International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 41
Box-Counting Dimension
of Fractal Urban Form:
Stability Issues and Measurement Design
Shiguo Jiang, Department of Geography, The Ohio State University, Columbus, OH, USA
Desheng Liu, Department of Geography and Department of Statistics, The Ohio State
University, Columbus, OH, USA
ABSTRACT
The difficulty to obtain a stable estimate of fractal dimension for stochastic fractal (e.g., urban form) is an
unsolved issue in fractal analysis. The widely used box-counting method has three main issues: 1) ambiguities in setting up a proper box cover of the object of interest; 2) problems of limited data points for box sizes;
3) difficulty in determining the scaling range. These issues lead to unreliable estimates of fractal dimensions
for urban forms, and thus cast doubt on further analysis. This paper presents a detailed discussion of these
issues in the case of Beijing City. The authors propose corresponding improved techniques with modified
measurement design to address these issues: 1) rectangular grids and boxes setting up a proper box cover of
the object; 2) pseudo-geometric sequence of box sizes providing adequate data points to study the properties
of the dimension profile; 3) generalized sliding window method helping to determine the scaling range. The
authors’ method is tested on a fractal image (the Vicsek prefractal) with known fractal dimension and then
applied to real city data. The results show that a reliable estimate of box dimension for urban form can be
obtained using their method.
Keywords:
Box-Counting Method, Fractal Dimension, Pseudo-Geometric Sequence, Scaling Range,
Sliding Window Method, Urban Form
1. INTRODUCTION
Fractal dimension is a useful landscape metric in that it can capture the irregularity and
complexity of landscape patterns (Hargis et
al., 1998; Herold et al., 2005; Imre & Bogaert,
2004; Pincheira-Ulbrich et al., 2009). The
usefulness of fractal in characterizing urban
form is shown by various researchers (Batty,
1985; Batty & Longley, 1994; Benguigui et
al., 2000; Frankhauser, 1994; Thomas et al.,
2008; Thomas et al., 2007). Mostly inspired
by the studies on coast line (Mandelbrot, 1967;
Richardson, 1961), early studies of urban form
mainly focus on city boundaries (Batty & Longley, 1987; Batty & Longley, 1988; Longley &
Batty, 1989a, 1989b). Later fractal urban form
studies extend to urban surface, i.e., urban land
use (Batty & Longley, 1994; Benguigui et al.,
2000; Shen, 2002; Thomas et al., 2008; White &
DOI: 10.4018/jalr.2012070104
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
42 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
Engelen, 1993). It seems to be widely accepted
that urban boundary as well as urban surface are
both fractals at least in certain stages (Batty &
Longley, 1994; Batty & Xie, 1996; Benguigui
et al., 2000).
Fractal analysis of urban form relies heavily on the calculation of fractal dimension – the
main scaling exponent to describe a fractal
set. The fractal dimension of a deterministic
fractal (e.g., the Vicsek fractal) can usually be
estimated analytically. However, the dimension
of a stochastic fractal (e.g., urban form) needs
to be estimated numerically. Three numerical
methods are popular in the research community:
the perimeter-area relation method (Batty &
Longley, 1988; Batty & Longley, 1994), the
area-radius method (Frankhauser, 1994; White
& Engelen, 1993), and the box-counting method
(Benguigui et al., 2000; Lu & Tang, 2004; Shen,
2002). Due to its simple algorithm and equal
effectiveness to point sets, linear features, areas,
and volumes, the box-counting method enjoys
a wide popularity across various disciplines
such as physics (Lovejoy et al., 1987), earth
sciences (Walsh and Watterson 1993), biology
(Foroutan-pour et al., 1999), ecology (Halley et
al., 2004), and urban studies (Benguigui et al.,
2000; Feng & Chen, 2010; Lu & Tang, 2004;
Shen, 2002; Verbovsek, 2009). In the case of
urban studies, box-counting dimension is an
indicator of compactness for the distribution
of built-up areas. Despite its popularity in the
research community, several issues of the boxcounting method remain unsolved.
The first issue is concerned with the ambiguities in setting up a proper box cover of the
object. This issue has two aspects. The first
aspect is related to the shape of the grids and
boxes in the box-counting method. Theoretically, the shape of the grid and box does not
influence the estimate of the box dimension.
However, in practice, the object does not have
infinite detail, thus different covering schemes
may lead to different estimates. For simplicity
of practical calculation, the conventional boxcounting method is performed based on square
boxes (Shen, 2002; Verbovsek, 2009). Although
the square box is widely used, it is not necessarily
the only caliber. The choosing of square boxes
cannot always efficiently cover the object. The
second aspect is more related to the analysis of
urban form. Due to the scale-free characteristics
of urban form, it is difficult to find the exact
boundary of a city and there is no agreement
on a theoretically correct definition of urban
boundary (Benguigui et al., 2000; Berry et al.,
1968). In urban studies, we often need to make
a subjective decision about the city boundary.
As we will illustrate later, the fractal dimension
of the same city changes with the study area.
Therefore, the box dimension of a city should
be given along with the study area. We should
be cautious in comparing the fractal dimension
of different cities as the calculated dimension
may not be comparable.
The second issue in the box-counting
method is the problem of dimension estimation due to the limited number of data points
for regression (Pruess, 1995). The issue comes
from the use of a dyadic sequence. Box size
approaches zero quickly and thus provides only
a few data points for regression. Take the unit
box size as level one, and the box size changes
as 1/2, 1/4… In the 10th level of box division,
the box size is 1/29, and the total number of
boxes is 218. This is the lower bound for box
division in most literature, which provides 10
data points of box size (Benguigui et al., 2000;
Grau et al., 2006; Lu & Tang, 2004). Besides
the dyadic sequence, other choices have been
proposed, such as the odd number sequence
(Bisoi and Mishra 2001; Chen et al., 1993),
modified arithmetic sequence (Buczkowski
et al., 1998; Foroutan-pour et al., 1999), and
modified dyadic sequence (Shen, 2002).
The third issue is the difficulty in determining the scaling range of box sizes over
which the fractal analysis is applied (Saucier &
Muller, 1998). It is widely accepted that fractal
properties only exist on a certain scaling range
for stochastic fractals (Goodchild, 1980; Lam &
Quattrochi, 1992; Roy et al., 2007). However,
there is no generally agreed method to specify
the upper and lower bounds of the scaling range
(Foroutan-pour et al., 1999; Huang et al., 1994;
Liebovitch & Toth, 1989). Most studies suggest
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 43
that the largest box size used should be no larger
than one half or one fourth of the shorter side
of the image. When the box size is large, all
the boxes are likely to be occupied, producing
a slope of 2 on the scatter plot (Foroutan-pour
et al., 1999; Shen, 2002). However, the empirical choice of this upper bound has no sound
justification. As to choosing the lower bound
of the box size, there are mainly five methods
in the literature.
1. Use the image resolution as the lower
bound without considering the scale break
(Benguigui et al., 2000).
2. Eliminate or add the data points of the
box size one by one to observe the best fit
(Foroutan-pour et al., 1999).
3. Use a sliding window method to calculate
the local fractal dimension (Brewer & Di
Girolamo, 2006; Dubuc et al., 1989). For
example, we can compute the fractal dimension for the first 5 values of box size, then
for 2 to 6, and so forth. Thus we can find
the lower bound that will result a stable
estimation of fractal dimension.
4. Use the condition ds / dr → 0 to get the
cutoff lower bound of the box size (Roy et
al., 2007). Here, s is the standard deviation
from a linear regression equation fitted to
log(N) versus log(r) with data for r < rcutoff
sequentially excluded. However, this
method is not well justified because the
best fit does not necessarily guarantee the
best estimate (Huang et al., 1994).
5. Use configuration entropy (Andraud et al.,
1997) to determine the lower bound of the
box size in multifractal analysis (Dathe et
al., 2006; Tarquis et al., 2006).
