CSIS Discussion Paper No. 149

Descriptive measures of point distributions summarized over the entire spatial scale

Yukio Sadahiro

July 2017

Center for Spatial Information Science, The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa-shi, Chiba 277-8568, Japan
Phone: +81-471-36-4310
Fax: +81-3-5841-8521
sada@csis.u-tokyo.ac.jp

Abstract
Keywords: point distributions, spatial scale, summary measures, visualization
Visualization is a first step of exploratory point pattern analysis. It is often recommended to visualize
point distributions at various spatial scales from local to global, because it permits us to detect spatial
patterns of different scales that are useful for building research hypotheses. Visual analysis, however,takes a considerable amount of time especially when numerous maps are explored. Five measures aredeveloped in this paper that summarize the properties of point distributions over the entire range of spatial
scales. The maps of the measures help us to capture efficiently the overall spatial pattern of point
distributions. Numerical experiments and applications to real data analysis are performed to test thevalidity of the proposed measures. The results reveal the effectiveness of the measures as well as theirshortcomings to be revolved in future research.

11. Introduction
Visualization is a first step of exploratory point pattern analysis (Kovalerchuk and Schwing(2005); Oyana and Margai (2015)). Dot maps indicate the exact location of points using map symbols.
Quadrat method generates lattices, counts the number of points in each cell, and visualizes them as grid
maps. Kernel smoothing converts point distributions into smooth surfaces represented by hill-shaded andcontour maps (Silverman (1986); Scott (2015)). These maps provide us a lot of useful information on thespatial pattern of point distributions that are helpful for building research hypotheses.
Spatial scale plays a key role in quadrat method and kernel smoothing. Large cells and windows
yield smooth maps that reveal the global structure of point distributions. Small ones are appropriate for
exploring the local pattern since they emphasize the details of point distributions. Analysts arerecommended to try various spatial scales from local to global to capture spatial patterns of a wide varietyof scales.
Recent GIS software permit us to easily visualize point distributions with changing the spatial
scale of visualization. They generate grid maps and kernel surfaces of different scales very rapidly. Visual
analysis, however, takes a considerable amount of time and is inevitably subjective when treating
numerous maps generated at various scales. It is not easy to evaluate and memorize many spatial patterns
in a consistent and objective way. The difficulty increases when many distributions are analyzed
simultaneously as is often seen in ecology and biology.
One approach to this problem is to choose an optimal scale in visual analysis. The optimal
window width is proposed in kernel smoothing that minimizes the difference between point distribution
and kernel surface, or more precisely, approximately minimizes the mean integrated square error indensity estimation (Silverman (1986); Sheather (2004); Scott (2015), Sheather (2004)). Spatial scanstatistics define the optimal window as maximizing the difference in point density between inside andoutside the window (Kulldorff (1997); Tango and Takahashi (2005)). Visualization at an optimal scale,
unfortunately, inevitably conceals the spatial patterns observed at the other scales. Point clusters observed
at a certain scale may not be visualized at the optimal scale. Moreover, the optimal scale depends on thespatial pattern of point distributions. Different distributions have different optimal scales so that the maps
generated at their own optimal scales are not comparable with each other.
Summary measures of point distributions are an alternative to optimal scaling that also reduce
the time of exploratory analysis. χ2-statistic used in quadrat method evaluates the nonuniformity of point
distributions. K-function evaluates the clustering degree of points (Ripley 2005). Spatial median and thenumber of peaks are basic measures in kernel smoothing. These measures summarize the properties ofpoint distributions at a certain scale represented as cell size, circle radius, and window width. Measurescalculated at various scales are visualized as graphs whose x-axis is spatial scale that permit us to grasp
efficiently the properties of the distributions over different scales.
One of the drawbacks of summary measures is that they do not convey directly the spatial2

information on point distributions. For instance, summary measures do not indicate the location, size, andshape of point clusters. They are not effective to discuss the spatial relationship between point
distributions. Some measures do not distinguish the same spatial pattern at different location since they
are insensitive to the congruent transformation. To complement these measures, this paper proposes a new
method of visualizing point distributions for exploratory spatial analysis. The method calculates summary
measures that outline the properties of point distributions over the entire range of scales at each locationand visualizes their spatial distributions as maps. These measures and the aforementioned ones arecomplementary to each other, i.e., the former encapsulate the properties of point distributions over theentire range of scales at each location, while the latter summarize the properties over the whole area at
each spatial scale. Section 2 defines summary measures of point distributions. Measures for evaluating asingle distribution of points are proposed, which are further extended to treat multiple distributions.
Section 3 performs numerical experiments to test the validity of the proposed measures. Section 4 applies
the measures to the analysis of real datasets. Section 5 summarizes the conclusions with discussion.

