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# Multifractal portrayal of the Swiss population

Représentation multifractale de la population suisse
Carmen Delia Vega Orozco, Jean Golay and Mikhail Kanevski

## Abstracts

Fractal geometry is a fundamental approach for describing the complex irregularities of the spatial structure of point patterns. The present research characterizes the spatial structure of the Swiss population distribution in the three Swiss geographical regions (Alps, Plateau and Jura) and at the entire country level. The analyses were carried out by means of a fractal measure (the box-counting dimension) and two multifractal measures (the Rényi generalized dimensions and the multifractal spectrum) for point patterns, which enabled the estimation of the spatial degree of clustering of a distribution at different scales. The Swiss population dataset is presented on a grid of points and thus it can be modelled as a realization of a "point process" where each point is characterized by its spatial location (geometrical support) and a number of inhabitants (measured variable). Results showed that the four patterns are multifractals and their population distribution present different clustering behaviours. Thus, applying multifractal and fractal methods at different geographical regions and at different scales allowed us characterising the degree of clustering of the population distribution in Switzerland and quantifying their dissimilarities. This paper is the first Swiss geodemographic study applying multifractal methods using high resolution data.

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# 1. Introduction

1The spatial clustering of real patterns is an important subject in many fields and its characterization can be assessed by an ample number of indices (Cressie, 1993; Kanevski and Maignan, 2004; Illian et al., 2008). Here, clustering is defined as the spatial non-homogeneity of a point pattern distributed in a geographical space (Tuia and Kanevski, 2008), and variability is related to the variation of the point density. Among these indices, fractal measures are widely developed. Their mathematical framework yields a useful tool for describing the irregularity or complexity of spatial phenomena and imparts great advantages over other traditional statistical methods.

2Introduced by Mandelbrot (1967), the word fractal was first coined to describe sets with abrupt and tortuous edges. A set of points whose any scale portion is statistically identical to the original object (statistical self-similarity) is fractal and it can be characterized by a fractal dimension which refers to the invariance of the probability distributions of the set under geometric changes of scale (Rodríguez-Iturbe and Rinaldo, 1997). In the case of multifractal point sets, all the moments of the probability distribution do not scale equivalently and an entire spectrum of generalized fractal dimensions is required (Grassberger and Procaccia, 1983; Hentschel and Procaccia, 1983; Paladin and Vulpiani, 1987; Tél et al., 1989; Borgani et al., 1993; Perfect et al., 2006; Seuront, 2010; Golay et al., 2014), i.e. the sparser and denser regions of a spatial distribution might have different scaling behaviours.

3Many investigations have demonstrated that environmental, ecological and natural data are fractals. Bunde and Havlin (1994) discussed in detail fractals in biology, chemistry and medicine. Burrough (1981) showed that many data of environmental variables and landscapes display a certain degree of statistical self-similarity over many spatial scales. Goodchild and Mark (1987) presented the relevance of fractals to geographic phenomena. Turcotte and Malamud (2004) showed that landslides, forest fires and earthquakes presented a fractal distribution. Telesca et al. (2001, 2004); Telesca and Lasaponara (2006); Telesca et al. (2007) characterized the spatial and temporal clustering behaviour of earthquakes and forest fire sequences using fractal measures. Frontier (1987) and Seuront (2010) applied fractal theory to ecology and aquatic ecosystems.

4In physical and human geography, fractal analyses have been carried out in many cases. A large literature on urban geography mentions the use of fractals to study the geometry and the creation of central places (Arlinghaus, 1985), the town and city systems (François et al., 1995; Sambrook and Voss, 2001; Chen and Zhou, 2004; Chen, 2014), the irregularities of city morphologies (Batty and Longley, 1994; Frankhauser, 1994), urban growth models (Batty and Longley, 1986; Batty et al., 1989), intra-urban built-up patterns (Batty and Xie, 1996; Frankhauser, 1998; De Keersmaecker et al., 2003), and the dynamics of population growth (Le Bras, 1998; Ozik et al., 2005). Appleby (1996) applied multifractal methods to characterise the distribution pattern of the human population in the United States and Great Britain. Adjali and Appleby (2001) analysed the multifractal behaviour of the distribution of human population in 10 countries around the world suggesting that the multifractal properties of their population distribution could be related to demographic and economic factors. These works have proved that the implementation of the fractal and multifractal formalism is relevant to urban studies.

5In Switzerland, a fractal analysis has been carried out by Frankhauser (2004) who compared the morphology of urban patterns in Europe; Tannier and Pumain (2013) who applied fractal measures to study the urban space structure and the delimitation of built-up areas in Basle; and Kaiser et al. (2009) who applied the lacunarity index to study the clustering urban areas at different scales. Nonetheless, there are not known works concerning any structure analysis of the population distribution using multifractal measures at a local level.

