Multifractal portrayal of the distribution of the Swiss population

Multifractal portrayal of the distribution of the Swiss population Carmen Vega Orozco Jean Golay, Mikhail Kanevski Colloque SAGEO November 7, 2012

Outline 1 Introduction 2 Measures of clustering Topological measures Statistical measures Fractal measures 3 Multifractal analysis Case study 4 Conclusions

Outline 1 Introduction 2 Measures of clustering Topological measures Statistical measures Fractal measures 3 Multifractal analysis Case study 4 Conclusions

Introduction The present research studies the spatial pattern of the population distribution in Switzerland. The description is carried out using a wide variety of global spatial structural analysis tools such as topological, statistical and fractal measures, enabling the estimation of the spatial degree of clustering of a point pattern. A particular attention is given to the analysis of the multifractality to characterise the spatial structure of the distribution of the Swiss population at different scales.

Spatial clustering Most of environmental and socio-economic events are observed locations distributed over the geographical space in a non-homogeneously (clustered) manner. To quantify the level of structuring of spatial patterns, many measures of clustering can be used, such as: Topological measures: Voronoï polygons Statistical measures: the Morisita Index, Ripley s K -function (Multi) fractal measures: Sandbox-counting, Box-counting, Rényi s dimensions CSR distributions and validity domain (VD) are used along with the clustering measures in order to assess whether real patterns are clustered or randomly distributed. The clustering of spatial patterns has many important consequences from data collection to exploratory analysis, geostatistical modelling and spatial prediction.

Spatial clustering Most of environmental and socio-economic events are observed locations distributed over the geographical space in a non-homogeneously (clustered) manner. To quantify the level of structuring of spatial patterns, many measures of clustering can be used, such as: Topological measures: Voronoï polygons Statistical measures: the Morisita Index, Ripley s K -function (Multi) fractal measures: Sandbox-counting, Box-counting, Rényi s dimensions CSR distributions and validity domain (VD) are used along with the clustering measures in order to assess whether real patterns are clustered or randomly distributed. The clustering of spatial patterns has many important consequences from data collection to exploratory analysis, geostatistical modelling and spatial prediction.

Spatial clustering Most of environmental and socio-economic events are observed locations distributed over the geographical space in a non-homogeneously (clustered) manner. To quantify the level of structuring of spatial patterns, many measures of clustering can be used, such as: Topological measures: Voronoï polygons Statistical measures: the Morisita Index, Ripley s K -function (Multi) fractal measures: Sandbox-counting, Box-counting, Rényi s dimensions CSR distributions and validity domain (VD) are used along with the clustering measures in order to assess whether real patterns are clustered or randomly distributed. The clustering of spatial patterns has many important consequences from data collection to exploratory analysis, geostatistical modelling and spatial prediction.

Spatial clustering Most of environmental and socio-economic events are observed locations distributed over the geographical space in a non-homogeneously (clustered) manner. To quantify the level of structuring of spatial patterns, many measures of clustering can be used, such as: Topological measures: Voronoï polygons Statistical measures: the Morisita Index, Ripley s K -function (Multi) fractal measures: Sandbox-counting, Box-counting, Rényi s dimensions CSR distributions and validity domain (VD) are used along with the clustering measures in order to assess whether real patterns are clustered or randomly distributed. The clustering of spatial patterns has many important consequences from data collection to exploratory analysis, geostatistical modelling and spatial prediction.

Spatial clustering Most of environmental and socio-economic events are observed locations distributed over the geographical space in a non-homogeneously (clustered) manner. To quantify the level of structuring of spatial patterns, many measures of clustering can be used, such as: Topological measures: Voronoï polygons Statistical measures: the Morisita Index, Ripley s K -function (Multi) fractal measures: Sandbox-counting, Box-counting, Rényi s dimensions CSR distributions and validity domain (VD) are used along with the clustering measures in order to assess whether real patterns are clustered or randomly distributed. The clustering of spatial patterns has many important consequences from data collection to exploratory analysis, geostatistical modelling and spatial prediction.

