CSIS Discussion Paper No. 25
Spatio-Temporal Analysis of Polygon Distributions:
Event-Based Approach
Yukio Sadahiro and Mitsuru Umemura
APRIL, 2000

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

April 5, 2000
Spatio-Temporal Analysis of Polygon Distributions: Event-Based Approach

Yukio Sadahiro* and Mitsuru Umemura***Center for Spatial Information Science and Department of Urban Engineering**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.jp

Spatio-Temporal Analysis of Polygon Distributions: Event-Based Approach
Abstract
This paper develops a method for analyzing changes in polygon distributions. Fourtypes of topological events are proposed to describe the change: 1) generation, 2)
disappearance, 3) union, and 4) division. Any change of polygon distributions can be
decomposed into a combination of these events. From polygon distributions of two times
a set of events causing the change is estimated. Me asures summarizing the change ofpolygon distributions are then proposed. They are essentially based on the number of
events, thus represent the complexity of change. The method is applied to the analysis of
the spatial competition between the major and small chains of convenience stores inTokyo, Japan. The empirical study reveals the spatial structure of competition in both
local and global scales.

1. Introduction
Polygons, as well as points and lines, are most important spatial objects in GIS.
Geological maps describe the spatial configuration of soils by polygons. Landuse maps
contain polygons representing commercial areas, residential areas, parking lots, urban
parks, and so forth. Polygons are also created from points and lines by spatial
operations. Thiessen polygons are constructed from a set of points. Buffer operation
generates polygons approximating curved figures from points and lines.
Polygons often change their shape and location over time. Consider, for instance,
spatio-temporal changes in a city in which geographers have a great interest. A residential
area expands with urban growth. A business district appears at the city center. Emergenceof supermarkets causes declination of shopping streets; old stores are closed and retailclusters shrink. These changes are represented as the spatio-temporal change of polygons
in GIS.
Geography, especially spatial analysis, is concerned with such changes, and GIS
provides several functions to analyze them. One is overlay operation, which
superimposes polygon distributions of different times on a base map to investigatechanges between them. Another option is to see the change by animation. Since recent
GIS can generate animation from sectional data, this method is now widely available.
The above methods are too naive as methodology of spatial analysis, though they
are essential in exploratory data analysis. A more sophisticated method is to calculate
summary statistics for individual times and analyze their temporal change. The averageand total area of polygons are widely used. Such measures, however, fail to take spatialchanges into account because they are based on the sectional data. Therefore, they cannot
detect even basic transformations of polygons such as translation and rotation, though
they are significant changes in the real world.
In GIS field, spatio-temporal changes have been studied chiefly by computerscientists. Galton (1997), for instance, analyzes various types of continuous change inpolygons. Using several measures of separation between polygons, he examines howchange processes are represented. Egenhofer and Al-Taha (1992) discuss the gradualchange of a polygon from the viewpoint of spatial relations. They propose a graphrepresenting the closest topological relationship between spatial relations, and describe thechange of spatial relation between two polygons caused by the change of one of thepolygons. Hornsby and Egenhofer (1997, 1998) introduce a framework for representing
the spatio-temporal change of spatial objects in GIS. Using a qualitative representation ofchange, they classify various types of change and operation. The above studies provide
computational models to represent spatio-temporal changes. However, since they focus
on spatial information theory rather than spatial analysis, they are not applicable directly1

