. Kropp
90
5
10 Eliminated
15
20
variable
Figure 4 Reconstruction of the trained distribution if a component from the measurement is excluded. See Table 1 for definitions of the numbered variables on the x-axis
vector
recalculated and the result is compared with the distribution found after the training process. Figure 4 indicates that the network has completely stored the underlying information of the following variables: total water consumption, gas consumption by households, net tax yield and trade tax yield. On the other hand, the number of single-room apartments and social expenditure are the most prominent components. If they are missing, the associated observed system can be assigned to its feature class only with less certainty. The procedure discussed here has been applied to 37 measurements that each have between one and six missing components. As displayed in Figure 5, these fragmented feature spectra can be fitted to the phenomenological feature class whose spectrum is most similar. This method enables us to fill in the incomplete measurement vectors with high-quality data. Other examples of cities with missing components that are not included l.o.~O:~‘~‘~~‘~~~“’
5
Node
1
10 Vorioble
Node
‘.
15
20
5
3
10
15
20
Vorioble
Figure 5 Relating cities to their phenomenological type by using the information stored in the network. The solid line shows the averaged feature spectra obtained after the training process (Node I: Ahlen, Arnsberg, Bergheim. Dorsten, Grevenbroich, Lippstadt Menden, Unna, Wesel; Node 3: Hildesheim, Hof, Koblenz, Osnabruck, Wilhelmshaven). The dashed line represents the feature spectra yielded after the fitting of the fragemented measurement vectors (Node I: Kerpen, Node 3: Neumtinster, Pirmasens). The components fetched from the network are labelled with diamonds. Kerpen and Neumunster are denoted by one and Pirmasens by six missing components. The numbered variables on the x-axis are listed in T&e 1
A
neural network in the analysis of city systems: J. Kropp
91
in Figure 3 include Saarbrttcken and Bremen (both associated with node 17) and Mtinster and Passau (both associated with node IO). Note that this procedure only yields good results if the number of missing components is not larger than the number of known components.
Selected
German city prototypes
Table 3 focuses
on six of the 24 clearly distinguishable city types noted in Figure 3. population size are here grouped together in the same functional classes suggests that urban functions, as first formulated by Christaller (1933), are at least partly independent of city size. Cities associated with node 9 (e.g. Mainz and Regensburg) are examples of typical medium-sized service centres. Their most prominent industries are local government, broadcasting companies and universities. Areas available for building are larger than in bigger cities (apart from Cuxhaven) and households are small, as evidenced by a high number of single- and double-room apartments (see also Frankfurt). Finally, although tax revenues are low in these cities, so too are levels of social expenditure. Cities falling in node 11 are mainly tourist centres, including spas and architectural treasures (e.g. Baden Baden, Cuxhaven, Wolfensbtittel, Hameln). Populations are mobile and these places have a good deal of open space. Large family apartments are common and industry plays only a minor role in the local economies. Node 15 cities are located in the Ruhr. Their tax yields are relatively low and their social expenses are high, probably reflecting recent structural changes in the local economy that have resulted in industrial closures and significant unemployment. The traditional steel and coal industries of the Ruhrgebiet are now in decline. New industrial investment in mechanical engineering, power generation and services have not compensated for these losses. Apartment sizes are typically in the three-four-room range, indicative of working-class families. Node 16 cities are mainly located in northern Germany. They are usually harbour cities (e.g. Hamburg) or cities associated with industrial fairs and exhibitions (e.g. Hannover), sometimes also car manufacturing centres and machine engineering industries. Mobility of the residents, as expressed by the number of motor vehicles per capita, is extremely low. Tax yields are low but social expenses are high. Node 20 cities (e.g. Sindelfingen, Bad Homburg) have a relatively small number of residents but healthy economies. Compared with the larger cities in node 21 these places have poorly diversified economies dependent on only a few large industrial plants or governmental agencies. Apartment sizes suggest the dominance of families and social expenses are extremely low. Node 21 cities are primarily large industrial and service centres. Frankfurt, for example, is the home of Germany’s largest stock exchange, the federal bank and major chemical industries. Dusseldorf is the government centre of Germany’s largest state, as well as the home of chemical and heavy industries. All the cities in this group possess large international airports and healthy economies that generate large tax receipts. They are heavily built up and lacking in open space. Apartments in high-rise buildings are the dominant form of housing. Single-person households and non-German residents are common. Social expenditures are within the means of city budgets.
