Before we go any further, you need to read a portion of the chapter associated with this lesson from the course text:
- Chapter 4, "Spatial patterning," pages 73 - 99.
- Chapter 2.2, "Approaches to local adaptation," pages 23-27.
There is a brief section (page 81) discussing the Joins Count approach to measuring spatial autocorrelation. This approach is useful for non-numeric data. However, it is only infrequently used, and so, although the concepts introduced are useful, they are not central to a modern treatment of spatial autocorrelation.
Other measures have been developed for numerical data, and, in practice, these are much more widely used. These are discussed in Section 4.3, with a particular focus on Moran's I.
While equation 4.8 (page 82) for Moran's I looks intimidating, it makes a great deal of sense. It consists of:
- a measure of similarity,
- a mechanism that includes only those map units that are near to one another in the calculation, and
- a weighting factor that scales the resulting calculation so that it is in a standard numerical range.
In the case of Moran's I, the similarity measure is the standard method used in correlation statistics, namely the product of the differences of each value from the mean. This produces a positive result when both the value and neighboring values are higher or lower than the mean, and a negative result when the value and neighboring values are on opposite sides of the mean (one higher, the other lower).
The difference measure is summed over all neighboring pairs of map units (this is where the wij values from a weights matrix come in) and then adjusted so that the resulting index value is in a standard numerical range.
Weight Matrices, pages 24-25
The inclusion of spatial interaction weights between pairs of map units in the formulas for calculating I means that it is possible to experiment with a wide variety of spatial autocorrelation measures by tailoring the particular choice of interaction weights appropriately.