to the spatio-temporal analysis of polygon distributions.
This paper develops a method for analyzing the spatio-temporal change of the
distribution of polygons. We focus on polygons that do not move in space, say,residential lots and parks. In section 2, we propose four types of topological events todescribe the change of polygon distributions. From polygon distributions of two times
we estimate a set of events that cause the change. We also propose several measures tosummarize the change of polygon distributions. In order to test the validity of the method,
section 3 empirically analyzes the spatial competition between the major and small chainsof convenience stores in Tokyo, Japan. Section 4 summarizes the conclusions withdiscussion.
2. Methodology
Spatio-temporal distribution of polygons can be represented in several ways incurrent GIS (Langran, 1992). One is the layer-based approach in which each layer
indicates a snapshot of polygon distributions at a certain time. Another method describes
temporal information as attributes of polygons, say, the time of birth and death (objectoriented approach). This paper assumes the former representation, that is, we analyze thespatio-temporal change of polygon distributions using sectional data. Note, however, thatthe method we propose can be applied to object-oriented data with a slight modification.
Suppose n sets of polygons Γ1, Γ2, ..., Γn, where Γi represents the distribution ofpolygons at a time ti. Let #(Γ) and A(Γ) be the number and total area of polygons in apolygon set Γ, respectively. We first propose a method for analyzing the change between
two times, from t1 to t2, and then extend it to the case of n(>2) times.
We adopt the definition of a polygon as an open set of points; the boundary does
not belong to the polygon. It allows us to treat adjacent polygons as separated (Figure 1).
In order to distinguish boundaries shared by two polygons from outer boundaries, wecall the former partition and the latter boundary hereafter.
Figure 1 An example of polygon sets Γ1 (left) and Γ2 (right).
2.1 Classical summary statistics
Given two sets of polygons Γ1 and Γ2, we can calculate several summary statistics
as below.
1) Number ratio: the ratio of the number of polygons in Γ2 and Γ1.
# (Γ2 )rNUMBER =# (Γ1 )

2(1)

2) Total area ratio: the ratio of the area of polygons in Γ2 and Γ1.
A(Γ2 )rAREA _ TOTAL =A(Γ1 )
3) Average area ratio: the ratio of the average area of polygons in Γ2 and Γ1.
A(Γ2 ) /# (Γ2 )rAREA _ AVERAGE =A(Γ1 ) /# (Γ1 )r= AREA _ TOTALrNUMBER(2)(3)

These statistics partly describe how a polygon distribution changes; whether thenumber, total area, and average area of polygons increase. However, since they arecalculated only from sectional data, they provide little information about spatial changes.
They cannot detect the change from Γ 1 to Γ 2 , for instance, if Γ 2 is obtained by atranslation of Γ1. More detailed description of spatio-temporal changes is necessary inspatial analysis.
2.2 Event-based method
In order to take spatial changes into account more explicitly, we consider four typesof topological events that occur in polygon distributions: 1) generation, 2)
disappearance, 3) union, and 4) division (Figure 2). Any change of polygon
distributions can be described by a combination of these events as shown later.
Figure 2 Topological events. (a) Generation, (b) disappearance, (c) union, and (d)division.
The set Γ1 changes into Γ2 by a sequence of topological events. It is not known,
however, what events occurred between t1 and t2. We thus estimate the topologicalevents that occurred from t1 to t2. Note that we do not estimate the change process itself;
we are interested in the combination of events rather than their order.
There are numerous possible combinations of topological events causing thechange from Γ1 and Γ2. To choose the most plausible one we follow two principles. The
principle of minimum events is that a set of fewer events is more plausible, that is, wechoose the smallest number of events. Similar ideas are often used in spatial reasoning(for example, see Egenhofer and Al-Taha, 1992). The second principle is change without
redundancy, which assumes that no two changes occur at the same location between twotimes. For instance, the change shown in Figure 3a is inhibited while that in Figure 3b is
acceptable. The points on the boundary of two right polygons are generated at the first3

step and then disappeared in Figure 3a.
Figure 3 Two changes causing the same result. (a) The inhibited change, (b) thepermissible change.
Following the above two principles we can estimate the topological events that
occurred from t1 to t2. Figure 4 shows some examples of the change and the estimatedcombination of events. Concerning the order of events, there may be more alternatives
other than those shown in Figure 4.
Figure 4 Changes of polygon distributions and the estimated combinations of events.
Computational procedure of estimation is as follows. The polygon sets Γ1 and Γ2
are overlaid to generate a new set of polygons. Each polygon is labelled two variables(S1, S2), where Si indicates the state of the polygon at ti. The variables take either E or ∅:E stand for Existing and ∅ for Not-existing. For instance, a polygon whose elements(points) exist in both Γ 1 and Γ 2 is labelled (E, E). The label (∅ , E) implies thegeneration of a polygon, while (E, ∅) is a disappearance event. Figure 5b shows anexample of labels.
The polygons generated by overlay operation are classified by label into threesubsets Ω 0, Ω 1, and Ω 2. The set Ω 0 is a set of polygons labelled (E, E). The sets Ω 1and Ω2 consist of polygons labelled (∅, E) and (E, ∅), respectively.
The arcs generated by overlay operation are also labelled two variables (S1, S2).
The variables take one of the four values: B, P, I, or ∅. Arcs that compose the boundaryand partition of polygons are labelled B and P, respectively. Arcs belonging to theinterior of polygons have I, which appear when a boundary changes into a part of apolygon by a union event. The empty set ∅ represents the absence of arcs. The variables(S1, S2) enable us to classify arcs into twelve groups: (B, B), (B, P), (B, I), (B, ∅), (P,B), (P, P), (P, I), (P, ∅), (I, B), (I, P), (∅, B), and (∅ P). Figure 5c depicts the labelof arcs representing their state at t1 and t2.
Figure 5 Labels representing the state of polygons and arcs at t1 and t2. (a) A change ofpolygons, (b) labels for polygons, (c) labels for arcs.
Using the label of polygons and arcs we can estimate the smallest number of
topological events that yield the change from Γ 1 to Γ 2. Concerning generation anddisappearance events, estimation is straightforward. The smallest number of these events