6Thus, the present research aims at characterising the spatial distribution pattern of the population in Switzerland at different scales and in three geographical regions (Alps, Plateau and Jura) through the fractal and multifractal formalism. These areas present different topographical features. The fractal dimension of the Swiss population distribution (SPD) for the three regions and for the entire country was quantified by means of the box-counting method, while the multifractal dimensions were estimated with both the Rényi generalized dimensions and the multifractal spectrum. The application of these methods allowed quantifying the degree of clustering of the studied patterns and quantifying their dissimilarities. Another innovation of this paper lies in the fact that the multifractal analysis of the population distribution was applied to high resolution data which enabled carrying out analyses from an intra-city to the country-size levels.

7Section 2 describes the dataset and the implemented methodology; section 3 provides the results and section 4 presents the conclusions of our findings.

# 2. Theory/calculation

## 2.1. Data

8Switzerland is a landlocked country located in Western Europe (Figure 1). It borders France, Germany, Austria, Liechtenstein and Italy and it covers an area of 41,285 km2. The census of the year 2000 counted 7,351,900 permanent residents and, since then, this amount has increased to 7,954,700 inhabitants (Office, 2010). The population in Switzerland has more than doubled since the beginning of the 20th century, starting from 3.3 million (1900) to 7.95 million (2011). In the period after World War II (1950-1970), the country underwent an important population growth with an annual average rate of about 1.4%. It slowed down (0.6%) from the 1970s to 1990s as a result of immigration restrictions and because of the economic recession. This growth was mostly concentrated in smaller centres and in agglomeration belts; while some larger urban centres experienced population decline. But, since then, the rate of population growth has increased again to 0.8% (Office, 2010) while the population concentration has experienced a reversal trend. Nowadays, Switzerland can be considered as a densely populated country with an average population density of around 193 inhabitants per square kilometre.

Figure 1: Geographical regions in Switzerland

9Geographically, Switzerland can be divided into three main regions: the Swiss Alps, the Plateau and the Jura (Figure 1). Each region presents different geological and topographical features. Likewise, demographically, they also support dissimilar population distributions as presented in Figure 2. This figure displays the SPD of the year 2000 and a 3D visualization of the dataset where an inhomogeneous land-occupation structure is clearly detected with clusters of different sizes.

Figure 2: Population distribution in Switzerland, year 2000.

(a-c) Population distribution in the three geographic regions: Jura, Plateau and Alps, respectively. (d) 3D visualization of the ln-transformed data.

Source: Swiss census 2000

10For instance, while the Alps occupy 60% of the total country territory, only 23% of the population lives in this highly mountainous region (average altitude of 1700 m). The Plateau, which is the economic epicentre, covers 30% of the country’s surface area and it concentrates 2/3 of the total population, most of the Swiss industries and farmlands, as well as the major cities such as Geneva, Bern and Zurich. There are few regions in Europe that are more densely populated than the Plateau (450 people per km2) and, in some areas such as the main cities, the population density can surpass 1000 people per km2 [1]. The Jura constitutes 10% of the country and hosts 9% of the population.

11The population database used in the present study is the Swiss census of the year 2000. These high-resolution data are upheld and managed by the Swiss Federal Statistical Office (Office, 2010) and they can be visualized through the 325,951 nodes (i.e. points) of a grid superimposed onto Switzerland, each of which is associated to the number of people living in a hectare (i.e.100 x 100 m).

12Reference patterns were generated to create confidence intervals of spatial randomness (Illian et al., 2008). This procedure was done by creating a large number of random permutations of the measured value (i.e. number of inhabitants) in order to destroy the dependence existing between the number of inhabitants and the spatial location of each grid node (or point). For this, the number of inhabitants were shuffled and then assigned randomly to the original set of location points (or support grid), ensuring that the marginals of the number of inhabitants remained unchanged. This procedure was iterated 999 times. Figure 3 illustrates the real distribution (left) and one of the simulated random patterns (right). The multifractal methods were also applied to these simulated samples and their results were compared with the original structure of the Swiss data to evaluate the departure of the real pattern from a random structure.

Figure 3: The original population distribution in Switzerland (left) and one of the 999 random permutation distributions simulated for comparisons (right)

## 2.2. Methodology

13A point process is represented as sets of random points (events) generated within a space. Frequently, such processes exhibit a scaling behaviour indicating a high degree of point clustering over all scales (Lowen and Teich, 1995). Fractal and multifractal tools can be used to characterise the intensity of the spatial clustering at a wide range of scales (Lowen and Teich, 1995; Cheng and Agterberg, 1995).

### 2.2.1. Fractal dimension

14According to Lovejoy et al. (1986); Salvadori et al. (1997); Tuia and Kanevski (2008), a fractal dimension can be used to analyse the clustering properties (non-homogeneity) across scales of point process realizations. If the studied distribution is embedded within a 2D space (e.g. geographical space), its fractal dimension ranges from 0 (i.e. the topological dimension of a point) to 2 (i.e. the dimension of geographical space). If the points are dispersed or randomly distributed within the 2D study area, the corresponding fractal dimension is equal to 2. But this value decreases as the level of clustering increases and it can reach 0 if all the points are superimposed at one single location. Thus, fractal dimensions allow detecting the appearance of clustering as a departure from a dispersed or random situation.

