Pattern analysis

Algorithm

Data in a point map represent spatial occurrence of a particular phenomenon. To acquire knowledge about the occurrence of the phenomenon, the spatial distribution of the points in the map can be examined. Point pattern analysis is a technique that is used to obtain information about the arrangement of point data in space to be able to make a statement about the occurrence of certain patterns.

Measures of dispersion:

The mean distance between RNNs are tested against the expected distances in a CRS.

First a number of different distance figures are defined. Then from distance 0 to the upper limit of each distance figure, the probability of finding one or more other points is calculated for output columns Prob1Pnt, Prob2Pnt, ..., Prob6Pnt where:

Column Distance lists distances. These distances should be considered as the distance from any point in the input point map, i.e. a kind of search radius;
Column Prob1Pnt lists the probability that within a certain distance (column Distance) of any point, at least one other point will be found, i.e. the distance at which any point's nearest neighbour can be found;
Column Prob2Pnt lists the probability that from any point, at least two other points will be found;
The same goes for columns Prob3Pnt, Prob4Pnt, Prob5Pnt and Prob6Pnt;
For a dataset of n points, ProbAllPnt is the summation of Prob1Pnt + Prob2Pnt + ... + Probn-1Pnt, divided by (n - 1).

The formula to calculate probabilities within certain distances reads:

Measures of arrangement:

For every order the expected number of reflexive neighbours is calculated. Multiplying the probability that a point in a CSR pattern is a reflexive neighbour by the number of points gives you the expected number. Note that a reflexive point must be a member of a pair, so the observed value will always be an even number.

To calculate the mean distance to nearest neighbour, the nearest neighbour distance (d) for each of the points is summed and divided by the number of points. The formula reads:

where:

d_i	is the nearest neighbour distance for a point in the map
n	is the number of points.

Note that the mean distance to nearest neighbour is not corrected for boundary effects.

Reference:

Boots, B.N. & A. Getis, 1988, Point Pattern Analysis. Sage University Scientific Geography Series no. 8. Sage Publications, Bevery Hills.