This section will provide some deeper insights into concepts of interactive and dynamic graphics mentioned in the previous sections. [23] contains a taxonomy of interactive data visualization based on the notions of focusing, linking, and arranging views of data. [170] discusses some of the main concepts in the context of interactive graphics software.
Perhaps the most basic concepts for statistical graphics are
scatterplots
(see Figs. 10.1, 10.2,
10.3, and 10.4). In a simple
scatterplot, we place different symbols (sometimes also called glyphs)
at 
- and 
-positions in a two-dimensional plot area. These
positions are determined by two of the variables. The type, size, and
color of the symbols may depend on additional variables. Usually,
explanatory information such as axes, labels, legends, and titles are
added to a scatterplot. Additional information such as a regression
line or a smoothed curve can be added as well.
![\includegraphics[width=11.7cm]{text/2-10/fig1.eps}](img2921.gif)
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![\includegraphics[width=11.7cm]{text/2-10/fig2.eps}](img2922.gif)
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If the data consist of more than two variables (e.g., somewhere
between three to ten), the data can be displayed by a scatterplot
matrix
(see Figs. 10.2 and 10.3) that shows all
pairwise scatterplots of the variables. The essential property of
a scatterplot matrix is that any adjacent pair of plots have one of
their axes in common. When plotting the full array of all
pairwise scatterplots, each plot in the upper triangle of plots
has a matching plot in the lower triangle of plots, with the exception
that the axes in these pairs of plots have been flipped. Therefore,
sometimes only the upper or lower triangle of scatterplots is
displayed; thus gaining plotting speed and allowing each individual
plot to be somewhat larger. Early examples of scatterplot matrices
can be found in [46] and [48] for example.
[46] initially called an array of pairwise scatterplots for
three variables a draftsman's display and for four (or more) variables
a generalized draftsman's display. In their (generalized) draftsman's
display, each point is plotted with the same symbol. When encoding
additional information through the use of different plotting symbols,
[46] speak of symbolic (generalized) draftsman's displays.
Today, we hardly make any distinction of these different types of
displays and just speak of scatterplot matrices.
[121] and [171] discuss features good scatterplots and related interactive software should provide, e.g., meaningful axes and scales, features for rescaling and reformatting, good handling of overlapping points and missing data, panning and zooming, and querying of points. [33] describe techniques for scatterplot matrices particularly useful for large numbers of observations.
Brushing, as introduced in [13] and [14], initially was considered as a collection of several dynamic graphical methods for analyzing data displayed in a scatterplot matrix. The central idea behind brushing is a brush, usually a rectangular area on the computer screen, that is moved by the data analyst to different positions on the scatterplot or any other graphical display. Four brushing operations were introduced in [13]: highlight, shadow highlight, delete, and label. The most commonly used brushing technique is highlighting - often in the context of linked brushing, i.e., for linked views. All points that are inside the brush in the currently selected display are highlighted, i.e., marked with a different symbol or color. Simultaneously, points that correspond to those points are automatically highlighted with the same symbol/color in all linked views.
A very useful brushing technique is the transient paint mode. As the brush is moved, the new points that come inside the brush are highlighted while points that move outside the brush are no longer highlighted.
While brushing initially was only developed for scatterplot matrices, it quickly has been adapted to other types of linked graphical displays. Linked brushing among different displays is one of the most useful techniques used within dynamic and statistical graphics. Linked brushing can be applied to graphical representation of continuous data, summary data such as histograms ([140]), or even displays of categorical data such as mosaic plots ([89], [90]). All dynamic statistical graphics software packages support linked brushing among different types of graphical displays these days.
When dealing with massive data sets, it is often beneficial to focus on particular subgroups of the data and also be able to quickly return to a previous stage of the analysis. Selection sequences ([165,91]) are an extension of the conventional linked-highlighting paradigm as they store the whole hierarchical path of a selection and allow an easy editing, redefinition, and interrogation of each selection in the path of the analysis. In a selection sequence, we can easily jump from one branch of the hierarchic selection tree to another.
Focusing techniques, as introduced in [25], are based on the idea that it often might be easier for a human analyst to understand several individual displays, each focused on a particular aspect of the underlying data, rather than looking at the full data set. Focusing techniques include subset selection techniques, e.g., panning and zooming or slicing, and dimensionality reduction techniques, e.g., projection. Methods for focusing can be automatic, interactive, or a combination of both. While focusing shows only part of the data at a time, it is important to display multiple linked views of the data, perhaps each focusing on a different aspect of the data, to maintain the full picture of the data.