15A variety of fractal measures have been proposed such as the box-counting method (Russell et al., 1980; Lovejoy et al., 1986; Tuia and Kanevski, 2008), the sandbox-counting method (Grassberger and Procaccia, 1983; Daccord et al., 1986; Feder, 1988; Tél et al., 1989; Vicsek, 1990; Tuia and Kanevski, 2008) and the information dimension (Balatoni and Rényi, 1956; Hentschel and Procaccia, 1983; Seuront, 2010). The fractal dimension of the SPD was estimated by applying the box-counting method.

#### The Box-Counting Method

16The box-counting method examines how the observed pattern changes with the scale. The computation of this method consists on breaking the fractal object (point set) into pieces by superimposing a regular grid of boxes of size δ1 on the region under study. Then the number of boxes, N(δ1), necessary to cover the fractal object is counted; that is, one counts all occupied boxes despite the number of points in each box. Next, the linear size of the boxes is reduced, δ2 (< δ1) and the number of boxes, N(δ2), is counted again. The algorithm goes on until a minimum size δmin is reached (see example in Figure 4). For a fractal pattern, the scales (δ) and the number of boxes (N(δ)) follow a power law:

where dfbox is the fractal dimension measured with the box–counting method (Tuia and Kanevski, 2008). Theoretically, as introduced formerly, self-similarity is a property present over an infinite range of scales, however, in natural fractals (also named by Mandelbrot as pre-fractals), this property is only encountered statistically over a finite range of scales (Abramenko, 2008) for which it is possible to consider -dfbox as the slope of the linear regression fitting the data of the plot which relates log(N(δ)) to log(δ).

### 2.2.2. Multifractality

17As stated by Mandelbrot (1988), the notion of self-similarity can be extended to measures (spreading mass or probability) distributed on a Euclidean support (e.g. a point set). In this context, fractal sets might be described by not just one fractal dimension (as exposed in the previous subsection 2.2.1), but rather by a function (Stanley and Meakin, 1988) or a spectrum of interlinked fractal dimensions. Such fractal sets are said to be multifractal.

Figure 4: Schematic illustration of the box counting method for the fractal dimension computation: a regular grid of boxes of size δ is superimposed on the studied region. Then the number of boxes of size δ containing a part of the fractal object (in red), N(δ), is counted

18Two different approaches were used to conduct multifractal analysis: (1) Rényi’s generalized dimensions (Hentschel and Procaccia, 1983; Grassberger, 1983; Paladin and Vulpiani, 1987; Tél et al., 1989; Borgani et al., 1993; Perfect et al., 2006; Seuront, 2010) which can be viewed as a global parameter (Chen 2014) examining how different densities are distributed in the space; and (2) the multifractal singularity spectrum (Halsey et al., 1986; Meakin et al., 1986; Stanley and Meakin, 1988; Chhabra and Jensen, 1989), a local parameter (Chen 2014) which examines the regularity of the distribution of regions with similar scaling indices. Although, each of these two methods presents a different approach, they both describe the same information. They are both based on the box-counting method, where a regular grid of boxes of size δ is superimposed on the point set and for both of them a normalized measure (probability distribution) is computed for all boxes (Lopes and Betrouni, 2009) (see Figure 5).

#### The Rényi Generalized Dimensions

19The spectrum of generalized dimensions, Dq, is estimated by computing the Rényi information, Iq(δ), of qth order (Rényi, 1970):

where pi(δ) is the probability population distribution in the ith box of size δ, q ∈ Z, and the sum is taken for all non-empty boxes.

Figure 5. Schematic illustration of the box weighting procedure for the multifractal computation by means of the box counting method with a regular grid of 8 x 8 boxes: the probability distribution is computed over all boxes and weighted by q. This procedure is applied to a wide range of box-size δ. The darker the box (dark red), the higher the measure within the box. For q=0, all boxes are equally weighted (gray); while for q>0, the boxes with relative higher normalized measures gradually gains more importance and their contribution to the entropy will dominate

20When q → 1, Iq(δ) is defined as:

21Then, when applied to multifractal sets, Iq(δ) follows a power law as:

22And the Rényi generalized dimensions are defined as (Hentschel and Procaccia, 1983; Grassberger, 1983; Paladin and Vulpiani, 1987):

23The Dq spectrum is obtained by the slope of the plot relating log(Iq(δ)) to log(1). For monofractal sets, Dq is equal for all q order moments, whereas, for multifractal sets, Dq depends on q and decreases as q increases (Hentschel and Procaccia, 1983; Golay et al., 2014) characterizing the variability of the measure (pi(δ)). For q = 0, D0 is equivalent to the box–counting dimension, dfbox, corresponding to the capacity dimension of the support since it is not sensitive to the density of points because all boxes are equally weighted (= 1). For q > 0, the measure within the boxes gradually gains more importance in the overall box contribution to the Rényi information. That is, the larger the measure within a box, the higher the weight of the box.