2. Methods
2.1 Evaluation of a single distribution of points
Suppose there are n points Π={P1, P2, ..., Pn} in region S. Let zi be the location of ith point.
Function F0(x, w0; Π) is the density of points within distance w0 at location x:1∑ δ ( x − z i , w0 ) ,π w02 iF0 =( x, w0 ; Π )(1)where δ(|x-z|, w0) is a binary function given by

1

if x − z i ≤ w0.
otherwise0δ ( x − z i , w0 ) =(2)Function F0(x, w0; Π) becomes large where points are clustered, while it is small where points are sparse.
Distance w0 works as a scale parameter at which F0(x, w0; Π) is evaluated. A large w0 considers points ina wide area, and consequently, gives a global view on point distributions. A small w0 focuses on the local
variation of point distribution. Function F0(x, w0; Π) becomes smooth if w0 is large while a small w0 yields
a rough function.
Standardized form of F0(x, w0; Π) is defined as

F0 ( x, w0 ; Π )f 0 ( x, w0 ; Π ) =∫x∈S

F0 ( x, w0 ; Π ) dx

∑δ ( x − z , w )=∫ ∑ δ ( x − z , w ) dx
i

0

i

x∈S

i

i

3

0

.

Function f0(x, w0; Π) indicates the relative density of points while F0(x, w0; Π) is the absolute density.
The former is useful for comparing the spatial patterns between different distributions.
Both F0(x, w0; Π) and f0(x, w0; Π) are the functions of location x and scale w0. Given a certain
w0, we can create the maps of these functions. If location x is specified, we can visualize the functions asgraphs whose x-axis is w0. Figure 1 shows seven distributions of points labelled ΠA-ΠG in square region
S. Figure 2a illustrates the graphs of F0(x, w0; Π) as functions of w0, where x is the center of the region.
Points ΠA-ΠC clustered at the center in Figure 1 yield monotonically decreasing functions in Figure 2a.
Points in ΠD and ΠE are distributed close to the boundaries, and consequently, F0(x, w0; Π) monotonicallyincreases with w0. Donut-shaped distributions of ΠF and ΠG generate unimodal and bimodal functions inFigure 2a, respectively.
Figure 2b shows f0(x, w0; Π) of the distributions in Figure 1. The functions almost inherit the
topological properties of F0(x, w0; Π), though the two figures may look quite different. Points in ΠA-ΠC
have monotonically decreasing functions in both in Figures 2a and 2b. Points in ΠF and ΠG have one
and two peaks, respectively, in both figures.

(a) Distribution ΠA(b) Distribution ΠB(c) Distribution ΠC(e) Distribution ΠE(f) Distribution ΠF(g) Distribution ΠG

Figure 1 Seven distributions of points.

4(d) Distribution ΠD

500

250

400

200

ΠA

ΠB
ΠA
300

150

ΠF

200

ΠC

ΠC

ΠD
100

ΠB

ΠE

ΠG

100

50

0

0

ΠG

ΠF

ΠE

w0

w

ΠD
1

(c)

(a)
250

1.0

ΠA

ΠB
200

0.8

ΠB

ΠA
150

0.6

ΠC

100

50

0.2

ΠG

ΠE

ΠE

ΠD
0

w0

ΠF

ΠG

0.4

ΠF

0

ΠC

w

ΠD
1

(d)

(b)

Figure 2 Density functions of points in Figure 1. (a) F0(x, w0; Π), (b) f0(x, w0; Π), (c) f(x, w; Π), (d)function f’(x, w; Π).
Functions F0(x, w0; Π) and f0(x, w0; Π) indicate the degree of point clustering around x at
various scales from local to global. Summary measures of these functions with respect to w0 are useful in
exploratory analysis since we can visualize the measures as maps. The maps permit us to grasp the spatial
pattern of point distributions efficiently, and consequently, save our time of visual analysis with
continuously changing the spatial scale. Suppose, for instance, the mean of f0(x, w0; Π) with respect to w0.
It becomes large where point clusters are observed consistently over a wide range of scales. Detection ofsuch clusters is important and often critical in point pattern analysis and spatial statistics (Ord and Getis(1995); Anselin et al. (2008)).
One difficulty in the calculation of summary measures lies in the infinite range of scaleparameter w0, from zero to infinity. This prohibits us from calculating the mean of f0(x, w0; Π) in section[0, ∞) (details are discussed later in Appendix). We thus introduce a new parameter w ranging from zero
to one that has a one-to-one correspondence with w0:

e w0 − 1w = w0.
e= 1 − e − w0(3)
Parameter w0 is given by=
w0 log (1 − w ) .

