Clustering Contextual Neural Gas: A New Approach for Spatial Planning and Analysis Tasks - Computational Approaches for Urban Environments - page 78

the ordering of the l spatially closest neurons depends on attribute similarity. Hence,
spatial closeness is less important for the final ordering and less spatial dependence
is being incorporated. If l equals the total number of neurons, the spatial ordering
does not matter for the final ordering, because all neurons are totally reordered in
the second step by similarity of attributes. Consequently, no spatial dependence is
incorporated at all.
CNG has several advantages over other spatial clustering algorithms: Like the
GeoSOM, CNG enforces spatial proximity between observations and neurons by
means of neural distance, defined by either the map's topology or the rank ordering
or neurons. Consequently, it is not necessary to weigh or scale spatial proximity and
attribute similarity in the data space. Furthermore, the neurons are basically local
averages. Thus, the process of incorporating spatial dependence is less sensitive to
random variations in the input data. Finally, the parameter l restricts the mapping
of observations; all observations are always mapped to one of its l spatially closest
neurons. Hence, the mapping maintains a certain degree of spatial closeness, even
for observations whose attributes are very different from those of their spatial
neighbors (spatial outliers).

4.2.2 Competitive Hebbian Learning

Competitive Hebbian Learning (CHL; Martinetz and Schulten 1991 ; Martinetz
1993 ) forms a topology on a set of neurons by creating a number of connections
between neighboring neurons. More specifically, the algorithm can be described
as follows: For each input vector, the two closest neurons are determined and
a connection between these is added to the total set of connections. Thereby,
closeness is typically measured by Euclidean distance. After all input vectors have
been presented, the set of connections represents the topology of the underlying
data.
The resulting graph optimally preserves the topology in a very general sense
(Martinetz 1993 ). In particular, each connection between two neurons belongs to the
Delaunay triangulation corresponding to the neurons in data space. The theoretical
foundations of CHL in terms of topology preservation have been provided by
Edelsbrunner and Shah ( 1997 ).
CHL is especially useful for NG and other vector quantization algorithms which
do not define a topological structure. It can be applied concurrently to the training
of NG or as a post-processing step. However, in the first case, the movement
of neurons during the training may make previously learned connections invalid.
Therefore, it is necessary to constantly adapt the topology to these movements, e.g.,
by removing outdated connections (Martinetz and Schulten 1991 ). In the latter case,
NG is trained before CHL is applied, and hence, the topology is not affected by
the movement of the neurons. For simplicity, this study applies CHL as a post-
processing step.