=Paper=
{{Paper
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|title=Fractal Dimension Calculation for CORINE Land-Cover Evaluation in GIS - A Case Study
|pdfUrl=https://ceur-ws.org/Vol-706/papersg02.pdf
|volume=Vol-706
|dblpUrl=https://dblp.org/rec/conf/dateso/PasztoMT11
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==Fractal Dimension Calculation for CORINE Land-Cover Evaluation in GIS - A Case Study==
https://ceur-ws.org/Vol-706/papersg02.pdf
Fractal
Fractal Dimension Calculation
Dimension Calculation for CORINE
for CORINE Land-
Cover Evaluation
Land-Cover Evaluationin GIS −A–
in GIS Case Study Study
A Case
1 and Pavel Tuček 1,2
Vı́t Vít Pászto , LukášMarek
Marek, and Pavel Tuček ,
1 1 1,2
1
Pászto , Lukáš
1 1Department of Geoinformatics, Faculty of Science, Palacký University in Olomouc, Tř.
Department of Geoinformatics, Faculty of Science, Palacký University in Olomouc
Tř. Svobody
Svobody26,
26,771
7714646Olomouc,
Olomouc,Czech Republic
Czech Republic
2 2 Department of Mathematical Analysis and Applied Mathematics, Faculty of Science,
Department of Mathematical Analysis and Applied Mathematics, Faculty of Science
Palacký
Palacký University
University in Olomouc,
in Olomouc, 17.17. listopadu,12,
listopadu 771771
46 Olomouc, Czech
46 Olomouc, Republic
Czech Republic
vit.paszto@gmail.com
vit.paszto@gmail.com, , lukas.marek@upol.cz, pavel.tucek@upol.cz
lukas.marek@upol.cz, pavel.tucek@upol.cz
www.geoinformatics.upol.cz
www.geoinformatics.upol.cz
Abstract. Together with rapid development in GI science recent decades, the
fractal geometry represents a powerful tool for various geographic analyses and
studies. This study points the land-cover areas with extreme values of fractal
dimension in Olomouc region. This leads, together with consequent statistical
analyses, to result that according to fractal dimension it is possible to
distinguish (or at least to assume) the origin of areas. General fractal calculation
method is used in the case study. Statistical methods are also applied to test
mean values of land-cover areas fractal dimension (Student’s t-test and analysis
of variance). Using non-integer, fractal dimension, one can analyze complexity
of the shape, explore underlying geographic processes and analyze various
geographic phenomena in a new and innovative way.
Keywords: fractal geometry, GIS, land-cover, fractal dimension,
geocomputation, shape metrics.
1 Introduction
When Weierstrass’s continuous nonwhere-differentiable curve appeared in 1875, it
was called by other mathematicians as “regrettable evil” and these types of object
were known as mathematical “monsters” [8, 18]. Nobody imagined that fundamentals
of fractal geometry were just established. However, since Mandelbrot’s published its
basics in [13], fractal geometry and fractal dimension (non-integer dimension, e.g.
1.32 D) is well known as a valuable tool for describing the shape of objects. It gained
great popularity in geosciences [1, 7] (among other disciplines), where the measures
of object’s shape are essential.
Complex and detailed information about fractal geometry is in [8, 11, 14, 15, 18,
19]. Books provide the broad view of the underlying notions behind fractals and, in
addition, show how fractals and chaos theory relate to each other as well as to natural
phenomena. Especially introduction of fractals to the reader with the explicit link to
V. Snášel, J. Pokorný, K. Richta (Eds.): Dateso 2011, pp. 196–205, ISBN 978-80-248-2391-1.
Fractal Dimension Calculation for CORINE Land-Cover Evaluation in GIS 197
natural sciences, such as ecology, geography (demography), physical geography,
spatio-temporal analyses and others is in [8]. Some papers concerning topics
investigated in this paper (land-use/land-cover pattern) were published yet, e.g. Batty
and Longley’s book [1] as pioneer work. Many other studies, such as [2, 5, 6, 16, 24,
26], applied different fractal methods for description of city morphology and
connected issues. Fractal analyses applied especially on land-use/land-cover pattern
are described as well, such as in [5, 9, 10, 17, 22, 27, 28].
