Spatial weight matrix

<p>The concept of a spatial weight is used in <a href="/facts/Spatial_analysis/R6rlDzRl">spatial analysis</a> to describe neighbor relations between regions on a map. If location 
  
    
      
        i
      
    
    {\displaystyle i}
  
 is a neighbor of location 
  
    
      
        j
      
    
    {\displaystyle j}
  
 then 
  
    
      
        
          w
          
            i
            j
          
        
        ≠
        0
      
    
    {\displaystyle w_{ij}\neq 0}
  
 otherwise 
  
    
      
        
          w
          
            i
            j
          
        
        =
        0
      
    
    {\displaystyle w_{ij}=0}
  
. Usually (though not always) we do not consider a site to be a neighbor of itself so 
  
    
      
        
          w
          
            i
            i
          
        
        =
        0
      
    
    {\displaystyle w_{ii}=0}
  
. These coefficients are encoded in the spatial weight matrix
</p>

W
        =
        
          
            (
            
              
                
                  
                    w
                    
                      11
                    
                  
                
                
                  
                    w
                    
                      12
                    
                  
                
                
                  …
                
                
                  
                    w
                    
                      1
                      N
                    
                  
                
              
              
                
                  
                    w
                    
                      21
                    
                  
                
                
                  
                    w
                    
                      22
                    
                  
                
                
                  …
                
                
                  
                    w
                    
                      2
                      N
                    
                  
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
              
              
                
                  
                    w
                    
                      N
                      1
                    
                  
                
                
                  
                    w
                    
                      N
                      2
                    
                  
                
                
                  …
                
                
                  
                    w
                    
                      N
                      N
                    
                  
                
              
            
            )
          
        
      
    
    {\displaystyle W={\begin{pmatrix}w_{11}&w_{12}&\ldots &w_{1N}\\w_{21}&w_{22}&\ldots &w_{2N}\\\vdots &\vdots &\vdots &\vdots \\w_{N1}&w_{N2}&\ldots &w_{NN}\\\end{pmatrix}}}

<p>Where 
  
    
      
        N
      
    
    {\displaystyle N}
  
 is the number of sites under consideration. The spatial weight matrix is a key quantity in the computation of many spatial indices like <a href="/facts/Moran%2527s_I/qdIqxfTJ">Moran's I</a>, <a href="/facts/Geary%2527s_C/qBAuuG42">Geary's C</a>, <a href="/facts/Getis-Ord_statistics/3H3ysNMK">Getis-Ord statistics</a> and Join Count Statistics.
</p>

Spatial weight matrix open-in-new

Spatial weight matrix