considering autocorrelation in predictive models - Department of ...

30 Definition of the Problem
Figure 2.4: Spatio-temporal autocorrelation. An example of the effect of the spatio-temporal autocorrelation
on an observed variable in a map. The different colors indicate different values of the observed
variable. The different maps represent the observed variable in 1980 and 1984.
searchers in these fields have proven that systems of different nature can be represented as networks
(Newman and Watts, 2006). For instance, the Web can be considered as a network of web-pages, which
may be connected with each other by edges representing various explicit relations, such as hyperlinks.
Social networks can be seen as groups of members that can be connected by friendship relations or can
follow other members because they are interested in similar topics of interests. Metabolic networks can
provide insight about genes and their possible relations of co-regulation based on similarities in their
expressions level. Finally, in epidemiology, networks can represent the spread of diseases and infections.
Regardless of where we encounter them, networks consist of entities (nodes), which may be connected
to each other by edges. The nodes in a networks are generally of the same type and the edges
between nodes express various explicit relations. Information on the nodes is provided as a set of properties
(attributes) whose values are associated to each node in the network. The edges reflect the relation
or dependence between the properties of the nodes.
Network (relational) autocorrelation is defined as the property that a value observed at a node depends
on the values observed at neighboring nodes in the network (Doreian, 1990). It can be seen as a
special case of spatial (multi-dimensional and multi-directional) autocorrelation, but with different measures
of relationship applied. In social analysis, it is recognized by the homophily’s principle, that is the
tendency of nodes with similar values to be linked to each other (McPherson et al, 2001).
Figure 2.5 shows an example of a data network where different colors represent different node labels.
We can observe that there is a higher tendency of nodes having the same color to be connected to each
other, than the tendency of nodes having different colors to be connected to each other.
Formally, a network is a set of entities connected by edges. Each entity is called a node of the
network. A number (which is usually taken to be positive) called weight is associated with each edge.
In a general formulation, a network can be represented as a (weighted and directed) graph, i.e., a set of
nodes and a ternary relation which represents both the edges between nodes and the weights associated to
those edges. The network is represented by an adjacency matrix W, whose entries are positive (wi j > 0)
if there is an edge connecting i to j, and equal to 0 (wi j = 0), otherwise. In practice, when the original
data come in the form of a network, the weight wi j represents the strength of the connection from one
node to another.
Generalizing the Moran’s definition of spatial autocorrelation (Moran, 1950) of a random variable xn
to match n dimensions, we deal with a n-dimensional vector and the autocorrelation function is expressed
as the covariance between the values xd = (x1,d,...,xn,d) and xd ′ = (x1,d ′,...,xn,d ′), where τ = d′ −d is the