Zooming is a technique that can be used for inspecting details of the data when overplotting arises. Zooming can be done via some kind of a magnifying glass or by manually selecting subsections of the visible axes, e.g., via sliders. The main idea behind zooming is that when several points overplot in the full display, it may indeed turn out that these points are exactly the same when zooming into the neighborhood of these points - or, what most frequently happens, that these points have a particular structure and are not exactly the same.
Panning is closely related to zooming. An analyst should know which subset of the data is currently visible. Therefore, an information plot should reveal the current location on which subregion we have zoomed.
Slicing, as described in [79] and [80], is a technique that takes sections (or slices) of a high-dimensional data set. While slicing (and projections) are useful means for an exploratory data analysis, these techniques also have their limitations. However, these limitations may be overcome by combining slicing and projections in so-called prosections ([80]). An extension of individual prosection views is the prosection matrix ([168]), some kind of a density plot summarizing multi-dimensional volumetric information. The prosection matrix is a useful representation for engineering design, allowing an analyst to interactively find a design that leads to a maximal manufacturing yield.
Rescaling
is a technique that allows a user
to quickly change the scale of the displayed variables,
e.g., by taking the log, square root, standardize, or by mapping to
a 
scale. When looking at multiple variables, it might also be
beneficial to have a common scale (from the minimum across all
variables to the maximum across all variables). By interactively
rescaling variables, an analyst may identify useful transformations
for a follow-up modeling step of the data.
Reformatting
includes features as simple as swapping and axes in
a scatterplot or changing the order of coordinate axes in a parallel
coordinate plot.
[171] provides more details on several of the techniques described above.
Rotation, as introduced in [73] and later refined in [14], is a very powerful tool for understanding relationships among three or more variables. The familiar planar scatterplot is enhanced by rotation to give the illusion of a third dimension. We typically rotate plots in search of some interesting views that do not align with the plot axes and therefore cannot be seen in a scatterplot matrix. Usually, a three-dimensional point cloud representing three of the variables is shown rotating on a computer screen. The rotation shows us different views of the points and it produces a 3D effect while moving, allowing us to see depth. Basic rotation controls with a mouse have been introduced in [14].
Mathematically speaking, each rotation within a 3D space onto a 2D computer screen is based on a projection. Obviously, it is mathematically possible to project high-dimensional data onto low-dimensional subspaces and gain insights into the underlying data through dynamic visualizations of such projections. One particular example of a continuous sequence of projections, the grand tour, will be discussed in the next section. [52] discuss methods how to manually control high-dimensional data projections. [50] provides a variety of training data sets that help new users to get a visual feeling of the underlying high-dimensional data set when seen as a projection into low-dimensional space.
Often, simple plot rotation, as discussed in the previous section, does not suffice to see all interesting views of the data. To produce a plethora of possible interesting views, the grand tour has been introduced in [7] and [21]. In [7], the grand tour has been described as ''a method for viewing multivariate statistical data via orthogonal projections onto a sequence of two-dimensional subspaces. The sequence of subspaces is chosen so that it is dense in the set of all two-dimensional subspaces.'' Some of the features the grand tour can be used for are examining the overall structure and finding clusters or outliers in high-dimensional data sets.
In the context of the grand tour, an alternating sequence of brushing, looking at additional projections from the grand tour, brushing, and so on, is referred to as the brush-tour strategy in the remainder of this chapter. We can only be sure that a cluster visible in one projection of the grand tour really is a cluster if its points remain close to each other in a series of projections and these points move similarly when the grand tour is activated. If points move apart, we probably found several subclusters instead of one larger cluster.
[181] discusses a form of the grand tour for general
-dimensional space. The algorithms for computing a grand tour are
relatively computationally intensive. [190] discuss an
approximate one- and two-dimensional grand tour algorithm that was
much more computationally efficient than the Asimov winding
algorithm. That algorithm was motivated in part by a discussion of the
Andrews (multidimensional data) plot, discussed in
Sect. 10.3.9, which can also be regarded as
a highly restricted pseudo tour.
![\includegraphics[width=11.7cm]{text/2-10/fig3.eps}](img2925.gif)
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Parallel coordinate plots ([102,180])(see Fig. 10.3) are a geometric device for
displaying points in high-dimensional spaces, in particular, for
dimensions greater than three. The idea is to sacrifice orthogonal
axes by drawing the axes parallel to each other resulting in a planar
diagram where each -dimensional point
is
uniquely represented by a continuous line. The parallel coordinate
representation enjoys some elegant duality properties with the usual
Cartesian coordinates and allows interpretations of statistical data
in a manner quite analogous to two-dimensional Cartesian
scatterplots. This duality of lines in Cartesian plots and points in
parallel coordinates extends to conic sections. This means that an
ellipse in Cartesian coordinates maps into a hyperbola in parallel
coordinates. Similarly, rotations in Cartesian coordinates become
translations in parallel coordinates.