Fig. 1. Example of fractal coast and scale-invariance principle (in six steps/scales) [18].
One of the major principles in chaos theory and descriptive fractal geometry is
self-similarity and self-affinity. The most theoretical fractal objects, such as
Mandelbrot set, are self-similar – this means that any part of the object is exactly
similar to the whole. But these types of fractals are rarely used to approximate objects
or shapes from the real world. And thus, another type of fractals is suitable for real-
world object description – self-affine ones. These fractals are in fact self-similar too,
but transformed via affine transformation (e.g. translation, rotation, scaling, shear
mapping) of the whole or the part of fractal object [1, 8, 11, 15, 18, 19]. This
observation is closely related to scale-invariance (Fig. 1), which means that object has
same properties in any scale, in any detail. In other words, if characteristics of some
fractal object are known in certain scale, it is possible to anticipate these
characteristics of another fractal object in different scale. The very typical example of
this object is land-cover and/or urban forms with theirs dynamics.
Concept of fractality was described in detail in many publications [4, 7, 15, 18,
23]. Fractal dimension is a measure of complexity of the shape, based on irregularity,
scale dependency and self-similarity of objects [2]. The basic property of all fractal
structures is their dimension. Although there is no exact definition of fractals, the
publicly accepted one, coming from Mandelbrot himself: “A fractal is by definition a
set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological
dimension” [15]. Hausdorff-Besicovith dimension is therefore a number, which
198 Vı́t Pászto, Lukáš Marek, Pavel Tuček
describes the complexity of an object and its value is non-integer. The bigger the
value of Hausdorff-Besicovitch dimension, the more complex the shape of object and
the more fills the space. In sense of Euclidean geometry, dimension is 1 for straight
line, 2 for circle or square and 3 for cube or sphere, all. For real objects in plane,
Hausdorff-Besicovitch dimension (fractal dimension) has values greater than 1 and
less than 2. It obvious that Euclidean, integer, dimension is extreme case of fractal,
non-integer, dimension. So it is claimed that regions with regular and less complex
shape has lower fractal dimension (approaching to 1) and vice versa – the more
irregular and complex shapes, the higher fractal dimension (approaching to 2). Values
of fractal dimension of land-cover regions vary between 1 and 2 because of the fact
that area represented in the plane space without vertical extend is in fact enclosed
curves. And fractal dimension of curves lies between 1 and 2.
As depicted in mentioned publications, fractals provide tool for better
understanding the shape of given object. Furthermore, fractal geometry brings very
effective apparatus to measure object’s dimension and shape metrics in order to
supply or even substitute other measurable characteristics of the object.
Next paragraphs do not intent to completely identify socio-economical,
demographical and geographical aspects of land-cover current state in Olomouc
region. The case study demonstrates the opportunity and power of fractal analyses of
geographical data. Particularly, objectives are: land-cover pattern and its geometric
representation in GIS. Land-cover pattern fractal analyses, among others, identify
areas with maximal and minimal fractal dimension to evaluate complexity of such
areas.
2 Methods, Data and Study Region
There is a number of methods for estimating fractal dimension and as [20] shows,
results obtained by different methods often differ significantly. Also not only the
method itself, but also the software, which calculates the fractal dimension may
contribute to the differences [20]. In this case, ESRI ArcGIS 9.x software was used.
It has to be mentioned that in the case study statistical testing was accomplished.
For this purpose, R-project statistical software was used. Because of the well-known
formulas and characteristics, detailed description of the methods is not stated,
excluding the program code in R-project environment. The methods were, namely,
analysis of variance (hereafter as ANOVA) and Tukey's honest significant difference
test and t-test.
2.1 General Calculation of Fractal Dimension
As mentioned above, there exists a plenty of methods to calculate fractal dimension
[3, 6, 8, 18, 25]. For this purpose, one of the most general fractal dimension formula
is used:
Fractal Dimension Calculation for CORINE Land-Cover Evaluation in GIS 199
2 ⋅ log P
D= . (0)
log A
Where P is the perimeter of the space being studied at a particular length scale, and
A is the two-dimensional area of the space under investigation [26]. Formula (1) was
used to calculate fractal dimension for land-cover regions classified by Level 1 of the
hierarchy. Formula (1) can be easily computable directly within GIS, thus ESRI
ArcGIS 9.x was used in order to obtain fractal dimension values.
