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it. As a result, such methods can drastically distort initial data structures (Duch et al.
2000) that were relevant to the user’s task. For instance, a formal proximity measure
such as the Euclidean metric can contradict meaningful similarity of n-D points known
in the given domain. Domain experts can know that n-D points a and b are closer to
each other than n-D points c and d,|a, b|<|c, d|, but the formal externally imposed
metric F may set up an opposite relation, F(a, b)>F(c, d). In contrast, lossless data
displays presented in this book provide opportunity to improve interpretability of
visualization result and its understanding by subject matter experts (SME).
The common expectation of metric approaches is that they will produce relatively
simple clouds of 2-Dpointson the plane with distinct lengths, widths, orientations,
crossings, and densities. Otherwise, if patterns differ from such clouds, these methods
do not help much to use other unique human visual perception and shape recognition
capabilities in visualization (Grishin 1982; Grishin et al. 2003). Together all these
defic iencies lead to a shallow understanding of complex n-D data.
To cope with abilities of the vision system to observe directly only 2-D/3-D spaces,
many other common approaches such as Principal Components Analysis (PCA) also
project every n-D data point into a single 2-D or 3-D point. In PCA and similar
dimension reduction methods, it is done by plotting the two main components of these
n-D points (e.g., Jeong et al. 2009). These two components show only a fraction of all
information contained in these n-D points. There is no way to restore completely n-D
points from these two components in general beyond some very special datasets. In
other words, these methods do not provide an isomorphic (bijective, lossless, rever-
sible) mapping between an n-D dataset and a 2-D dataset. These methods provide only
a one-way irreversible mapping from an n-D dataset to a 2-D data set.
Such lossy visualization algorithms may not find complex relations even after
multiple time-consuming adjustments of parameters of the visualization algorithms,
because they cut out needed information from entering the visualization channel. As
a result, decisions based on such truncated visual information can be incorrect
.
Thus, we have two major types of 2-D visualizations of n-D data available to be
combined in the hybrid approach:
(1) each n-D point is mapped to a 2-D point (P-to-P mapping), and
(2) each n-D point is mapped to a 2-D structure such as a graph (we denote this
mapping as P-to-G), which is the focus of this book.
Both types of mapping have their own advantages and disadvantages.
Principal Component Analysis (PCA) (Jolliffe 1986;Yin2002), Multidimensional
Scaling (MDS) (Kruskal and Wish 1978), Self-Organized maps (SOM) (Kohonen
1984), RadVis (Sharko et al. 2008) are examples of (1), and Parallel Coordinates
(PC) (Inselberg 2009), and General Line Coordinates (GLC) presented in this book are
examples of (2). The P-to-P representations (1) are not reversible (lossy), i.e., in general
there is no way to restore the n-D point from its 2-D representation. In contrast, PC and
GLC graphs are reversible as we discuss in depth later.
The next issue is preserving n-D distance in 2-D. W hile such P-to-P repre-
sentations as MDS and SOM are specifically designed to meet this goal, in fact,
they only minimize the mean difference in distance between the points in n-D and
1.2 Visualization: From n-D Points to 2-D Points 3