Analysing spatial point patterns in R - CSIRO

78 Exploring intensity
12 Exploring intensity
When we analyse numerical data, we often begin by taking the sample mean. The analogue of
the mean or expected value of a random variable is the intensity of a point process.
‘Intensity’ is the average density of points (expected number of points per unit area). Intensity
may be constant (‘uniform’ or ‘homogeneous’) or may vary from location to location
(‘inhomogeneous’). Investigation of the intensity should be one of the first steps in analysing a
point pattern.
12.1 Uniform intensity
12.1.1 Theory
If the point process X is homogeneous, then for any sub-region B of two-dimensional space, the
expected number of points in B is proportional to the area of B:
E[N(X ∩ B)] = λarea(B)
and the constant of proportionality λ is the intensity. Intensity units are numbers per unit
area (length −2 ). If we know that a point process is homogeneous, then the empirical density of
points,
λ =
n(x)
area(W)
is an unbiased estimator of the true intensity λ.
12.1.2 Implementation in spatstat
To compute the estimator λ in spatstat, use summary.ppp:
> data(swedishpines)
> summary(swedishpines)
Planar point pattern: 71 points
Average intensity 0.0074 points per square unit (one unit = 0.1 metres)
Window: rectangle = [0, 96]x[0, 100]units
Window area = 9600 square units
Unit of length: 0.1 metres
The estimated intensity is λ = 0.0074 points per square unit. To extract this intensity value,
type
> lamb lamb
[1] 0.007395833
The units are decimetres, so this is 0.74 points per square metre.
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