Page 89 - Contributed Paper Session (CPS) - Volume 2

CPS1440 Avner Bar-Hen et al.
observed repartition. For bivariate point processes, many tools based, using
first-order characteristics of the point processes, may be used to quantify
departure from independence (see [4] for example).
Under the assumptions of stationarity and isotropy, the intertype Kij-
function is a bivariate extension of Ripley’s K-function, proposed in [8], and
defined as

− 1
() = λ j (number of points of type within distance of a (3)

randomly chosen point of type i)
where the intensity parameters λi and λj correspond to the expected numbers
of type and type j points per unit area, respectively. While Ripley’s K-function
characterizes the spatial structure of a univariate pattern at various scales, the
intertype -function characterizes the spatial structure of a bivariate pattern,

and more precisely the spatial relationship between two types of points
located in the same study area. The intertype -function is defined so that

() is the expected number of type j points in a circle of radius r centered

on an arbitrary type point of the pattern. Symmetrically, we can define an
intertype -function so that () is the expected number of type points

in a circle of radius r centered on an arbitrary type j point.
The estimator of the intertype -function can be defined by:

̂ ̂
̂
() = ( ) −1 ∑ , { , <} (4)

where , is the distance between the th location of type point and the

th location of type point, A is the area of the region of interest and and
̂

are the estimated intensities.
̂

3. Result
a. Spatial CART method
The key idea is to take into account in the splitting strategy, the spatial
dependency of the data. It is done by modifying the original impurity loss,
which is usually the entropy index. We introduce a dissimilarity index based
on Ripley’s intertype K-function quantifying the interaction between two
populations within a rectangular window of finite area A.
Let focus on the impurity loss associated with as defined in Equation 4.
̂

For a node t and a split s splitting t into two child nodes tL and tR, we define
(), the estimation of the Ripley’s intertype K function restricted to node t:
̂

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