4

can be easily obtained by counting the number of polygons labelled (E, ∅) or (∅, E).
The polygons labelled (∅, E) give the smallest set of generation events, while thenumber of (E, ∅) polygons is the smallest number of disappearance events. Let G and
Dis be the sets of estimated generation and disappearance events, respectively. Each
event is denoted by its own number: the ith generation event, for instance, is denoted by
G i. Let P(G i) be the polygon generated by G i. Similarly, P(Dis i) is the polygon
disappeared at the ith disappearance event.
Estimation of union events is somewhat complicated. We examine all the arcs tofind labels (B, I) and (P, I) where unions must have occurred (Figure 6a). We thenextract the nodes shared by only (B, I) or (P, I) arcs, which define connecting groups ofarcs as shown in Figure 6b. For each group of arcs we calculate the number of polygonsthat are at least partly bounded by the arcs. Subtracting one from the number we obtainthe smallest number of union events that occurred at the arc group. Division events can be
estimated in a similar way.
Union and division events are estimated at an aggregated level so that it is
impossible to identify individual events; for instance, there are three ways of combining
polygons shown in Figure 6b. We thus denote U i as the ith set of union events thatcannot be decomposed further into individual events. Let PO(Ui) and NUi be the set andnumber of polygons combined by Ui, respectively. The resultant polygon is denoted by
PR(Ui). Similarly, Divi indicates the ith set of division events and NDi is the number ofresultant polygons. The original polygon and the set of resultant polygons involved with
Divi are denoted by PO(Di) and PR(Divi), respectively.
Figure 6 Estimation of union events in the change of polygons shown in Figure 5. (a)Arcs where unions must have occurred (thick broken lines) and a node shared by only
(B, I) or (P, I) arcs (the white circle), (b) a connecting group of arcs (thick solid lines)
and the polygons bounded by the arcs (gray-shaded regions).
The estimated topological events and the polygons involved with the events fully
describe the change from Γ 1 to Γ 2 . However, the information they provide is tooabundant for direct use in spatial analysis. We thus propose several measures tosummarize the change from Γ1 to Γ2.
One simple but useful measure is the total number of events given by# (U )# ( Div )

i =1

i =1

MEVENT = # (G)+ # ( Dis) + ∑ NUi + ∑ NDi ,(4)

where #(S) indicates the number of elements in a finite set S. This measure shows a large
value if many events occur from t1 to t2.

5However, this measure is somewhat inconvenient if we compare the spatiotemporal change between different sets of polygons; the number of events necessarily
increases with that of original and resultant polygons. We thus standardize MEVENT by
dividing it by the number of polygons in Γ1 and Γ2:
mEVENT =# (U )# ( Div )

i =1

i =1

# (G)+ # ( Dis) + ∑ NUi + ∑ NDi# (Γ1 )+ # (Γ2 ).

(5)

We call this measure the event measure. The event measure represents the degree ofchange in terms of events, in other words, the complexity of change. For instance, alarge mEVENT implies that many events occurred from t1 to t2, that is, the change wasquite complicated. The measure mEVENT shows zero if and only if the polygons in Γ1 andΓ2 are exactly the same.
A more general form of the measure is# ( Dis )# ( G )1
mGENERAL =fP
G+fDis ( P( Disi ))
(
)
(
)
∑
∑ G
i
g(Γ1 ) + g(Γ2 )  i =1
i =1
.
(6)
# (U )# ( Div )+ ∑ fU ( PO(Ui ), PR(Ui )) + ∑ fDiv ( PO( Divi ), PR( Divi ))
i =1
i =1

