This vignette gives a few examples on how to create different spatial weights matrices using the SDPDmod package.

Introduction

A spatial weights matrix is an \(N \times N\) non-negative matrix, where \(N\) is the size of the data set. The elements of the spatial matrix \(W\), \(w_{ij}\) are non-zero if \(i\) and \(j\) are considered to be neighbors and zero otherwise. Since unit \(i\) can not be a neighbor to itself, the diagonal elements of the spatial weights matrix are zero, i.e. \(w_{ij}=0\).

Data

Data on German districts and distances between district’s centroids in meters are included in the SDPDmod package and are used for the examples.

library("sf")
ger <- st_read(system.file(dsn = "shape/GermanyNUTS3.shp",
                         package = "SDPDmod"))

data(gN3dist, package = "SDPDmod")

Types of spatial weights matrices

Contiguity based

Spatial Contiguity Weights Matrix

\[ w_{ij} = \begin{cases} 1,&i &\text{and} &j &\text{have a shared boundary}\\ 0,& \text{otherwise} \end{cases} \]

    W_1 <- mOrdNbr(ger) ## first order neighbors

Higher Order Contiguity

\[ w_{ij} = \begin{cases} 1,&i &\text{and} &j &\text{are neighbors of order} &m\\ 0,& \text{otherwise} \end{cases} \]

   W_2n <- mOrdNbr(sf_pol = ger, m = 2) ## second order neighbors
   W_3n <- mOrdNbr(ger, 3) ## third order neighbors

Shared Boundary Spatial Weights Matrix

\[ w_{ij} = \begin{cases} len,&i &\text{and} &j &\text{have a shared boundary}\\ 0,& \text{otherwise} \end{cases} \] \(len_{ij}\) - length of the boundary between units \(i\) and \(j\)

   ls <- ger[which(substr(ger$NUTS_CODE,1,3)=="DE9"),] ## Lower Saxony districts
   W_len_sh <- SharedBMat(ls)

Based on distance

k-Nearest Neighbor

\[ w_{ij} = \begin{cases} 1,& \text{if unit} &j &\text{is one of the} &k &\text{nearest neighbor of} &i\\ 0,& \text{otherwise} \end{cases} \]

    W_knn <- mNearestN(distMat = gN3dist, m = 5) ## 5 nearest neighbors

Inverse Distance

\[ w_{ij} = d_{ij}^{-\alpha} \] \(d_{ij}\) - distance between units \(i\) and \(j\), \(\alpha\) - positive exponent

    W_inv1 <- InvDistMat(distMat = gN3dist) ## inverse distance no cut-off
    W_inv2 <- InvDistMat(distMat = gN3dist, distCutOff = 100000) ## inverse distance with cut-off 100000 meters
    gN3dist2 <- gN3dist/1000 ## convert to kilometers
    W_inv3 <- InvDistMat(distMat = gN3dist2, distCutOff = 100) ## inverse distance with cut-off 100 km 
    W_inv4 <- InvDistMat(gN3dist2, 200, powr = 2) ## inverse distance with cut-off 200km and exponent 2

Exponential Distance

\[ w_{ij} = exp(-\alpha d_{ij}) \] \(d_{ij}\) - distance between units \(i\) and \(j\), \(\alpha\) - positive exponent

    W_exp1 <- ExpDistMat(distMat = gN3dist) ## Exponential distance no cut-off
    W_exp2 <- ExpDistMat(distMat = gN3dist, distCutOff = 100000) ## Exponential distance with cut-off 100000 meters
    gN3dist2 <- gN3dist/1000 ## convert to kilometers
    W_exp3 <- ExpDistMat(gN3dist2, 100) ## Exponential distance with cut-off 100 km 
    W_exp4 <- DistWMat(gN3dist2, 100, type = "expo") ## Exponential distance with cut-off 100 km
    all(W_exp3==W_exp4)
#> [1] TRUE
    W_exp5 <- ExpDistMat(gN3dist2, 200, expn = 0.001) ## Exponential distance with cut-off 200 km and exponent 0.001

Double-Power Distance

\[ w_{ij} = \begin{cases} (1-(\frac{d_{ij}}{D})^p)^p,&0 \leq d_{ij} \leq D \\ 0,& d_{ij} \geq D \end{cases} \] \(d_{ij}\) - distance between units \(i\) and \(j\), \(p\) - positive exponent, \(D\) - distance cut-off

    W_dd1 <- DDistMat(distMat = gN3dist) ## Double-Power distance no cut-off, exponent 2
    W_dd2 <- DDistMat(distMat = gN3dist, distCutOff=100000) ## Double-Power distance with cut-off 100000 meters, exponent 2
    gN3dist2 <- gN3dist/1000 ## convert to kilometers
    W_dd3 <- DDistMat(gN3dist2, 100) ## Double-Power distance with cut-off 100 km 
    W_dd4 <- DistWMat(gN3dist2, 100, type = "doubled") ## Double-Power distance with cut-off 100 km
    all(W_dd3==W_dd4) 
#> [1] TRUE
    W_dd5 <- DDistMat(gN3dist2, 200, powr = 3) ## Double-Power distance with cut-off 200km and exponent 3

Normalization

Row normalization

\[ w_{ij}^{normalized} =w_{ij}/\sum_{j=1}^N w_{ij} \]

   W_2n_norm <- mOrdNbr(sf_pol = ger, m = 2, rn = T) ## second order neighbors
   W_2n_norm2 <- rownor(W_2n)
   all(W_2n_norm==W_2n_norm2)
#> [1] TRUE

Scalar normalization

\[ w_{ij}^{normalized} =w_{ij}/\lambda_{max} \] \(\lambda_{max}\) maximum eigenvalue of \(W\)

  W_inv1_norm <- InvDistMat(distMat = gN3dist, mevn = T) ## Inverse distance
  W_inv1_norm2 <- eignor(W_inv1)
  all(W_inv1_norm==W_inv1_norm2)
#> [1] TRUE

Spatial matrices