24Thus, higher q order moments capture the scaling behaviour of regions where the measure is clumped. It is also important to notice that D1 and D2 correspond to the information dimension and the correlation dimension respectively (Grassberger and Procaccia, 1983; Halsey et al., 1986).

#### Multifractal Singularity Spectrum

25The multifractal singularity spectrum is a local parameter (Chen 2014) which provides an alternative way to describe the scaling behaviour of a pattern through an interlinked set of Hausdorff dimensions, f(α), associated to a singularity strength α (Halsey et al., 1986; Meakin et al., 1986; Chhabra and Jensen, 1989). This singularity strength is an index providing information about the local scaling behaviour of the measure (degree of regularity). Thus, αi is first calculated for each ith box of size δ by:

where pi(δ) is the probability population distribution inside the ith box of size δ. Then, the number of boxes, Nα(δ), having a singularity strength in the neighbourhood of α (α + dα), can be related as:

where f(α) is the fractal dimension of the set of boxes with the same α (Halsey et al., 1986; Meakin et al., 1986; Chhabra and Jensen, 1989). By changing α, one must obtain a spectrum of f(α). Although, the singularity spectrum can be computed directly from the data, it is related to the Rényi generalized dimensions through a Legendre transform as:

for more details see (Halsey et al., 1986; Meakin et al., 1986; Mandelbrot, 1988; Seuront, 2010). Chhabra and Jensen (1989) proposed a method for determining the multifractal singularity spectrum as a function of the q orders without the application of the Legendre transform. Let μ(q) be the normalized measure of the probabilities in the boxes of size δ, such as:

where, again, q provides a tool for exploring denser and rarer regions of the singular measure (Seuront, 2010). Then, α(q) and f(α(q)) can be computed as:

and

26The multifractal spectrum is obtained by plotting the singularity spectrum f(α(q)) vs. the singularity exponent α(q). For q = 0, f(α(0)) takes its maximum value and is equal to D0 and dfbox. For q = 1, f(α(1)) = α(1) = D1 is the information dimension.

# 3. Results and discussion

27As it was exposed in section 2, the Swiss population can be divided into three subsets according to the main geographic regions: the Swiss Alps, the Plateau and the Jura. In this study, the analysis of the Swiss population was first conducted from a global perspective and was then narrowed down to each of the three subsets.

28Figure 6 shows the fractal dimensions of the SPD in 2000 in both the entire country and the three geographic regions. It also exhibits the dependence between the logarithm of the number of boxes necessary to cover the pattern and the logarithm of the box size. The error values indicate the standard deviation of the slopes from the scales ≥ 4 km. The supports of all four patterns are fractals. These results bring to light three main fractal behaviours characterized at different scale ranges: one fractal behaviour at scales ≤ 1 km (~log2(δ)=10), a second at scales ranging from 1 to ~ 4 km (~log2(δ)=12) and the third, from which the rest of this paper is focused on, at scales ≥ 4 km. Certainly, these behaviours are the result of the influence of socioeconomic and topographic factors structuring the human settlement process in Switzerland (FOH 2006).

29The closer to 0 the values of dfbox, the stronger the aggregation of the population, whereas, values of dfbox close to 2 indicate a homogeneous/random distribution. However, there are no differences in the dfbox values of the four patterns from those of their corresponding simulated samples meaning that the random patterns present the same degree of clustering as for the real patterns. This is due to the fact that the random patterns were generated on the same support of the original database; instead the number of inhabitants was shuffled. Therefore, since in the computation of the fractal dimension the number of inhabitants is not taken into account, there are no differences between the original patterns and their simulated samples. In addition, Figure 6 does not reveal significant differences between the estimated dfbox values of the different regions. These drawbacks lead to the conclusion that a comprehensive analysis of the SPD requires more than a single fractal dimension to quantify and characterise the irregularities of the three different regions.

30To overcome the limitation of the fractal dimension in differentiating the four studied patterns, a multifractal analysis was carried out by means of the Rényi generalized dimensions, Dq, and the multifractal spectrum, f(α(q)). This multifractal formalism was limited to only positive values of q because, when working with finite datasets (pre-fractals), the boxes with very small populations will dominate in the entropy. This generates instability in the statistics computation of the box and will dramatically affect the interpretation of the real scaling behaviour of the low density areas. Thus, we focused on the characterisation of the populated areas in Switzerland with relative high density values.

Figure 6: The box-counting fractal dimension of the SPD in 2000 for the entire country (grey dots), the Jura region (gold diamonds), the Plateau region (orange squares), the Alp region (green triangles) and their corresponding simulated patterns (black empty dots). The slopes were computed for scales ≥ 4 km

31Multifractal analyses were performed by inspecting the Swiss population patterns at different levels and by changing the values of q. The probability distribution, pi, was calculated as the fraction of the population in the ith box of size δ (pi=ni/N, where ni is the number of inhabitants falling in the box i and N is the total population in the country).

