# Spatial weight matrix

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{{Short description|Neighbor Matrix}}

The concept of a '''spatial weight''' is used in [spatial analysis](/source/spatial_analysis) to describe neighbor relations between regions on a map.<ref name="clifford">
{{cite book
|  title=Spatial Processes: Models & Applications
|  author=Cliff, A.D. and Ord, J.K.
|  isbn=9780850860818
|  url=https://books.google.com/books?id=Mi0OAAAAQAAJ
|  year=1981
|  publisher=Pion
}}</ref> If location <math>i</math> is a neighbor of location <math>j</math> then <math>w_{ij} \neq 0</math> otherwise <math>w_{ij} = 0</math>. Usually (though not always) we do not consider a site to be a neighbor of itself<ref name="geoda1">{{cite web | url=https://geodacenter.github.io/workbook/4a_contig_weights/lab4a.html | title=Contiguity-Based Spatial Weights }}</ref> so <math>w_{ii} = 0</math>. These coefficients are encoded in the '''spatial weight matrix'''
:<math>
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}
</math>
Where <math>N</math> is the number of sites under consideration. The spatial weight matrix is a key quantity in the computation of many spatial indices like [Moran's I](/source/Moran's_I), [Geary's C](/source/Geary's_C), [Getis-Ord statistics](/source/Getis-Ord_statistics) and [Join Count Statistics](/source/Join_Count_Statistics).

== Contiguity-Based Weights ==

thumb|Common connectivity patterns described on a regular grid. Non-zero values of <math>w_{Aj}</math> are indicated by the red lines. For example in the rook's case, for the <math>A</math> row of the weight matrix only <math>w_{AN}, w_{AS}, w_{AE}, w_{AW} \neq 0</math> and similarly in the other cases.

This approach considers spatial sites as nodes in a [graph](/source/Graph_(discrete_mathematics)) with links determined by a shared boundary or vertex.<ref name="fortindale">Dale MR, Fortin MJ. Spatial analysis: a guide for ecologists. Cambridge University Press; 2014 Sep 11.</ref> The elements of the spatial weight matrix are determined by setting <math>w_{ij} = 1</math> for all connected pairs of nodes <math>ij</math> with all the other elements set to 0. This makes the spatial weight matrix equivalent to the [adjacency matrix](/source/adjacency_matrix) of the corresponding network. It is common<ref name="geoda1"/> to row-normalize the matrix <math>W</math>,
:<math>w_{ij} \rightarrow w_{ij}/\sum_j w_{ij}</math>
In this case the sum of all the elements of <math>W</math> equals <math>N</math> the number of sites.

thumb|African quadripoint. Using Rook neighbors, Zimbabwe is neighbours with Zambia and Botswana. Using Queen neighbors, Zimbabwe is also a neighbor of Namibia. Bishop neighbors are rarely used for polygonal data.
There are three common methods for linking sites<ref name="fortindale" /> named after the [chess](/source/chess) pieces which make similar moves:
* Rook: sites are neighbors if they share an edge
* Bishop: sites are neighbours if they share a vertex
* Queen: sites are neighbours if they share an edge or a vertex

In some cases statistics can be quite different depending on the definition used, especially for discrete data on a grid.<ref name="fortindale" /> There are also other cases  where the choice of neighbors is not obvious and can affect the outcome of the analysis. [Bivand](/source/Roger_Bivand) and Wong<ref>Bivand RS, Wong DW. Comparing implementations of global and local indicators of spatial association. Test. 2018 Sep;27(3):716-48.</ref> describe a situation where the value of spatial indices of association (like [Moran's I](/source/Moran's_I)) depend on the inclusion or exclusion of a ferry crossing between counties. There are also cases where regions meet in a [tripoint](/source/tripoint) or [quadripoint](/source/quadripoint) where Rook and Queen neighborhoods can differ.

