CSIS Discussion Paper No. 31
Event-Based Analysis of Changes in Surfaces
Yukio Sadahiro
OCTOBER, 2000

Center for Spatial Information Science
University of Tokyo
7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan

August 11, 2000
Event-Based Analysis of Changes in Surfaces

Keywords: surface, change, primitive event
Short title: analysis of changes in surfaces

Yukio Sadahiro
Center for Spatial Information Science and Department of Urban Engineering
University of Tokyo
7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Phone: +81-3-5841-6273
Fax: +81-3-5841-8521
E-mail: sada@okabe.t.u-tokyo.ac.jp1Event-Based Analysis of Changes in Surfaces
Abstract
This paper develops a method for analyzing changes in a surface, a scalar function
defined over a two-dimensional region. The method is based on the topological method
for static surfaces that uses critical points (peaks, bottoms, and cols) and their connecting
lines. To extend the topological method to spatio-temporal domain, four types ofprimitive events are proposed: 1) generation, 2) disappearance, 3) movement, and 4)
switch. The change of a surface is described by a combination of these primitives. Fromsurfaces of two times a set of primitive events that causes the change is deduced. They are
stored in GIS as spatial objects with attributes, and their spatio-temporal pattern isvisually analyzed. To test the validity of the method, the change of a retail cluster inShinjuku and Shibuya area in Tokyo is analyzed. The empirical study yielded someinteresting findings that help us understand changes in the spatial structure of retailing.

2

1. INTRODUCTION
Surface, a scalar function defined over a two-dimensional region, is one of themost important objects in GIS. Continuous variables such as terrain elevation andatmospheric temperature are approximately modelled by TIN or grid in GIS. Surfaces arealso generated by conversion of other spatial objects; point distributions are aggregatedinto some spatial units and smoothed to represent the density distribution of points.
Scalar fields modelled as surfaces in GIS are analyzed not only visually but also
mathematically and even statistically. To this end there have been proposed numerous
methods in the literature, and they can be classified into three types as follows. The first
type of method is so called geostatistics including variogram, correlogram, and kriging(Isaaks and Srivastava 1989; Cressie 1993). It is used for spatial analysis as well as
spatial interpolation, and plays a major role in surface analysis. For instance, variogram
cloud and its related concepts such as sill and range are useful for describing spatial
characteristics of surfaces. The second type of method focuses on modelling surfaces bymathematical functions. It includes trend surface analysis (Tobler 1964; Bailey andGatrell 1995) and its extensions (Hagget 1968; Griffith 1981). Trend surface analysis issuitable for modelling simple surfaces, say, a unicentered population distribution.
However, since they assume the form of surface functions a priori, operationaldifficulties arise in modelling multicentered distributions. The third type of method deals
with the topological structure of surfaces, so they can be called topological or qualitative
method. Warntz (1966), Warntz and Waters (1975), and Okabe and Masuda (1984)
extract a graph consisting of peaks, bottoms, cols, and their connecting links from asurface, in order to describe its global spatial structure. Topological method ischaracterized by its simplicity; we can easily understand the rough form of a surface by
the graph.
These methods are basically developed for analyzing static aspects of surfaces, thatis, the state of surfaces at a fixed time. However, with the advance of GIS technology,
demand for dynamic analysis of surfaces is now rapidly growing. Spatial data acquisition
tools such as GPS and RS now provide us a large amount of spatio-temporal data ofsurfaces, and we can easily see the change of surfaces over time through GIS. One
method to analyze the change is to apply the existing methods to snapshot data anddiscuss the results on the temporal axis. Unfortunately, since the existing methods do notconsider the dynamics of surfaces explicitly, they often fail to detect even basic
transformations of surfaces such as translation and rotation. In spatio-temporal analysis it
is obviously necessary to take dynamics into account explicitly in its methodology.
To answer this demand, the present paper proposes a new method for analyzing thechange of surfaces. The method is based on the topological method mentioned above, that3