5(4)Functions F0(x, w0; Π) and f0(x, w0; Π) become1∑ ρ ( x − zi , log (1 − w) )π w02 iF=
( x, w; Π )

(5)
and

F ( x, w; Π )f ( x, w; Π ) =∫x∈SF ( x, w; Π ) dx

∑ ρ ( x − z , log (1 − w) ) ,=∫ ∑ ρ ( x − z , log (1 − w) ) dx
i

i

x∈S

i

i

(6)
respectively. Figure 2c shows f(x, w; Π) of the distributions in Figure 1. The function still keeps the
topological properties of F0(x, w0; Π) with the finite range of parameter w.
Another difficulty in the calculation of summary measures is caused by the undefinability ofF(x, w; Π) and f(x, w; Π). They are not defined at the locations of points, where the numerator of Equation(5) is positive while the denominator is equal to zero. Integral of f(x, w; Π) with respect to w is
incomputable, on which basic measures including the mean and variance are defined. One method toresolve the problem is to transform f(x, w; Π) into another function of a finite range:f ' ( x, w; Π ) = 1 − e − f ( , w;Π ) .
x(7)Figure 2d shows f'(x, w; Π) that ranges from zero to one. The mean of f'(x, w; Π) is given by=µ ( x; Π )

∫

1

w=0

f ' ( x, w; Π ) dw .
(8)

The mean indicates the average degree of point clustering around location x. The variance of f'(x, w; Π)is also a basic statistic:=σ 2 ( x; Π )∫ { f ' ( x, w; Π ) − µ} dw .
1

2

w=0

(9)
Variance evaluates the stability in the degree of point clustering. If µ(x; Π) is large and σ2(x; Π) is small,
it is highly plausible that a large and dense cluster of points exist around x. Small µ(x; Π) and σ2(x; Π)
indicate that few points exist around x.
Both mean and variance concern the degree of point clustering at each location. In contrast,
6

range represents the spatial extent of point distribution around each location. It is defined as the averageof w weighted by f'(x, w; Π):∫ wf ' ( x, w; Π ) dw .
r ( x; Π ) =∫ f ' ( x, w; Π ) dw
1

w=0
1

w=0

(10)
Range r(x; Π) shows a large value when points are located away from x such as ΠD and ΠE in Figure 1.
Distributions ΠA and ΠB have small r(x; Π) because points are clustered around x.
The above three measures are defined based on f'(x, w; Π) that was introduced to avoid the
undefinability of f(x, w; Π). An alternative to resolve the problem is to define summary measures that canbe calculated directly from f(x, w; Π). The median of f(x, w; Π), which is denoted by m(x; Π), substitutes
for the mean of f'(x, w; Π). It is defined in an implicit form as

∫

1

w=0

,
ξ ( f ( x, w; Π ) , m ( x; Π ) ) dw =

1
2

(11)
where ξ( f(x, w; Π), s) is a binary function:

1 if f ( x, w; Π ) ≥ s.
otherwise0ξ ( f ( x, w; Π ) , s ) =(12)Median m(x; Π) ranges from zero to infinity. Similar to mean µ(x; Π), m(x; Π) indicates the average
degree of point clustering around location x. It becomes large where clusters are observed consistently
across different scales.
Extent e(x; Π) is an alternative to range r(x; Π) defined based on f(x, w; Π). This measureevaluates the relationship between w and f(x, w; Π) by extending the Spearman’s rank correlation
coefficient. Suppose we divide section [0, 1] into m subsections, in each of which a representative w is
defined at the center:352m − 1  1
Wm = 
w,
w,
w, ...,
w .
2m
2m
2m
 2m

(13)
The set of f(x, w; Π) values corresponding set Wm is given byΨm ={ψ 1 , ψ 2 , ψ 3 , ..., ψ m }
1
3
5
 
 
 

 2m − 1 .
= f  x,
w; Π  , f  x,
w; Π  , f  x,
w; Π  , ..., f  x,
w; Π  2m
  2m  2m  2m(14)
Let R(ψi; Ψm) be the rank of ψi in set Ψm in ascending order. The Spearman’s rank correlation coefficient7

between Wm and Ψm is
m

6∑ { R (ψ i ; Ψ m ) − i}

ρ (Tm , Ψ m ) =1 −

2

i =1

.

m3 − m(15)
Increasing m to infinity, we define e(x; Π):e ( x; Π ) =− lim ρ (Tm , Ψ m )m →∞m

6∑ { R (ψ i ; Ψ m ) − i}

= lim

2

−1

i =1

m3 − m

m →∞

.

(16)
Similar to the Spearman’s rank correlation coefficient, e(x; Π) ranges from -1 to 1. Extent e(x; Π) showsa large value when points decrease with an increase of w around x as seen in ΠA and ΠB in Figure 1. Points
in ΠD and ΠE increase with w, and consequently, e(x; Π) takes a small value.
The five measures proposed above evaluate the degree of point clustering from different
perspectives. Mean µ(x; Π) and median m(x; Π) focus on the average degree of point clustering acrossdifferent scales. Range r(x; Π) and extent e(x; Π) concern the spatial extent of point distribution at eachlocation. Variance σ2(x; Π) indicates the stability in the degree of point clustering with respect to the scaleof analysis. The maps of these measures reveal the spatial patterns of point distributions evaluated acrossdifferent scales, including important point clusters observed consistently over the entire range of scales.