The fact that cities of widely different
ffi
++
++
+t
Trade taxes:
Net taxes:
Trade taxes:
Net taxes:
Trade taxes:
Net taxes:
Trade taxes:
Net taxes:
Trade taxes:
<60,000
>15o,ooo
<65O,OCKI
>200,000
<1.7-106
>5o,ooo
~60,000
>500,000
<1.2-106
~-
--
~ -
Net taxes:
>50,000
-
Trade taxes:
<180,000
--
Motor vehicles: Number flats: Flats, 2 Flats, 5 Motor vehicles: Number flats: Fiats, 2 Flats, 5 Motor vehicles: Number flats: Flats, 2 Flats, 5 Motor vehicles: Number flats: Flats, 2 Flats, 5 Motor vehicles: Number flats: Flats, 2 Flats, 5 Motor vehicles: Number flats: Flats, 2 Flats, 5
Net taxes:
>70,000
rooms: rooms
of
rooms: rooms:
of
rooms: rooms:
of
rooms: rooms:
of
rooms: rooms:
of
rooms: rooms:
of
the different variable
Lifestyle
representing
Economic
Selected variables
Residents
Table 3
+++ +t _-_
_
tt
+t
++
f
* +
f
Social expenditure:
Built-up
Social expenditure:
Built-up
Foreigners: Social expenditure:
Area: Built-up
area:
Foreigners:
Area: area:
Foreigners:
Area: area:
Social expenditure:
Built-up
Social expenditure:
Foreigners:
Social expenditure:
Foreigners: area:
area:
area:
Area:
Built-up
Area:
Built-up
Foreigners:
Area:
J
++t
--~
+
i+
_
+
_
_
obtained after the training process Social
examples
Physical
groups for six pbenomenological
Frankfurt Mtinchen
Sindelfingen Bad Homburg
Hamburg Kiel
Gelsenkirchen Dortmund
Cuxhaven Baden Baden
Mainz Regensburg
Examples
W h,
A neural network in the analysis of city systems: J. Kropp
Sensitivities
93
to change in city systems
City planners and developers are interested in the dynamics of urban systems. They would like to know which factors are now operating to bring about change and how cities are likely to evolve in the future. Here, the neural network approach may be of help because it can simulate successional patterns and hierarchies. The example of Hamburg is used as illustration. Hamburg is the second-largest city of Germany with approximately 1.6 million inhabitants. Incremental changes to existing data about Hamburg are fed into the training network and the results observed. As displayed in Figure 6, this procedure muves Hamburg from feature domain (node) 16 rowards node 15 or node 22, rather than towards other adjacent nodes such as 4, 13 and 17. The same effect can be observed by comparing the associated feature spectra of adjacent sites (Figure 7). When this procedure is repeated for other cities, the results show two distinct patterns. Cities such as Frankfurt, Gelsenkirchen, Mainz, Oldenburg and Stuttgart do not migrate in the direction of new feature domains, whereas others are sensitive to incremental data changes (e.g. Iserlohn, Niimberg). Roughly speaking, the first group of cities form the representative core of the system and the second group are cities that cannot be definitely assigned to any single feature domain. They exhibit features that occur in several feature domains. The possibilities of this approach for improving the understanding of changes in city systems are many but it must be remembered that the examples given here are merely ‘snapshots’ of a much more complex process. In order to gain mare complete and sophisticated insights, different combinations of variables and time series also must be taken into account.
Figure 6 Detail of Figure 3 showing the neighbourhood relationships of the city of Hamburg. The shaded domain represents the transition possibilities of Hamburg due to a variable change
94
A neural
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in the analysis
of city systems:
J. Krnpp
A neural network
in the analysis
of city systems:
J. Kropp
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Conclusion Self-organizing map algorithms are non-linear devices that can handle more complex systems and situations than traditional quantitative analytical tools. For this reason they appear well suited to applications seeking to understand the patterns of human geography in cities and urban regions. Such algorithms make it possible to isolate characteristic functional types of cities and to identify their relationships with others. In addition, they help analysts to foresee the paths by which one type of city is transformed into another in response to various change agents. The technique that has been discussed in this paper is still incomplete. For example, only data about a selection of German cities have been subject to analysis. But the approach of using available urban datasets for systems analysis and modelling is promising because it can be applied to very large and complex entities such as megacities. The vast body of available data on human-nature interactions in such places could be interpreted to provide a global assessment of cities and city systems that are vulnerable to change. Eventually it will be necessary to undertake time series analyses of long-term changes that are still in progress, hopefully with the aid of improved data and reliable indicators of change. This is the direction in which future work should move.