2.2 Statistical Evaluation of Fractal Dimension
After the fractal dimensions of land-cover shapes were calculated, the statistical
evaluation was done. Firstly, the differences of fractal dimension among objects of
various origins were testing using ANOVA. A null hypothesis, stated as ”there are no
differences among mean values of the classes”, was formulated. In the case that null
hypothesis was denied, the differences among groups were statistically significant and
Tukey's honest significant difference test (hereafter as TukeyHSD) was performed.
Although TukeyHSD is weaker test than ANOVA [21], it allowed multiple
comparison procedure and statistical testing for finding particular differences among
class couples. Thus, it could be helpful in the evaluation of fractal analysis [12]. If the
criterion of TukeyHSD was on the boundary between a distinction and non-
distinction, the t-test was also performed.
Example of a program commands used in R-project for an analysis of variance:
setwd("C:/Data/")
fd=read.csv2("fd.txt")
anova<-aov(fd[,1]~fd[,2])
summary(anova)
TukeyHSD(anova)
plot(TukeyHSD(anova))
2.3 Data and Study Region
For the case study, territory of Olomouc region was used (Fig. 2). Its area is
approximately 804 km2 and every single type of LEVEL1 land-cover classification is
represented. It is necessary to note that CORINE Land-Cover dataset from year 2000
was examined. Olomouc region is mainly covered by the agricultural areas, but the
north-east part is almost completely covered by forests, because of military area
occurrence. Despite this fact, Olomouc region is the most typically agricultural region
with a great number of dispersed villages (Fig. 3).
200 Vı́t Pászto, Lukáš Marek, Pavel Tuček
Fig. 2. Position of Olomouc region within Czech Republic (brown filled area).
3 Case Study: Fractal Analysis of Land-Cover within Olomouc
Region
Visualization of land-cover in Olomouc region, which has fractal structure typical for
landscape, is shown in Fig. 3. Areas with maximal and minimal fractal dimension,
both for artificial areas and natural areas, are also outlined. From Artificial areas, the
maximal fractal dimension (D=1.393) has town Hlubočky (Mariánské Údolí) and the
minimal value of fractal dimension has part of Bystrovany municipality (D=1.220).
In the first case, the maximal fractal dimension is caused by the topography of the
town. Hlubočky (Mariánské Údolí) was built in steep valley on both sides of the river
and thus is forced to follow highly irregular topography, which results into observed
fractal dimension.
Fractal Dimension Calculation for CORINE Land-Cover Evaluation in GIS 201
Fig. 3. Olomouc region land-cover in 2000 and highlighted areas with minimal and maximal
fractal dimension.
On the contrary, part of Bystrovany municipality represents distinct regular shape
– almost square. There were no landscape borders or limitations when the settlement
was built and regular fabrication of the build-up area (agricultural facility) was,
probably, the most logical one. From natural areas, maximal fractal dimension has the
Wetland area of Bystřice river (D=1.396), which is part of highlands with almost
intact landscape. Very regular shape has forest southern from Olomouc called Les
Království and its fractal dimension (D=1.193) corresponds with that fact.
At last, join of all areas within class was accomplished and overall fractal
dimension calculated. Results are shown in Table 1.
Table 1. Overall fractal dimension of particular land-cover classes in Olomouc region.
Class index Land-Cover Class Fractal Dimension
A Artificial areas 1.438574
B Agricultural areas 1.385772
C Forests and seminatural areas 1.350355
D Wetlands 1.395799
E Water bodies 1.263722
202 Vı́t Pászto, Lukáš Marek, Pavel Tuček
It is clear from Table 1 that highest fractal dimension have Artificial areas, which
represents in the very most cases man-made build-up areas (villages, towns, various
facilities). Although knowledge how to plan and build up the settlement more
properly was known long ago, urban sprawl emerged and has great influence on the
irregular shape of artificial areas. Wetlands are very specific class, which are fully
determined by natural processes and its fractal dimension is the highest among natural
areas. On the other hand, Water bodies have the lowest overall fractal dimension. It is
necessary to note that line objects, which would fall into this class (rivers, streams,
channels, etc.), are excluded due to CORINE classification methodology. And that is
why the water bodies have this overall fractal dimension – only man-made or man-
regulated water bodies were identified by the classification process and consequently
analyzed.