2.2 Analysis of the relationship between multiple distributions of points
This subsection extends the proposed measures for the analysis of the relationship between two
and more distributions of points. The former consists of the evaluation of the change of a single
distribution and the comparison of two different distributions.
We first discuss the change of point distribution from Π to Π'. Our primary interest lies in theareas where points significantly increase or decrease. The difference between the distributions isrepresented by

F ( x, w; Π ') − F ( x, w; Π )f ∆ ( x, w; Π, Π ') =∫x∈SF ( x, w; Π ') dx + ∫x∈SF ( x, w; Π ) dx.

(17)
A variant with a finite range is defined asf '∆ ( x, w; Π, Π ') = 1 − e − f∆ ( , w;Π ,Π ') .
x

(18)
8

These functions indicate the increase in point density from Π to Π'. They permit us to evaluate the changeof point distribution by extending the five measures proposed in the previous subsection. The mean,variance, and range of f'∆(x, w; Π, Π') are given byµ=∆ ( x; Π , Π ' )

∫

1

w=0

f '∆ ( x, w; Π, Π ') dw ,(19)

∫ { f ' ( x, w; Π, Π ') − µ} dw ,2σ=
∆ ( x; Π , Π ' )

1

2

w=0

∆

(20)
and

∫ wf ' ( x, w; Π, Π ') dw ,r ( x; Π, Π ') =∫ f ' ( x, w; Π, Π ') dw
1

∆

w=0
1

∆

w=0

∆

(21)
respectively. Median m∆(x; Π, Π') is defined based on f∆(x, w; Π, Π') as

∫

1

w=0

ξ ( f ∆ ( x, w; Π, Π ' ) , m ( x; Π , Π ') ) dw = .
1
2

(22)
Extent e∆(x; Π, Π') is defined similarly according to Equation (16). Mean µ∆(x; Π, Π') and median m∆(x;
Π, Π') show large positive values where points greatly increase, while the decrease of points yields
negative values. Range r∆(x; Π, Π') and extent e∆(x; Π, Π') behave in the opposite way, i.e., they become
negative where points increase. Variance σ2∆(x; Π, Π') is always positive and behave independently ofthe other measures.
Comparison of two different types of points Π1 and Π2 can be performed in two different ways.
If a focus is on the absolute difference between Π1 and Π2, we can follow the above procedure by replacing
Π and Π' with Π1 and Π2. If an interest lies in the difference in the relative spatial patterns of Π1 and Π2,
we need to standardize each density function before comparison. The difference between Π1 and Π2 isrepresented asΠ 2 ) f ( x, w; Π 2 ) − f ( x, w; Π1 )f R∆ ( x, w; Π1 ,==F ( x, w; Π 2 )∫x∈SF ( x, w; Π 2 ) dx−

F ( x, w; Π1 )∫x∈SF ( x, w; Π1 ) dx
(23)

and

f 'R∆ ( x, w; Π1 , Π 2 ) = 1 − e − f R∆ ( x , w;Π1 ,Π 2 ) .

9

(24)
Summary measures are defined based on these functions. Mean, for instance, is given byµ R ∆=( x; Π1 , Π 2 )

∫

1

w=0

f 'R∆ ( x, w; Π1 , Π 2 ) dw .
(25)

The above measures are applicable to the analysis of the relationship between more than two
distributions of points. If we have numerous distributions, classification is also effective because we canfocus on the comparison between a smaller number of groups rather than the comparison between
individual distributions. Classification utilizes the integrals of the above measures over the whole area.
For instance, the integral of µ∆(x; Π1, Π2) is given byd µ (=
Π1 , Π 2 )∫x∈Sµ R∆ ( x; Π1 , Π 2 ) dx .
(26)

Measure dµ(Π1, Π2) works as a distance measure since it becomes large if Π1 and Π2 have different spatial
patterns. Calculating dµ(Π1, Π2) between every pair of distributions, we obtain a distance matrix. This
gives us a basis for classifying the distributions by using cluster analysis methods (Everitt et al. (2011);
Hennig et al. (2015)).

3. Numerical experiments
This section tests the performance of the proposed measures through numerical experiments.
Evaluation is based on whether they visualize the properties of point distributions clearly andappropriately, especially whether they are helpful for the detection of point clusters. Subsections 3.1 and3.2 treat a single distribution of points and the changes in a point distribution, respectively. Points are
distributed in a square region of side 1.0 in all the experiments. Every distribution consists of point clusters,
each of which follows a two-dimensional Gaussian distribution. We call the center of Gaussian
distribution a seed. Variables n and ρ denote the number of points and the standard deviation of Gaussian
distribution, respectively.