The three issues discussed are common in
numerical analysis of stochastic fractals and lead
to an unstable estimate of fractal dimension. For
example, the calculations of the same fractal
set by different people often result in different
fractal dimensions (Huang et al., 1994; Hunt,
1990; Li et al., 2009; Sarkar & Chaudhuri, 1994).
Although various methods have been proposed
to search for reliable calculations (Buczkowski
et al., 1998; Foroutan-pour et al., 1999; Li et
al., 2009; Sarkar & Chaudhuri, 1994; Walsh &
Watterson, 1993), there still lacks a systematic
exploration of the problems. In this paper, we
thoroughly address these issues and provide our
solutions. By using rectangular grids and boxes
with efficient cover of the object, we propose
a method to estimate the fractal dimension of
urban form. We develop an algorithm to create
a pseudo-geometric sequence which provides
enough data points for the dimension estimation. We also develop a generalized sliding
window method to determine the upper and
lower bound of the scaling range (box size).
Our method is first tested on a fractal image
(the Vicsek prefractal) with known dimension.
It is then applied in analyzing the fractal urban
form of Beijing City.
In most literature, only the point estimate
of fractal dimension is given. However, empirical determination of a particular quantity is
not error free (Taylor, 1997). The range of the
measurement quantity should be given along
with the point estimate. In this paper, we will
compare the results of our method with those
from traditional method in two aspects: the
point estimate and the measurement range (95%
confidence interval).
The rest of this paper is organized as follows. Section 2 is a simple introduction of the
data used in this study. Section 3 introduces
the box-counting method and our modification.
Section 4 presents the results of the calculation.
Section 5 discusses the results as well as the
problems in fractal urban analysis. Section 6
concludes the paper. For convenience, we will
treat urbanized area, city, and urban form as
interchangeable terms in this paper.
2. DATA
2.1. Prefractal Image with
Known Dimension
We estimate the fractal dimension of the Vicsek
prefractal to test our method. The Vicsek fractal
(also known as Vicsek snowflake or box fractal;
Vicsek, 1983) is generated by decomposing a
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
44 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
basic square into nine smaller squares in the
3-by-3 grid. The basic square is called the initiator (Figure 1(a)). Number the nine squares as 1,
2…9, from left to right and from top to bottom.
Squares 2, 4, 6, 8 are removed, and squares 1,
3, 5, 7, 9 are left. The result of this procedure is
also called the generator (Figure 1(b)), because
it specifies a rule that is used to generate a new
form. Repeat the process recursively for each of
the five remaining sub-squares and the Vicsek
fractal is obtained at the limit of this procedure.
In practice, due to the limitation of computer
power and resolution, the fractal object is usually
generated using finite number of iterations, say
the nth iteration, the result of which is called
prefractal (Feder, 1989). For convenience,
we denote the size of the squares in the last
iteration (i.e., nth) as unit square whose size is
1 (unit one). In remote sensing literature, the
unit square is generally referred to as image
pixel. The size of the initiator is thus 3n. Figure
1 shows the result of three iterations. The size
of the initiator is 33 = 27.
Vicsek fractal is a typical deterministic
fractal, which is created by a rule of some sort.
Properties like the Hausdorff dimension of a
deterministic fractal can be accounted accurately. The Hausdorff dimension of the Vicsek
fractal is log 5 / log 3 ≈ 1.4649 .
2.2. Real Data of Beijing City
As this paper aims at the measurement methods
in fractal analysis, we only chose two Landsat-5
TM (WRS-2 Path 123 Row 32) images for
demonstration purposes. The acquisition dates
for the two images are 3 October 1984 and 29
October 1999. The two images are rectified to
a common UTM coordinate system based on
1:50000 topographic maps of Beijing (provided by the National Bureau of Surveying and
Mapping). The two images are well aligned
with RMSE ≤ 15 meters. Other reference data
include an administrative boundary map of
Beijing and a district map of Beijing (both
provided by the Beijing Municipal Government
Planning Commission). Unsupervised classification is performed first, and is improved by
intensive human editing aided by supplemental
data and field investigation. The final map is
validated using land use maps (prepared by the
Beijing Municipal Government Planning Commission) and aerial photos with an accuracy of
95.5%. This highly reliable urban map product
is used in our study (Figure 2).
As discussed before, it is difficult to find
the exact boundary of a city. In practice, we
can set the study area based on administrative
boundary, development of the city, and road
system, etc. The ring road system of Beijing
serves as a reference to boundary for the city
in different periods. The old Beijing before the
1950s was confined in walls around the city. In
the 1950~1960s, the walls were demolished and
Ring Road 2 was built. With the expansion of
the city, Ring Road 3 (completed in 1994), Ring
Road 4 (2001), Ring Road 5 (2003), and Ring
Road 6 (2009) have been built in the past fifty
years. Three study areas are chosen based on
ring road system, the administrative boundaries, and the development of the city (Figure 2).
Although Ring Road 4, 5, and 6 were completed
Figure 1. Three iterations of the Vicsek fractal: (a) the basic square and initiator; (b) the generator and 1st iteration; (c) 2nd iteration; (d) 3rd iteration
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 45
Figure 2. Maps of Beijing City: (a) ring and radial road system: 2~6, ring roads; (b), (c) land
cover map in 1984, 1999: 1, region inside the Ring Road 5; 2, urbanized area; 3, region inside
Ring Road 6
after 1999, part of them had already been in
use before the completion of the whole rings.
Therefore, they provide a good reference limit
to define the study area.
Study Area 1: It is the main part of the region
bounded by Ring Road 5. It includes four
central districts (Dongcheng, Xicheng,
Xuanwu, Chongwen) and the main parts
of three inner suburban districts (Haidian,
Chaoyang, Fengtai). This is the densely
developed and closely connected area of
Beijing before 1990s. Study area 1 has a
size of 29*25 =725km2.
Study Area 2: It is based on the traditional
“Cheng Baqu” (i.e., eight main districts
of Beijing), including four central districts
and four inner suburban districts of Beijing:
Dongcheng, Xicheng, Xuanwu, Chongwen,
Haidian, Shijingshan, Chaoyang, Fengtai.
We regard this study area as the urbanized
area of Beijing. Study area 2 has a size of
40*39 =1560km2.
Study Area 3: It is the region bounded by Ring
Road 6. Four outer suburban cities (Mengtougou, Fangshan, Daxing, and Shunyi) are
included in addition to study area 2. This
is the metropolitan area of Beijing. Study
area 3 has a size of 57*54 =3078km2.
3. BOX-COUNTING METHOD
AND MODIFICATION
3.1. Traditional BoxCounting Method
“Box-counting” is a sampling process to find
the complexity, irregularity, and heterogeneity
of the object of interest at different scales (Barnsley, 1988). It is a numerical method developed
independently by many authors from a description by Mandelbrot (Mandelbrot, 1983; Pruess,
1995). Here we present an intuitive and descriptive definition of the box-counting method.
For a precise technical definition, the reader is
urged to read the book by Falconer (2003). The
basic procedure is to systematically lay a series
of grids composed of boxes in decreasing size
over an image and then record data (counting)
for each successive box size. The objective is
to find fractal dimensions characterizing the
spatial structure of the object. Box-counting
dimension is the dimension calculated based
on box-counting method.
The procedures of box-counting method
can be explained as follows (Figure 3(a)).
First, use a square grid of only one box to
cover the city image so that the urbanized area
of the city is well covered by the grid. The side
length of the box, r1, equals the side length of
the basic grid, L. Count the number of distinct
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
46 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
Figure 3. Box-counting methods with different covering grids: (a) square grid with square
boxes; (b) rectangular gird with square boxes, L2 = kL1, k is integer; (c) rectangular grid with
square boxes, L2 = αL1, α is rational number; (d) rectangular grid with rectangular boxes.