Acknowledgements Discussions with Matthias Liideke, Gerhard Petschel-Held and H.-J. Schellnhuber inspired this study. For comments on a previous draft of this paper I am indebted to Fritz Reusswig.
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Kohonen, T. (1993) Physiological interpretation of the self-organizing map algorithm. Neural Networks 6, 895-905. Kohonen, T. (1995) Self-Organizing Maps. Springer Series in Informations Sciences. Springer, New York. Kropp, J. and Petschel-Held, G. (1996) Prototyping and topological analysis of German city classes by application of self-organizing feature maps. In Self-Organization of Complex Structures: From Individual to Collective Dynamics, ed. F. Schweitzer, pp. 559-567. Gordon and Breach, London. Li, X., Gasteiger, J. and Zupan, J. (1993) On the topology distortion in self-organizing feature maps. Biological Cybernetics 70, 189-l 98. Liebert, W., Pawelzik, K. and Schuster, H. G. (1991) Optimal embeddings of chaotic attractors from topological considerations. Europhysics Letters 14(6), 521626. Makse, H., Havlin, S. and Stanley, E. (1995) Modelling urban growth patterns. Nature 377, 608-612. Ritter, H., Martinetz, T. and Schulten, K. (1990) Neuronale Netze: Eine Einfiihrung in die Neuroittformatik selbstorganisierender Netzwerke. Addison Wesley, Bonn. Ritter, H. and Schulten, K. (1988) Convergence properties of Kohonen’s topology conserving maps: fluctuations, stability and dimension selection. Biological Cybernetics 60, 59-7 1. Sarle, W. S. (1994) Neural networks and statistical models. In Proceedings of the 19th Annual SAS Users Group International Conference, pp. 1538-1550. Carey, NC. Schweitzer, F. and Steinbrink, J. (1996) Analysis and computer simulation of urban cluster distributions. In From Niirdlingen to Las Angeles: Fractal Laws of Urbanization, ed. K. Humpert, S. Becker and K. Brenner. Gordon and Breach, London. Schweizer, M. (1995) Wissensanalyse und -erhebung mit Kohonen-Netzen am praktischen Beispiel der Lawinenprognose. PhD thesis, University of Zurich. Straussvogel, D. (1991) Modelling suburbanization as an evolutionary system dynamic. Geographical Analysis 23(l), l-24. Voronoi, G. (1908) Nouvelles applications des parametres continus a la theorie des formes quadratiques: Recherches sur les parallelloedres primitifs Journal fir reine angewandte Mathematik 134, 198-287. Wegener, M. (1994) Operational urban models. Journal of the American Planning Association 80( 1). 17-29. White, A. (1986) Ecological approaches to urban systems: retrospect and prospect. Nature and Resources 12, 13-20. White, R. and Engelen, G. (1993) Cellular automata and fractal urban form: a cellular modelling approach to the evolution of urban land use patterns. Environment and Planning A 25, 1175-l 199. Wittig, H. and Sukopp, R. (1993) Was ist Stadtokologie? In Stadtokologie, ed. H. Wittig and R. Sukopp, pp. l-9. Fischer, Stuttgart.
