14-17 sept 2023

Outline

  • General context and particularities of Switzerland
  • Data sets and Swiss popular votes
  • Political and geographical features
  • Quantitative treatments
  • Spatial treatments

General context I

  • 26 cantons
    • Divided in 2158 municipalities

  • Four languages
    • German
    • French
    • Italian
    • Romansh

  • Political system: popular initiative and referendum

General context II

  • 26 cantons
    • Divided in 2158 municipalities

  • Four languages
    • German
    • French
    • Italian
    • Romansh

  • Geographical barriers

Dataset: spatial structure and political features

  • \(n=2158\) Swiss communes: shapefile, main spoken language
  • \(p=352\) federal votes from 1971 to 2021: proportion of ‘yes’
    • \(n \times p\) dataset
  • Relative weights \(f\) of the municipalities are proportional to the cumulated number of all votes for \(n\) regions.

Let \(f_i>0\) with \(\sum_{i=1}^n f_i=1\)

##   noMuni       municipality sumVotes marriageAll capitalTax polTerrorism
## 1      1    Aeugst am Albis   196982        65.8       27.8         53.6
## 2      2 Affoltern am Albis   954228        64.8       33.7         60.9
## 3      3         Bonstetten   499275        70.2       33.7         59.5
## 4      4    Hausen am Albis   366969        68.1       31.7         52.0
## 5      5           Hedingen   350420        67.4       29.3         58.2
## 6      6    Kappel am Albis   111520        61.6       26.5         50.2

Votes results example

  • 1971: Federal Decree on the introduction of women’s suffrage in federal matters
  • 2020: Federal Decree on the procurement of new combat aircraft

Political feature and Geographical distances

Kernels

We consider two types of distances:

  • 1. Political Configuration Distances
    These distances measure dissimilarities between political vote results:
    • Political \(\tilde{\chi}^2\) distances
  • 2. Geographical Distances
    These distances capture the geographical characteristics:
    • Road distance
    • Road time
    • Geodesic distance
    • Difference in elevation distance
    • Distance extracted from spatial weights
    • Heat (spatial) kernel based on adjacency \(A\) matrix

    Note: All distances are represented as dissimilarity matrices \(\textbf{D} = (d_{ij} ) ∈ \mathbb{R}^{n×n}\) with the following properties: \({d_{ij} ≥ 0, d_{ij}=d_{ji}, d_{ii}=0}\)

Distances comparison

\(2158 \times 2158\) relations

  • Road distance vs time distance


  • Road distance vs geodesic distance

Accessibility

Averege distance [km] to reach a commune \(j\) from a commune \(i\) by road

Spatial Kernels, based on geographical distances

  1. From the five geographical distances \(\mathbb{D}=(d_{ij})\), spatial kernels can directly be defined as

\[ \mathbf{K}_{\mathbb{D}}=-\frac12\sqrt{\bf{\Pi}}\,\mathbf{H}\, \mathbb{D} \mathbf{H}^\top\sqrt{\bf{\Pi}} \] where \(\mathbf{\Pi = diag(f)}\) and \(\mathbf{H=I_n-1_n f^\top}\), and which defines a perfectly valid index of spatial autocorrelation

\[ \delta=\frac{\mbox{Tr}(\mathbf{K}_{\mathbb{D}}\, \mathbf{K_X})}{\mbox{Tr}(\mathbf{K_X})} \] without needing to construct the spatial weights matrix \(\mathbf{W}\).

Spatial Kernels, based on spatial weights

  1. From (adjusted, reversible) spatial weights $\mathbf{W}$ and the affinity matrix $\mathbf{A}$ (based on the adjacency of the 2158 municipalities), computing the *weighted Laplacian*.

$$ \mathbf{K_{Lap} = \bf{\Pi}^{-\frac{1}{2}} [diag(A1)-A]\bf{\Pi}^{-\frac{1}{2}} } $$ Set a diffusive time $t>0$ (here $t=10$). $$ \mathbf{K_{heat}}(t) = \exp(-t\mathbf{K_{Lap}}) - \mathbf{\sqrt f} \mathbf{\sqrt f^\top} $$ The associated Markov transition matrix $W(t)$ (spatial weights) with stationary distribution $f$ is given by $$ \mathbf{W(t) = \Pi^{-\frac{1}{2}} \exp(−tK_{Lap})\Pi^{\frac{1}{2}}} $$ Mathematically, the matrix of spatial weights $\mathbf{W}$ is the matrix of a reversible Markov chain with stationary distribution $f$. $$ \mathbf{K_W}=\bf{\Pi}^{\frac12}\mathbf{W}\bf{\Pi}^{-\frac12}-\sqrt{\mathbf{f}}\sqrt{\mathbf{f}}^\top $$ which defines the second index of spatial autocorrelation $$ \delta=\frac{\mbox{Tr}(\mathbf{K}_{\mathbf{W}}\mathbf{K_X})}{\mbox{Tr}(\mathbf{K_X})} $$

Weighted MDS

From the kernels \(\mathbf{K_\mathbb{D}}\), one extracts the regional coordinates \(\mathbf{\tilde{X}}\) by weighted MDS:

\[\mathbf{\tilde{X}=-\frac{1}{\sqrt{f}}U\sqrt{\Lambda}}\] where \(\Lambda\) contains the Eigenvalues and \(U\) the Eigenvectors of \(\mathbf{K_\mathbb{D}}\).

MDS, political chi-squared distance

Mapping MDS coordinates

MDS, road distances

Mapping MDS coordinates

MDS, time distances

Mapping MDS coordinates

MDS, geodesic distance

Mapping MDS coordinates

MDS, difference of elevation distance

Mapping MDS coordinates

MDS, weights size distance

Mapping MDS coordinates

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Multivariate, weighted auto-correlation index \(\delta\) and \(RV\) coefficient

With a feature kernel \(K_X\) and a spatial kernel \(K_W\), we can express:

\[\mathbf{ \delta = \frac{Tr(K_W K_X)}{Tr(K_X)} }\] and the \(RV\) index:

\[ \mathbf{ RV=\frac{Tr(K_W K_X)}{\sqrt{Tr(K_W^2)Tr(K_X^2)}} \in [-1,1] } \]

Z-score from \(\delta\) between all configurations

\[\mathbf{ \delta = \frac{Tr(K_W K_X)}{Tr(K_X)} }\]

\[ z=\frac{\delta-\mathbb{E}(\delta) }{\sqrt{\mathbb{V}\mbox{ar}(\delta)}} = \frac{\mbox{RV}-\mathbb{E}(\mbox{RV}) }{\sqrt{\mathbb{V}\mbox{ar}(\mbox{RV})}} \]

  • DX: Political configuration, \(\tilde{\chi}^2\) distances

  • DY: Road distance

  • DZ: Road time

  • DW: Euclidean distance

  • DV: Diff. of elevation distance

  • Df: Kernel with weights themselves

\[ \small{Note: \alpha=0.001\%: z > 4.26} \]

Moran

Mapping Moran

MDS, Bern - Valais

Moran, Bern - Valais

Mapping Moran, Bern - Valais

Conclusion

  • Multiple distances
  • Depends of the size of the sample
  • Road distance can explain relationships in mountain regions
  • Other feature than political
  • Vote: could be interesting to also classify votes to predicts new results

Thank you for your attention