Spatial weight matrix

<h2 id="contiguity-based-weights">Contiguity-Based Weights</h2>

This approach considers spatial sites as nodes in a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> with links determined by a shared boundary or vertex.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> The elements of the spatial weight matrix are determined by setting 
 
 
 
 
 w
 
 i
 j
 
 
 =
 1
 
 
 {\displaystyle w_{ij}=1}
 
 for all connected pairs of nodes 
 
 
 
 i
 j
 
 
 {\displaystyle ij}
 
 with all the other elements set to 0. This makes the spatial weight matrix equivalent to the <a href="/facts/Adjacency_matrix/jYXNhYGs">adjacency matrix</a> of the corresponding network. It is common<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> to row-normalize the matrix 
 
 
 
 W
 
 
 {\displaystyle W}
 
,

w
          
            i
            j
          
        
        →
        
          w
          
            i
            j
          
        
        
          /
        
        
          ∑
          
            j
          
        
        
          w
          
            i
            j
          
        
      
    
    {\displaystyle w_{ij}\rightarrow w_{ij}/\sum _{j}w_{ij}}

In this case the sum of all the elements of 
 
 
 
 W
 
 
 {\displaystyle W}
 
 equals 
 
 
 
 N
 
 
 {\displaystyle N}
 
 the number of sites.

There are three common methods for linking sites<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> named after the <a href="/facts/Chess/47ykvkHF">chess</a> pieces which make similar moves:

<ul><li>Rook: sites are neighbors if they share an edge</li>
<li>Bishop: sites are neighbours if they share a vertex</li>
<li>Queen: sites are neighbours if they share an edge or a vertex</li></ul>
In some cases statistics can be quite different depending on the definition used, especially for discrete data on a grid.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> There are also other cases where the choice of neighbors is not obvious and can affect the outcome of the analysis. <a href="/facts/Roger_Bivand/SXJV13Sv">Bivand</a> and Wong<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> describe a situation where the value of spatial indices of association (like <a href="/facts/Moran%2527s_I/qdIqxfTJ">Moran's I</a>) depend on the inclusion or exclusion of a ferry crossing between counties. There are also cases where regions meet in a <a href="/facts/Tripoint/j9IFVW6w">tripoint</a> or <a href="/facts/Quadripoint/lzw2zK7N">quadripoint</a> where Rook and Queen neighborhoods can differ.

<h2 id="distance-based-weights">Distance-Based Weights</h2>
Another way to define spatial neighbors is based on the distance between sites. One simple choice is to set 
 
 
 
 
 w
 
 i
 j
 
 
 =
 1
 
 
 {\displaystyle w_{ij}=1}
 
 for every pair 
 
 
 
 (
 i
 ,
 j
 )
 
 
 {\displaystyle (i,j)}
 
 separated by a distance less than some threshold 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> Cliff and Ord<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> suggest the general form

w
          
            i
            j
          
        
        =
        g
        (
        
          d
          
            i
            j
          
        
        ,
        
          β
          
            i
            j
          
        
        )
      
    
    {\displaystyle w_{ij}=g(d_{ij},\beta _{ij})}

Where 
 
 
 
 g
 
 
 {\displaystyle g}
 
 is some function of 
 
 
 
 
 d
 
 i
 j
 
 
 
 
 {\displaystyle d_{ij}}
 
 the distance between 
 
 
 
 i
 
 
 {\displaystyle i}
 
 and 
 
 
 
 j
 
 
 {\displaystyle j}
 
 and 
 
 
 
 
 β
 
 i
 j
 
 
 
 
 {\displaystyle \beta _{ij}}
 
 is the proportion of the perimeter of 
 
 
 
 i
 
 
 {\displaystyle i}
 
 in contact with 
 
 
 
 j
 
 
 {\displaystyle j}
 
. The function

w
          
            i
            j
          
        
        =
        
          d
          
            i
            j
          
          
            −
            α
          
        
        
          β
          
            i
            j
          
          
            b
          
        
      
    
    {\displaystyle w_{ij}=d_{ij}^{-\alpha }\beta _{ij}^{b}}

is then suggested. Often the 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 term is not included and the most common values for 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 are 1 and 2.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> Another common choice for the distance decay function is<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

w
          
            i
            j
          
        
        =
        exp
        ⁡
        (
        −
        
          d
          
            i
            j
          
        
        )
      
