  Self-organizing maps (SOMs) are a technique that has been used with
high-dimensional data vectors to develop an archetypal set of states (nodes)
that span, in some sense, the high-dimensional space. Noteworthy applications
include weather states as described by weather variables over a region and
speech patterns as characterized by frequencies in time. The SOM approach is
essentially a neural network model that implements a nonlinear projection from
a high-dimensional input space to a low-dimensional array of neurons. In the
process, it also becomes a clustering technique, assigning to any vector in the
high-dimensional data space the node (neuron) to which it is closest (using,
say, Euclidean distance) in the data space. The number of nodes is thus equal
to the number of clusters. However, the primary use for the SOM is as a
representation technique, that is, finding a set of nodes which
representatively span the high-dimensional space. These nodes are typically
displayed using maps to enable visualization of the continuum of the data
space. The technique does not appear to have been discussed in the statistics
literature so it is our intent here to bring it to the attention of the
community. The technique is implemented algorithmically through a training set
of vectors. However, through the introduction of stochasticity in the form of a
space--time process model, we seek to illuminate and interpret its performance
in the context of application to daily data collection. That is, the observed
daily state vectors are viewed as a time series of multivariate process
realizations which we try to understand under the dimension reduction achieved
by the SOM procedure.