3.1 Visualization of a single distribution
Figure 3a shows a point distribution consisting of one large (CC) and four small (CR, CL, CT,
CB) clusters. The small clusters share the same spatial extent (ρ) with different densities (n/ρ2). Figure 3bd show the kernel density distributions of different window widths generated from the point distribution.
Figure 3e-i show the distributions the proposed measures. Mean µ and median m are large wherepoints are clustered while variance σ2, range r, and extent e are small in those areas. Measures µ and mhave rough surfaces in Figures 3e and 3f where the global pattern and point clusters are not clear. Measuresm and e, on the other hand, successfully displays all the small clusters. Median m is close to kernel surface
10K1, though the peaks of small clusters are located a little inside of their seeds. Range r looks similar to K3,
in both of which only the smallest cluster CR is not clearly recognizable. Extent e is close to K1 except its
rough contour lines.
Measures µ and m aim to indicate the density of points, while r and e intend to represent thesize of point clusters. Figure 3g shows that m successfully visualizes the difference in point density among
the small clusters. Measures r and e, on the other hand, do not fully achieve their purpose since theyvisualize denser clusters as larger ones. These measures are affected by point density at least to someextent.

CT

CL

CC

CC
CR
CL
CT
CB

CR

n
10000
200
400
600
800

ρ
0.30
0.10
0.10
0.10
0.10

Seeds
PointsCB(a) Point distribution(b) Kernel surface K1(c) Kernel surface K2(d) Kernel surface K3(e) Mean µ(f) Variance σ2(g) Median m(h) Range r

Small

Large

(i) Extent e

Figure 3 Point distribution and its summary measures.

Figures 4 and 5 focus on the effect of the size and density of point clusters on the performanceof the proposed measures. Four clusters in Figure 4a share the same size (ρ) but differ in point density(n/ρ2). Surfaces of µ and σ2 are both smoother than those in Figure 3 because the points are more tightly
clustered. Measures µ, σ2 and m successfully visualize the difference in point density among the clusters.
Measures r and e are expected to visualize all the clusters in the same size, which is not fully attained.
Though e yields better result than r, both visualize the clusters in different sizes.
Four clusters in Figure 5a consist of the same number of points with different spatial extents,
and consequently, they are different in both size and point density. Measures µ, σ2, and m correctly
visualize the difference in point density among the clusters, and e indicates the difference in cluster size.
Range r visualizes the four clusters almost in the same way, which implies that r tends to represent the
number of points rather than the spatial extent of point clusters.

11

CR
CL
CT
CB

CT

ρ
0.05
0.05
0.05
0.05

n
250
500
1000
2000

CR

CL

Seeds
PointsCB(a) Point distribution(b) Kernel surface K1(c) Kernel surface K2(d) Kernel surface K3(e) Mean µ(f) Variance σ2(g) Median m(h) Range r

Small

Large

(i) Extent e

Figure 4 Point distribution and its summary measures.

CR
CL
CT
CB

CT

CR

CL
CB

n
1000
1000
1000
1000

ρ
0.025
0.050
0.100
0.200

Seeds
Points(a) Point distribution(b) Kernel surface K1(c) Kernel surface K2(d) Kernel surface K3(e) Mean µ(f) Variance σ2(g) Median m(h) Range r

Large

Small

(i) Extent e

Figure 5 Point distribution and its summary measures.

Figures 6 and 7 consider the cases where one large and five small clusters are arranged rather
irregularly. The small clusters share the same size and density in each figure. Points are more tightly
clustered in Figure 6a than those in Figure 7a so that point clusters are more easily detectable in the former.
Mean µ and variance σ2 have rough surfaces where clusters are rather undistinguishable. Median m also
fails to visualize the clusters clearly in both figures; some are combined into one while others are coveredby the larger cluster. Measures r and e perform better than the other measures. Extent e displays small
12clusters more separately than r, especially in Figure 6.

nρLarge cluster 2000 0.30
Small clusters 500 0.10

Seeds
Points(a) Point distribution(b) Kernel surface K1(c) Kernel surface K2(d) Kernel surface K3(e) Mean µ(f) Variance σ2(g) Median m(h) Range r

Small

Large

(i) Extent e

Figure 6 Point distribution and its summary measures.

nρLarge cluster 2000 0.60
Small clusters 500 0.20

Seeds
Points(a) Point distribution(b) Kernel surface K1(c) Kernel surface K2(d) Kernel surface K3(e) Mean µ(f) Variance σ2(g) Median m(h) Range r

Small

Large

(i) Extent e

Figure 7 Point distribution and its summary measures.

3.2 Visualization of changes in a point distribution
This subsection discusses the changes in a point distribution. We assume unmovable points, i.e.,their change is limited to generation and disappearance. Other dynamic changes such as movement andintegration are out of scope of this paper.
13

Figures 8 and 9 treat the cases where only the cluster generation is observed. One large andthree small clusters are generated (red circles) where one large and five small clusters already exist (bluecircles). Measures m∆, r∆, and e∆ successfully detect the generation of large cluster in both figures, while
it is not clear in µ∆ and σ∆2. Small clusters are also clear in e∆ and r∆, although the latter to a lesser extent
in Figure 9. Small clusters in m∆ are detectable but their location is different from their actual ones.
Measures µ∆ and σ∆2 fail to visualize the small clusters clearly except the one at the lower-left corner inFigure 8.