Pergamon
PII: SO143-6228(97)00048-9
A neural network approach to the analysis of city systems J. Kropp Department for Integrated Systems Analysis, Potsdam Institute for Climate Impact Research, PO Box 601203, 14412 Potsdam, Germany This study describes a method for analysing systems of cities and for assessing their sensitivity to change. It is based on the premise that the macroscopic appearance of a city is a result of a larger set of underlying processes which can he indicated by useful variables. Herein, a neural approach makes use of Kohonen’s self-organizing maps (SOM) to create a phenomenological model of the (West) German city system. SOMs can display hidden patterns in input data as well as neighbourhood relations among the cities that make up the system. The 171 measurement vectors and 21 variables comprising the city system dataset can be reduced to just four dimensions that represent all relevant features of the system. The SOM technique permits classification of German cities into 24 groups that share common characteristics. By inputting a sequence of small changes to the data about a given city it is possible to observe whether and how it evolves towards the characteristics of another group. Some cities (e.g. Frankfurt, Stuttgart) are relatively insensitive to these data manipulations, whereas others respond quickly (e.g. Ntirnberg). It is believed that the former are core representatives of discrete city types. With further refinement and broader application to global datasets, this technique may be useful for identifying cities that are susceptible to perturbations of human-nature interactions, including those that involve environmental hazards and disasters. 0 1998 Elsevier Science Ltd. All
rights reserved Keywords: city systems, Kohonen urban modelling
maps, neural
networks,
systems analysis,
Despite recent progress in the field of urban analysis and modelling (Becker et al., 1994; Wegener, 1994), a comprehensive integrated model of anthropogenic structures in urban systems remains elusive. In the pursuit of one, various theoretical paradigms have been explored; most of these have been transferred from modem physics. Cellular automaton techniques and the concept of fractility have been employed to portray urban fragmentation and the evolution of land use patterns (White and Engelen, 1993; Batty and Longley, 1994; Makse et al., 1995). Bifurcation and self-organizing theory have been used to investigate suburbanization processes (Straussvogel, 1991; Becker et al., 1994; Schweitzer and Steinbrink, 1996). Sociological and economic approaches to urban analysis and modelling, partly based on dynamic decision theory, have also been common (Haag, 1989). Perhaps the most promising approach has been the application of systems theory and ecological theory to the analysis of urban evolution and flows of material 83
A neural network in the analysis of city systems: J. Kropp
84
through urban environments. For example, in the UNESCO Man and the Biosphere programme (MAB), urban agglomerations are conceived as organisms with their own circulatory systems that excrete waste and demand resources. However approaches which capture the complexity of large urban systems, and efforts to integrate the various themes are rare (see, e.g. Duvigneaud and Denayer-de Smet, 1975; Boyden et al., 1981; Giacomini, 1981). One major reason for the paucity of comprehensive integrative modelling is the overwhelming number of interdependencies between the subsystems of existing models (White, 1986; Wittig and Sukopp, 1993). Today, modelling assessments concentrate on the building of operational models, composed of up to eight sectoral subsystems (e.g. population, housing, land use, travel, etc.), either loosely coupled or tightly integrated, but without an implicit interface to city environments (Wegener, 1994). In the long term, the kinds of integrative models that are aspired to in the MAB programme are probably the only way to quantify the interactive processes of urban systems. In the short run, inductive methods can supply valuable insights of a qualitative nature, and the interrogation of existing datasets can offer limited understanding about the behaviour of urban systems (Kropp and Petschel-Held, 1996) The approach presented here uses available data about city systems and a neural network model. Assuming that data are representative of the system under investigation, complex signal processing tasks can be undertaken by an appropriate encoding of the relevant information from the underlying dataset (Ferran and Ferrara, 1992; Finch and Chater, 1992; Schweitzer, 1995; Cottrell et al., 1996). This provides the motivation for the application of neural network techniques in qualitative systems analysis. These techniques have several advantages over classical methods, such as their ability to handle noisy data and to use datasets in which the number of variables that pertain to a given unit exceed the number of measurements (observations). Neural networks also immanently discover salient features in a data distribution and categorize each pattern according to these features. However, most of the neural network models are related to or are identical to classical statistical models (e.g. primary component analysis; Joliffe, 1986). Instead, this analysis employs the well-known Kohonen algorithm, which is a form of non-linear dimensionality reduction that has no statistical analogue (Sarle, 1994). If such a method is applied carefully, additional knowledge can be extracted from data and a phenomenological model can be built, as shown by the case study of the German city system that follows. For readers who are not familiar with the methods used, two topicsthe concept of self-organizing maps and the measurement of topological ordering-are briefly reviewed (for further details see Kohonen, 1982, 1993, 1995; Ritter et al., 1990; Bauer and Pawelzik, 1992; Bauer et al., 1996b)
Methods Self-organizing
networks
This section gives a short account of Kohonen’s SOM, which is inspired by a biological example-the brains of mammals. It has been found that in areas of the neocortex the neurons are organized in ways that reflect some physical characteristics of the signals that stimulate them (Bauer et al., 1996a, b). Self-organizing networks work in the same manner. In a self-supervised (or from the external point of view, unsupervised) way, a network extracts structural information from numerical data instead of memorizing it all.