(Note: Method (a)-(c) are commonly used in previous literature although they are not consistent
with the minimum cover rule of box-counting dimension. In our paper, we used method (d) which
provides minimum cover of the object. The box dividing is for demonstration purpose. It does
not reach the maximum number of dividing or the pixel resolution level, therefore the boxes does
not necessarily either filled completely or completely empty)
occupied boxes N1 that can cover the city completely. Here, as we only have one box, N1 = 1.
Second, divide the grid by 2×2 to get a
grid composed of four boxes (quadrants) with
the size r2 =L/2. Count the number of occupied
boxes N2. Continue the division process for n
iterations. Record the corresponding Nn. As
the box size becomes smaller and smaller, the
total area Nnrn2 of those occupied boxes should
approach the actual urban area closely.
The box-counting dimension is then estimated as:
D = lim
rn → 0
log N n
log(1 / rn )
. (1)
In practice, the dimension is estimated as
the slope of the scatter plot of logNn against
log(1/rn). Thus, we estimate the following
linear equation,
ln N (r ) + D ln(1 / r ) + b1 = 0. (2)
Equation (1) defines the capacity dimension (Nayfeh, 1995; Peitgen, 1986; Weisstein,
2003). In practice, we usually start with the
smallest box size and increase the box size by
sequentially aggregating boxes together.
In previous literature, three versions of grid
cover and box division method are generally
found as follows.
1. Use a square grid to cover the object (Shen
2002). The side length of the grid,
L = 2n ε, where ε is the image resolution
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 47
(Figure 3(a)). The smallest box size equals
the size of the image resolution, ε.
2. Use a rectangular grid to cover the object,
and the length and width of the grid are of
integer ratio (Saa et al., 2007). Let the width
of the grid, L1 = 2n ε, then the length of
the grid L2= kL1, where k is an integer
number (Figure 3(b)).
3. Use a rectangular grid to cover the object,
and the length and width of the grid are of
any rational number (Lu and Tang 2004).
Let the width of the grid L1 = 2n ε, then
the length of the grid L2 = αL1, where α
is a rational number (Figure 3(c)).
Although the methods in Figure 3 (a), (b),
(c) are widely used, they violate the minimum
cover rule of box-counting dimension (Falconer,
2003). The minimum cover rule means that
when we measure the fractal dimension, we
should use minimum number of open box to
cover the object. In the case of box-counting
dimension, we should use minimum number of
rectangular boxes to cover the object. Strictly
speaking, these methods do not provide estimate
of box-counting dimension per se.
3.2. Improved Box-Counting
Method with Modified
Measurement Design
Our improved box-counting method can be
illustrated in Figure 4. Detailed explanation is
introduced in the following three sub-sections
3.2.1-3.2.3.
3.2.1. Grid Dividing with
Rectangular Boxes
In order to overcome the minimum cover
problem in traditional box-counting method,
we use rectangular grid and boxes to cover the
object (Figure 3(d)). The image preparation and
processing is shown as follows.
1. If the source map of the object is of vector
format, we can encompass it with a minimum rectangular grid, called basic grid
(initiator). Suppose the short side length
of the grid is L. Divide the grid into amax
by amax rectangular boxes whose size is L/
amax. Here, amax is the number of boxes per
row/column. In practice, amax is usually
determined by the software threshold of
record storage. In our processing, the Create
Fishnet ArcToolbox in ESRI ArcGIS is
used, which can be accessed through ArcToolbox → Data Management Tools →
Feature Class → Create Fishnet (ESRI,
2010). The limit is arrived when the basic
grid is divided into 211 × 211 boxes, i.e.,
amax = 211 = 2048.
2. If the source map of the object is of raster
format, there are two equivalent methods
to process the map. The first method is to
convert the raster image into a vector image
and then process it as in (1). The second
method is to resample the raster image into
amax by amax pixels of rectangular shape.
3.2.2. Pseudo-Geometric
Sequence of Box Size
The definition of box dimension does not necessarily require a dyadic sequence of box size.
Dyadic sequence is a special case of the geometric sequence rn = r1b −n +1 with factor b =
2, where rn is the box size in the nth division,
r1 is the largest box size. The use of a geometric sequence (sometimes it is referred to as
logarithmic sequence) is well justified in that
fractal is a hierarchy with cascade structure.
The data points based on a geometric sequence
are distributed evenly in the log-log plot. In
general, we can use a geometric sequence of
any factor, for example, b = 1.05, 1.10, 1.15 …
1.95, 2.00, 3.00, etc. When non-integer factors
are used, rounding is applied to the box size,
resulting in pseudo-geometric sequences.
The general algorithm to derive the
pseudo-geometric sequence is as follows. For
consistency, we use a1 to represent amax.
Suppose the geometric sequence has n
elements thus producing n grids. Grid 1 is
obtained by dividing the basic grid into a1 × a1
boxes (for vector map) or by resampling the
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
48 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
Figure 4. Flow chart of the improved box-counting method
image into a1 × a1 pixels (for raster data). Each
box can be treated as a unit box, or an abstract
pixel in general sense. The size of the unit box
is,
r1 =
L
, a1
(3)
where
L is the side length of the basic grid,
a1 is the number of boxes per row/column, in
our case, a1=2048.
Grid i is obtained by grouping a1 × a1 unit
boxes into ai × ai blocks. Each block is composed of K i × K i unit boxes which are then
aggregated into one large box with size ri = Kir1
(Figure 5). Partial blocks at the boundary (if
there are any) are padded with empty boxes
(Figure 5 (c)).
Ki and ai are determined by Equation (4)
and (5),
K i = round (b i −1 ) ≤ a1, (4)
 a 
ai = ceil  1 ,  K i 
(5)
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 49
Figure 5. Generation of example grids by aggregating unit boxes at different levels: (a), grid 1
with unit boxes, a1 = 8, K1 = 1; (b) grid 2, a2 = 4, K2 = 2; (c) grid 3, partial blocks at boundary
are padded with empty boxes (dashed boxes), a3 = 3, K3 = 3; (d) grid 4, a4 = 2, K4 = 4. (Note:
For the meanings of notations ai, Ki, please refer to the notations below Equation (5).)
where
Ki is the number of unit boxes per row/column
in a block to be aggregated at level i,
ai is the number of blocks per row/column with
size ri = Kir1,
b is the exponent factor, e.g., 1.05, 1.1 … 2, 3 …
i is the division level, i = 1, 2 ... n,
round(A) function rounds A to the nearest
integer,
ceil(A) function returns the nearest integer that
is greater than or equal to A.
Note that both Ki and ai should be integer.
The actual sequence is created as follows.
First, the raw number of elements in the
pseudo-geometric sequence, i.e., the raw number of box size is calculated as
 log(a )
1 
n = round 
  log(b) 
(6)
Second, calculate Ki using Equation (4).
If Ki = Ki+1 =…= Ki+s, only Ki+s is accounted
in order to remove the redundant values. The
corresponding ai, ai+1 … ai+s-1 calculated from
Equation (5) are also dropped. Update the value
of n and re-code the sequences as K1, K2 …
Kn and a1, ak … an. For example, if m pairs of
elements for Ki and ai are removed, the value
of n is reduced by m.
Third, if aj = aj+1 =…= aj+t, only aj is accounted. The corresponding Kj, Kj+1 … Kj+t-1
are removed. Update the value of n again and
re-code the sequences as K1, K2 … Kn and a1,
ak … an.
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
50 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
Fourth, reduce the over coverage. When
partial blocks exist at the boundary of the grid
as shown in Figure 5(c), the over cover rate
(OCR) is defined in Equation (7),
OCR = [Ki × ai – a1] / a1 × 100%. (7)
If Ki > ceil (a1/ai), replace Ki with ceil (a1/ai).
This algorithm minimizes the over cover rate.