32Figure 7 shows the Rényi generalized dimensions for the four studied patterns and for 0 ≤ q ≤ 10. The dependence between Dq and q is non-linear for each considered SPD, which highlights their multifractal nature. Dq values denote the degree of clustering of the distribution (non-homogeneity). The width of the spectrum of Dq values (Dmax Dmin) is an indicator of the heterogeneity of the densities (complexity) in the distribution; where a wider spectrum indicates a more heterogeneous distribution of the irregularities of the pattern (Tarquis et al., 2006). The slope of the Dq curve is an indicator of how clustered the population distribution is; where the steeper the slope is, the more unevenly distributed the population densities are and the more clustered the high density areas are (Dauphiné, 2011).

Figure 7: The Rényi generalized dimensions of the SPD in 2000 for the entire country (grey dot lines), the Jura region (orange dot lines), the Plateau region (gold dot lines), the Alp region (green dot lines) and their corresponding simulated patterns (coloured shadowed areas)

33The Dq curves of the four SPD patterns do not fall inside the shadowed areas of the simulated patterns and their curves decline faster than those from the simulated ones. This substantial departure from the unstructured distributions indicates the significance of the clustering of the SPD patterns. Furthermore, the Dq curves of the SPD in the Alps and Jura regions decrease faster than those of the Plateau and the entire country, exposing a higher clustering of the population distribution in these regions (Alps and Jura).

34Regarding the width of the Dq spectrum (Dmax-Dmin), the four curves revealed a strong irregularity between the high and low densely populated areas for each of the considered regions; though, it is more accentuated in the mountainous regions (Alps and Jura). This can also be depicted by comparing the differences of D1D0. The D0 values indicate how the support of the SPD (where the population is counted) occupies the land and it is equivalent to the fractal dimension of the box-counting method presented before. The Dq>0 values determine how the population is distributed on the support with D1 measuring the pure state of the population distribution. The slope of the Dq curves also exposes the degree of clustering of all of the patterns.

35The multifractal spectrum of the four SPDs is illustrated in Figure 8. The multifractal spectrum of the four SPDs is significantly different from that of the corresponding shuffled patterns. The asymmetry of the original distributions is more skewed to the left than for those of the shuffled data confirming that the SPDs present a significant degree of clustering. This reflects the uneven distribution of the irregularities in the patterns.

Figure 8: The multifractal spectrum of the SPD in 2000 for the entire country (grey dot lines), the Jura region (orange dot lines), the Plateau region (gold dot lines), the Alp region (green dot lines) and their corresponding simulated patterns (coloured shadowed areas)

36The computed multifractal spectrum is in agreement with the estimations of the Rényi generalized dimensions. The SPD of the entire country has a multifractal behaviour similar to that of the population of the Plateau, but its heterogeneity is slightly higher as shown by the width of the αmax and αmin values. This similarity can be interpreted as the strong contribution of the Plateau to the whole structure when analyses are conducted at the entire country level. This is due to the fact that the Plateau concentrates two-thirds of the total population as well as the major cities of the country such as Zurich, Geneva, Basel, Bern and Lausanne.

37The multifractal spectrum obtained for the Alps and the Jura are even more skewed to the left than the previous ones (Swiss and Plateau), which is an indicator of highly clustered distributions. This is also depicted by their f(α(q)) curves which are lower and wider than those of the Plateau and the entire country. Although, their multifractal natures are quite similar, the irregularities in the Alps are more heterogeneous and unevenly distributed than the Jura region.

38The significant clustering behaviour observed in the Alps and Jura regions can be mainly explained by the influence of the topography. Topographic features in these mountainous regions are the major factors constraining the land occupation which force populations to live in specific areas of the main valleys.

# 4. Conclusions

39A fractal and multifractal analysis was carried out to characterise the spatial structure of the population distribution in Switzerland and in the three Swiss geographical regions (Jura, Plateau and Alps). The fractal dimension computed from the box-counting method showed that the support of the population distribution is fractal. However, it provided weak information to characterise the dissimilarities between the population distributions in the three geographical regions. By applying the multifractal formalism we were able to both quantify the degree of clustering of each pattern and quantify the dissimilarities of their clustering properties. The Rényi generalized dimensions and the multifractal spectrum of the four spatial population distributions (SPD) showed different scaling behaviours between the high and low densely populated areas revealing the multifractal behaviour of the patterns. The used of the generated random patterns allowed quantifying the significance of the degree of clustering of each original pattern. By comparing the observed pattern with these references patterns, there is no need to recourse to statistical tests for statistical significance.

40Analyses highlighted that the SPD of the Plateau region is more homogeneous than those of the Alps and Jura regions. Additionally, the study pointed out the importance of multifractal analyses at different scale levels and at different geographic subregions, which helps discerning between distinct underlying processes. The distribution of the Swiss population has certainly been shaped by the socio-economic history and the complex geomorphology of the country. These factors have revealed a highly clustered, inhomogeneous and very variable population distribution at different spatial scales.