is, its extension to spatio-temporal domain. Following Okabe and Masuda (1984), we
first outline the topological method in section 2. To extend their method, we then propose
four types of local changes in surfaces which we call primitive events. The change of asurface is described by a combination of primitive events; given surfaces of two times,
we deduce a set of primitives that causes the change. The primitive events deduced arestored in GIS as spatial objects with attributes, and their spatio-temporal pattern isvisually analyzed. In order to test the validity of the method, section 3 empiricallyanalyzes the change of a retail cluster in Shinjuku and Shibuya area in Tokyo, Japan.
Section 4 summarizes the conclusions with discussion.
2. EVENT-BASED METHOD FOR ANALYZING CHANGES IN SURFACES
2.1 Topological method for analyzing surfaces
The method we will propose is based on the topological method for analyzing
surfaces originally developed by Warntz (1966) and Warntz and Waters (1975), andextended by Okabe and Masuda (1984). We thus outline the topological method before
we extend it to spatio-temporal domain.
Suppose a continuous function f(x) defined over a connected region S, which isusually assumed to be smooth and differentiable at any point in S. Approximating the
local surface around a point in S by a quadratic function, we can classify every point in S
into one of the four types of points: 1) peak, 2) bottom, 3) col, and 4) slope (Figure 1).
The first three points are called critical points (Iri et al., 2000). The Poincaré index
theorem assuresn p + nb − nc = 1,
(1)where np, nb, and nc are the numbers of peaks, bottoms, and cols, respectively.
FIG. 1. Four types of local surfaces.
The integral curves of the steepest ascent vectors of f(x) compose a set of curvesconnecting peaks and bottoms. Some curves pass through cols; at each col four integral
curves meet. We call these curves corridors hereafter. Critical points and corridors
provide a graph representation of a surface which we call the structural graph, as shownin Figure 2b. The graph shows the global structure of a surface, so that we can easily
understand its rough form. Okabe and Masuda (1984) uses this representation of surfaces
to compare population distributions of various cities in Japan.
FIG. 2. Structural graph of a surface. a) Original surface, b) critical points and corridors.

4

The topological method detects all peaks and bottoms no matter how their height or
depth is small. In spatial analysis, however, only significant critical points are of interest.
To remove insignificant peaks and bottoms, a threshold value is often used for defining
peaks and bottoms. For example, the height of a peak is defined as its relative heightcompared to that of its connecting cols, and peaks whose height is smaller than athreshold value are neglected.
2.2 Primitive events
To extend the topological method outlined above, we use the concept of primitive
events defined for critical points and corridors (Sadahiro and Umemura, 2000). Supposea spatio-temporal function f(x, t) defined over a connected region S for a certain time
period, which is smooth and differentiable with respect to x at any point in S.
We first define primitive events for peaks. Generation of a peak is defined as thechange of a point on f(x, t) from a slope to a peak which is caused by the rise of the point(Figure 3a) or the fall of its surrounding surface. In contrast, disappearance of a peak isthe change from a peak to a slope (Figure 3b). Movement of a peak is defined as thecontinuous locational change of a peak which is caused by the smooth change of a surface(Figure 3c). Though spatial objects that compose a surface may not actually move, it isconvenient in spatial analysis to represent such changes as if peaks actually move.
FIG. 3. Three types of primitive events for peaks. a) Generation, b) disappearance, and c)movement.
The three primitive events can be classified further by considering the change off(x, t) when the event occurs. As mentioned above, peaks are generated by either the riseof a slope point or the fall of its surrounding surface. We distinguish these changes using
the derivative of f(x, t) with respect to t:f ' ( x, t ) =∂f ( x, t ) .
∂t(2)

Generation+, abbreviated as G+, is the generation of a peak when f'(x, t)>0, that is,
generation by the rise of a slope point. Similarly, generation- (G-) is the generation of apeak when f'(x, t)≤0. Disappearances are also classified further into two types: D+ (f'(x,t)≥0) and D- (f'(x, t)<0). The former indicates the disappearance of a peak by the rise its
surrounding surface, not by its own fall. Movements are classified into three types: M+,

