Mapping Associations - Mapping Scientific Frontiers: The Quest for Knowledge Visualization - page 123

Fig. 3.34 SSCI weighted individual differences scaling display (Reproduced from Morris and
McCain 1998 )
difference between Euclidean distances and geodesic distances is explained in the
following example. For a passenger on a particular line of the London Underground,
the geodesic distance between two stations is measured along the rail tracks, which
form a curved or wiggled one-dimensional data. The geodesic distance is how far
the train has to cover. For a passenger on a hot-air balloon, on the other hand, the
distance between the two stations could be measured along a straight line connecting
the two stations. The straight-line distance is the Euclidean distance, which is often
shorter than the geodesic distance. In classic PCA and MDS, there is no built-in
mechanism to distinguish geodesic distances and Euclidean distances. Manifold
scaling algorithms, also known as non-linear MDS, are designed to address this
problem. Because they are more generic than standard PCA and MDS, and given
the popularity of PCA and MDS, manifold scaling algorithms have a potentially
broad critical mass of users.
The basic idea is linear approximation. When we look the railway track
immediately underneath our feet, they are straight lines. On the other hand, if we
look far ahead, the track may bend smoothly in distance. An important step in linear