Substituting(7)fG ( P(Gi )) = fDis ( P( Disi )) = 1,
fU ( PO(Ui ), PR(Ui )) = NUi ,(8)

fDiv ( PO( Divi ), PR( Divi )) = NDi ,(9)g(Γ ) = # (Γ )

(10)

and
we obtain equation (5).
Equation (5) implies that the four kinds of events are equivalent in the sense ofcomplexity. If this assumption is not acceptable, we can weight each type of event with
its complexity:fG ( P(Gi )) = α G ,(11)fDis ( P( Disi )) = α Dis ,

and

(12)

fU ( PO(Ui ), PR(Ui )) = α U NUi ,(13)

fDiv ( PO( Divi ), PR( Divi )) = α Div NDi .

(14)

Substituting these equations (10)-(14) into equation (6) we have the weighted event
measure6

mWEIGHTED =# (U )# ( Div )

i =1

i =1

α G # (G) + α Dis # ( Dis) + α U ∑ NUi + α Div ∑ NDi# (Γ1 )+ # (Γ2 )

.

(15)

The event and weighted event measures are both based only on the topology ofevents; they do not incorporate metric information about events. However, we can takemetric factors into account by adopting metric functions in equation (6). For instance, if
we consider that the generation of a large polygon is a greater change than that of a small
polygon, we may use(16)fG ( P(Gi )) = A( P(Gi ))andfDis ( P( Disi )) = A( P( Disi )) .

In addition to these equations, we substitutefU ( PO(Ui ), PR(Ui )) = fDiv ( PO( Divi ), PR( Divi )) = 0

(17)
(18)

and
g(Γ ) = A(Γ )into equation (6) to obtain
mAREA =

A(Ω1 ) + A(Ω 2 ).
A(Γ1 ) + A(Γ2 )

(19)

(20)

We call this measure the area measure.
2.3 The case of n(>2) times
Classical summary statistics and event-based measures describe the change ofpolygon distributions between two times. This subsection extends the methods to thecase of n(>2) times --- the change from Γ1 to Γn.
Extension of the classical statistics, that is, the number ratio, total area ratio, andaverage area ratio, is straightforward; we calculate these statistics for the first and lasttimes, say,# (Γn )rNUMBER =.
(21)# (Γ1 )The event-based measures are calculated as follows. We first estimate thetopological events that occurred between every neighboring times using the method
proposed in the previous subsection. Enumerating the events we obtain a set of plausible
events from t1 to tn, and classify them into the four groups: G, Dis, U, and Div. The
measures are calculated on the basis of these events and the sets of polygons at t1 and tn,
say,

7

mGENERAL =# ( Dis )# ( G )1f
P
G+ ∑ G ( ( i )) ∑ fDis ( P( Disi ))g(Γ1 ) + g(Γn )  i =1
i =1
# (U )# ( Div )+ ∑ fU ( PO(Ui ), PR(Ui )) + ∑ fDiv ( PO( Divi ), PR( Divi ))
i =1
i =1


.

(22)

3. Empirical study
In the previous section we proposed a method for analyzing the spatio-temporalchange of polygon distributions. To test the validity of the method, this section
empirically analyzes the spatial competition among convenience stores in the Tokyo 23-ku
area in Japan.
Most convenience stores belong to retail chains in Japan; one third of stores belong
to one of three major chains (7-Eleven, Family Mart, and Lawson). There is keencompetition among chains, especially between the major chains and other small chains.
We are concerned with the spatial competition between the major and small chains.
Spatial data used in the analysis are based on the list of convenience stores in the
NTT telephone directory which is published every month. We obtained the address of
convenience stores in September every year from 1990 to 98 in ASCII format, andconverted it into geocoded data by address matching. Figure 7 shows the number ofstores in the Tokyo 23-ku area from 1990 to 98.
Figure 7 The number of convenience stores in the Tokyo 23-ku area, 1990-98.
Figure 7 indicates that the stores of the major chains increased rapidly from 1990 to
94 while those of the small chains increased moderately. Since 1994, however, both the
major and small chains grew at a similar speed. We thus analyzed the spatial competition
between the chains separately for two periods: 1990 to 94 and 94 to 98.
Analysis is based on the market area of chains which is estimated from the point
data of convenience stores. As for the consumer behavior, we assume that they go to thenearest stores, and that they do not use convenience stores located further than 400
meters. These assumptions are reasonable in urban areas in Japan. Market area is thengiven by the Voronoi diagram of convenience stores clipped by a buffer region of
distance 400 meters. Table 1 shows the number of stores and the size of market area.
Table 1 The number of stores and the size of market area, 1990-98.
The total market area of all chains was stable during 1990-98. This is because in8