5M-, and M0. If a peak moves and its height grows, the movement is denoted by M+.
When a peak moves keeping its height, the movement is denoted by M0.
Generation and disappearance of peaks are described in relation to their neighboringpeaks. As shown in Figure 2, each col has its two neighboring peaks. This and equation(1) assure that generation of a peak is always accompanied by generation of its
neighboring col, and that the col has another neighboring peak (Figure 4). To state thisexplicitly, in Figure 4, we say that the peak B was generated from the peak A which isconnected to the newly generated col C. This representation is reasonable because thepeak B belonged to the slope of A before its generation; it looks as if the mountainrepresented by the peak A born its child mountain represented by the peak B. Similarly,
for disappearance of peaks, we say that a peak was absorbed into the peak connected to
the vanished col. This is because disappearance of a peak reflects that the mountainrepresented by the peak is combined into its neighboring mountain.
FIG. 4. Generation of a peak. We say "the peak B was generated from the peak A."
Generation, disappearance, and movement are defined for bottoms and cols as wellas peaks. Generation and disappearance of a bottom are again related to its neighboringbottom. Generation and disappearance of a col is accompanied by those of a peak or abottom, so that they are described together. For example, we say that a col vanished withthe disappearance of its neighboring peak.
Primitive events for corridors are somewhat richer than those for critical points.
Generation and disappearance are defined in connection with those of peaks and bottoms;
they are always accompanied by those of peaks or bottoms. Movement, which does notchange the topology of the structural graph, is also defined for corridors. In addition tothese events, an event which we call switch can occur on corridors. Every corridor has acol on one end, and a peak or a bottom on the other. Keeping the connection with a col, acorridor can change the critical point on the other end (Figure 5).
FIG. 5. Switch event.
Among primitive events defined above, generation, disappearance and switch arecalled topological events in this paper, since they change the topology of the structuralgraph. In contrast, movement is called non-topological event.
Any change of the structural graph can be classified into one or a set of primitive
events. This implies that the structural change of a surface can be described by theprimitive events, which provides a means of visualizing and analyzing changes in6

surfaces.
2.3 Significance of critical points
As well as the static topological method, our extension to spatio-temporal analysis
is concerned with the significance of critical points. To extract significant critical points,
we define the relative height and depth of a col at x ash( x ) = f ( x ) − max{ f ( x B1 ), f ( x B 2 )}(3)andd ( x ) = min{ f ( x P1 ), f ( x P 2 )} − f ( x ) ,(4)

respectively, where xPi and xBi are the locational vectors of peaks and bottoms directly
connected with the col, respectively. If these values are very small, we find the col is not
significant. Formally, a col at x is significant if(5)h( x ) ≥ T
andd (x) ≥ T ,(6)where T is a threshold value that is given extrinsically. Insignificant cols are removed
from the structural graph with its connecting corridors and neighboring peaks or bottoms
if necessary.
The choice of the threshold value T depends on the circumstances. One method is
to adopt a value around which the resultant structural graph is stable for a wide range of T
(detailed discussion can be found in Okabe and Masuda 1984 and Okabe and Masuyama
2000). Otherwise, we can choose a value which is suitable for the purpose of analysis. If
we are interested in the global structure of a surface, we should adopt a large value for T.
If we want to analyze details of a surface, a small value or even zero value is desirable.
2.4 Deduction of primitive events
Change of a surface is mathematically defined as the change of a continuousfunction from f(x, t1) to f(x, t2). If the set of all primitive events between t1 and t2 is
explicitly recorded in GIS data, we can easily visualize and analyze them, which helps us
understand the spatio-temporal structure of the surface f(x, t). However, in most cases,
we can obtain only the snapshots of f(x, t), for example f(x, t1) and f(x, t2). We thus haveto deduce primitives events between t1 and t2 from f(x, t1) and f(x, t2).
To this end, we first interpolate f(x, t) from t1 to t2 by linear interpolation. The
surface function at t is given byt −t
t − t1f ( x, t ) = 2
f ( x, t1 ) +f ( x, t 2 ) .
(7)
t2 − t1
t2 − t1
Though linear interpolation may not be always valid, it is a good approximation of f(x, t)