The above algorithm generates optimal
cover of the object using a pseudo-geometric
sequence with different factors. Table 1 provides
an example of our pseudo-geometric sequence
with 20 data points (b = 1.45). If we choose b
= 1.05, we get 73 data points, the average over
cover rate (AOCR) is 0.42%.
3.2.3. Generalized Sliding
Window Method
There is no theoretical method to determine the
scaling range of stochastic fractals. Among the
existing empirical methods, the sliding window
method provides a better choice (Brewer &
Di Girolamo, 2006; Dubuc et al., 1989). In
previous research, the sliding window method
was only used to find the lower bound of the
scaling range, while taking half or one fourth
of the shorter side of the image as the upper
bound. It is natural to extend the algorithm
to find the upper bound. In the following, we
present a generalized sliding window method
to find the scaling range.
Let Oi be points on the scatter plot, i = 1,
2…n. O1 corresponding to the minimum box
size. Let k be the size (length) of the window,
k = 3, 4…n. The generalized sliding window
method is composed of three ways of sliding
windows (Figure 6). The scaling range can be
determined by inspecting the the dimension
profiles (Figure 7, Figure 9) and scatter plot
(Figure 8, Figure 10). The generalized sliding
window method includes three parts as follows.
Through method (1) we found that the window
size should be at least 20. We then identify the
lower and upper bound of the scaling range
through method (2) and (3). Method (1)-(3)
are combined together to obtain the appropriate
scaling range.
1. Window Size Fixed, Slide the Window
Through Data Points: For window size
k, using O1, O2…Ok to fit a straight line,
we get dimension Dk,1 and coefficient of
determination R2(k,1). Then use O2, O3…
Ok+1 to get Dk,2 and R2(k,2). Continue the
sliding window until the last k points, On-k+1,
On-k+2…On, and the dimension, Dk,n-k+1, R2(k,
Table 1. Example pseudo-geometric sequence for box size with 20 data points (b = 1.45)
i
Ki
ai
Ki × ai
OCR (%)
i
Ki
ai
Ki × ai
OCR
(%)
1
2048
1
2048
0.00
11
41
50
2050
0.10
2
1024
2
2048
0.00
12
28
74
2072
1.17
3
683
3
2049
0.05
13
20
103
2060
0.59
4
512
4
2048
0.00
14
13
158
2054
0.29
5
342
6
2052
0.20
15
9
228
2052
0.20
6
256
8
2048
0.00
16
6
342
2052
0.20
7
171
12
2052
0.20
17
4
512
2048
0.00
8
121
17
2057
0.44
18
3
683
2049
0.05
9
86
24
2064
0.78
19
2
1024
2048
0.00
10
59
35
2065
0.83
20
1
2048
2048
0.00
Note: Box size ri = Kir1. For meanings of notations Ki, ai, ri, please refer to the explanation below Equation (5).
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 51
Figure 6. Three ways of sliding windows: (a) Window size fixed, slide the window through data
points; (b) Lower bound fixed, increase window size; (c) Upper bound fixed, increase window size
n-k+1) (Figure 6(a)). The scatter plot of
Dk,i shows the variability of the dimension with the scaling range of the length
k, where i = 1, 2...n-k+1. Figure 7(a)-(b)
shows the influence of window size on
the estimation of box dimension and R2.
As we can see, when the window size is
small, the local slope (fractal dimension,
Figure 7(a)) and the R2 (Figure 7(a)) vary
greatly. In other words, the fractal dimension estimated using small window size is
not stable and thus are not likely valid. As
the window size increases to 20, the box
dimension curve becomes smooth, and the
R2 becomes stable. Therefore, we should
choose a window size larger than 20 where
stable estimation can be obtained.
2. Lower Bound Fixed, Increase Window
Size: Start from point Oi, using Oi, Oi+1,
Oi+2 to fit a straight line, we get a slope and
dimension Di,1. Then use Oi, Oi+1, Oi+2, Oi+3
to get Di,2. Continue increasing window size
until all Oi, Oi+1…On are used, and we get
the dimension, Di,n-i+1(Figure 6(b)). Figure
7(c) is the dimension profile indicating the
variability of dimension across scaling
ranges with a fixed lower bound at point
Oi. It shows that we can get a stable fractal dimension estimation when the lower
bound is 4 and the window size is between
20~30.
3. Upper Bound Fixed, Increase Window
Size: Start from point Oj, using Oj, Oj-1,
Oj-2 to fit a straight line, we get a slope and
dimension Dj,1. Then use Oj, Oj-1, Oj-2, Oj-3
to get Dj,2. Continue increasing window
size until all Oj, Oj-1…O1 are used, and
we get the dimension, Dj,n-j+1(Figure 6(c)).
Figure 7(d) is the dimension profile indicating the variability of dimension across
scaling ranges with a fixed upper bound at
point Oj. It shows that we can get a stable
fractal dimension estimation when the
upper bound is 41 and the window size is
between 20~30.
3.3. Confidence Interval
of the Point Estimate
The range of the confidence interval for an
empirical estimate is a good index to measure
the accuracy of estimation. In this paper, the
fractal dimension is estimated through log-linear
regression based on points in the scaling range.
The confidence interval is defined as
D ± tα/2δ. (8)
where:
D is the point estimate of the fractal dimension,
tα/2δ is the margin of error,
tα/2 is the t value providing an area of α / 2 in
the upper tail of a t distribution with n - 2
degrees of freedom,
δ is the standard error of the estimate.
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
52 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
Figure 7. Dimension profile of the Vicsek prefractal (generated with 5 iterations). Note: (1) The
upper two subplots indicate the influence of window sizes: (a) shows the influence on the estimated box dimension, (b) shows the influence on R2. r(min) is the lower bound of the window,
and the number labels in the legend of (a) and (b) represent window size. (2) The lower two
subplots indicate the influence of lower and upper bound on fractal dimension estimation: (c)
shows the influence of lower bound, (d) shows the influence of upper bound.(3) We only show
some sample curves here. (4) For better view, please refer to the online digital figure with colors
In our method, the 95% confidence interval will be calculated, i.e., α = 0.05 .
while the number of boxes (Ni) grows with the
order of 5. The fractal dimension can be simply
represented as
4. RESULTS
4.1. Results for the
Vicsek Prefractal
The fractal dimension of the Vicsek prefractal
can be estimated using both analytical and
numerical methods. The analytical method is
based on the rule to generate the Vicsek fractal.
The box sizes (ri) decrease with the order of 3,
D =−
log 5i
−i
log 3
=
log 5
= 1.4649.
log 3
In order to examine the method presented
in this paper, we give two numerical estimates
of the Vicsek prefractal using the box-counting
method with different sequences of box sizes.
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 53
Figure 8. Scatter plot for the Vicsek prefractal using two methods: (a) Our method, (b) Traditional
method. Note: The Vicsek prefractal is generated with 5 iterations
The measuring procedure consists of three
steps. First, generate a Vicsek prefractal with n
iterations. As defined in Section 2.1, the size of
the squares in the nth iteration is 1 (unit one),
and the size of the initiator is 3n. Second, generate grids whose box sizes follow a geometric
sequence of factor b. In our experiment, two
values are tested for factor b: 3 and 1.05. The
exact fractal dimension of the Vicsek prefractal
can be estimated using a geometric sequence of
factor 3 since it is the common ratio of the sizes
of the fractal copies in different orders. The other
factor, 1.05 is chosen to test our method. When
b = 3, box size ri = 3i. When b = 1.05, the box
size is determined using the method in Section
3.3. Third, overlay the Vicsek prefractal with
the grids, and calculate the number of occupied
boxes Ni for each grid. The two sequences ri and
Ni are used to estimate the fractal dimension of
the Vicsek prefractal.