nρLarge clusters 800 0.20
Small clusters 200 0.10
Seeds Points
Remained
Generated(a) Point distribution(b) Kernel surface K1(c) Kernel surface K2(d) Kernel surface K3(e) Mean µ∆(f) Variance σ∆2(g) Median m∆(h) Range r∆

Small

Large

(i) Extent e∆

Figure 8 Cluster generation in a point distribution. The distribution originally consists of one large andfive small clusters (blue circles). One large and three small clusters are newly generated (red circles).

nρ

Large clusters 800 0.30
Small clusters 200 0.15
Seeds Points
Remained
Generated(a) Point distribution(b) Kernel surface K1(c) Kernel surface K2(d) Kernel surface K3(e) Mean µ∆(f) Variance σ∆2(g) Median m∆(h) Range r∆

14

Small

Large

(i) Extent e∆

Figure 9 Cluster generation in a point distribution. The distribution originally consists of one large andfive small clusters (blue circles). One large and three small clusters are newly generated (red circles).

Figures 10 and 11 treat the cases where both cluster generation and disappearance occur. A point
distribution initially consists of two large and eight small clusters. One large and three small clusters
disappear, while the same number of clusters are generated. Measures µ∆ and m∆ become positive (redshades) while r∆ and e∆ become negative (navy shades) where clusters are generated. Measures r∆ and e∆
detect point clusters almost successfully except the cases where cluster generation and disappearance
closely occur as observed at the center in both figures. Median m∆ is similarly effective in Figure 10, but
tends to locate point clusters inward in Figure 11 where points are more dispersed in each cluster.
Measures µ∆ and σ∆2 are not effective due to their rough distributions.

ρnLarge clusters 800 0.20
Small clusters 200 0.10
Seeds Points
Remained
Generated
Vanished(a) Point distribution(b) Kernel surface K1(c) Kernel surface K2(d) Kernel surface K3(e) Mean µ∆(f) Variance σ∆2(g) Median m∆(h) Range r∆

Negative

Positive(i) Extent e∆

Figure 10 Cluster generation and disappearance in a point distribution. The distribution originallyconsists of two large and eight small clusters (blue and yellow circles). One large and three small
clusters disappear (yellow circles), while one large and three small clusters are generated (red circles).

15

ρ

n

Large clusters 800 0.30
Small clusters 200 0.15
Seeds Points
Remained
Generated
Vanished(a) Point distribution(b) Kernel surface K1(c) Kernel surface K2(d) Kernel surface K3(e) Mean µ∆(f) Variance σ∆2(g) Median m∆(h) Range r∆

Negative

Positive(i) Extent e∆

Figure 11 Cluster generation and disappearance in a point distribution. The distribution originallyconsists of two large and eight small clusters (blue and yellow circles). One large and three small
clusters disappear (yellow circles), while one large and three small clusters are generated (red circles).

4. Application to real data
This section applies the proposed measures to the analysis of real data. Subsection 4.1 evaluates
the change of the distribution of convenience stores in Tokyo, Japan. Subsection 4.2 analyzes the change
in the number of children in the Greater Tokyo Area by using spatially-aggregated census data. Subsection
4.3 classifies the distributions of commercial facilities in Chiba, Japan, by using the distance measures
proposed in Subsection 2.2.

4.1 Distribution of convenience stores in Tokyo
Figures 12a and 12b show the distributions of convenience stores in Tokyo listed in thetelephone directory of the NTT TownPage cooperation. Stores had increased from 2070 in 1990 to 3382
in 2000, during which period 1949 new stores opened (Figure 12c) and 637 stores were closed (Figure12d). Closed stores are distributed rather uniformly while new stores are clustered in some areas.
Figure 12e-Figure 12g show the distribution of calculated measures. We omit µ∆ and σ2∆ due tothe space limitations. Measures m∆ and r∆ are positive while e∆ is negative over the whole area, because
stores had increased as a total during this period. Measures m∆ and r∆ clearly indicate the global patternof new stores, i.e., new stores opened more in the central area than in the surrounding area. Extent e∆reveals smaller clusters of new stores that are not easily recognizable in Figure 12c. Figure 13 shows the
distributions of new and closed stores overlaid on the distribution of e∆. A close relationship exists
between e∆ and new stores in Figure 13a. The relationship, on the other hand, is not clear in Figure 13b.
16

One reason for this is that closed stores are fewer than new ones. The spatial pattern of new stores masksthat of closed stores. Another reason is that closed stores are distributed rather uniformly as seen in Figure
12d. They do not contain large clusters that greatly reduce the value of e∆.

(a) Stores in 1990(b) Stores in 2000(c) Opened until 2000(e) Median m∆(f) Range r∆(g) Extent e∆(d) Closed until 2000

Negative

Figure 12 Convenience stores in Tokyo, Japan, in 1990 and 2000.

17

Positive(a) New stores(b) Closed stores

Negative

Positive

Figure 13 New and closed convenience stores in Tokyo, Japan, between 1990 and 2000.

