Spatial Patterning
4.3 Spatial autocorrelation: Global measures
The term spatial autocorrelation refers to the correlation of a variable with itself. The term spatial dependence has been defined as referring to the lack of independence in data at locations which are close together (173). So, a measure of spatial autocorrelation may suggest spatial dependence or spatial independence. In this section, the focus is on global measures of spatial autocorrelation. Cliff and Ord (87) discuss some key issues in the analysis of spatial autocorrelation; another introduction is given by Griffith (166).
Properties such as elevation and precipitation tend to vary smoothly (although there are, of course, exceptions), and they are usually positively spatially autocorrelated (at least at some scales of measurement). That is, values at locations close together tend to be similar. In contrast, grey scales in a remotely-sensed image may be negatively autocorrelated if, for example, there are neighbouring fields in an agricultural image that have very different
∗http://cran.r-project.org/web/packages/spgwr/index.html
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0 10 20 Kilometres GW skewness
-1.68 - -0.76 -0.75 - -0.01 0.00 - 0.75 0.76 - 1.31 Inland water
FIGURE 4.7: Geographically weighted skewness for a Gaussian function with a 2km bandwidth: log-ratio of Catholics/Non-Catholics in Northern Ireland in 2001 by 1km grid squares. Northern Ireland Census of Population data —
c
° Crown Copyright.
characteristics. Getis (141) provides a discussion about the concept and measurement of spatial autocorrelation.
Various forms of local measures have been developed for use in cases where local variation in spatial autocorrelation is suspected. This section introduces global measures, and Section 4.4 discusses some local variants. In the case of areal data, the centroids of areas can be used to measure distances between areas. Alternatively, simple connectivity between areas could be used, as discussed below.
The joins-count statistic provides a simple measure of spatial autocorrela-tion. The method is used to assess structure in areas coded black (B) or white (W). Tests can be conducted based on the number each of the possible type of joins: BB, WW, BW, and WB (304). Comparisons between areas can be made using only neighbours that share edges (i.e., boundaries) (termed rook’s case, as detailed further below) or neighbours that share edges or corners (or vertices) (termed queen’s case and also detailed further below).
Two measures of spatial autocorrelation frequently encountered in the GISystems literature are Moran’s I and Geary’s c. Both global and local versions of these statistics are detailed below.
82 Local Models for Spatial Analysis Moran’s I is obtained using:
I = nPn
where wij is an element of the spatial proximity matrix, and the attribute values yi have the mean y. The number of zones is given by n. When spatial proximity is represented using contiguity,Pn
i=1
Pn
j=1wijis twice the number of adjacent zones. Note that y is used here as z is later used to refer to the deviation of y from its mean.
Where I is positive this indicates clustering of similar values, whilst where I is negative this indicates clustering of dissimilar values; a value of zero indicates zero spatial autocorrelation.
The most simple definition of the weights uses binary connectivity whereby wij has a value of 1 if regions i and j are contiguous and 0 if they are not.
The weights may also be a function of the distance between regions i and j (the inverse squared distance is sometimes used). A simple example is given using the small grid below:
47 48 44 42 40 38 43 37 35
Approaches using various combinations of neighbours are common. When all common boundaries and vertices are used this is termed queen contiguity.
As defined above, when only observations sharing an edge (rather than a corner or vertex) with a given observation are used this is termed rook contiguity. Queen contiguity is used in the example below — neighbours are pixels that are horizontally, vertically, or diagonally connected to another pixel. The appropriate values calculated are inserted into Equation 4.8 giving:
I = 9 × 167.012
158.222 × 40 = 1503.111
6328.889 = 0.237
Note that 40 is twice the number of adjacent zones. Where the weights are row-standardised (they sum to one with respect to the neighbourhood of each cell — e.g., where a cell has four neighbours, the weights are 0.25), I = 0.286.
I is not forced to be within the (–1,1) range, but the equation could be modified to ensure that this is the case. A further worked example of Moran’s I is provided by O’Sullivan and Unwin (304).
Geary’s contiguity ratio, c, is estimated with:
c = (n − 1)Pn
Small values of c indicate positive spatial autocorrelation, while large values of c indicate negative spatial autocorrelation. Both I and c can be generalised
Spatial Patterning 83 easily to enable measurement of spatial autocorrelation at several lags (that is, observations separated by a given distance or range of distances — directions could be considered too). Moran’s I can be estimated at spatial lag h with:
I(h) = nPn where wij(h) is an element of the spatial proximity matrix for lag h. The exploration of spatial autocorrelation is discussed further in Chapter 7 where the covariance function, correlogram (related to Moran’s I), and variogram (related to Geary’s c) are defined.
The Moran scatterplot provides one way of examining variation in spatial autocorrelation for a given dataset (11). The Moran scatterplot is a plot which relates the values of some property in one zone, i, to values in neighbouring zones. As for the Moran’s I measure itself, neighbouring zones may be those that are adjacent, or a function of separation distance may be specified. The scatterplot may be particularly useful for identifying outliers that correspond to local anomalies (e.g., a value markedly different to its neighbours). This topic is explored further in Section 4.4.1.2.
Bivariate measures of spatial autocorrelation are used to assess spatial association between two different variables. The GeoDaTMsoftware (12), (13), (14) provides functionality to compute bivariate local indicators of spatial association (LISAs; as defined in Section 4.4.1). Section 7.4.3 describes the cross-variogram which is used to characterise spatial covariation.
4.3.1 Testing for spatial autocorrelation
A test for spatial autocorrelation can be constructed where there are a sufficiently large number of observations. By assuming that the yi are drawn independently from a normal distribution (they are observations on random variables Yi) then if Yi and Yj are spatially independent (for i 6= j) I has a sampling distribution that is approximately normal with expected value of I given by (32):
84 Local Models for Spatial Analysis
it is possible to test the observed value of I against the percentage points of the approximate sampling distribution. Where the value of I is “extreme”, spatial autocorrelation is indicated (32).
An alternative approach is a random permutation test. The approach is based on the idea that, if there are n observations over a particular region, n! permutations of the data are possible with different arrangements of the data over the region. The value of I may be obtained for any one of the permutations. An empirical distribution of possible values of I can then be constructed with random permutations of the data. Where the observed value of I is extreme with respect to the permutation distribution, this may be considered evidence of significant spatial autocorrelation. In practice, it is not usually possible to obtain n! permutations, and a Monte Carlo approach may be used instead to approximate the permutation distribution.
An appropriate number of values can then be drawn randomly from among the n! permutations (32). If we assume that the process generating the observed data is random and the observed pattern is one of many possible permutations, then the variance of I is given by:
Var(I) = nS4− S3S5 Further details about tests of spatial autocorrelation are provided by Fotheringham et al. (128). The topic is explored further in Section 4.4.1.2.
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