Figure 8 compares the scatter plot for the
Vicsek prefractal using two methods. Figure
8(a) shows the result from our method, where
the pseudo-geometric sequence with factor b
= 1.05 is used, and the scaling range is determined through the generalized sliding window
method. Based on the analysis in Section 3.3
and referring to Figure 7, the lower bound of
the scaling range is the 4th smallest box size
while the upper bound is the 41st largest box
size. Therefore, by removing the 40 largest box
sizes, and 3 smallest box sizes, the remaining 30
points fitted a line with a slope of D = 1.4669.
Figure 8(b) shows the result from the traditional
method, where the geometric sequence of factor
b = 2 is used, and the scaling range is determined
through observing the scatter points. There are
12 points corresponding to box sizes r(i) = 20,
2-1…2-11. The points corresponding to the three
largest box sizes (20, 2-1, 2-2) are removed since
they fall into a line with the slope of 2. The
point corresponding to the smallest box size
(2-11) is also removed because it deviates from
the other points. The remaining 8 points fitted
a line with a slope of D = 1.5199.
Table 2 compares the detailed parameters of
the two estimates. There are mainly three findings. First, the point estimate of our method (D
= 1.4669) is smaller than that of the traditional
method (D = 1.5199), and our point estimate is
also closer to the real fractal dimension of the
Vicsek fractal (D = 1.4649). Second, the 95%
confidence interval of our estimate, [1.4496,
1.4841], covers the real value, while that from
the traditional method, [1.4936, 1.5463], does
not. Third, the 95% confidence intervals of both
methods do not overlap with each other, and
the range of our estimate (0.0345) is smaller
than that of the traditional method (0.0527).
The experiment on the Vicsek prefractal shows
that our method can provide a more accurate
estimate of the real fractal dimension.
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
54 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
Table 2. Comparing fractal dimension estimates of the Vicsek prefractal
D
R2
df
δ
Lower 95%
Upper 95%
Range
Our method
1.4669
0.9990
28
0.0084
1.4496
1.4841
0.0345
Traditional method
1.5199
0.9997
6
0.0108
1.4936
1.5463
0.0527
Note: (1) The Vicsek prefractal is generated with 5 iterations. (2) The true dimension of the Vicsek fractal is 1.4649.
(3) Degree of freedom (df) is defined as the number of data points involved in the least squares calculation minus 2. (4)
δ is the standard error of the estimate. (5) Range is the difference of the Upper 95% minus Lower 95%.
4.2. Results for Beijing City
The scaling range can be obtained by observing the dimension profiles (Figure 9). Similar
to Figure 7, when the window size is small,
the variability of the box dimension and R2 are
both very high. Ideally, the window size should
be greater than 20, i.e., at least 20 data points
should be used to get a stable estimate of fractal
dimension (Figure 9(a), (b)). Figure 9(c) is the
dimension profile showing the variability of
dimension across scaling ranges with a fixed
lower bound. For the first several curves, i.e.,
curves 1-6, there is large variability of the estimated fractal dimension. Starting from curve 7,
there roughly exists a plateau showing a stable
dimension when window size is greater than
20. Thus the box size corresponding to the 7th
smallest point might be the lower bound of the
box size. Figure 9(d) is the dimension profile
showing the variability of dimension across scaling ranges with a fixed upper bound at point Oj.
Curve 1 is very smooth and there is a plateau of
2 for the first several window sizes as expected.
All the curves are smooth if the window size is
greater than 20. Starting from curve 31, there is
roughly a plateau when window size is greater
than 20. This plateau means that the dimension
estimation is stable when the upper bound of
the box size is the value corresponding to the
31th largest box size.
Based on the scatter plot (Figure 10) and
dimension profiles (Figure 9), the 7th smallest
box size is chosen as the lower bound of the
scaling range, and the 31st largest box size is
the upper bound of the scaling range. Starting
from the 7th sample point, the box dimension
has less variability for the first 30 window
sizes. Starting from the 31th largest sample
point, the box dimension has less variability
for the 20~30 window sizes. Figure 10 shows
an example of regression on the scaling range
(the middle scale range).
Table 3 gives the OLS regression results
for the three scale ranges as well as that for
all the points. There is a significant difference
between the dimension on scale range 2 and the
dimension on scale range 1 or 3 because their
95% confidence intervals are well separated.
The real dimension is the slope of the fitted
straight line on scale range 2, i.e., D = 1.672.
As expected, the slope of the fitted line on scale
range 1 is close to 2 while the slope of the fitted
line on scale range 3 is pulled down from 2 by
the true dimension. The R2 for all data points
is smaller than those on separate scale ranges,
although they are all very high.
We examine Beijing City in two years
for method demonstration purpose. The box
dimension of Beijing City is shown in Table 4.
5. DISCUSSION
5.1. Implications for
the Beijing City
We compared the results from our method with
that from the traditional method (Table 4).
Similar to the results for the Vicsek prefractal,
three main findings can be found from the comparison. First, the point estimates of our method
are smaller than those of the traditional method
for all study areas. Second, the 95% confidence
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 55
Figure 9. Dimension profile Beijing City (Note: Same as Figure)
intervals of both methods do not overlap with
each other except study area 1 in 1999. Third,
the ranges of 95% confidence intervals from
our methods are smaller than those from the
traditional method.
The box dimensions increase from 1984 to
1999 in all three study areas, which confirms
previous research on fractal trend of urban
growth (Batty and Longley 1994; Benguigui
et al., 2000; Feng and Chen 2010). Beijing is
an international city growing with a fast speed.
Box dimension is an indicator of the distribution
of built-up areas. High box dimension usually
indicates high density of a built-up environment
and less open space. As is shown in a research
by Feng and Zhou (2003), the population of
Beijing in different zones increased during
1982-2000 (Table 5).
The box dimension decreases with the
expansion of the study area. This is true for
urban area, as cities usually develop from a
center (or several centers) and expand outside.
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
56 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
Figure 10. Example of the log linear regression on the scale ranges. Note: (1) three scale ranges
are given here, denoted as scale range 1, 2, 3, where scale range 2 is the true scaling range. (2)
The scatter plot is based on vector data, which are different from that based on raster data (i.e.,
pixels). When raster data (pixels) is used, the regression line of the data points in scale range
3 will approach zero
Table 3. Regression results for three scale ranges
Parameter/Statistic
Scale Range 1
Scale Range 2
Scale Range 3
All Points
D
1.977
1.672
1.804
1.802
R2
0.9999
1.0000*
0.9997
0.9977
δ
0.0044
0.0016
0.0132
0.0102
df
29
33
5
71
Lower 95%
1.968
1.668
1.768
1.782
Upper 95%
1.986
1.675
1.841
1.822
* The real value round to five digits is 0.99997.
Note: (1) The calculation is based on vector data, which are different from that based on raster data (i.e., pixels). When
raster data (pixels) is used, the slope of the regression line of the data points in scale range 3 will approach zero, not as large
as the 1.804 shown here. (2) Only scale range 2 is the true scaling range on which D is the estimated fractal dimension.
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 57
Table 4. Box dimension estimate of Beijing urban area using two methods
Range
Our method
Traditional
Method
Parameter
/Statistic
Study Area 1
1984
Study Area 2
1999
1984
Study Area 3
1999
1984
1999
D
1.779
1.880
1.672
1.772
1.545
1.670
R2
1.0000*
1.0000**
1.0000***
0.9999
0.9998
0.9998
δ
0.0023
0.0027
0.0016
0.0032
0.0037
0.0046
df
32
32
33
27
25
28
Lower 95%
1.7743
1.8745
1.6687
1.7654
1.5374
1.6606
Upper 95%
1.7837
1.8855
1.6753
1.7786
1.5526
1.6794
Range
0.0094
0.011
0.0066
0.0132
0.0152
0.0188
D
1.831
1.898
1.738
1.820
1.652
1.750
R2
0.9998
0.9999
0.9997
0.9998
0.9992
0.9996
δ
0.0085
0.0068
0.0120
0.0087
0.0175
0.0134
df
6
6
6
6
6
6
Lower 95%
1.8105
1.8813
1.7090
1.7985
1.6088
1.7170
Upper 95%
1.8522
1.9148
1.7678
1.8411
1.6944
1.7828
Range
0.0417
0.0335
0.0588
0.0426
0.0856
0.0658
Note: The values round to five digits are, *: 0.99996; **: 0.99995; ***: 0.99997. Range is the difference of the dimension in the upper 95% and lower 95% bounds of the estimates.