== Distance-Based Weights ==

Another way to define spatial neighbors is based on the distance between sites. One simple choice is to set <math>w_{ij} = 1</math> for every pair <math>(i,j)</math> separated by a distance less than some threshold <math>\delta</math>.<ref name="geoda2">{{cite web | url=https://geodacenter.github.io/workbook/4b_dist_weights/lab4b.html | title=Distance-Band Spatial Weights }}</ref> Cliff and Ord<ref name="clifford"/> suggest the general form
:<math>
w_{ij} = g(d_{ij}, \beta_{ij})
</math>
Where <math>g</math> is some function of <math>d_{ij}</math> the distance between <math>i</math> and <math>j</math> and <math>\beta_{ij}</math> is the proportion of the perimeter of <math>i</math> in contact with <math>j</math>. The function 
:<math>
w_{ij} = d_{ij}^{-\alpha} \beta_{ij}^{b}
</math>
is then suggested. Often the <math>\beta</math> term is not included and the most common values for <math>\alpha</math> are 1 and 2.<ref name="fortindale" /> Another common choice for the distance decay function is<ref name="geoda3">{{cite web | url=https://geodacenter.github.io/workbook/4c_distance_functions/lab4c.html | title=Spatial Weights as Distance Functions }}</ref>
:<math>
w_{ij} = \exp( - d_{ij} )
</math>
though a number of different [Kernel](/source/Kernel_(statistics)) functions can be used. The exponential and other Kernel functions typically set <math>w_{ii} = 1</math> which must be considered in applications.

It is possible to make the spatial weight matrix a function of 'distance class':<ref>Legendre P, Legendre L. Numerical ecology. Elsevier; 2012 Jul 21.</ref> <math>w_{ij} \rightarrow w_{ij}(d)</math> where <math>d</math> denotes the 'distance class', for example <math>d=1,2,3,\ldots</math> corresponding to first, second, third etc. neighbors. In this case, functions of the spatial weight matrix become distance class dependent. For example, [Moran's I](/source/Moran's_I) is
:<math> 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}  </math>
This defines a type of spatial [correlogram](/source/correlogram), in this case, since Moran's ''I'' measures spatial autocorrelation, <math>I(d)</math> measures how the autocorrelation of the data changes as a function of distance class. Remembering [Tobler's first law of geography](/source/Tobler's_first_law_of_geography), "everything is related to everything else, but near things are more related than distant things" it usually decreases with distance.

Common distance functions include<ref name="geoda2" /> [Euclidean distance](/source/Euclidean_distance), [Manhattan distance](/source/Manhattan_distance) and [Great-circle distance](/source/Great-circle_distance).

== Spatial Lag ==

One application of the spatial weight matrix is to compute the '''spatial lag'''<ref>{{cite web | url=https://geodacenter.github.io/workbook/4d_weights_applications/lab4d.html | title=Applications of Spatial Weights }}</ref>
:<math>
[Wx]_i = \sum_j w_{ij} x_j
</math>
For row-standardised weights initially set to <math>w_{ij} = 1</math> and with <math>w_{ii} = 0</math>, <math>[Wx]_i</math> is simply the average value observed at the neighbors of <math>i</math>. These lagged variables can then be used in [regression analysis](/source/regression_analysis) to incorporate the dependence of the outcome variable on the values at neighboring sites.<ref>Anselin L, Griffith DA. Do spatial effecfs really matter in regression analysis?. Papers in Regional Science. 1988 Jan 1;65(1):11-34.</ref>
The standard regression equation is
:<math>
y_i = \sum_k x_{ik} \beta_k + \epsilon_i
</math>
The ''spatial lag model'' adds the spatial lag vector to this
:<math>
y_i = \rho\sum_j w_{ij}y_j + \sum_k x_{ik} \beta_k + \epsilon_i
</math>
where <math>\rho</math> is a parameter which controls the degree of autocorrelation of <math>y</math>.<ref>Seya H, Yoshida T, Yamagata Y. Spatial econometric models. InSpatial Analysis Using Big Data 2020 Jan 1 (pp. 113-158). Academic Press.</ref> This is similar to an [autoregressive model](/source/autoregressive_model) in the analysis of [time series](/source/time_series).

== See also ==

* [Spatial Analysis](/source/Spatial_Analysis)
* [Moran's I](/source/Moran's_I)
* [Geary's C](/source/Geary's_C)
* [Join Count Statistics](/source/Join_Count_Statistics)

== References ==
{{reflist}}

Category:Spatial analysis
Category:Covariance and correlation

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Adapted from the Wikipedia article [Spatial weight matrix](https://en.wikipedia.org/wiki/Spatial_weight_matrix) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Spatial_weight_matrix?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.