Spatial Weights Details
A traditional way to represent neighbor relationships is as a spatial weights matrix. It is an n x n matrix, with n being the number of objects or locations in the geography. There is a row and column for each geographic object (for example, state or county). The value in each cell represents whether the location in the column header is a neighbor of the row location, termed "ego". Cells with nonzero weights are considered the neighbors of that ego.
Weights can be binary (1 or 0, neighbor or not), or have continuous values, such as an inverse distance between points or the length of a shared border between polygons. Weights sets can include ego, may or may not be symmetric, and can be standardized.
In SpaceStat, spatial weights sets are created by first defining what a neighbor is and then assigning weight values to neighbors. A single geography could have several different weights sets. Methods differ for points and polygons.
As an example, consider a simple geography with 3 locations (IDs: 1, 2, & 3). The spatial weight matrix is shown in the table below where each of the non-header cells (gray) holds a value of the weight for the column object to the row object.
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ID |
1 |
2 |
3 |
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1 |
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2 |
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3 |
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Include Ego as Neighbor?
Usually, the diagonal of the matrix is zeroes, as there is no weight for location 1 on location 1. For the Gi* statistic, however, the weight set should include ego in the neighbor set, as shown in the version on the right.
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Ego not included
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Ego is included
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Symmetry in Weights
Some weights sets are symmetric. For instance, if two locations neighbor each other, they have the same weights in each row. Non-symmetric weights occur, however, when using a standardized weight set for locations that have different numbers of neighbors or when using a set number of neighbors for a point geography. A symmetric matrix can be efficiently represented as the lower half of the matrix.
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Symmetric
Symmetric, short
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Asymmetric
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Standardized Weights
If you standardize by neighbor count or shared border length, SpaceStat will normalize the weights for each location (each "ego") by its neighbor count. The sum of the weights for each location (each "ego") is 1. Because many spatial statistics sum up the weight multiplied by a value (such as a z-score), those locations with a higher weight sum have a larger influence on the statistic. Standardizing the weights gives each location the same vote.
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Standardized
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Not Standardized
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