85
A neural network in the analysis of city systems: J. Kropp
During this learning process information from an m-dimensional continuous input space V is mapped (a) onto an n-dimensional discrete space A, representing the structural information of the input data, and where normally n < m. The discrete space is defined by the network, whose geometry can be either rectangular or hexagonal. Kohonen’s algorithm can be formulated in a shortened form as follows (Figure 1): (1) Initialize the synaptic weights (reference vectors) w at random on the unity sphere. (2) At each step t a stochastically chosen sensory input v is presented to the network, where all nodes compete to represent the input pattern. (2.1) Compute the Euclidean distance to all nodes (neurons): IIv - wi(( 5 min+J(v
- will,
and select as output node i that which is most similar to v (best matching unit; BMU), whereby similarity means that the vectors are adjacent in the spatial sense. (2.2) Modify the weights of the winner and of its neighbours according to the update rule: w;(t + 1) = w,(t) + e(t).hij.[v(t)
- w,(t)]
where e(f) is a decreasing to zero learning parameter and hi,, a time-dependent neighbourhood function (Gaussian) which defines the vicinity of the mesh in which other nodes learn from the same input stimulus. (2.3) Calculate the average change rate of the map or the mapping error, and if
Figure 1 Schematic representation
of the learning process of a self-organizing feature map (after Ritter et al., 1990). The grey ‘towel’ on the left denotes the m-dimensional input space; the ndimensional grid on the right represents the neural network. Both are connected via a set of socalled reference vectors. The dark-grey shaded circles label the neighbourhood of the neuron activated by an input stimulus. These nodes are also affected by the following learning step
A
86
neural
network in the analysis of city systems: J. Kropp
it is greater than a defined threshold training process.
go to step (2); otherwise
terminate
the
Artificial neural networks are adaptive models. They generalize what they have learned from input data and can process non-linear relations of multidimensional data. This technique offers a convenient way of reducing the amount of information in a complex environment (e.g. megacity) and provides an implicit model, without the necessity of creating a traditional physical model of the underlying problem. Topological
ordering
The aggregation of similar objects onto a neuron, during the training process, is a topology-preserving representation of the input data. Mapping divides the input space in a kind of Voronoi segmentation (Voronoi, 1908). Each node represents one of the gravity centres of the partially segmented input space V. If the dimensionality of the input data and the network differ, it is impossible to describe the data in a simple way and with optimal quality (Li et al., 1993). In addition, if the training parameters are inappropriate the topological map may be distorted (Figure 2). Therefore it is necessary to include a measure that estimates the quality of the mapping. This can be done by calculating the topographic product P (Bauer and Pawelzik, 1992). It measures the preservation of the neighbourhood between the neural units i in A and their weight vectors wi lying on V. If the preservation of the neighbourhood relations is achieved, P = 0; P > 0 indicates an embedding dimension that is too large, and P < 0 a dimension that is too small. Adjustment of the learning parameters is an additional precondition necessary to minimize lP(. Only in this case the error caused by the stochastic training process can also be minimized and thereby the network converges to a topographic map (Ritter and Schulten, 1988). Computational
methods
The dutuset. The dataset used here is for cities in the former Federal Republic of Germany (West Germany) that have populations between 45,000 and 2.2 million. It includes 171 measurement vectors of 21 variables (components; Table I). For some vectors (37 items), certain variables are not determined. Assuming that the data are representative
Figure 2 Example of two different topologies: a well-adapted (left) and a distorted (twisted) network topology due to an unsuitable training process (right)
A neural network in the analysis of city systems: J. Kropp Table 1
The 21 variables
in the dataset
1 2 3 4 5 6 7 8 9 10
Non-German residents Total city area Built-up area Number of motorcycles Total power consumption Total gas consumption Total water consumption Gas consumption by households Gas consumption by authorities Water consumption by households 11Number of flats
12 13 14 15 16 17 18 19 20 21
Single-room flats Double-room flats Triple-room flats Hats with 4 rooms Flats with 5 rooms Flats with 6 rooms Flats with > 6 rooms Net tax yield Trade tax yield Social expenditure
of the German city system, a subset of 134 measurements (including all items with a complete component vector) is used for the training process. Some of the variables involve physical data (e.g. city area; area occupied by buildings); others reflect lifestyles (e.g. total number of motor vehicles; size of apartments); a few variables represent social characteristics (e.g. number of non-German residents; social expenditure); and the remainder pertain to the economy of cities (e.g. net tax yield; trade tax yield). All variables are normalized with respect to city population sizes. Data preprocessing.