Beijing is a very centralized city with economic
activities densely concentrated in the core area.
The change pattern of box dimensions for different study areas is expected. This change
trend is also confirmed by research on other
cities (Benguigui et al., 2000).
5.2. Difficulty in Determining
the Scaling Range
Theoretically, there should be three scale ranges
on the scatter plot of the urban form data (Figure
10). Scale range 1: when the box size is very
large, i.e., all the boxes are very likely to be
occupied, and thus we get a regression line with
the slope of 2. Scale range 3: When the box
size is very small, we are just dividing inside
pixels (which are vectorized as patches), and
the slope of the regression line is also 2. Scale
range 2: Between scale range 1 and scale range
3 is the true scaling range that should be used
to estimate the fractal dimension. It should be
noticed that, when raster data or point data are
used, the scatter points in scale range 3 will
approach to a line with the slope of 0.
It is difficult to determine the scaling range.
One may want to determine the scaling range
by using the cutoff slope value of 2. However,
we cannot do so in practice due to three reasons.
First, there usually exists a transition region
between two scale ranges. In the transition
region between scale range 1 and scale range
2, the box dimension changes from 2 to the
true dimension, while in the transition region
between scale range 2 and scale range 3, the box
dimension changes from the true dimension to
2. Second, the points on the scatter plot deviate
from their theoretical values due to the existence
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
58 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
Table 5. Population growth in different zones of Beijing during 1982-2000
(1982-1990)
Population
Change
(Persons)
Total
Change
Rate (%)
(1990-2000)
Change
Rate Per
Year (%)
Population
Change
(Persons)
Total
Change
Rate (%)
Change
Rate Per
Year (%)
Core city and inner
suburbs
1067508
37.08
3.91
2177663
50.65
3.83
Outer suburbs
521236
13.12
1.55
572124
12.73
1.21
Metropolitan area
1446204
18.02
2.09
2710174
28.62
2.55
Source: Compiled from Feng and Zhou (2003). The exact extents of the three regions (Core city and inner suburbs,
Outer suburbs, Metropolitan area) are not the same as our three study areas but they show similar increasing trends
during 1982-2000 as the dimensions during 1984-1999.
of over coverage of the object by boxes. The
cutoff slope value is not exactly equal to 2.
Third, due to the limit of computation power,
the box sizes are not small enough. Most of
the points on scale range 3 lie in the transition
region. The slope of the regression line on scale
range 3 is significantly influenced by the true
dimension. It might have a value very close to
the true dimension, which makes it difficult to
separate scale range 2 and 3.
5.3. Measurement Design
and Fractal Dimension
The properties of scaling range and dimension
profiles of urban form can be explored in our new
measurement design: 1) setting up a proper box
cover of the city; 2) using grids of sufficient box
sizes; 3) determining the correct scaling range.
The choosing of study area, image type (raster
or vector), and box shape all have influences
on building a proper box cover.
The difference of dimensions as to the
study areas poses a problem for comparison
across cities or the same city in different years.
Ideally, we should cover the target cities with
study areas of the same size and shape. For
example, Shen (2002) correctly used images of
the same size (width by height in 1000 × 1000
pixels) in a study of 20 US cities. The strength
of comparison study turns weak if images of
different sizes are used. In a study of Hangzhou
city, China, Feng and Chen (2010) compared
the box dimensions calculated using different
study areas. The extents of the study areas in
the two compared years (1980 and 1996) are
different. Their conclusions would have been
stronger if they kept the study areas consistent
in both years. When the same study area is used
to calculate the fractal dimension of the same
city from different years, the rural areas inside
the study area should be removed in order to
get accurate results.
The problem of observing too few scales
in most research raises issues of incorrect or
inaccurate estimates for fractal dimension
(Halley et al., 2004; Hamburger et al., 1996).
Using pseudo-geometric sequence with small
factors, we can get much more data points of
box size than from the dyadic sequence. The
generalized sliding window method provides
new views of looking at the data, and can help
the researchers to find proper scaling range.
5.4. How Confident
the Analysis is?
Compared to the traditional method, our method
can result a more accurate estimate of fractal
dimension. However, limitation still exists in
determining the scaling range. In our method,
the scaling range is determined by observing the
dimension profiles (Figure 7 or Figure 9) and
scatter plot (Figure 8 or Figure 10). Observing
the existence of plateau in the dimension profile
can be subjective, especially when shapes of
the dimension profile for different objects are
different. The subjectivity is reduced by com-
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 59
paring it with the scatter plots. The existence
of a transition region between scale ranges also
makes a fuzzy boundary between scale ranges.
The results may be improved with more data
points by reducing the factor of the geometric
sequence. There is a tradeoff between the factor
and the noise (over coverage rate) introduced.
For factors smaller than 1.05, the over coverage rate increases quickly, which influences the
dimension profile and reduces the accuracy of
dimension estimation. As to Beijing City in the
two years we examined, b = 1.05 provide a good
alternative to get reliable dimension estimation.
When applied to other cities, different factors
may be selected.
Another problem is concerned with the
length of the scaling range. Some researchers
argue that for genuine fractal pattern, the scaling range should have a length larger than two
orders of magnitude (Halley et al., 2004). This
criterion is not generally conformed to in practice. The factor of magnitude generally refers
to 10, thus the criterion requires a scaling range
across 103. For natural fractals, the scaling range
is generally small. As to Beijing City shown in
Figure 10, the scaling range lies between 0.0034
~ 0.0308, which is only one order of magnitude
( 0.0308 / 0.0034 ≈ 101 ). Two modifications
can be made to the criterion. One is to extend
the magnitude factor of 10 to any integer number. In Figure 10, the scaling range 2-8.20 ~ 2-5.02
is more than three orders of magnitude 2. Another alternative is to treat the length of the
scaling range itself as a property of fractal pattern. A larger scaling range means a more
mature fractal structure.
The third problem is due to the regression method used in box-counting method.
As is shown by numerous researchers (Batty
& Longley, 1994; Feng & Chen, 2010; Halley
et al., 2004; Shen, 2002), the goodness of fit
(R2) for a straight line on the scatter plot is very
high. Although the variability might be large,
as shown in Figure 7 (b) and Figure 9(b), R2 is
well above 0.9 or even greater than 0.95 on the
selected scaling range. The standard error and
confidence intervals are generally small. It is
argued that irregular patterns of scatter points
are not easy to be disguised on logarithmic axes
(Halley et al., 2004). Here comes the critical
question: does the unanimously good fit of
the scatter plot guarantee a fractal pattern? In
order to answer this difficult and fundamental
question, we should look for the evidence of
processes that can produce a natural fractal.
Accurate estimation of fractal dimension
is important in practice although we cannot
completely eliminate the uncertainties in the
computation process. Inaccurate calculation of
fractal dimension may lead to misunderstanding
of our study objects. As discussed in Section
1, due to the issues related to the box-counting
method, different people often have different
estimates of the dimension for the same fractal
set (Huang et al., 1994; Hunt, 1990; Li et al.,
2009; Sarkar & Chaudhuri, 1994). It makes
no sense to compare the fractal dimension of
different cities if the estimation is inaccurate
or even wrong. The further interpretation based
on comparison of fractal dimension would be
misleading.
