2.4.2 ‘Slicing’ the n-Dimensional Data Space
2.4.2.3 Prosection matrix
A variant of thescatterplot matrixmethod is calledprosection matrix[198]. It essentially works in the same way as the originalscatterplot matrix with the added bonus of allow-ing the user to define the range of each variable. Only observations whose variables lie within their corresponding range (or section) is projected on the scatterplot matrix (see
Figure 2.2-a). Hence the name prosection, which, in fact, was originally introduced by Furnas and Buja [76].
(a) (b)
Figure 2.2: Visualization of 3D parameter space with prosection matrix (pictures taken from [198]). Picture (a) shows a section defined on parameterp3and projected down on the plane defined by parameters p1/p2. Note the tolerance box as a dark grey square on that plane.
Picture (b) shows an actual prosection matrix visualization of a four parameter space. In that the different grey areas represent the levels of satisfaction for performance requirements, with the darkest grey level meaning the best satisfaction. In this case the tolerance region is reduced to the white dot in the centre of each scatterplot.
The prosection matrix was originally applied to the exploration of parameter space in a design simulation of lamp bulbs. A design simulation usually requires the definition of acceptable ranges for each parameter of the design and a tolerance range within which a design is considered valid. Note that the parameter space in this case is defined over a con-tinuous range. Thus the data for a single scatterplot are obtained from the design model, by providing the continuous acceptable ranges of the two parameters associated with the scatterplot, while the other parameters are randomly sampled over their corresponding tolerance range. Consequently if the tolerance range is very small all the input parame-ters of the design model are set to a single point (no random sampling), thus yielding a smooth output; whereas if the tolerance range (which is a square region) is interactively expanded on the prosection matrix, so is the scatterplots’ degree of fuzziness due to the random sampling applied on the corresponding increased parameters (see Figure 2.2-b).
2.4.2.4 HyperSlice
Sometimes the dimensional object is a function defined over a continuous high-dimensional domain, for which case it is a common approach to project slices of the
function instead of the whole high-dimensional domain. Functions of this nature are called multidimensional functions or scalar functions of many variables and are frequently used to describe models and simulations in Engineering and Mathematics applications. A typical application example occurs in parameter optimization problems, where we wish to visualize the value of an objective function, F = ( f1) in terms of a large number of control parameters, X= (x1, x2, . . .xn), say.
In 1993 van Wijk and van Liere proposed a technique called hyperslice [199, 200]
to tackle this type of function. Hyperslice looks at all 2D orthogonal subspaces of X , and presents a grid of contour maps. Each of these subspaces is a slice of the original data obtained by fixing the value of (n − 2) parameters to the corresponding coordinates of a focal point, c= (c1, c2, . . . , cn); and varying the remaining two parameters within a specified region. Thus it reduces from one n-dimensional space to m 2D spaces, where m= n(n − 1)/2. In fact the subspace visualizations are laid out in a symmetric n x n grid, with the diagonal showing n 1D visualizations, where only one parameter varies (so a line graph is drawn).
Figure 2.3 shows thehypersliceconcept applied to a 3D ellipsoid-type function:
f(x, y, z) =(x − 1.0)2
(0.8)2 +(y − 1.0)2
(0.5)2 +(z − 1.0)2 (0.2)2
Note how the different focal points of Figure 2.3-b and Figure 2.3-e yield different visu-alizations of the same function.
Hyperslice’s strongest point is the direct relation between screen space and data space, which is lost only if n-dimensional rotations in data space are applied. This relation made it possible to design a user interface that affords direct manipulation of operations such as navigation, identification of extrema, and definition of paths in n-dimensional space.
Although the authors decided to represent the function visually by two-dimensional slices, they recognised that three-dimensional techniques seem to be the most natural choice for the basic representation of the multidimensional function. The authors argue that the 3D representation has the advantage of encoding as many dimensions as possible simultaneously. Notwithstanding, they decided not to use 3D as the basic representation for thehyperslice for three reasons: (1) the techniques for volume rendering, at that time, were too slow for direct manipulation; (2) the interpretation of a 3D representation is more difficult than simpler two- and one-dimensional forms; and, (3) the interaction in 3D is not trivial.
Figure 2.3: Hyperslice technique applied to a 3D ellipsoid expressed by the function f(x, y, z) = (x−1.0)(0.8)22 +(y−1.0)(0.5)22 +(z−1.0)(0.2)22, with f restricted to the interval [0.0, 1.0]. Picture (a) shows an isosurface f = 1.0 of the function. In (b) the focal point is set toc1= (1, 1, 1) lo-cated where the three orthogonal slicing planes intersect, which generates the contours in (c). Picture (d) shows a hyperslice-type arrangement of those planes. In (e) the focal point was moved toc2= (0.4, 0.7, 0.85), generating the contour planes in (f), which in turn, yields the arrangement shown in (g).
Although thehyperslicemethod has proved to be a useful tool in visualizing multidi-mensional functions, we have identified three
shortcomings:-• Sometimes a single collection of two-dimensional slices is not enough to readily distinguish special features in a function. Instead we need to investigate several of those slices, taken from different locations in the n-dimensional space or even navigate through that space to mentally ‘visualize’ such a feature.
In contrast if we had a collection of three-dimensional ‘slices’ we might have iden-tified the same feature (e.g. a minimum point) without requiring navigation.
• Becausehyperslicerepresents the multidimensional function as a matrix of orthog-onal two-dimensiorthog-onal slices simultaneously, all the data corresponding to those sub-sets must be present before drawing. Thus the method is constrained in terms of performance by both the dimensionality of the function and the number of samples necessary to plot each slice.
• Hyperslice provides only a single focal point. Hence the only way of comparing different regions in the n-dimensional space is by moving the focal point through a user defined path. However, moving the focal point has an inevitable side effect:
the loss of the initial visualization.