When the data are encoded at different scales it is useful to transform them into comparable sets. However, data standardization presents problems (Kohonen, 1995). During the learning process the map orientates itself in the direction of the components with the largest variance. It is easy to see that this can depend on the normalization method. Here, the clear-cut separation of feature domains should be the major goal, and because the similarity measure loses identity of component differences via summation, the components must contribute as much as possible to the similarity measure. Consequently the data were scaled to the unit interval by using the minimum/maximum values of each component of the measurement vector. Network simulations.
The source code of the network simulator, including the algorithm for the calculation of the topographical product, was developed in the programming language C. The programs used for the analysis of the trained networks and the graphical presentations were generated in PV-WAVE. Numerical calculations were carried out on an IBM RS60001410 workstation. Because the learning is a Markovian process, five runs each were performed for the different network geometries and chosen embedding space dimensions.
Results and discussion The results of simulating the German city system by this technique three headings: reduction of information; classification of cities; change. Dimensionality
are discussed under and sensitivities to
reduction
In order to quantify the quality of the mapping, the topographical product was averaged for each geometry and for numbers of network nodes (Table 2). The simulations show
A neural network in the analysis of city systems: J. Kropp
88 Table 2 metries
Topographic
product for the map of the 21.dimensional
P
Dimension
input space onto different network geo-
Geometry
4d 3d 2d
0.0 176900 + 0.003448 -0.016001 k 0.002371 -0.020414 + 0.001561
2X2X2X2 4x2x2 4x4
Id
-0.090070 k 0.004376
16
4d 3d 2d Id
-0.021756 + 0.002762 -0.031520 + 0.001240 -0.089628 f 0.009103
5X2X2 5x4 20
4d 3d 2d Id
-0.002568 -0.018259 -0.028588 -0.092010
3x2~2~2 4x3x2 6x4 24
Nodes
16
_
+ 0.000374 rt0.001540 f 0.001724 + 0.005483
20
24
that the 21-dimensional data manifold can be mapped onto a 4-dimensional network representing the relevant features of the underlying dataset. This subspace is defined by a 3 x 2 x 2 x 2 geometry. A network of this dimensionality is used for further analysis. After the termination of the training process, each of the nodes represents a phenomenological city type, which can be characterized by an associated feature spectra. In addition, the method yields an upper limit of the degrees of freedom of the dynamic system under investigation (Liebert et al., 1991). This number is also equivalent to the minimal number of equations needed for modelling the underlying systems dynamic. Extraction
of phenomenological
city types
Trained city distribution. The classified city distribution obtained from the training process makes intuitive sense (Figure 3). Cities with a sound economic basis, mainly dominated by services, are grouped together (node 21: Munich, Frankfurt, Dtisseldorf). Cities containing large universities also show up on the same node (node 6: Heidelberg, Freiburg, Tubingen). Cities that are major tourist centres cluster together (node 11: Baden Baden, Cuxhaven, Hameln), as do cities in the Ruhr that have undergone recent economic restructuring (node 15: Essen, Dortmund, Duisburg). It is possible to obtain similar groupings by using clustering algorithms but the network methodology applied here allows us a more holistic view on city systems, and to observe how the system might change. ClassiJcation of incomplete information. Normally measurement vectors containing unknown components cannot be subjected to statistical examination because a network stores information learned in a content-addressable way. It is possible to forecast a missing variable by seeking the appropriate BMU. Predicted values are fetched from the associated reference vector. (In this case the network behaves as an adaptive reference table for a qualitative system decryption.) In order to estimate the quality of such forecasts we have computed the extent to which training measurements can be reconstructed from the information stored within the network (Figure 4). This is done by successively excluding single components from the dataset. Then the BMU for each measurement is
A neural network in the analysis of city systems: J. Kropp
A neural network in the analysis of city systems: J. Kropp
90
5
10 Eliminated
15
20
variable
Figure 4 Reconstruction of the trained distribution if a component from the measurement is excluded. See Table 1 for definitions of the numbered variables on the x-axis
vector
recalculated and the result is compared with the distribution found after the training process. Figure 4 indicates that the network has completely stored the underlying information of the following variables: total water consumption, gas consumption by households, net tax yield and trade tax yield. On the other hand, the number of single-room apartments and social expenditure are the most prominent components. If they are missing, the associated observed system can be assigned to its feature class only with less certainty. The procedure discussed here has been applied to 37 measurements that each have between one and six missing components. As displayed in Figure 5, these fragmented feature spectra can be fitted to the phenomenological feature class whose spectrum is most similar. This method enables us to fill in the incomplete measurement vectors with high-quality data. Other examples of cities with missing components that are not included l.o.~O:~‘~‘~~‘~~~“’
5
Node
1
10 Vorioble
Node
‘.