41This investigation is the first Swiss geodemographic analysis which applies the multifractal formalism to high resolution data carrying out analysis from intra- to inter-“city” levels without making any hypotheses on the definition of the “urban zone”. The analysis of the SPD is an interesting and challenging task and constitutes a fundamental approach for demography studies.

# 5. Acknowledgements

42This work was partly supported by the Swiss National Science Foundation Project No. 200021-140658, "Analysis and Modelling of Space-Time Patterns in Complex Regions".

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## Bibliography

Abramenko V., 2008, “Multifractal nature of solar phenomena”, in Wang P. (ed.), Solar physics research trends, New York, Nova, Ch.2, 95-136.

Adjali I., Appleby S., 2001, “The multifractal structure of the human population distribution”, in Tate N., Atkinson P. (eds.), Modelling scale in geographical information science, Chichester, Wiley, Ch.4, 69-85.

Appleby S., 1996, “Multifractal characterization of the distribution pattern of the human population”, Geographical Analysis, Vol.28, No.2, 147-160.

Arlinghaus S.L., 1985, “Fractals take a central place”, Geografiska Annaler. Series B, Human Geography, Vol.67, No.2, 83-88.

Balatoni J., Rényi A., 1956, “On the notion of entropy”, Publications of the Mathematical Institute of the Hungarian Academy of Science, Vol.1, 5-40, English translation in Selected Papers of Alfred Rényi, Vol.1 (1976) 558-584, Akademiat Kiado, Budapest.

Batty M., Longley P., 1986, “The fractal simulation of urban structure”, Environment and Planning A, Vol.18, No.9, 1143-1179.

Batty M., Longley P., 1994, Fractal cities, London, Academic Press.

Batty M., Longley P., Fotheringham S., 1989, “Urban growth and form: scaling, fractal geometry, and diffusion-limited aggregation”, Environment and Planning A, Vol.21, No.11, 1447-1472.

Batty M., Xie Y., 1996, “Preliminary evidence for a theory of the fractal city”, Environment and Planning A, Vol.28, No.10, 1745-1762.

Borgani S., Plionis M., Valdarnini R., 1993, “Multifractal analysis of cluster distribution in two dimensions”, The Astrophysical Journal, Vol.404, No.1, 21-37.

Bunde A., Havlin S., 1994, Fractals in Science, Berlin, Springer, 2nd Edition.

Burrough P., 1981, “Fractal dimensions of landscapes and other environmental data”, Nature, Vol.294, No.5838, 240-242.

Chen Y., Zhou Y., 2004, “Multi-fractal measures of city-size distributions based on the three-parameter Zipf model”, Chaos, Solitons & Fractals, Vol.22, No.4, 793-805.

Chen Y., 2014, “Multifractals of central place systems: models, dimension spectrums, and empirical analysis”, Physica A, Vol.402, 266-282.

Cheng Q., Agterberg F., 1995, “Multifractal modeling and spatial point processes”, Mathematical Geology, Vol.27, No.7, 831-845.

Chhabra A., Jensen R., 1989, “Direct determination of the f(α) singularity spectrum”, Physical Review A, Vol.62, No.12, 1327-1330.

Cressie N., 1993, Statistics for spatial data, New York, Wiley.

Daccord G., Nittman J., Stanley H., 1986, “Radial viscous fingers and diffusion-limited aggregation: fractal dimension and growth sites”, Physical Review Letters, Vol.56, No.4, 336-339.

Dauphiné A., 2011, Géographie fractale, Paris, Lavoisier.

De Keersmaecker M., Frankhauser P., Thomas I., 2003, “Using fractal dimensions for characterizing intra-urban diversity: the example of Brussels”, Geographical Analysis, Vol.35, No.4, 310-328.

Feder J., 1988, Fractals (Physics of solids and liquids), New York, Plenum Press.

FOH – Federal Office for Houssing, 2006, “Human settlement in Switzerland, spatial development and housing”, Swiss Confederation, Housing Bulletin, Vol.78.

Frankhauser P., 1994, La fractalité des structures urbaines, Paris, Anthropos.

Frankhauser P., 1998, “The fractal approach: a new tool for the spatial analysis of urban agglomerations”. Population, Vol.1, 205–240, an English Selection, Special issue New methodological Approaches in the Social Sciences.

Frankhauser P., 2004, “Comparing the morphology of urban patterns in Europe: a fractal approach”, in Borsdorf, A., Zembri, P. (eds.), European Cities Insights on Outskirts. Structures, Brussels, 2, COST Action C10, Urban Civ. Eng., 79-105.

François N., Frankhauser P., Pumain D., 1995, “Villes, densité et fractalité. Nouvelles représentations de la répartition de la population”, Annales de la recherche urbaine, Vol.67, 55-63.

Frontier S., 1987, “Applications of fractal theory to ecology”, in Legendre, P., Legendre, L. (eds.), Developments in Numerical Ecology, Berlin, Springer Verlag, 335-378, nATO ASI Series G: Ecological Sciences Vol.14.