6. CONCLUSION
In this paper, we explored three main issues of
the conventional box-counting method in fractal
urban analysis. Corresponding techniques with
improved measurement designs were proposed
to address these issues.
First, we proposed a new algorithm to
extract land use information using rectangular
grids and boxes to measure the object. Rectangular boxes can cover the object more efficiently.
As the urban boundary is generally fractal,
different study areas of the same city can have
different box dimensions. It is suggested that
the fractal dimension of a city should be given
along with the study area.
Second, we developed an algorithm to
generate a pseudo-geometric sequence for box
size. The factor of the geometric sequence can
be any value specified by the researchers. Our
experiment on the urbanized area of Beijing
shows that b = 1.05 is a reliable factor, which
produces sufficient points on the scatter plot
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
60 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
and introduces less noise (over coverage rate,
OCR). When a different object is used, the
readers may need to try different factors.
Third, we introduced a generalized sliding
window method to determine the scaling range.
Sliding the window with fixed window size on
different starting points show that the minimum
window size should be no less than 20 in order
for a stable estimate of box dimension. Sliding
the window from the points for the smallest box
size with changing window size can be used to
determine the lower bound of the scaling range,
while sliding the window from the points for
the largest box size can help to find the upper
bound of the scaling range. Integrating the
generalized sliding window method with the
scatter plot, we can get a proper scaling range.
Our method is first experimented on the
Vicsek prefractal with known dimension. It
is then applied to Beijing City. It is found that
the dimensions of the city in three study areas
increased during 1984-1999, which is consistent
with the population growth trend in 1982-2000.
The dependence of the dimensions on the study
areas suggests that the comparison among cities
should be bounded in images of the same size
and shape. Our measurement design provides
an alternative method to explore the properties
of dimension profile and scaling range. There
is still room for better methods to determine an
accurate scaling range.
ACKNOWLEDGMENT
The authors wish to thank Dr. Yanguang Chen
from Peking University, Dr. Chandana Gangodagamage from Los Alamos National Lab, and
Professor Yixing Zhou from Peking University
for valuable discussions; Mrs Carolyn Pickard
for editorial help. The research was supported
by the International Affairs Grant to Shiguo
Jiang from the Office of International Affairs
at The Ohio State University.
REFERENCES
Andraud, C., Beghdadi, A., Haslund, E., Hilfer,
R., Lafait, J., & Virgin, B. (1997). Local entropy
characterization of correlated random microstructures. Physica A. Statistical Mechanics and Its
Applications, 235, 307–318. doi:10.1016/S03784371(96)00354-8.
Barnsley, M. (1988). Fractals everywhere. Boston,
MA: Academic Press.
Batty, M. (1985). Fractals-geometry between dimensions. New Scientist, 106, 31–35.
Batty, M., & Longley, P. A. (1987). Urban shapes as
fractals. Area, 19, 215–221.
Batty, M., & Longley, P. A. (1988). The morphology of urban land use. Environment and Planning.
B, Planning & Design, 15, 461–488. doi:10.1068/
b150461.
Batty, M., & Longley, P. A. (1994). Fractal cities:
A geometry of form and function. London, UK:
Academic Press.
Batty, M., & Xie, Y. (1996). Preliminary evidence
for a theory of the fractal city. Environment &
Planning A, 28, 1745–1762. doi:10.1068/a281745
PMID:12292847.
Benguigui, L., Czamanski, D., Marinov, M., & Portugali, Y. (2000). When and where is a city fractal?
Environment and Planning. B, Planning & Design,
27, 507–519. doi:10.1068/b2617.
Berry, B. J. L., Goheen, P. J., & Goldstein, H. (1968).
Metropolitan area definition: A re-evaluation of
Concept and Statistical Practice. Washington, DC:
US Bureau of the Census.
Bisoi, A. K., & Mishra, J. (2001). On calculation
of fractal dimension of images. Pattern Recognition Letters, 22, 631–637. doi:10.1016/S01678655(00)00132-X.
Brewer, J., & Di Girolamo, L. (2006). Limitations
of fractal dimension estimation algorithms with implications for cloud studies. Atmospheric Research,
82, 433–454. doi:10.1016/j.atmosres.2005.12.012.
Buczkowski, S., Kyriacos, S., Nekka, F., & Cartilier,
L. (1998). The modified box-counting method:
Analysis of some characteristic parameters. Pattern
Recognition, 31, 411–418. doi:10.1016/S00313203(97)00054-X.
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 61
Chen, S. S., Keller, J. M., & Crownover, R. M. (1993).
On the calculation of fractal features from images.
IEEE Transactions on Pattern Analysis and Machine
Intelligence, 15, 1087–1090. doi:10.1109/34.254066.
Dathe, A., Tarquis, A. M., & Perrier, E. (2006).
Multifractal analysis of the pore- and solid-phases
in binary two-dimensional images of natural porous
structures. Geoderma, 134, 318–326. doi:10.1016/j.
geoderma.2006.03.024.
Dubuc, B., Quiniou, J. F., Roquescarmes, C., Tricot, C., & Zucker, S. W. (1989). Evaluating the
fractal dimension of profiles. Physical Review A.,
39, 1500–1512. doi:10.1103/PhysRevA.39.1500
PMID:9901387.
ESRI. (2010). ArcGIS Desktop: Release 10.
Redlands, CA: Environmental Systems Research
Institute.
Falconer, K. J. (2003). Fractal geometry: Mathematical foundations and applications. Chichester, UK:
Wiley. doi:10.1002/0470013850.
Feder, J. (1989). Fractals. New York, NY: Plenum.
Feng, J., & Chen, Y. (2010). Spatiotemporal evolution
of urban form and land-use structure in Hangzhou,
China: evidence from fractals. Environment and
Planning. B, Planning & Design, 37, 838–856.
doi:10.1068/b35078.
Feng, J., & Zhou, Y. X. (2003). The latest development
in demographic spatial distribution in Beijing in the
1990s. City Planning Review, 27, 55–63.
Foroutan-pour, K., Dutilleul, P., & Smith, D. L.
(1999). Advances in the implementation of the
box-counting method of fractal dimension estimation. Applied Mathematics and Computation, 105,
195–210. doi:10.1016/S0096-3003(98)10096-6.
Frankhauser, P. (1994). La fractalité des structures
urbaines. Paris, France: Economica.
Goodchild, M. F. (1980). Fractals and the accuracy of
geographical measures. Journal of the International
Association for Mathematical Geology, 12, 85–98.
doi:10.1007/BF01035241.
Grau, J., Mendez, V., Tarquis, A. M., Diaz, M. C., &
Saa, A. (2006). Comparison of gliding box and boxcounting methods in soil image analysis. Geoderma,
134, 349–359. doi:10.1016/j.geoderma.2006.03.009.
Halley, J. M., Hartley, S., Kallimanis, A. S., Kunin,
W. E., Lennon, J. J., & Sgardelis, S. P. (2004). Uses
and abuses of fractal methodology in ecology.
Ecology Letters, 7, 254–271. doi:10.1111/j.14610248.2004.00568.x.
Hamburger, D., Biham, O., & Avnir, D. (1996). Apparent fractality emerging from models of random
distributions. Physical Review E: Statistical Physics,
Plasmas, Fluids, and Related Interdisciplinary Topics, 53, 3342–3358. doi:10.1103/PhysRevE.53.3342
PMID:9964642.
Hargis, C. D., Bissonette, J. A., & David, J. L. (1998).
The behavior of landscape metrics commonly used in
the study of habitat fragmentation. Landscape Ecology, 13, 167–186. doi:10.1023/A:1007965018633.
Herold, M., Couclelis, H., & Clarke, K. C. (2005). The
role of spatial metrics in the analysis and modeling
of urban land use change. Computers, Environment
and Urban Systems, 29, 369–399. doi:10.1016/j.
compenvurbsys.2003.12.001.