15
20
5
3
10
15
20
Vorioble
Figure 5 Relating cities to their phenomenological type by using the information stored in the network. The solid line shows the averaged feature spectra obtained after the training process (Node I: Ahlen, Arnsberg, Bergheim. Dorsten, Grevenbroich, Lippstadt Menden, Unna, Wesel; Node 3: Hildesheim, Hof, Koblenz, Osnabruck, Wilhelmshaven). The dashed line represents the feature spectra yielded after the fitting of the fragemented measurement vectors (Node I: Kerpen, Node 3: Neumtinster, Pirmasens). The components fetched from the network are labelled with diamonds. Kerpen and Neumunster are denoted by one and Pirmasens by six missing components. The numbered variables on the x-axis are listed in T&e 1
A
neural network in the analysis of city systems: J. Kropp
91
in Figure 3 include Saarbrttcken and Bremen (both associated with node 17) and Mtinster and Passau (both associated with node IO). Note that this procedure only yields good results if the number of missing components is not larger than the number of known components.
Selected
German city prototypes
Table 3 focuses
on six of the 24 clearly distinguishable city types noted in Figure 3. population size are here grouped together in the same functional classes suggests that urban functions, as first formulated by Christaller (1933), are at least partly independent of city size. Cities associated with node 9 (e.g. Mainz and Regensburg) are examples of typical medium-sized service centres. Their most prominent industries are local government, broadcasting companies and universities. Areas available for building are larger than in bigger cities (apart from Cuxhaven) and households are small, as evidenced by a high number of single- and double-room apartments (see also Frankfurt). Finally, although tax revenues are low in these cities, so too are levels of social expenditure. Cities falling in node 11 are mainly tourist centres, including spas and architectural treasures (e.g. Baden Baden, Cuxhaven, Wolfensbtittel, Hameln). Populations are mobile and these places have a good deal of open space. Large family apartments are common and industry plays only a minor role in the local economies. Node 15 cities are located in the Ruhr. Their tax yields are relatively low and their social expenses are high, probably reflecting recent structural changes in the local economy that have resulted in industrial closures and significant unemployment. The traditional steel and coal industries of the Ruhrgebiet are now in decline. New industrial investment in mechanical engineering, power generation and services have not compensated for these losses. Apartment sizes are typically in the three-four-room range, indicative of working-class families. Node 16 cities are mainly located in northern Germany. They are usually harbour cities (e.g. Hamburg) or cities associated with industrial fairs and exhibitions (e.g. Hannover), sometimes also car manufacturing centres and machine engineering industries. Mobility of the residents, as expressed by the number of motor vehicles per capita, is extremely low. Tax yields are low but social expenses are high. Node 20 cities (e.g. Sindelfingen, Bad Homburg) have a relatively small number of residents but healthy economies. Compared with the larger cities in node 21 these places have poorly diversified economies dependent on only a few large industrial plants or governmental agencies. Apartment sizes suggest the dominance of families and social expenses are extremely low. Node 21 cities are primarily large industrial and service centres. Frankfurt, for example, is the home of Germany’s largest stock exchange, the federal bank and major chemical industries. Dusseldorf is the government centre of Germany’s largest state, as well as the home of chemical and heavy industries. All the cities in this group possess large international airports and healthy economies that generate large tax receipts. They are heavily built up and lacking in open space. Apartments in high-rise buildings are the dominant form of housing. Single-person households and non-German residents are common. Social expenditures are within the means of city budgets.