Golay J., Kanevski M., Vega Orozco C., Leuenberger M., 2014, “The multipoint Morisita index for the analysis of spatial patterns”, Physica A, Vol.406, 191-202.

Goodchild M., Mark D., 1987, “The fractal nature of geographic phenomena”, Annals of the Association of American Geographers, Vol.77, No.2, 265-278.

Grassberger P., 1983, “Generalized dimensions of strange attractors”, Physics Letters A, Vol.97, No.6, 227-230.

Grassberger P., Procaccia I., 1983, “Measuring the strangeness of strange attractors”, Physica D, Vol.9, No.1-2, 189-208.

Halsey T., Jensen M., Kadanoff L., Procaccia I., Shraiman B., 1986, “Fractal measures and their singularities: the characterization of strange sets”, Physical Review A, Vol.33, No.2, 1141-1151.

Hentschel H., Procaccia I., 1983, “The infinite number of generalized dimensions of fractals and strange attractors”, Physica D, Vol.8, No.3, 435-444.

Illian, J., Penttinen, A., Stoyan, H., Stoyan, D., 2008, Statistical analysis and modelling of spatial point patterns, Chichester, Wiley, 1st Edition, page 52.

Kaiser, C., Kanevski, M., Da Cunha, A., Timonin, V., 2009, “Emergence of Swiss metropole and scaling properties of urban clusters”, International Conference on Emergence in Geographical Space S4, Paris, 23-25 November, pp. 23-25.

Kanevski M., Maignan M., 2004, Analysis and modelling of spatial environmental data, Lausanne, EPFL Press.

Le Bras H., 1998, La planète au village: migrations et peuplement en France, France, Editions de l’Aube.

Lopes R., Betrouni N., 2009, “Fractal and multifractal analysis: a review”, Medical Image Analysis, Vol.13, No.4, 643-649.

Lovejoy S., Schertzer D., Ladoy P., 1986, “Fractal characterization of inhomogeneous geophysical measuring networks”, Nature, Vol.319, No.6048, 43-44.

Lowen S., Teich M., 1995, “Estimation and simulation of fractal stochastic point processes”, Fractals, Vol.3, 183-210.

Mandelbrot B., 1967, “How long is the coast of Britain? Statistical self-similarity and fractional dimension”, Science, Vol.156, No.3775, 636-638.

Mandelbrot B., 1988, “An introduction to multifractal distribution functions”, in Stanley H., Ostrowsky N. (eds.), Random fluctuations and pattern growth: experiments and models, Dordecht, Kluwer Academic, pp. 279-291, nATO ASI Series E: Aplied Sciences Vol.157.

Meakin P., Coniglio A., Stanley H., Witten T., 1986, “Scaling properties for the surfaces of fractal and nonfractal objects: an infinite hierarchy of critical exponents”, Physical Review A, Vol.34, No.4, 3325-3340.

Office S.F.S., 2010, “Population”, Panorama, Switzerland, http://www.bfs.admin.ch/bfs/portal/en/index/themen/01/01/pan.html.

Ozik J., Hunt B., Ott E., 2005, “Formation of multifractal population patterns from reproductive growth and local resettlement”, Physical Review E, Vol.72, No.046213, 1-15.

Paladin G., Vulpiani A., 1987, “Anomalous scaling laws in multifractal objects”, Physics Reports (Review Section of Physics Letters), Vol.156, No.4, 147-225.

Perfect E., Gentry R., Sukop M., Lawson J., 2006, “Multifractal Sierpinski carpets: theory and application to upscaling effective saturated hydraulic conductivity”, Geoderma, Vol.134, No.3-4, 240-252.

Rodríguez-Iturbe I., Rinaldo A., 1997, Fractal river basins: chance and self-organization, UK, Cambridge University Press.

Russell D., Hanson J., Ott E., 1980, “Dimension of strange attractors”, Physical Review Letters, Vol.45, No.14, 1175-1178.

Salvadori G., Ratti S. P., Belli G., 1997, “Fractal and mutlifractal approach to environmental pollution”, Environmental Science and Pollution Research, Vol.4, No.2, 91-98.

Sambrook R., Voss R., 2001, “Fractal analysis of US settlement patterns”, Fractals, Vol.9, No.3, 241-250.

Seuront L., 2010, Fractals and multifractals in ecology and aquatic science, Boca Raton, CRC Press.

Stanley H., Meakin P., 1988, “Multifractal phenomena in physics and chemistry”, Nature, Vol.335, No.6189, 405-409.

Tannier C., Pumain D., 2013, “Fractals in urban geography: a theoretical outline and an empirical example”, Cybergeo: http://cybergeo.revues.org, No.307, 20 April 2005.

Tarquis A., McInnes K., Key J., Saa A., García M., Díaz M., 2006, “Multiscaling analysis in a structured clay soil using 2D images”, Journal of Hydrology, Vol.322, No.1-4, 236-246.

Telesca L., Amatucci G., Lasaponara R., Lovallo M., Rodrigues M., 2007, “Space-time fractal properties of the forest-fire series in central Italy”, Communications in Nonlinear Science and Numerical Simulation, Vol.12, No.7, 1326-1333.

