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Fig. 3.21
An alluvial map of popular tweet topics identified as Hurricane Sandy approaching
Fig. 3.22
An alluvial map of co-occurring patterns of chemical compound fragments
3.3.1
Geometry of Similarity
The strength of association is often measured in terms of proximity, similarity,
or relatedness. Multidimensional scaling (MDS) is a commonly used method to
reduce the dimensionality of such data and to depict data points in a two- or three-
dimensional spatial configuration. The principal assumption of MDS is that the
similarity data can be transformed to inter-point distances in a metric space by a
linear or monotonic decreasing function. The stronger the similarity between two
data points in the source data, the closer the corresponding points are in the metric
space. The key concern is how well such mapping preserves the structure in the
original data. The goodness of fit is most commonly measured by a stress value.
While symmetric similarity measures are common, similarity measures can
be asymmetric in nature. Amos Tversky ( 1977 ) questioned both the metric and
dimensional assumptions of similarity data. He proposed a feature - matching model
in an attempt to establish that the common features tend to increase the perceived
similarity of two concepts, and distinct features tend to diminish perceived simi-
larity. Furthermore, Tversky's model claims that our judgments of similarity are
asymmetric. Common features tend to have more influencing power than distinct
features over the way we gauge similarity conceptually.
Carol Krumhansl ( 1978 ) proposed a distance-density model in response to the
objections to geometric models raised by Tversky. She suggested that the similarity
between objects is a function not only of inter-point distance in a metric space but
also the spatial density of points in the surrounding configuration. In short, the
density of the metric space reduces the strength of the perceived similarity. Two
points in a relatively dense region of a stimulus space would appear to have a smaller
similarity measure than two points of equal inter-point distance but located in a less
dense region of the space.
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