7

when no additional information on f(x, t) is available.
We then deduce primitive events between t1 and t2 from f(x, t). Let us briefly show
a procedure for deducing events for peaks. Assume that the surface is represented by a setof scalar values measured at sample locations and that they are spatially interpolated byplanes, as frequently done in TIN and grid. We first examine all the sample points andtheir neighboring points from t1 to t2 to know whether they could be critical points during
the period, setting temporarily the threshold value T to zero. For candidate points we thencheck whether they could be significant with respect to the true threshold value T,
comparing their height with that of all the candidates of cols. From this we obtain a list of
possible generations and disappearances in temporal order. Following the list, we
reproduce the surface and its structural graph to examine whether topological events could
really occur, and specify all the generations and disappearances of peaks. In contrast to
topological events, movement is defined for intervals rather than instants. Therefore, we
discretely trace the location of peaks between topological events.
2.5 Analysis of events
Once primitive events are deduced, they can be stored in GIS as spatial objects withattributes. Therefore, one method to analyze the change of a surface is to visualize the
distribution of primitive events and the structural graph through GIS. The spatial
distribution of events can be visualized for any time interval; the structural graph and its
original surface can also be visualized for instants.
Primitive events are defined for critical points and corridors. However, this does
not imply that all the primitive events are equally important in any application; analysts
should focus on primitive events closely related to their interest. In fact, we often pay
attention to the change of peaks and bottoms when analyzing surfaces, since their spatial
arrangement characterizes the global spatial structure of surfaces. In addition, as Okabeand Masuda (1984) noted, cols and either peaks or bottoms are redundant descriptors of
the structural graph. This is because of two reasons: 1) a col is always directly connected
to two peaks and two bottoms; 2) bottoms are always encircled by corridors connecting
peaks and cols, and so are peaks by those connecting bottoms and cols. Because of theseproperties and equation (1), most of the topological events of cols and corridors areaccompanied by those of peaks or bottoms. Even if we remove bottoms, cols, andcorridors connecting bottoms and cols, we can restore the original graph from theremaining peaks and corridors. Consequently, to focus on either peaks or bottoms is arational choice.
However, when we analyze the population distribution, not only peaks and bottoms
but also cols and corridors are of interest because the doughnut phenomenon is of great

8

importance in population research. The doughnut phenomenon is represented as a circuitwith a col in the structural graph, and its appearance is accompanied by generation of abottom or switch of a corridor.
Spatio-temporal statistics is also useful when analyzing the spatio-temporal
distribution of topological events of critical points. For example, Knox (1964) and
Mantel (1967) propose statistical tests for analyzing clusters in spatio-temporal point
distributions. Openshaw (1994) develops an exploratory method for detecting spacetime-attribute patterns in point distributions. Jacquez (1996) extends the K-function
method to spatio-temporal analysis. These methods are useful for analyzing spatiotemporal pattern in the distribution of topological events, especially for examining
whether they are clustered in the spatio-temporal dimension.
3. EMPIRICAL STUDYThis section empirically analyzes the change of a surface in order to test the validityof the method proposed in the previous section. We investigate the change of retail
activity in Shinjuku and Shibuya area in Tokyo, Japan (Figure 6). The study region
contains the biggest retail cluster in Japan, surrounded by urban parks, residential areas
and business districts. The Yamanote Line and Meiji Street pass through the region, andthere are five major railway stations: Shinjuku, Yoyogi, Harajuku, Shibuya, and Ebisu.
FIG. 6. Shinjuku and Shibuya area in Tokyo, Japan.
Spatial data used in the analysis are based on the list of retail stores and restaurants
in the NTT telephone directory which is published monthly. We extracted their addresses
in September every other year from 1990 to 98 in ASCII format, and converted them into
spatial data by geocoding. Table 1 shows the number of retail stores and restaurants inthe study region.
TAB. 1. The number of retail stores and restaurants, 1990-98.
From the point data we calculate the density distribution of stores using thefollowing equation:  x − yi  2 
f ( x ) = ∑ exp −
(8)
 ,
i
  R  
where yi is the locational vector of store i and R is the distance decay parameter. We
adopt two values for R, 100m and 200m, and perform analysis in order to look at thechange in both local and global scales. Figure 7 shows the density distribution function in9