Huang, Q., Lorch, J. R., & Dubes, R. C. (1994). Can
the fractal dimension of images be measured? Pattern Recognition, 27, 339–349. doi:10.1016/00313203(94)90112-0.
Hunt, F. (1990). Error analysis and convergence of capacity dimension algorithms. SIAM
Journal on Applied Mathematics, 50, 307–321.
doi:10.1137/0150018.
Imre, A. R., & Bogaert, J. (2004). The fractal
dimension as a measure of the quality of habitats. Acta Biotheoretica, 52, 41–56. doi:10.1023/
B:ACBI.0000015911.56850.0f PMID:14963403.
Lam, N. S. N., & Quattrochi, D. A. (1992). On the
issues of scale, resolution, and fractal analysis in the
mapping sciences. The Professional Geographer,
44, 88–98. doi:10.1111/j.0033-0124.1992.00088.x.
Li, J., Du, Q., & Sun, C. X. (2009). An improved
box-counting method for image fractal dimension
estimation. Pattern Recognition, 42, 2460–2469.
doi:10.1016/j.patcog.2009.03.001.
Liebovitch, L. S., & Toth, T. (1989). A fast algorithm to determine fractal dimensions by box
counting. Physics Letters. [Part A], 141, 386–390.
doi:10.1016/0375-9601(89)90854-2.
Longley, P. A., & Batty, M. (1989a). Fractal
measurement and line generalization. Computers
& Geosciences, 15, 167–183. doi:10.1016/00983004(89)90032-0.
Longley, P.A., & Batty, M. (1989b). On the fractal measurement of geographical boundaries. Geographical
Analysis, 21, 47–67. doi:10.1111/j.1538-4632.1989.
tb00876.x.
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
62 International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012
Lovejoy, S., Schertzer, D., & Tsonis, A. A.
(1987). Functional box-counting and multiple elliptical dimensions in rain. Science, 235,
1036–1038. doi:10.1126/science.235.4792.1036
PMID:17782251.
Saucier, A., & Muller, J. (1998). Multifractal approach to textural analysis. In M. M. Novak (Ed.),
Fractals and beyond, complexities in the sciences
(pp. 161–171). London, UK: World Scientific Publishing Co..
Lu, Y. M., & Tang, J. M. (2004). Fractal dimension
of a transportation network and its relationship with
urban growth: A study of the Dallas-Fort Worth area.
Environment and Planning. B, Planning & Design,
31, 895–911. doi:10.1068/b3163.
Shen, G. Q. (2002). Fractal dimension and fractal
growth of urbanized areas. International Journal of
Geographical Information Science, 16, 419–437.
doi:10.1080/13658810210137013.
Mandelbrot, B. (1967). How long is the coast of
Britain? Statistical self-similarity and fractional
dimension. Science, 156, 636–638. doi:10.1126/
science.156.3775.636 PMID:17837158.
Tarquis, A. M., McInnes, K. J., Key, J. R., Saa, A.,
Garcia, M. R., & Diaz, M. C. (2006). Multiscaling
analysis in a structured clay soil using 2D images.
Journal of Hydrology (Amsterdam), 322, 236–246.
doi:10.1016/j.jhydrol.2005.03.005.
Mandelbrot, B. (1983). The fractal geometry of nature
(Revised and enlarged edition ed.). New York, NY:
W.H. Freeman and Co.
Taylor, J. (1997). An introduction to error analysis:
The study of uncertainties in physical measurements.
Mill Valley, CA: University Science Books.
Nayfeh, A. H., & Balachandran, B. (1995). Applied
nonlinear dynamics: Analytical, computational,
and experimental methods. New York, NY: Wiley.
doi:10.1002/9783527617548.
Thomas, I., Frankhauser, P., & Biernacki, C. (2008).
The morphology of built-up landscapes in Wallonia
(Belgium): A classification using fractal indices.
Landscape and Urban Planning, 84, 99–115.
doi:10.1016/j.landurbplan.2007.07.002.
Peitgen, H.-O., & Richter, D. H. (1986). The beauty
of fractals: Images of complex dynamical systems.
New York, NY: Springer-Verlag.
Pincheira-Ulbrich, J., Rau, J. R., & Pena-Cortes, F.
(2009). Patch size and shape and their relationship
with tree and shrub species richness. Phyton-International Journal of Experimental Botany, 78, 121–128.
Pruess, S. A. (1995). Some remarks on the numerical estimation of fractal dimension. In C. C. Barton,
& P. R. L. Pointe (Eds.), Fractals in the earth sciences (pp. 65–75). New York, NY: Plenum Press.
doi:10.1007/978-1-4899-1397-5_3.
Richardson, L. F. (1961). The problem of contiguity:
an appendix of Statistics of deadly quarrels. General
Systems Yearbook, 139-187.
Roy, A., Perfect, E., Dunne, W. M., & McKay, L. D.
(2007). Fractal characterization of fracture networks:
An improved box-counting technique. Journal of
Geophysical Research. Solid Earth, 112, 9.
Saa, A., Gasco, G., Grau, J. B., Anton, J. M., &
Tarquis, A. M. (2007). Comparison of gliding box
and box-counting methods in river network analysis.
Nonlinear Processes in Geophysics, 14, 603–613.
doi:10.5194/npg-14-603-2007.
Sarkar, N., & Chaudhuri, B. B. (1994). An efficient
differential box-counting approach to compute
fractal dimension of image. IEEE Transactions
on Systems, Man, and Cybernetics, 24, 115–120.
doi:10.1109/21.259692.
Thomas, I., Frankhauser, P., & De Keersmaecker, M.L. (2007). Fractal dimension versus density of builtup surfaces in the periphery of Brussels. Papers in
Regional Science, 86, 287–308. doi:10.1111/j.14355957.2007.00122.x.
Verbovsek, T. (2009). BCFD - a visual basic program
for calculation of the fractal dimension of digitized
geological image data using a box-counting technique. Geological Quarterly, 53, 241–248.
Vicsek, T. (1983). Fractal models for diffusion
controlled aggregation. Journal of Physics. A,
Mathematical and General, 16, L647–L652.
doi:10.1088/0305-4470/16/17/003.
Walsh, J. J., & Watterson, J. (1993). Fractal analysis
of fracture pattern using the standard box-counting
technique: Valid and invalid methodologies.
Journal of Structural Geology, 15, 1509–1512.
doi:10.1016/0191-8141(93)90010-8.
Weisstein, E. W. (2003). Concise encyclopedia of
mathematics (2nd ed.). Boca Raton, FL: Chapman
and Hall/CRC.
White, R., & Engelen, G. (1993). Cellular automata
and fractal urban form: A cellular modelling approach
to the evolution of urban land-use patterns. Environment & Planning A, 25, 1175–1199. doi:10.1068/
a251175.
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
International Journal of Artificial Life Research, 3(3), 41-63, July-September 2012 63
Shiguo Jiang is a PhD student (2007-) of Geography at The Ohio State University. He received
B.S. degree in Urban and Regional Planning (2001), B. A. degree in Economics (2001), and M.S.
degree in Urban Studies (2004) from Peking University, China. He was a research associate and
urban planner at Beijing Tsinghua Urban Planning and Design Institute from 2004 to 2007. His
research interests include remote sensing of urban and natural environment, spatial statistics,
land use and land cover change, GIS and spatial modeling. Recently he has been involved with
a project on classification confidence of remote sensing imagery.
Desheng Liu is an Associate Professor of Geography and Statistics at The Ohio State University.
He obtained his M.A. (2004) in Statistics and his M.S. (2003) and PhD (2006) in Environmental
Science from the University of California at Berkeley. His research focuses on understanding
forest ecosystem dynamics, land use and land cover change, and human-environment systems
through the integrated use of geospatial information technologies (GIS, remote sensing, and
GPS) and spatial statistical methods.
Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.