The fact that cities of widely different
ffi
++
++
+t
Trade taxes:
Net taxes:
Trade taxes:
Net taxes:
Trade taxes:
Net taxes:
Trade taxes:
Net taxes:
Trade taxes:
<60,000
>15o,ooo
<65O,OCKI
>200,000
<1.7-106
>5o,ooo
~60,000
>500,000
<1.2-106
~-
--
~ -
Net taxes:
>50,000
-
Trade taxes:
<180,000
--
Motor vehicles: Number flats: Flats, 2 Flats, 5 Motor vehicles: Number flats: Fiats, 2 Flats, 5 Motor vehicles: Number flats: Flats, 2 Flats, 5 Motor vehicles: Number flats: Flats, 2 Flats, 5 Motor vehicles: Number flats: Flats, 2 Flats, 5 Motor vehicles: Number flats: Flats, 2 Flats, 5
Net taxes:
>70,000
rooms: rooms
of
rooms: rooms:
of
rooms: rooms:
of
rooms: rooms:
of
rooms: rooms:
of
rooms: rooms:
of
the different variable
Lifestyle
representing
Economic
Selected variables
Residents
Table 3
+++ +t _-_
_
tt
+t
++
f
* +
f
Social expenditure:
Built-up
Social expenditure:
Built-up
Foreigners: Social expenditure:
Area: Built-up
area:
Foreigners:
Area: area:
Foreigners:
Area: area:
Social expenditure:
Built-up
Social expenditure:
Foreigners:
Social expenditure:
Foreigners: area:
area:
area:
Area:
Built-up
Area:
Built-up
Foreigners:
Area:
J
++t
--~
+
i+
_
+
_
_
obtained after the training process Social
examples
Physical
groups for six pbenomenological
Frankfurt Mtinchen
Sindelfingen Bad Homburg
Hamburg Kiel
Gelsenkirchen Dortmund
Cuxhaven Baden Baden
Mainz Regensburg
Examples
W h,
A neural network in the analysis of city systems: J. Kropp
Sensitivities
93
to change in city systems
City planners and developers are interested in the dynamics of urban systems. They would like to know which factors are now operating to bring about change and how cities are likely to evolve in the future. Here, the neural network approach may be of help because it can simulate successional patterns and hierarchies. The example of Hamburg is used as illustration. Hamburg is the second-largest city of Germany with approximately 1.6 million inhabitants. Incremental changes to existing data about Hamburg are fed into the training network and the results observed. As displayed in Figure 6, this procedure muves Hamburg from feature domain (node) 16 rowards node 15 or node 22, rather than towards other adjacent nodes such as 4, 13 and 17. The same effect can be observed by comparing the associated feature spectra of adjacent sites (Figure 7). When this procedure is repeated for other cities, the results show two distinct patterns. Cities such as Frankfurt, Gelsenkirchen, Mainz, Oldenburg and Stuttgart do not migrate in the direction of new feature domains, whereas others are sensitive to incremental data changes (e.g. Iserlohn, Niimberg). Roughly speaking, the first group of cities form the representative core of the system and the second group are cities that cannot be definitely assigned to any single feature domain. They exhibit features that occur in several feature domains. The possibilities of this approach for improving the understanding of changes in city systems are many but it must be remembered that the examples given here are merely ‘snapshots’ of a much more complex process. In order to gain mare complete and sophisticated insights, different combinations of variables and time series also must be taken into account.
Figure 6 Detail of Figure 3 showing the neighbourhood relationships of the city of Hamburg. The shaded domain represents the transition possibilities of Hamburg due to a variable change
94
A neural
network
in the analysis
of city systems:
J. Krnpp
A neural network
in the analysis
of city systems:
J. Kropp
95
Conclusion Self-organizing map algorithms are non-linear devices that can handle more complex systems and situations than traditional quantitative analytical tools. For this reason they appear well suited to applications seeking to understand the patterns of human geography in cities and urban regions. Such algorithms make it possible to isolate characteristic functional types of cities and to identify their relationships with others. In addition, they help analysts to foresee the paths by which one type of city is transformed into another in response to various change agents. The technique that has been discussed in this paper is still incomplete. For example, only data about a selection of German cities have been subject to analysis. But the approach of using available urban datasets for systems analysis and modelling is promising because it can be applied to very large and complex entities such as megacities. The vast body of available data on human-nature interactions in such places could be interpreted to provide a global assessment of cities and city systems that are vulnerable to change. Eventually it will be necessary to undertake time series analyses of long-term changes that are still in progress, hopefully with the aid of improved data and reliable indicators of change. This is the direction in which future work should move.
Acknowledgements Discussions with Matthias Liideke, Gerhard Petschel-Held and H.-J. Schellnhuber inspired this study. For comments on a previous draft of this paper I am indebted to Fritz Reusswig.
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