1998 for R=200m.
FIG. 7. Density distribution function of retail stores and restaurants, 1998 (R=200m).
Figure 7 shows the global structure of the retail cluster. There are two big
subclusters, one at the northeast of Shinjuku, and the other at the west of Shibuya. In
addition, there are smaller subclusters at the east of Harajuku and the southwest of Ebisu.
These four subclusters form a big retail belt running from north to south between the
Yamanote Line and Meiji street.
From the density functions we calculate critical points and corridors. For thethreshold value T, which are used for extracting significant critical points, we triedvarious values ranging from 0.01 to 1.0. Here we adopt 0.2 for R=100m and 0.1 for
R=200m, respectively, because the structural graph was relatively stable around thesevalues. The graph is calculated for every other year from 1990 to 98, and primitive eventsare deduced for each two-year period. In the following discussion we focus on thechange of peaks because they represent subcenters of the retail cluster which characterize
the basic structure of the cluster.
3.1 Global analysis
We first discuss the change of global structure of the retail cluster using the density
function of R=200m. The number of peaks and topological events are shown in Table 2.
The structural graphs in 1990 and 1998 are shown in Figure 8.
TAB. 2. The number of peaks and topological events, 90-98 (R=200m).
FIG. 8. The structural graphs of the density function of retail stores and restaurants, 1990
and 98 (R=200m).
In Figure 8 we notice that the graph had been considerably stable during the period;
since 1990 only two peaks appeared and one peak disappeared. We should note that
topological events shown in Table 2 include two pairs of generations and disappearances(94-96 and 96-98) that occurred at the same peak located at the west of Shinjuku. Twopeaks that disappeared until 1992 were absorbed into the peaks at Harajuku and Ebisu
which represent big subclusters. From this we may say, at the global scale, that small
subclusters were integrated into larger ones during 90-92. However, in general, thespatial structure of the retail cluster had been quite stable from 1990 to 98.
Movement of peaks during 90-98 is shown in Figure 9. The figure indicates that10

peaks had generally moved away from railway stations. This is probably because of thefollowing reason. There remains only little space available for new stores and restaurants
near stations such as Shinjuku and Shibuya. This promotes expansion of retail clusters
from stations to its surrounding area. However, clusters usually expand not uniformly
but in some specific directions, which results in movement of peaks.
FIG. 9. Movement of peaks, 90-98 (R=200m).
3.2 Local analysis
Let us then turn to local analysis where R=100m in equation (8). The number ofpeaks and topological events are shown in Table 3. The structural graphs in 1990 and1998 are shown in Figure 10.
TAB. 3. The number of peaks and topological events, 90-98 (R=100m).
FIG. 10. The structural graphs of the density function of retail stores and restaurants,
1990 and 98 (R=100m). For the sake of simplicity, bottoms, cols, and corridors
connecting bottoms and cols are omitted.
The results shown in Table 3 and Figure 10 are quite different from those of the
global analysis. The number of peaks fluctuates sharply from 1990 to 96. During 90-92
and 94-96 a large number of subcenters appeared mainly by G+, which implies the
opening of new stores. In contrast, nineteen peaks disappeared from 1992 to 94, nine by
D+ and ten by D-. Disappearance of peaks, that is, integration of subclusters, occurred
by both expansion and declination of existing clusters.
We then analyze the topological events in more detail in conjunction with
development projects carried out in this region.
1991 41994 7
1994 101995 1
1996 10

New Tokyo Metropolitan Government Office (office building)
Shinjuku Park Tower (office building)Ebisu Garden Place (office buildings and commercial facilities)
Shinjuku Island Tower (office building)
Shinjuku Takashimaya (department store)

Figure 11 shows that the openings of new office buildings have similar effect on thespatial structure of the retail cluster. From 1990 to 92, two peaks appeared around Tokyo
Metropolitan Government Office, which reflects an increase in demand of employees for11restaurants and retail stores. Similarly, during 94-96, two peaks were newly generated
near Shinjuku Park Tower and Shinjuku Island Tower. These three buildings are all very
high office buildings having 48, 52, and 44 floors. From this we can say that the openingof very high office buildings greatly affects the spatial structure of retailing.
FIG. 11. Topological events of peaks, 90-98 (R=100m).
Effect of the opening of Ebisu Garden Place was more drastic; it had totallychanged the spatial structure of retailing around Ebisu station as shown in Figures 10 and
11. From 1992 to 94 seven peaks disappeared; as a result, eleven peaks were integrated
into only four peaks. Note that this change does not imply the decline of retailing because
four out of seven disappearances are classified into D+; the number of stores had notnecessarily decreased. We should rather say that rapid growth of retailing had promoted
integration of subclusters in this region. On the other hand, after the opening of Ebisu
Garden Place in 1994, six peaks were newly generated by G+. However, the location ofpeaks is quite different from that of the peaks that had disappeared before 1994.
Consequently, we can conclude that Ebisu Garden Place had not only contributed to retail
activity around Ebisu station but also greatly changed the spatial structure of retailing inthat region.
In contrast to the above four projects, the opening of Shinjuku Takashimaya showsno significant effect in retail structure. We can find one generation (G+) at the south of
Takashimaya, but it does not seem to be the result of the opening of Takashimaya. The
effect of the opening of department stores may appear slowly compared to that of officebuildings.
We finally examine local movement of peaks. Figure 12 clearly shows regionalvariation of movement pattern; peaks were continuously moving around Shinjuku,
Shibuya, and Ebisu stations while rather stable around Yoyogi and Harajuku. Thissuggests that large retail clusters change their spatial structure more frequently than small
clusters. We also notice, in contrast to global analysis, that peaks had not necessarily
moved away from railway stations. Peaks had moved in various directions at the local
scale without showing any clear tendency.
FIG. 12. Movement of peaks, 90-98 (R=100m).
4. CONCLUSION
In this paper we have developed a method for analyzing the change of surfaces,
extending the topological method to spatio-temporal domain. The spatial structure of a12