The individual parallel coordinate axes represent one-dimensional projections of the data. We can isolate clusters by looking for separation between data points on any axis or between any pair of axes. Because of the connectedness of the multidimensional parallel coordinate diagram, it is usually easy to see whether or not this clustering propagates through other dimensions.
The use of parallel coordinate plots for a -dimensional grand tour
sequence, sometimes called a parallel coordinate grand tour, has been
described in [181] and [184]. By using such a parallel
coordinate grand tour, an analyst can find orientations where one or
more clusters are evident. The general strategy for detecting clusters
is the following: We begin with a static plot of the data in parallel
coordinates. If there are any gaps along a horizontal axis (which
incidentally does not need to coincide with the coordinate axes), then
we color the individual clusters with distinct colors. Once all
clusters are identified in the original coordinate system, we run the
grand tour until an orientation of the axes is found in which another
gap in one of the horizontal axes is found. Again we color the
individual subclusters with distinct colors. This procedure is
repeated until no further subclusters can be identified. This is
another example of the brush-tour strategy referred to in
Sect. 10.3.5. Indeed, when to stop is a matter
of judgement, since the procedure can be repeated until practically
every data point can be individually colored. The crucial issue, which
really depends on the dynamic graphics, is to see that clusters
identified in this manner track coherently with the grand tour
animation. That is, data points of the same color stay together as the
grand tour rotation proceeds. If they do not, then there are likely to
be substructures that can be identified through further grand tour
exploration.
Slopes of parallel coordinate line segments can also be used to
distinguish clusters. That is, if a group of line segments slopes,
say, at 45
to the horizontal and another group slopes at,
say, at 135 to the horizontal, then even though the lines
fully overlap in both adjacent parallel coordinate axes and there is
no horizontal gap, these sets of lines represent two distinct clusters
of points. Fortunately, when such indication of clustering exists, the
grand tour will also find an orientation of axes in which there is
a horizontal gap. Thus the general strategy is to alternate color
brushing of newly discovered clusters with grand tour rotations until
no further clusters can be easily identified.
In some software packages, the parallel axes in a parallel coordinate plot are drawn as horizontal lines (e.g., in ExplorN) while in other software packages they are drawn as vertical lines (e.g., in XGobi). While it may be argued that this makes no difference from a mathematical point of view, the wider aspect ratio in the horizontal mode coupled with a more usual sense of plotting data along an abscissa rather than along the ordinate tends to allow for an easier human interpretation. Detailed interpretations are given in [180].
While the grand tour, as discussed in Sect. 10.3.5, is a dynamic tool, projection pursuit ([110,78,92]), see also Chap. III.6, is a static tool. Projection pursuit results in a series of static plots of projections that are classified as ''interesting'' with respect to a particular projection pursuit index. Many projection pursuit indexes, e.g., the ones discussed in [105], [76], [83], [119], [120]), [53], and [129], are based on the idea to search for the most non-normal projections. Usually, each projection pursuit index, a function of all possible projections of the data, results in many hills and valleys. [76] suggests a projection pursuit algorithm that initially searches for relatively high values of the function and then starts derivative-based searches to find the global maximum.
The combination of grand tour and projection pursuit, called projection pursuit guided tour ([54]), helps to direct the grand tour towards ''interesting'' projections. This combination of the two methods into an interactive and dynamic framework not only shows the ''interesting'' projections but it maintains the motion so the user has a feeling how successive ''interesting'' projections have been obtained.
The idea of the pixel or image grand tour (IGT) evolved from an initial application of one-dimensional tours to image data. Multiple registered images can be regarded as a multidimensional image in which each pixel location has a vector attached to it. For example, ordinary red, green, and blue (RGB) color images are vector-valued images. The basic idea of the image tour is to apply the same one-dimensional grand tour to each vector for all pixel locations in an image. This combines the vectors into a scalar function of time which can be rendered as a time-varying gray-scale image. The [190] algorithm generalizes easily to two dimensions, so that an alternate approach to the IGT is to project the multidimensional vector into two dimensions and render the image as a false color image with two complementary colors such as red and cyan. It should be noted that red and cyan are complementary colors in the RGB color model used for most computer monitors whereas red and green are complementary colors in the conventional color model, introduced by the Commission Internationale de l'Éclairage (CIE) in 1931. A detailed comparison of these two and other color models can be found in [74], Chap. 13. The initial discussion of the IGT was given by [189]. Additional examples of the IGT can be found in [160].
