Geography Reference
In-Depth Information
most of the state of Ohio (OH) combines block groups whose mapping had included distinct
gaps (see region corresponding to Ohio in the block group visualization in Figure 8.6).
There are also a number of examples where the expected preservation of topological
relationships does not occur, and illustration of this is also made easier by a GIS-based
representation (Figure 8.8). Some states are represented by a number of disjoint polygons. For
example, Pennsylvania (PA) becomes represented by four main polygons in the spatialization
of states (left side of Figure 8.8). Notice, however, the distinct geographic organization of
these four parts as representing mainly the western and eastern portions of the state (parts
numbered 1 and 2). All four portions of Pennsylvania are themselves positioned next to
neighbouring states (parts numbered 1-4 positioned next to Ohio, New Jersey, New York
and Maryland, respectively).
There are also some decidedly odd neighbours in the SOM-based map of states. For
example, notice the polygons representing Kansas and Nebraska appearing as neighbours
of California (Figure 8.7). Zooming in on a part of that SOM region shows that there is
a large gap between block groups in northern California/southern Oregon and those in
southeastern Nebraska/northeastern Kansas (right side of Figure 8.8). Clearly, creating and
using a high-resolution SOM is a difficult proposition and much remains to be learned
about strategies for training such a SOM and for how to identify artefacts introduced by the
computational process.
This is greatly helped by specific knowledge of the computational method used for di-
mensionality reduction. For example, the SOM method is known to preserve the density of
input vectors. During the course of SOM training, the occurrence of multiple input vectors
from one region of attribute space will cause that region to be represented by a large number
of neurons, i.e. the region will be represented in expanded form. The opposite is true also,
so that thinly 'populated' attribute space regions become represented by few neurons, i.e.
attribute space appears compacted. In the case of census block groups this has pronounced
effects. Owing to the role played by total population numbers in the delineation of census
block groups (aiming at a total population of around 1500, as mentioned earlier), geo-
graphic areas exhibiting high population density will be represented by more block groups
and therefore more neurons, compared with regions with lower population density. In our
experiment, where state polygons are constructed from block group polygons, the SOM thus
acts as a type of area cartogram! That is why high-population states like California (CA)
and high-density states like Connecticut (CT) are represented as relatively large polygons,
while low-density states like North Dakota (ND) or Idaho (ID) remain small (Figure 8.7).
The reliability of this density preservation effect is, however, tempered by edge effects.
Edge neurons capture relatively large portions of attribute space and the size of state poly-
gons near the edges and especially corners (like Arizona (AZ) in the lower right corner of
Figure 8.7) should thus be treated with caution.
8.4.5 Mapping n-dimensional clusters onto the climate SOM
Readers will at this point appreciate the difficulty of judging whether two-dimensional pat-
terns and relationships visually observed in the SOM actually correspond to n -dimensional
structures. For instance, our climate experiment led to multiple examples of the SOM gener-
ating shared borders between states that are not neighbours in traditional geographic space.
It would be nice to more directly operate on n -dimensional data, while being able to project
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