4.2 Distribution of children in the Greater Tokyo Area
The proposed measures are applicable to point data aggregated by spatial units such as census
tracts and zip codes. This subsection analyzes the change in the number of children in the Greater Tokyo
Area, Japan (Figure 14a). Population had increased continuously from 1995 to 2005 in this area due tocentralization, while the birth rate had gradually decreased. Our interest lies in the spatial pattern of thechange in the distribution of children.
Figure 14b shows the change in the number of children from 1995 to 2005 aggregated at thecity level in the census data. Though we can see that children had increased in the central and somesurrounding areas, local variation of the distribution prevents us of grasping the overall pattern of thechange. Figure 14c-e show the distribution of calculated measures. Similar to Figure 12, m∆ and r∆ arepositive while e∆ is negative over the whole area in Figure 14, which implies that children had increased
as a whole during this period. The scale of spatial pattern revealed by m∆, r∆ and e∆ changes from global
to local in this order. Median m∆ indicates that children had increased more in the central area than in its
surroundings. Range r∆ adds local clusters in the north area, represented as cities such as Maebashi,
Utsunomiya and Mito. Extent e∆ visualizes the local clusters more clearly, and detects even smallerclusters including Oyama, Tsuchiura, Choshi and Mobara. Extent e∆ also reveals the detailed spatial
pattern of children increase in the central area of Tokyo.

18

Negative(a) Greater Tokyo Area(b) Change in the number of children

Utsunomiya

Utsunomiya

Utsunomiya

Hitachi
Maebashi Kiryu
Ashikaga
Isesaki

Hitachi
Maebashi Kiryu

Tochigi

Ashikaga
Mito

Oyama

Sano
Kumagaya

Isesaki

Mito

Oyama

(c) Median m∆

NaritaChoshiToganeTateyama

Kashima
Katori

NaritaChoshiMobaraTsuchiura

Kuki
Chichibu

Katori

Narita

Mito

Oyama

Sano
KumagayaKashimaChichibu

Katori

IsesakiTsuchiura

Kuki

KashimaChichibu

Tochigi

Sano

Hitachi
Maebashi Kiryu
Tochigi
Ashikaga

Kumagaya

Tsuchiura

Kuki

Kisarazu

Positive

ChoshiToganeKisarazu

MobaraToganeKisarazu

MobaraTateyama

Tateyama(d) Range r∆(e) Extent e∆

Figure 14 Change in the number of children between 1995 and 2005 in the Greater Tokyo Area. Whitelines in Figure 14a indicate railway lines.

4.3 Classification of the distributions of commercial facilities
This subsection applies the proposed measures to the classification of the distributions ofcommercial facilities in Chiba, Japan. The data are based on the list of shops and restaurants provided by
the NTT TownPage cooperation. We classify nineteen types of commercial facilities by their spatial
patterns. The k-medoids method (Everitt et al. (2011); Hennig et al. (2015)) uses dm, dr, and de as distancemeasures in classification.
Figure 15 shows the distributions of commercial facilities classified based on de where k=4. We
omit the other results since dm and dr yielded similar classifications. Group G1 is the largest groupcontaining convenience stores, vegetable stores and noodle restaurants that provide commodities andreasonable food. Shops are widely spread in both residential and commercial areas to satisfy the frequent
demand of residents and workers. Facilities in G2 are rather clustered around railway stations. Group G3
19

is smaller and more tightly gathered around railway stations. People visit G2 and G3 stores less frequently
such as pubs, bars, cosmetic shops and book stores. The stores form clusters around railway stations tooffer opportunities for comparison and multipurpose shopping. Group G4 consists of gas stations andsupermarkets that are dispersed over the whole area. Unlike other commercial facilities, gas stations andsupermarkets are not found around railway stations. They do not need to be close to railway stations where
the land price is relatively high because people usually visit these facilities by car.
The above classification looks generally reasonable. Each group consists of commercial
facilities that share similar spatial patterns as seen in Figure 15, which reflects the properties of goods andservices provided by the facilities.

N

N

00

Convenience stores
Vegetable stores
Drug stores
Liquor shops
Chinese noodle restaurants
Japanese noodle restaurants
Sushi restaurants
Laundry services

370
195
329
338
189
275
306
437

0

5km
5km

Japanese restaurants
Italian restaurants
Pubs
Bars
Cram schools(a) Group G1

853
165
113
669
247

(b) Group G2

N00

Cosmetic shops
Book storesJapanese clothing stores
Banks

173
107
102
200

5km

N

5km
5km

0

Gas stations
Supermarkets(c) Group G3

215
148

5km

(d) Group G4

Figure 15 Classification of commercial facilities in Chiba, Japan.