1990 convenience stores were already enough to cover the Tokyo 23-ku area with market
area. In spite of that, the number of stores increased in both the major and small chains,which reduced the average market area of stores.
Since the total market area was almost unchanged, it is sufficient to examine thechange of market area of the major chains in order to analyze the spatial competition
between the major and small chains. Table 2 shows the number and area of polygons
representing the market area of the major chains in 1990, 94, and 98. The ratios
discussed in Subsection 2.1 (the number, total area and average area ratios) were also
calculated.
Table 2 The number and area of polygons representing the market area of the major
chains.
The total area of polygons increased noticeably from 1990 to 94. This indicates thatthe major chains invaded the market area of the small chains. In contrast, the total area israther stable after 1994. It seems, at least from these figures, that the major and small
chains were in equilibrium in this period. We also calculated the ratios at the ward level,
but the results were quite similar.
We then move to the complexity of change of market area. Topological events
discussed in the previous section imply the spatial competition between the chains. We
thus calculated two event-based measures for each ward (Figure 8): the area measuremAREA and the event measure mEVENT . A large value of these measures indicates that themarket area changed considerably, that is, the major and small chains competed actively
with each other.
Figure 8 Twenty-three wards in the Tokyo 23-ku area.
The results are shown in Table 3. We notice that the total area ratio and the area
measure are positively correlated (r=0.68). This implies that a large value of the area
measure is yielded by the expansion of market area of the major chains. For instance, thearea measure is larger in 90-94 than 94-98 in most wards. This is chiefly because themarket area of the major chains expanded drastically from 1990 to 94. The event
measure, on the other hand, is larger in 94-98 than 90-94; more events occurred during
the latter period. From this we can say that the major and small chains competed keenly inboth periods: from 1990 to 94 the major chains clearly gained in strength; after 1994 thecompetition was still keen but almost even.

9Table 3 Event-based measures.
We then investigate the spatial structure of competition between the chains in a
global scale using the area and event measures. Figure 9 confirms that the area measuredecreased considerably from 90-94 to 94-98. Its spatial distribution slightly changed; thearea measure decreased more drastically in the north than in the south. Concerning theevent measure, its value increased equally in all the wards, keeping its global structure(Figure 10). Considering these results, we can say that the spatial structure ofcompetition was rather stable in a global scale during 1990-98; the situation became morecompetitive from 90-94 to 94-98 in all the wards; the expansion of the major chainsbecame somewhat slower in the north.
Figure 9 The area measure.
Figure 10 The event measure.
We finally examine the spatial competition in a local scale. Figure 11 shows thechange of market area of the major chains from 1990 to 1994 in Chuo, Taito, andShinagawa. Chuo is distinctive for its large area measure and small event measure, whichis confirmed by Figure 11a. There were only a few convenience stores in 1990 so that a
lot of space was left as potential market area. The competition was not so keen, thus thearea measure is large while the event measure is small. Taito and Shinagawa are similar in
the area measure but different in the event measure. Shinagawa has a large event
measure, which is confirmed by the drastic change of market area shown in Figure 11c.
Figure 12 shows the change of market area from 1994 to 1998 in Bunkyo,
Shibuya, and Itabashi. The total area ratio of these wards is almost the same as theaverage for all the wards. Variation in the area measure is so small that it can be ignored.
The difference among these wards lies in the event measure: Itabashi shows the largest
value among all the wards, while Bunkyo has quite a small value. The competition
between the major and small chains was much stronger in Itabashi than Bunkyo, as
depicted in Figure 12.
Figure 11 The change of the market area of the major chains from 1990 to 94 in (a)Chuo, (b) Taito, and (c) Shinagawa.
Figure 12 The change of the market area of the major chains from 1994 to 98 in (a)Bunkyo, (b) Shibuya, and (c) Itabashi.

