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CPS1440 Avner Bar-Hen et al.
cells defined by the partition are a realization of a spatial point process, where
the response variable Y can be seen as a mark of the point of position given
by X. Following this idea, the variant of the CART algorithm that we propose
takes into account the spatial dimension of the marked processes, and thus
makes it possible to provide the user a partition of the plan defining
homogeneous areas of interaction between the two marks.
While a classical assumption for CART is to consider a sample of i.i.d.
observations, our variant takes into account spatial dependence through the
quantification of the interaction between the two considered populations, i.e.
the link between the labels of the points. Indeed, if CART is often limited to
the one most commonly used and implemented, presented in [3], there exist
several ways to build CART type decision trees, by changing the family of
admissible splits, the cost function or the stopping rule. We consider here the
spatial dependence using the Ripley's intertype K-function to suitably redefine
the splitting criterion in CART. Instead of being based only on the respective
proportions of the marks, the splits will be selected according to the strength
of interaction between the two populations.
Among the various extensions of CART, Bel et al. ([2]) proposed a variant
for spatial data, but in the regression context. They consider the observations
as a sample of regionalized variables, which can be modeled as random fields
with spatial dependence. They propose various reweighting schemes to
equalize the contribution of each surface unit for the choices of splits. It is not
adapted to the classification case.

2. Methodology

a. Classical Classi_cation And Regression Trees (CART)
The data are considered as an independent sample of the random
1
variables (X ,...,X ,Y ), where the X s are the explanatory variables and Y is the
k
p
categorical response variable. CART is a rule-based method that generates a
binary tree through recursive partitioning that splits a subset (called a node)
of the data set into two subsets (called sub-nodes) according to the
minimization of a heterogeneity criterion computed on the resulting sub-
nodes. Each split involves a single variable. Some variables may be used
several times while others may not be used at all.
Let us consider a decision tree T. When Y is a categorical variable a class
label is assigned to each terminal node (or leaf) of T. Hence T can be viewed
as a mapping to assign a value = ( , … … , ) to each observation.

̂
1

The growing step leading to a deep maximal tree is obtained by recursive
partitioning of the training sample by selecting the best split at each node
according to some heterogeneity index, such that it is equal to 0 when there
is only one class represented in the node to be split, and is maximum when all
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