surface is represented by its structural graph that consists of critical points and corridors.
The change of a surface is described by a set of primitive events; generation,
disappearance, and movement are considered for critical points, and switch is additionallyintroduced for corridors. From surfaces of two times a set of primitive events that causes
the change is deduced. They are stored in GIS as spatial objects with attributes, and their
spatio-temporal pattern is visually analyzed. To test the validity of the method, weanalyzed the change of a retail cluster in Shinjuku and Shibuya area in Tokyo, Japan. The
empirical study yielded some interesting findings that help us understand the change ofthe spatial structure of retailing.
We finally discuss some limitations of our method for further research. First, we
adopt a graph-theoretical representation of surfaces to describe their change. Primitive
events are defined for the elements of the structural graph: peaks, bottoms, cols, andcorridors. Though they typically characterize the global structure of surfaces, some
applications may require more detailed description of surfaces. For example, one may beinterested in the direction of slope, its spatial distribution, and its change over time. To
deal with more details of surfaces, representation of surfaces and their change has to beextended to metric domain. Second, we interpolate surfaces between discrete times bylinear interpolation, assuming linear change of surfaces over time. This assumption,
however, may not be always valid. For example, let us consider a moving peak and its
snapshot data. From the data we may deduce the change between two times as a
combination of generation and disappearance, not movement. This misinterpretation can
be avoided by acquiring spatial data with high frequency as we did in the empiricalanalysis. However, it is certainly desirable to adopt a spatio-temporal interpolation
method suitable for the surface of analyst's interest. New spatio-temporal interpolation
methods that explicitly take the behavior of surfaces into account should be developed infuture research.

13

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Taylor & Francis).
Cressie, N., 1993, Statistics for Spatial Data (New York: John Wiley).
Griffith, D. A., 1981, Modeling Urban Population Density in a Multicentered City.
Journal of Urban Economics, 9, 298-310.
Hagget, P., 1968, Trend Surface Mapping in the Interregional Comparison of
Intraregional Structure. Papers of the Regional Science Association, 20, 19-28.
Iri, M., Shimakawa, Y., and Nagai, T., 2000, Extraction of Invariants from Digital
Elevation Data with Application to Terrain Topography. Proceedings of the
Symposium on Integrated Geographical Information Systems, 5, 33-46 (inJapanese).
Isaaks, E. H. and Srivastava, R. M., 1989, An Introduction to Applied Geostatistics(New York: Oxford University Press).
Jacquez, G. M., 1996, A K-nearest Test for Space-Time Interaction. Statistics in
Medicine, 15, 1935-1949.
Knox, E. G., 1964, The Detection of Space-Time Interaction. Applied Statistics, 13, 2529.
Mantel, N., 1967, The Detection of Disease Clustering and a Generalized Regression
Approach. Cancer Research, 27, 209-220.
Openshaw, S., 1994, Two Exploratory Space-Time-Attribute Pattern Analyzers Relevant
to GIS. In Spatial Analysis and GIS, edited by S. Fotheringham and P. Rogerson(London: Taylor & Francis), 83-104.
Okabe, A. and Masuda, S., 1984, Qualitative Analysis of Two-dimensional Urban
Population Distribution in Japan. Geographical Analysis, 16, 301-312.
Okabe, A. and Masuyama, A., 2000, An exploratory method for qualitative trend curve
analysis - a robust method against low quality data. International Journal of
Geographical Information Science, to appear.
Sadahiro Y, and Umemura, M., 2000, Spatio-temporal analysis of polygon distributions:
event-based approach. CSIS Discussion Paper Series, 25, Center for Spatial
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Tobler, W., 1964, A Polynomial Representation of the Michigan Population. Papers of
the Michigan Academy of Science, Arts, and Letters, 49, 445-452.
Warntz, W., 1966, The Topology of a Socioeconomic Terrain and Spatial Flows. Papersof the Regional Science Association, 17, 47-61.
Warntz, W. and Waters, N., 1975, Network Representations of Critical Elements ofPressure Surface." Geographical Review, 65, 476-492.