10

4. Concluding discussion
In this paper we have developed a method for analyzing the spatio-temporal changeof polygon distributions. Four types of topological events are introduced to describe thechange: 1) generation, 2) disappearance, 3) union, and 4) division. Any change ofpolygon distributions can be decomposed into a combination of these events. Frompolygon distributions of two times a set of events causing the change is estimated. Tosummarize the change of polygon distributions we proposed several measures, say, theevent measure. They reflect the number of events that occurred between two times, andthus represent the complexity of change. The general form of the measures, mGENERAL ,can take account of not only the topological information but also the metric informationsuch as the area of polygons. We then applied the method to the analysis of spatialcompetition between the major and small chains of convenience stores in Tokyo, Japan.
The empirical study revealed the spatial structure of competition in both local and global
scales.
The event-based approach is useful for spatio-temporal analysis of polygon
distributions. However, this does not assure that the method is applicable to any types ofpolygons. We finally discuss the limitations of the method for further research.
First, we proposed the four types of events as mentioned above, assuming thatpolygons are not movable. This implies that the method does not work well if polygons
can move; we additionally have to consider movement events (Galton, 1995). Taking
movement into account causes difficulties in estimating the change process, and thus theextension of the method is not straightforward. However, the difficulties have to be
overcome in future research to treat movable polygons appropriately in spatial analysis.
Second, the event-based method assumes that polygons are homogeneous; attributeinformation of polygons is not used. This does not matter at an early stage of spatial
analysis. At an advanced stage, however, it may be necessary to analyze the temporalchange in both spatial and aspatial aspects. An extension in this direction is also
important.

11

References
Cohn, A. G., Bennett, B., Gooday, J., and Gotts, N. M., 1997, Representing andReasoning with Qualitative Spatial Relations about Regions. In Spatial andTemporal Reasoning, edited by O. Stock (Dordrecht: Kluwer), 97-134.
Egenhofer, M. J. and Al-Taha, K. K., 1992, Reasoning about Gradual Changes ofTopological Relationships. In Theories and Methods of Spatio-Temporal Reasoningin Geographic Space, Lecture Notes in Computer Science 639, edited by A. U.
Frank, I. Campari and U. Formentini (Berlin: Springer), 196-219.
Egenhofer, M. J. and Golledge, R. G. 1998, Spatial and Temporal Reasoning in
Geographic Information Systems (New York; Oxford University Press).
Galton, A, 1995, Towards a Qualitative Theory of Movement. In Spatial Information
Theory: A Theoretical Basis for GIS. Proceedings of the International Conference
COSIT '95, Lecture Notes in Computer Science 988, edited by A. U. Frank and W.
Kuhn (Berlin: Springer), 377-396.
Galton, A, 1997, Continuous Change in Spatial Regions. In Spatial Information Theory:
A Theoretical Basis for GIS. Proceedings of the International Conference COSIT'97, Lecture Notes in Computer Science 1329, edited by S. C. Hirtle and A. U.
Frank (Berlin: Springer), 1-13.
Hornsby, K. and M. J. Egenhofer, 1997, Qualitative Representation of Change. In
Spatial Information Theory: A Theoretical Basis for GIS. Proceedings of theInternational Conference COSIT '97, Lecture Notes in Computer Science 1329,
edited by S. C. Hirtle and A. U. Frank (Berlin: Springer), 15-33.
Hornsby, K. and M. J. Egenhofer, 1998, Identity-Based Change Operations for
Composite Objects. Proceedings of the 8th International Symposium on Spatial
Data Handling, 202-213.
Langran, G, 1992, Time in Geographic Information Systems (London; Taylor &Francis).