To objectively prove the significant statistical difference among the land-cover
classes, the ANOVA was used. Before that, Shapiro-Wilk test (W=0.98) was
performed to check up the normality of data. Normality was confirmed and ANOVA
could be used. It was then proven that mean fractal dimension values are significantly
different among areas of the various origin and thus the classes are different too. One
can then claim that classes (Table 1) originate from diverse processes. To acquire
more detailed information, TukeyHSD was performed. This test allowed multiple
comparisons among classes and identified particular statistically significant
differences. TukeyHSD had two main outputs, tabular output, which mainly
contained p-value of difference between two classes, and also the graphical output,
which described the differences and is self-explanatory.
The graphical output (Fig. 4) clearly shows classes with higher variability or which
are more similar in chosen characteristics. Based on the TukeyHSD, it was possible to
state that Wetlands (D) were the most different from all classes. Furthermore, the
second class, which could have been recognizable is Forests and seminatural areas
(C), which were on the boundary of a distinction with regard to Artificial areas (A)
and Agricultural areas (B). T-tests proved that class Forests and seminatural areas
really differed from both, Artificial areas (p-value=0.006) and Agricultural areas (p-
value=0.015). Based on previous findings, one can claim that by calculating the
fractal dimension of land-cover areas and consequence statistics it is possible to
study, evaluate and interpret the processes lying underneath the current land-cover
appearance.
According to the CORINE Land-Cover classification system and acquisition of the
dataset in reference scale 1: 100.000, influence of generalization on the fractal
analysis needs to be taken into account. The more generalized areas in land-cover
classes, the more regular their shapes. And the results of fractal analyses are less
accurate (in sense of capturing objects as much realistically as it is possible).
Furthermore, formula (1) implies that the longer perimeter of the shape, the higher
fractal dimension as a result. And this is very important fact, when calculation using
formula (1) is used. Fractal analysis results are then influenced by the factors from
logic sequence:
Fractal Dimension Calculation for CORINE Land-Cover Evaluation in GIS 203
reference scale of map/dataset – generalization degree – perimeter of an area –
fractal dimension value
Fig. 4. Graphical output of TukeyHSD. Dotted vertical line is the mean, horizontal lines
describe comparison between two classes (meanings of the characters on y-axis are in Table 1).
5 Conclusion
The use of fractal geometry in evaluating land-cover areas was presented. Resulting
values of fractal dimension of such areas were commented using expert knowledge
of the Olomouc region. Geographical context was mentioned too and proper
visualization was made as well. Overall fractal dimension was calculated for
comprehension amongst land-cover classes. Finally, some important aspects
of generalization influence and CORINE classification system on the results were
mentioned.
The paper brings to the reader basics of fractal geometry and its possible usage in
geospatial analyses. Brief historical facts are also presented and plenty of publications
204 Vı́t Pászto, Lukáš Marek, Pavel Tuček
and papers noted. It is obligatory to introduce methodological frame of fractal
geometry apparatus, including formula by which the fractal dimension was calculated.
The original case study was carried out to demonstrate practical use of fractal
geometry and consequence analyses. As mentioned above, fractal analysis built its
stable position in various natural sciences, including geoinformatics and
geocomputation
Fractal analyses are very sufficient for measuring complexity or irregularity
of various objects, but there are other metric characteristics of the shape (e.g.
compactness, convexity, roundness, elongation and others) to evaluate objects, areas
in this case, respectively. But the main difference between fractal geometry and this
group of metric characteristics is in use of mathematical apparatus and, what is even
more important, in concept of fractal geometry and chaos theory. And that is why the
fractal geometry built its position in all kind of geospatial analyses.
Acknowledgments. This work was supported by the student project Research of
person movement at the intersection of urban and sub-urban area in Olomouc region
of the Palacký University (Integral Grant Agency, project no. PrF_2010_14).
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