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peak

bottomcol

Figure 1

slope

peak
bottomcolcorridor connecting
a peak and a colcorridor connecting
a bottom and a col

(a)

(b)

Figure 2

(a)

(b)

Figure 3

(c)

A

peak
bottomcol

B
C

Figure 4A

corridor connecting
a peak and a colcorridor connecting
a bottom and a col

peak
col
corridor connecting
a peak and a col

Figure 5

Shinjuku

Yoyogi

Yamanote
Line

Meiji
StreetHarajuku
railway
Yamanote Linerailway
Meiji Street
study region
urban parks

Shibuya

Ebisu

Figure 6

Shinjuku

Yoyogi

Harajuku
0 - 5050 - 100
100 - 200
200 - 300
300 - 500
500 - 700
700 - 1000
1000 - 1200
1200 -

Shibuya

Ebisu

Figure 7

railwayroadShinjuku

Shinjuku

Yoyogi

Yoyogi

Harajuku

Harajuku

Shibuyapeakbottomcolcorridor connecting
a peak and a colcorridor connecting
a bottom and a col

Shibuya

Ebisu

Ebisu

1990

1998

Figure 8

ShinjukuM+
0 - 10

Yoyogi10 - 20
20 - 30
30 - 40
40 - 50
50 - 60
60 -Harajuku

M0 - 10

Shibuya

10 - 20
20 - 30
30 - 40
40 - 50Ebisu

50 - 60
60 -Figure 9

Shinjuku

Shinjuku

Yoyogi

Yoyogi

Harajuku

Harajuku

peakShibuyaShibuyacorridor connecting
a peak abd a col

Ebisu

Ebisu

1990

1998

Figure 10

Figure 11
Ebisu

1992-94

1990-92

ShibuyaShibuya

Ebisu

Harajuku

YoyogiShinjuku

Harajuku

YoyogiShinjuku

Tokyo Metropolitan Government Office

Shibuya

Harajuku

Shinjuku Park Tower

Ebisu

1994-96

YoyogiShinjuku

Shinjuku Island Tower

Ebisu Garden Place

Shibuya

Harajuku

Ebisu

1996-98

YoyogiShinjuku Takashimaya

Shinjuku

disappearance of peaks

generation of peaks

D-

D+

G-

G+

Figure 12

Shibuya

Harajuku

1992-94

1990-92

ShibuyaShibuya

Ebisu

Harajuku

Harajuku

YoyogiShinjuku

Ebisu

YoyogiShinjuku

Ebisu

1994-96

YoyogiShinjuku

Shibuya

Harajuku

Shinjuku

1996-98

Ebisu

YoyogiM-

60 -

50 - 60

40 - 50

30 - 40

20 - 30

10 - 20

0 - 1060 -50 - 60

40 - 50

30 - 40

20 - 30

10 - 20

0 - 10M+year

number of stores

1990
1992
1994
1996
1998

17176
17486
17960
18520
18797

Table 1

1990
11

number of peaks

number of events

G+
GD+
D-

1992
9
90-92
1
1
1
1

1994
10
92-94
1
0
0
0

Table 2

1996
10
94-96
1
0
1
0

1998
10
96-98
0
1
0
1

1990
60

number of peaks

number of events

G+
GD+
D-

1992
70
90-92
11
5
2
4

1994
55
92-94
4
0
9
10

Table 3

1996
63
94-96
14
0
1
5

1998
64
96-98
4
1
0
4