---
title: "Spatial matrices"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Spatial matrices}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup}
library(SDPDmod)
```

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.
```{r, eval=T, echo=T, warning=FALSE, message=FALSE, results='hide'}
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
$$
w_{ij} =
\begin{cases}
 1,&i &\text{and} &j &\text{have a shared boundary}\\
 0,& \text{otherwise}
 \end{cases}       
$$

```{r}
    W_1 <- mOrdNbr(ger) ## first order neighbors
```


### Higher Order Contiguity
$$
w_{ij} =
\begin{cases}
 1,&i &\text{and} &j &\text{are neighbors of order} &m\\
 0,& \text{otherwise}
 \end{cases}       
$$

```{r}
   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 $j$

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

## Based on distance
### k-Nearest Neighbor
$$
w_{ij} =
\begin{cases}
 1,& \text{if unit} &j &\text{is one of the} &k &\text{nearest neighbor of} &i\\
 0,& \text{otherwise}
 \end{cases}       
$$

```{r}
    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

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

### Exponential Distance
$$
w_{ij} = exp(-\alpha d_{ij})     
$$
$d_{ij}$ - distance between units $i$ and $j$, $\alpha$ - positive exponent

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

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

# Normalization
## Row normalization
$$
w_{ij}^{normalized} =w_{ij}/\sum_{j=1}^N w_{ij}       
$$

```{r}
   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)
```

## Scalar normalization 
$$
w_{ij}^{normalized} =w_{ij}/\lambda_{max}       
$$
$\lambda_{max}$ maximum eigenvalue of $W$

 ```{r}
   W_inv1_norm <- InvDistMat(distMat = gN3dist, mevn = T) ## Inverse distance
   W_inv1_norm2 <- eignor(W_inv1)
   all(W_inv1_norm==W_inv1_norm2)
 ```