Spatial clustering Most of environmental and socio-economic events are observed locations distributed over the geographical space in a non-homogeneously (clustered) manner. To quantify the level of structuring of spatial patterns, many measures of clustering can be used, such as: Topological measures: Voronoï polygons Statistical measures: the Morisita Index, Ripley s K -function (Multi) fractal measures: Sandbox-counting, Box-counting, Rényi s dimensions CSR distributions and validity domain (VD) are used along with the clustering measures in order to assess whether real patterns are clustered or randomly distributed. The clustering of spatial patterns has many important consequences from data collection to exploratory analysis, geostatistical modelling and spatial prediction.

Spatial clustering Most of environmental and socio-economic events are observed locations distributed over the geographical space in a non-homogeneously (clustered) manner. To quantify the level of structuring of spatial patterns, many measures of clustering can be used, such as: Topological measures: Voronoï polygons Statistical measures: the Morisita Index, Ripley s K -function (Multi) fractal measures: Sandbox-counting, Box-counting, Rényi s dimensions CSR distributions and validity domain (VD) are used along with the clustering measures in order to assess whether real patterns are clustered or randomly distributed. The clustering of spatial patterns has many important consequences from data collection to exploratory analysis, geostatistical modelling and spatial prediction.

Outline 1 Introduction 2 Measures of clustering Topological measures Statistical measures Fractal measures 3 Multifractal analysis Case study 4 Conclusions

Topological measures Voronoï polygons The Voronoï polygons of a region are a partition of the original space, where each polygon s border is equidistante to the two nearest point. The histogram of the areas of Voronoï polygons describe the spatial irregularity of the pattern. Homogeneous sets follow approximatively Gaussian distribution; while clustered sets are skewed to the left.

Statistical measures The Morisita Index The region is divided into Q quadrats and the number of events n i within each quadrat i is counted. Q i=1 I = Q n i(n i 1) N(N 1) When the points are homogeneously distributed, every I D varies around 1. When the points are clustered, the empty quadrats at small scales increase the value of the index.

y Fractal measures Box-counting A regular grid of boxes of length δ is superimposed on the set and the number of boxes, N(δ), necessary to cover the set is counted. This procedure is repeated using different values of δ. The number of occupied boxes increases with decreasing box size, leading to the following power-law relationship: N(δ) = δ df box quadrats: 2 x 2 quadrats: 10 x 10 9 8 7 quadrats: 30 x 30 quadrats: 60 x 60 log(n(δ)) 6 5 4 Df = 1.64 3 2 1 0 11 12 13 14 15 16 17 log(δ)

Outline 1 Introduction 2 Measures of clustering Topological measures Statistical measures Fractal measures 3 Multifractal analysis Case study 4 Conclusions

Introduction Measures of clustering Case study Population distribution of Switzerland 2000 * Swiss Population Census (2000) * Hectometric resolution (100 x 100 m) Multifractal analysis Conclusions

Case study Rény s dimension Let N(δ) be the number of non-overlapping boxes of size δ needed to cover the fractal and p i the mass probability function in the ith box. The generalized dimension Dq is computed through the parameter q by: Dq = ( N(δ) 1 log (1 q) lim i=1 (p i) q) δ 0 log(1/δ) The D q is estimated from the slope of the linear regression fitting the data of the plot log(i q(δ)) vs. log(δ).

y Case study quadrats: 2 x 2 quadrats: 10 x 10 quadrats: 30 x 30 quadrats: 60 x 60

Case study

Case study 2 1.8 Dq 1.6 Legend Swiss Population In Alps population Out of Alps population Random patterns 1.4 1.2 1 0 1 2 3 4 5 6 7 8 9 10 q

Outline 1 Introduction 2 Measures of clustering Topological measures Statistical measures Fractal measures 3 Multifractal analysis Case study 4 Conclusions

Conclusions A characterisation of the spatial distribution of the Swiss population has been carried out implementing different measures of spatial clustering analysis, in particular the multifractality formalism. The generalized dimensions for the DSP showed a different scaling behaviour of the highly populated areas and less dense regions. Different researches concerning the scaling structure of the spatial pattern of the population distribution have only considered either a monofractal analysis of the occupied land or a multifractal characterisation of the population at the city level. This work goes beyond and deeper analysing the population distribution for Switzerland at the intra-city level using a high-resolution census data presented with population counts in a grid of 100 x 100 meters. Future research Multivariate analyses by taking into account both demographic and socioeconomic data.