12

Figure 1

(a)

(b)

(c)

Figure 2

(d)

(a)

(b)

Figure 3(a)generation

union

generation

union

(b)
division

disappearance

(c)
union

union

union

division

division

union

(d)

Figure 4

union

(a)

(E, E)(∅, E)(E, E)(∅, E)(E, E)(E, E)(E, E)(E, E)(E, E)(E, ∅)(∅, E)(b)(P, P)(P, I)(B, B)(∅, B)(B, B)(B, B)(P, I)(P, P)(∅, B)(B, I)(B, I)(B, P)(I, B)(B, I)(I, P)(∅, P)(B, P)(∅, B)(B, B)(B, B)(∅, B)(c)Figure 5(I, B)(B, ∅)

(P, I)(P, I)(B, I)(B, I)(B, I)(a)(b)

Figure 6

2000

1000
major chainssmall chains
0
1990

91

92

93

94

95

Figure 7

96

97

98
[year]

AdachiItabashi
Kita
NerimaKatsushika
Arakawa

ToshimaNakanoSuginami

BunkyoTaito Sumida

Shinjuku

Edogawa
ChiyodaShibuya

Chuo

Koto

MinatoSetagaya
MeguroShinagawa

N

Ota

0

Figure 85km

- 0.05- 0.10- 0.15- 0.20- 0.25- 0.30- 0.35- 0.400.40 -

1990-94

1994-98

Figure 9

- 2.2
- 2.6
- 2.8
- 3.0
- 3.2
- 3.4
- 3.6
- 3.8
3.8 -

1990-94

1994-98

Figure 10

Chuo

Taito

Shinagawatotal area ratio 1.679
area measure 0.435
event measure 2.083

total area ratio 1.720
area measure 0.197
event measure 2.385

total area ratio 1.308
area measure 0.199
event measure 3.424

(a)

(b)

(c)

Figure 11BunkyoShibuyaItabashitotal area ratio 1.019
area measure 0.074
event measure 2.828

total area ratio 1.020
area measure 0.093
event measure 3.677

total area ratio 0.958
area measure 0.040
event measure 4.375

(a)

(b)

(c)

Figure 12

number of stores
total market area
[km2]
average market area [km2]

all chains
1990
1994 1998

major chains
1990
1994 1998

small chains
1990
1994 1998

2069
431.2
0.204

683
144.4
0.211

1386
286.8
0.207

2757
461.2
0.167

3378
481.1
0.142

Table 1

1213
213.0
0.176

1484
231.6
0.156

1544
248.2
0.161

1894
249.5
0.132

number
total area
[km2]
average area [km2]

1990

1994

1998

90-94

94-98

414
144.4
0.349

447
213.0
0.477

498
231.6
0.465

1.08
1.48
1.37

1.11
1.09
0.98

Table 2

total area ratioarea measure

event measureward

90-94 94-98

90-94 94-98

90-94 94-98

ChiyodaChuo
MinatoShinjuku
BunkyoTaito
SumidaKoto
ShinagawaMeguroOta
SetagayaShibuyaNakanoSuginami
Toshima
Kita
Arakawa
Itabashi
NerimaAdachi
Katsushika
Edogawa

1.409
1.679
1.883
1.522
1.776
1.720
1.422
1.342
1.308
1.672
1.429
1.502
1.538
1.218
1.219
1.581
1.621
1.390
1.582
1.512
1.705
1.205
1.378

1.631
1.247
1.278
1.020
1.019
0.932
1.310
1.314
1.065
0.965
1.005
0.983
1.020
1.151
1.101
1.152
1.183
1.432
0.958
1.034
1.117
1.245
1.089

0.378
0.435
0.242
0.130
0.311
0.197
0.241
0.318
0.199
0.180
0.186
0.217
0.249
0.117
0.163
0.112
0.270
0.207
0.232
0.207
0.361
0.250
0.290

0.195
0.207
0.158
0.051
0.074
0.063
0.092
0.220
0.106
0.060
0.116
0.094
0.093
0.025
0.100
0.020
0.068
0.049
0.040
0.050
0.143
0.121
0.157

2.615
2.083
2.320
2.875
3.192
2.385
2.875
2.211
3.424
2.622
3.189
2.958
3.261
3.946
3.200
2.349
2.324
2.704
3.561
2.768
2.237
2.245
3.022

2.857
2.700
3.267
3.674
2.828
3.333
3.933
2.568
3.146
1.854
3.426
3.027
3.677
3.611
3.574
3.260
3.000
3.115
4.375
2.965
2.371
2.321
3.935

total

1.475

1.087

0.208

0.069

2.818

3.193

Table 3