Scatterplots
Figure 1
Scatterplots
are one of the oldest and most commonly used methods to project
high dimensional data to 2dimensions. In this method, N * (N
 1)/2 pairwise parallel projections are generated, each giving
the viewer a general impression regarding relationships within
the data between pairs of dimensions. The projections are
generally arranged in a grid structure to help the user remember
the dimensions associated with each projection. Many variations
on the scatterplot have been developed to increase the information
content of the image as well as provide tools to facilitate data
exploration. Some of these include rotating the data cloud
[TUK:88], using different symbols to distinguish classes
of data and occurrences of overlapping points, and using color
or shading to provide a third dimension within each projection.
Figure 1 presents
a seven dimensional data set using scatterplots. Note that
plotting each dimension against itself along the diagonal provides
distribution information on the individual dimensions. The
data set contains statistics regarding crime in Detroit between
1961 and 1973, and consists of 13 data points. The data set was
obtained via anonymous ftp from unix.hensa.ac.uk in the directory
/pub/statlib/datasets. Some dimensions of the original set have
been eliminated to facilitate display using scatterplots. Linear
structures within several of the projections indicate some correlation
between the two dimensions involved in the projections. Thus,
for example, there is a correlation between the number of fulltime
police, the number of homicides, and the number of government
workers (with a corresponding negative correlation in the percent
of cleared homicides).
One major
limitation of scatterplots is that they are most effective with
small numbers of dimensions, as increasing the dimensionality
results in decreasing the screen space provided for each projection.
Strategies for addressing this limitation include using three
dimensions per plot or providing panning or zooming mechanisms.
Other limitations include being generally restricted to orthogonal
views and difficulties in discovering relationships which span
more than two dimensions. Advantages of scatterplots include
ease of interpretation and relative insensitivity to the size
of the data set.
We have extended
flat scatterplots to hierarchical scatterplots. In hierarchical
scatterplots, clusters are displayed instead of individual data
points. In each plot, a cluster is presented by a point and a
colorful band around it. The point and the band indicate the mean
and the extend of the cluster. Movie
1 is a multiresolutional cluster display of hierarchical scatterplots.
References
[TUK:88]:
Tukey, J.W., Fisherkeller, M.S., Friedman, J.H.. PRIM9,
an interactive multidimensional data display and analysis system.
Dynamic Graphics for Statistics (W.S. Cleveland and M.E.
McGill, eds.), Wadsworth and Brooks, 1988.
[ward:94]:
M. Ward. Xmdvtool: Integrating multiple methods for visualizing
multivariate data. Proc. of Visualization '94, p.
32633, 1994.