Currently, the IGT software, written in C++ by Qiang Luo, is available
for Silicon Graphics, Inc., (SGI) workstations. To obtain a fast
rendering rate of large images, the software intensively uses SGI
hardware features such as the 
-channel hardware. There exists
also a MATLAB version of the IGT written by Wendy Martinez. Both
versions of the IGT software are not accessible through a Web site but
can be obtained from the corresponding software developers.
The Andrews (multidimensional data) plot, as introduced in [1] is based on a series of Fourier interpolations of the coordinates of multi-dimensional data points. Points that are close in some metric will tend to have similar Fourier interpolations and therefore will tend to cluster in the Andrews plot. Thus, the Andrews plot is an informative graphical tool most useful to detect clustering.
Ideas underlying the Andrews plot and the grand tour are quite similar. However, in contrast to the grand tour, the Andrews plot is a static plot while the grand tour is dynamic. Although dynamic renditions of the Andrews plot exist, and these sometimes also are (incorrectly) referred to as one-dimensional grand tour ([61]), the Andrews plot is not a grand tour since it cannot sweep out all possible directions as pointed out in [190]. Three-dimensional generalizations of the Andrews plot and other pseudo grand tours have been introduced in [190] as well.
[33] present techniques for visualizing data in scatterplots
and scatterplot matrices when the data consists of a large number of
observations, i.e., when overplotting of points frequently occurs
using standard techniques. A key idea to address in the visualization
of a large number of observations is based on the estimation and
representation of densities. For this purpose, the data is often
binned into an
matrix for two-dimensional representation
(or an
matrix for three-dimensional
representation). Possibilities to visualize the number of data points
in each bin can be based on gray-scale (or color) density
representations or by symbol area such as using differently sized
hexagon symbols, where the area of the plot symbol is proportional to
the count in each bin. [27] further extends these ideas and
presents additional low-dimensional displays for data that consist of
a large number of observations. [137] provides a general
overview on techniques for density estimation, including averaged
shifted histograms (ASH)
and kernel density estimators,
including possible visualization techniques via contour surfaces,
(transparent) -level contours,
and contour shells.
Further details on multivariate density estimation and visualization
can be found in Chap. III.4.
[187] use hue and saturation for plotting and brushing.
For each individual point, the hue is almost fully desaturated with
black. When points are overplotted, the hue components are added. The
saturation level should be interactively adjustable by the analyst.
If many points overplot, the pixel will be fully saturated. If fewer
points overplot, the pixel will be shown in a less saturated color.
Often, computer hardware devices such as the -channel
allow the blending of pixel intensities with no speed penalties. When
using saturation for parallel coordinate plots and the level of
saturation corresponds with the degree of overplotting, this creates
a kind of parallel coordinate density plot ([187], [188]).
![\includegraphics[width=11.7cm]{text/2-10/fig4.eps}](img2929.gif)
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Although categorical data are quite common in the real world, little research has been done for their analysis and visualization when compared to quantitative data. However, there exist useful interactive and dynamic graphics for categorical data ([127,166]). For example, brushing and linking of categorical data represented via bar charts and pie charts can be as useful as for quantitative data ([96]). Modified bar charts where the same height is used for each category and the width is varied according to the number of counts are called spine plots ([96]). When interactively highlighting a category of interest, spine plots allow the analyst to visually compare the proportions in the different subcategories by looking at the heights of the highlighted areas. Examples of interactive graphics for categorical data such as spine plots and interactive mosaic plots (see Fig. 10.4) can be found in Hofmann ([89], [90]). [175]) discuss spreadplots (and their implementation in ViSta), a method for laying out and simultaneously controlling graphics for categorical data.
[15] present a collection of papers dealing with the visualization of categorical data. Main topics include graphics for visualization, correspondence analysis, multidimensional scaling and biplots, and visualization and modeling. Several of these approaches benefit from interactive and dynamic graphics.
spatial locations is displayed in ArcView at the top. The two XGobi
windows at the bottom are showing scatterplots of Crime (horizontal)
vs. Education (vertical) (left) and Recreation (horizontal) vs. Arts
(vertical) (right). Locations of high Crime have been brushed and identified,
representing some of the big cities in the U.S. Also, locations of high
Education (above 
) have been brushed, mostly representing locations in
the northeastern U.S. All displays have been linked
, one group for a Population between
, and one group for a Population above
(left) and
above 
to