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Krumhansl analyzed the implications of the feature-matching model. She con-
cluded that when we judge the similarity between two entities, we actively seek
what they have in common. She suggested that the distance-density model is more
accurate than the feature-match model when it comes to accounting for variations
in similarity data.
In later chapters, we will explain the role of Pathfinder network scaling in
preserving salient structures with explicit links. This is related to Gestalt psychol-
ogy, which identifies pattern-inviting features such as proximity, similarity, and
continuity. However, the original Gestalt psychology overlooks the role of explicit
linkage in helping us recognize a pattern more easily. Pathfinder networks provide
representations that can enhance pattern recognition.
3.3.2
Multidimensional Scaling
PCA finds a low-dimensional embedding of the data points that best preserves their
variance as measured in the high-dimensional input space. Classic MDS finds an
embedding that preserves the pairwise point distances (Steyvers 2000 ). PCA and
MDS are equivalent if Euclidean distances are used.
Let us illustrate what MDS does with some real-world examples, including
distances between cities and similarities between concepts. In general, there are two
types of MDS: Metric and Non-metric MDS. Metric MDS assumes that the input
data is either ratio or interval data, while the non-metric model requires simply that
the data be in the form of ranks.
A metric space is defined by three basic axioms, which are assumed by a
geometric model:
1. Metric minimality: for the distance function d and any point x , the equation d ( x ,
x ) D 0 holds.
2. Metric symmetry: for any data points x and y, the equation d ( x , y ) D d ( y , x ) holds.
3. Metric triangle inequality: for any data points x , y ,and z , the inequality d ( x ,
y ) C d ( y , z ) d ( x , z ) holds.
Multidimensional scaling (MDS) is a standard statistical method used on mul-
tivariate data (See Fig. 3.23 ). In MDS, N objects are represented as d-dimensional
vectors with all pairwise similarities or dissimilarities (distances) defined between
the N objects. The goal is to find a new representation for the N objects as k-
dimensional vectors, where k < d such that the interim proximity nearly matches
the original similarities or dissimilarities. Stress is the most common measure of
how well a particular configuration reproduces the observed distance matrix.
Given a matrix of distances between a number of major cities from the back of
a road atlas or an airline flight chart, we can use these distances as the input data
to derive an MDS solution. Figure 3.6 shows the procedure of generating an MDS
map. When the results are mapped in two dimensions, the configuration should look
very close to a conventional map, except that you might need to rotate the MDS
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