Telesca L., Cuomo V., Lapenna V., Macchiato M., 2001 “Identifying space-time clustering properties of the 1983-1997 Irpinia-Basilicata (Southern Italy) seismicity”, Tectonophysics, Vol.330, No.1-2, 93-102.

Telesca L., Lapenna V., Macchiato M., 2004, “Mono- and multi-fractal investigation of scaling properties in temporal patterns of seismic sequences”, Chaos, Solitons and Fractals, Vol.19, No.1, 1-15.

Telesca L., Lasaponara R., 2006, “Emergence of temporal regimes in fire sequences”, Physica A, Vol.360, No.2, 543-547.

Tél T., Fülöp A., Vicsek T., 1989, “Determination of fractal dimensions for geometrical multifractals”, Physica A, Vol.159, No.2, 155-166.

Tuia D., Kanevski M., 2008, “Environmental monitoring network characterization and clustering”, in Kanevski M. (ed.), Advanced Mapping of Environmental Data: Geostatistics, Machine Learning and Bayesian Maximum Entropy, London, Iste/Wiley, Ch.2, pp. 19-46.

Turcotte D., Malamud B., 2004, “Landslides, forest fires, and earthquakes: examples of self-organized critical behaviour”, Physica A, Vol.340, No.4, 580-589.

Vicsek T., 1990, “Mass multifractals”, Physica A, Vol.168, No.1, 490-497.

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## Notes

1 http://www.swissworld.org/en/geography/ consulted in 01/2013

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## List of illustrations

Title Figure 1: Geographical regions in Switzerland
URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-1.jpg
File image/jpeg, 300k
Title Figure 2: Population distribution in Switzerland, year 2000.
Caption (a-c) Population distribution in the three geographic regions: Jura, Plateau and Alps, respectively. (d) 3D visualization of the ln-transformed data.
URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-2.jpg
File image/jpeg, 280k
Title Figure 3: The original population distribution in Switzerland (left) and one of the 999 random permutation distributions simulated for comparisons (right)
URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-3.jpg
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URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-4.png
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Title Figure 4: Schematic illustration of the box counting method for the fractal dimension computation: a regular grid of boxes of size δ is superimposed on the studied region. Then the number of boxes of size δ containing a part of the fractal object (in red), N(δ), is counted
URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-5.jpg
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URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-6.png
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Title Figure 5. Schematic illustration of the box weighting procedure for the multifractal computation by means of the box counting method with a regular grid of 8 x 8 boxes: the probability distribution is computed over all boxes and weighted by q. This procedure is applied to a wide range of box-size δ. The darker the box (dark red), the higher the measure within the box. For q=0, all boxes are equally weighted (gray); while for q>0, the boxes with relative higher normalized measures gradually gains more importance and their contribution to the entropy will dominate
URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-7.jpg
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URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-8.png
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URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-16.png
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Title Figure 6: The box-counting fractal dimension of the SPD in 2000 for the entire country (grey dots), the Jura region (gold diamonds), the Plateau region (orange squares), the Alp region (green triangles) and their corresponding simulated patterns (black empty dots). The slopes were computed for scales ≥ 4 km
URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-17.jpg
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Title Figure 7: The Rényi generalized dimensions of the SPD in 2000 for the entire country (grey dot lines), the Jura region (orange dot lines), the Plateau region (gold dot lines), the Alp region (green dot lines) and their corresponding simulated patterns (coloured shadowed areas)
URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-18.jpg
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Title Figure 8: The multifractal spectrum of the SPD in 2000 for the entire country (grey dot lines), the Jura region (orange dot lines), the Plateau region (gold dot lines), the Alp region (green dot lines) and their corresponding simulated patterns (coloured shadowed areas)
URL http://journals.openedition.org/cybergeo/docannexe/image/26829/img-19.jpg
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## References

### Electronic reference

Carmen Delia Vega Orozco, Jean Golay and Mikhail Kanevski, « Multifractal portrayal of the Swiss population », Cybergeo : European Journal of Geography [Online], Systems, Modelling, Geostatistics, document 714, Online since 09 March 2015, connection on 06 May 2021. URL : http://journals.openedition.org/cybergeo/26829 ; DOI : https://doi.org/10.4000/cybergeo.26829

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### Carmen Delia Vega Orozco

PhD student
Institute of Earth Surface Dynamics, Faculty of Geosciences and Environment, University of Lausanne, 1015 Lausanne, Switzerland
carmendelia.vegaorozco@unil.ch

### Jean Golay

PhD student
Institute of Earth Surface Dynamics, Faculty of Geosciences and Environment, University of Lausanne, 1015 Lausanne, Switzerland
jean.golay@unil.ch

### Mikhail Kanevski

Professor
Institute of Earth Surface Dynamics, Faculty of Geosciences and Environment, University of Lausanne, 1015 Lausanne, Switzerland
mikhail.kanevski@unil.ch

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