5. Concluding discussion
A new method of visualizing point distributions was proposed. Measures µ, σ2, m, r, and e andtheir extensions summarize the properties of point distributions evaluated over the entire range of scales.
The measures were developed for exploratory purposes, i.e., they are expected to help us building research
hypotheses. Visual analysis of these measures should be followed by confirmatory analysis that tests the20significance of the hypotheses, which is out of scope of the paper.
Measures µ and σ2 are extensions of descriptive measures widely used in general statistics.
Numerical experiments, unfortunately, showed that µ and σ2 are not necessarily effective for visual
analysis. Rough surfaces of these measures often too much emphasize the local variation of point
distributions so that the global pattern and point clusters are not easily recognizable. The strength of µ
and σ2, however, should not be underestimated. Their definition is clear and interpretation isstraightforward because they are based on popular measures in general statistics. We should resolve theshortcomings of these measures to improve their performance.
The other three measures showed good results in both numerical experiments and realapplications. They permit us to grasp the overall spatial pattern as well as detect point clusters and changes
in point distributions. This saves our time of visual analysis with continuously changing the spatial scale.
The maps of the three measures visualize the spatial patterns of different scales. Median m outlines theglobal pattern of point distributions, while r and e rather focus on the local details.
A weakness of m is that it tends to locate point clusters inward from their true locations as seenin Figures 8 and 9. It occurs primarily due to the edge effect, i.e., calculation of f(x, w; Π) and f'(x, w; Π)does not consider the points outside the study region that can potentially exist. All the measures sufferfrom the edge effect, though it was not evident in r and e in numerical experiments. We should try edge
correction methods proposed in spatial statistics and image processing such as periodic edge correction
and reflection correction methods in future research (Stoyan and Stoyan (1994); Wiegand and Moloney(2013)).
Extent e was found to be effective for the detection of point clusters. The primary objective ofe, however, is to represent the spatial extent of point distribution around each location. This has not been
fully attained because e is affected by the relative density of points in the neighborhood. Further
improvement is necessary for e to be more independent of point density.
This paper assessed the performance of the proposed measures by visually comparing their
spatial pattern with point distributions. We chose this approach to grasp the properties of the measuresfrom various perspectives. Further evaluation includes cartographic experiments, questionnaire survey,
and quantitative assessment. Experiments should test whether analysts can correctly understand the spatial
pattern of point distributions and detect point clusters in the maps of the proposed measures. The
performance of the measures can be assessed through questionnaire survey and the quantitative analysisof the results.
Dynamic changes such as movement, integration, and division of points were out of scope of
this paper. However, recent development of tracking technology such as GPS, laser and video tracking
systems permits us to obtain detailed point data that changes continuously. We should test the performanceof the proposed measures in the analysis of dynamic data as well as develop further measures that cancapture the properties of dynamic changes more appropriately.
21

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Appendix
Mean and variance are basic measures of continuous functions. The mean of density function
f(x, w0; Π) is defined as

1 w1
Mean
=
f ( x, w0 ; Π ) dw0 .
 f ( x, w0 ; Π )  lim
w1 →∞ w ∫w0 = 0
1(A1)
The mean defined above, however, is not meaningful since it is equal to one almost everywhere andundefined at the locations of points. The former occurs because f(x, w0; Π) is almost independent oflocation x when w0 is large enough, i.e.,f ( x, w0 ; Π ) ≈

1
.
w0
(A2)

22

Substituting this approximation into Equation A1, we obtain

1 w1=f ( x, w0 ; Π ) dw0
Mean f ( x, w0 ; Π )  lim
w1 →∞ w ∫w0 = 0
1
w1
1
1
lim ∫
dw0
w1 →∞ w ε → 0 + w0 =ε w
1
0

≈ lim

.

log w1
w1 →∞
w1

= lim
=1

(A3)
The second problem occurs because f(x, 0; Π) is the delta functions divided by n at the locations of points,i.e.,f ( x, 0; Π ) = 0 ( ∀i : x ≠ z i )
(A4)
and

1
∫ f ( x, 0; Π ) dx =x

(A5)
Density function f(x, w0; Π) is not integrable at w0=0 and thus its mean is not definable at the locations ofpoints.
The variance of density function f(x, w0; Π) is defined as
2
1 w1
f ( x, w0 ; Π ) -f ( x; Π )} dw0{∫
w1 →∞ w w0 = 0
1

Var= f ( x, w0 ; Π )  lim

2
2
1 w1
= lim
{ f ( x, w0 ; Π )} dw0 − { f ( x; Π )}w1 →∞ w ∫w0 = 0
1

.

(A6)
Substituting approximations A2 and A3 into the first term of the above equation, we obtain
2
2
1 w1
Var
,
;
d
;
x
x
=
Π
−
Π
f
w
w
f
 f ( x, w0 ; Π )  lim
(
)
(
)
{
}
{
}
0
0
w1 →∞ w ∫w0 = 01

1 w1 1dw0 − 1
w1 →∞ w ∫w0 = 0 w2
1
0

= lim

w1

1
1
=
− lim lim   − 1
w1 →∞ w ε →0 + w
1 0 ε1−1
w1 →∞ ε →0 + w ε
1= lim lim

23

.

(A7)
The first term of the right hand is undefinable, and consequently, Var[f(x, w0; Π)] is undefinable almost
everywhere.

24