    
    {\displaystyle w_{ij}=\exp(-d_{ij})}

though a number of different <a href="/facts/Kernel_(statistics)/otWLlBbA">Kernel</a> functions can be used. The exponential and other Kernel functions typically set 
 
 
 
 
 w
 
 i
 i
 
 
 =
 1
 
 
 {\displaystyle w_{ii}=1}
 
 which must be considered in applications.
It is possible to make the spatial weight matrix a function of 'distance class':<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> 
 
 
 
 
 w
 
 i
 j
 
 
 →
 
 w
 
 i
 j
 
 
 (
 d
 )
 
 
 {\displaystyle w_{ij}\rightarrow w_{ij}(d)}
 
 where 
 
 
 
 d
 
 
 {\displaystyle d}
 
 denotes the 'distance class', for example 
 
 
 
 d
 =
 1
 ,
 2
 ,
 3
 ,
 …
 
 
 {\displaystyle d=1,2,3,\ldots }
 
 corresponding to first, second, third etc. neighbors. In this case, functions of the spatial weight matrix become distance class dependent. For example, <a href="/facts/Moran%2527s_I/qdIqxfTJ">Moran's I</a> is

I
        (
        d
        )
        =
        
          
            N
            
              
                |
              
              W
              (
              d
              )
              
                |
              
            
          
        
        
          
            
              
                ∑
                
                  i
                  =
                  1
                
                
                  N
                
              
              
                ∑
                
                  j
                  =
                  1
                
                
                  N
                
              
              
                w
                
                  i
                  j
                
              
              (
              d
              )
              (
              
                x
                
                  i
                
              
              −
              
                
                  
                    x
                    ¯
                  
                
              
              )
              (
              
                x
                
                  j
                
              
              −
              
                
                  
                    x
                    ¯
                  
                
              
              )
            
            
              
                ∑
                
                  i
                  =
                  1
                
                
                  N
                
              
              (
              
                x
                
                  i
                
              
              −
              
                
                  
                    x
                    ¯
                  
                
              
              
                )
                
                  2
                
              
            
          
        
      
    
    {\displaystyle I(d)={\frac {N}{|W(d)|}}{\frac {\sum _{i=1}^{N}\sum _{j=1}^{N}w_{ij}(d)(x_{i}-{\bar {x}})(x_{j}-{\bar {x}})}{\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2}}}}

This defines a type of spatial <a href="/facts/Correlogram/jXkaMAq0">correlogram</a>, in this case, since Moran's I measures spatial autocorrelation, 
 
 
 
 I
 (
 d
 )
 
 
 {\displaystyle I(d)}
 
 measures how the autocorrelation of the data changes as a function of distance class. Remembering <a href="/facts/Tobler%2527s_first_law_of_geography/K4h37Fkg">Tobler's first law of geography</a>, "everything is related to everything else, but near things are more related than distant things" it usually decreases with distance.
Common distance functions include<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> <a href="/facts/Euclidean_distance/9qDoQKQe">Euclidean distance</a>, <a href="/facts/Manhattan_distance/zU7gGUPv">Manhattan distance</a> and <a href="/facts/Great-circle_distance/s6HNWn8e">Great-circle distance</a>.

<h2 id="spatial-lag">Spatial Lag</h2>
One application of the spatial weight matrix is to compute the spatial lag<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

[
        W
        x
        
          ]
          
            i
          
        
        =
        
          ∑
          
            j
          
        
        
          w
          
            i
            j
          
        
        
          x
          
            j
          
        
      
    
    {\displaystyle [Wx]_{i}=\sum _{j}w_{ij}x_{j}}

For row-standardised weights initially set to 
 
 
 
 
 w
 
 i
 j
 
 
 =
 1
 
 
 {\displaystyle w_{ij}=1}
 
 and with 
 
 
 
 
 w
 
 i
 i
 
 
 =
 0
 
 
 {\displaystyle w_{ii}=0}
 
, 
 
 
 
 [
 W
 x
 
 ]
 
 i
 
 
 
 
 {\displaystyle [Wx]_{i}}
 
 is simply the average value observed at the neighbors of 
 
 
 
 i
 
 
 {\displaystyle i}
 
. These lagged variables can then be used in <a href="/facts/Regression_analysis/n6z5Tf7K">regression analysis</a> to incorporate the dependence of the outcome variable on the values at neighboring sites.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
The standard regression equation is

y
          
            i
          
        
        =
        
          ∑
          
            k
          
        
        
          x
          
            i
            k
          
        
        
          β
          
            k
          
        
        +
        
          ϵ
          
            i
          
        
      
