Distance Decay Spatial Weight Matrix

Introduction

There are a number of methods that have been developed to determine the closeness or nearness between locations.

Among some of the more sophisticated calculate are the distance decay spatial weights matrices. These methods are based on the concept that areas that closer to your area of interest have more of an influence, or are more similar to them than those further away, and thus are weighted as such. This is a more nuanced approach than the Contiguous Spatial Weights Matrix or the Distance Spatial Weights Matrix in that, rather than counting all bordering areas, or all areas within \(d\) distance as the same in their ‘closesness’ (and those outside of these criteria as the same in their ‘farness’), the influence of an area decays with distance. This concept is illustrated in the figure below, which shows the decrease in influence of the areas (fading blue) on the central area (red)

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In AURIN two versions of distance decay spatial weights matrices are implemented. These are not particularly dissimilar from each other in their outcomes, but the differences are illustrated below

Inverse Distance Decay

\(w_{ij} = 1/d_{ij}\text{   if   }d_{ij} \leq d_{max}\)
\(w_{ij} = 0\text{   if   }d_{ij} > d_{max}\)
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Exponential Distance Decay

\(w_{ij} = exp(-\alpha*d_{ij})\text{   if   }d_{ij} \leq d_{max}\)
\(w_{ij} = 0\text{   if   }d_{ij} > d_{max}\)
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Inputs

To generate a distance decay spatial weights matrix, we will run the tool on a dataset of for the Darwin GCCSA. To do this:

  • Select Perth GCCSA as your area
  • Select SA2 Housing Transport as your dataset, selecting all variables
  • Spatialise the dataset that you have just created and rename the output, in this instance we have renamed it Spatialised SA2 Housing Transport

Once you have selected these parameters open the Distance Decay Spatial Weights Matrix tool (Tools → Spatial Statistics → Distance Decay Spatial Weights Matrix) and enter the parameters as shown below. (These are explained below the image as well)

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  • Spatial Dataset Input:  The dataset that you would like to calculate the spatial weights matrix – this must be spatialised in order for this tool to be run (either by spatialising an AURIN dataset with the spatialise tool, or by uploading a shapefile). In this instance our dataset is Spatialised SA2 Housing Transport
  • Type: The type of distance decay model that will be used – Inverse distance or exponential distance decay. In this instance we will select exponential distance decay
  • Threshold distance: This specifies the threshold distance, \(d_{max}\), beyond which areas will have a \(w\) value of zero. If you want all of the areas to be included, make sure that the distance you select is larger than the largest distance between areas in the dataset. In this instance we select (5km).
  • Exponential decay parameter: This specifies the value of \(\alpha\) in the exponential distance decay equation \(exp(\alpha*d_{ij})\). This value should normally be somewhere between 0 and 1. In this instance we will select 0.5. 
  • Style:  This specifies whether the standardisation will be binary, row-standardised (default) or globally-standardised. In this instance we will use row.standardised
  • Matrix Island Parameter:  This specifies how the matrix will deal with islands, which naturally have no contiguous neighbours 

Once you have entered these parameters click Add and Run

Outputs

Once you have executed the tool, you should see an output appear in your Data panel, named

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“0” scores within a cell indicate that the areas are not classified as “close”, and therefore do not affect each other. If a value is greater than zero, it indicates that the area specified by the column influences the area of the row by that much.

This spatial weights matrix can now be used for additional spatial statistical analysis, such as Moran’s I or Geary’s C.

References

O’Sullivan, D. & D. Unwin. 2010. Geographic Information Analysis. Wiley and Sons