    
    {\displaystyle y_{i}=\sum _{k}x_{ik}\beta _{k}+\epsilon _{i}}

The spatial lag model adds the spatial lag vector to this

y
          
            i
          
        
        =
        ρ
        
          ∑
          
            j
          
        
        
          w
          
            i
            j
          
        
        
          y
          
            j
          
        
        +
        
          ∑
          
            k
          
        
        
          x
          
            i
            k
          
        
        
          β
          
            k
          
        
        +
        
          ϵ
          
            i
          
        
      
    
    {\displaystyle y_{i}=\rho \sum _{j}w_{ij}y_{j}+\sum _{k}x_{ik}\beta _{k}+\epsilon _{i}}

where 
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
 is a parameter which controls the degree of autocorrelation of 
 
 
 
 y
 
 
 {\displaystyle y}
 
.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> This is similar to an <a href="/facts/Autoregressive_model/LpboAQ0W">autoregressive model</a> in the analysis of <a href="/facts/Time_series/fSXPR817">time series</a>.

<h2 id="see-also">See Also</h2>
<ul><li><a href="/facts/Spatial_Analysis/R6rlDzRl">Spatial Analysis</a></li>
<li><a href="/facts/Moran%2527s_I/qdIqxfTJ">Moran's I</a></li>
<li><a href="/facts/Geary%2527s_C/qBAuuG42">Geary's C</a></li>
<li>Join Count Statistics</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Cliff, A.D. and Ord, J.K. (1981). Spatial Processes: Models & Applications. Pion. ISBN 9780850860818.{{cite book}}: CS1 maint: multiple names: authors list (link) <a href="9780850860818" target="_blank">9780850860818</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Contiguity-Based Spatial Weights". <a href="https://geodacenter.github.io/workbook/4a_contig_weights/lab4a.html" target="_blank">https://geodacenter.github.io/workbook/4a_contig_weights/lab4a.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dale MR, Fortin MJ. Spatial analysis: a guide for ecologists. Cambridge University Press; 2014 Sep 11. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"Contiguity-Based Spatial Weights". <a href="https://geodacenter.github.io/workbook/4a_contig_weights/lab4a.html" target="_blank">https://geodacenter.github.io/workbook/4a_contig_weights/lab4a.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Dale MR, Fortin MJ. Spatial analysis: a guide for ecologists. Cambridge University Press; 2014 Sep 11. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Dale MR, Fortin MJ. Spatial analysis: a guide for ecologists. Cambridge University Press; 2014 Sep 11. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Bivand RS, Wong DW. Comparing implementations of global and local indicators of spatial association. Test. 2018 Sep;27(3):716-48. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">"Distance-Band Spatial Weights". <a href="https://geodacenter.github.io/workbook/4b_dist_weights/lab4b.html" target="_blank">https://geodacenter.github.io/workbook/4b_dist_weights/lab4b.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Cliff, A.D. and Ord, J.K. (1981). Spatial Processes: Models & Applications. Pion. ISBN 9780850860818.{{cite book}}: CS1 maint: multiple names: authors list (link) <a href="9780850860818" target="_blank">9780850860818</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Dale MR, Fortin MJ. Spatial analysis: a guide for ecologists. Cambridge University Press; 2014 Sep 11. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">"Spatial Weights as Distance Functions". <a href="https://geodacenter.github.io/workbook/4c_distance_functions/lab4c.html" target="_blank">https://geodacenter.github.io/workbook/4c_distance_functions/lab4c.html</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Legendre P, Legendre L. Numerical ecology. Elsevier; 2012 Jul 21. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">"Distance-Band Spatial Weights". <a href="https://geodacenter.github.io/workbook/4b_dist_weights/lab4b.html" target="_blank">https://geodacenter.github.io/workbook/4b_dist_weights/lab4b.html</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">"Applications of Spatial Weights". <a href="https://geodacenter.github.io/workbook/4d_weights_applications/lab4d.html" target="_blank">https://geodacenter.github.io/workbook/4d_weights_applications/lab4d.html</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Anselin L, Griffith DA. Do spatial effecfs really matter in regression analysis?. Papers in Regional Science. 1988 Jan 1;65(1):11-34. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Seya H, Yoshida T, Yamagata Y. Spatial econometric models. InSpatial Analysis Using Big Data 2020 Jan 1 (pp. 113-158). Academic Press. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
</ol>

Spatial weight matrix open-in-new

Spatial weight matrix