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RADVIZ |
An Investigation of Methods for Visualising Highly Multivariate Datasets## 3. The RADVIZ Approach to Visualisation}Like the previous approach, the RADVIZ method (Ankerst et al., 1996) maps a
set of S. Now suppose a set of
_{m}m springs are fixed at one end to each of these points, and that all of
the springs are attached to the other end to a puck, as in figure 7.Figure 7: The Physical System Basis for RADVIZ
Finally, assume the stiffness constant (in terms of Hooke's law) of the
i.
If the puck is released and allowed to reach an equilibrium position, the
coordinates of this position, (u,
_{i}v)_{i}^{T} say, are the projection in two
dimensional space of the point (x, ...,
_{i1}x)_{im}^{T} in m-dimensional space. This, if
(u, _{i}v)_{i}^{T} is
computed for i = 1 ... n, and these points are plotted, a visualisation
of the m-dimensional data set in two dimensions is achieved.
To discover more about the projection from R, consider the forces acting on the puck. For a given
spring, the force acting on the puck is the product of the vector spring
extension and the scalar stiffness constant. The resultant force acting on the
puck for all ^{2}m springs will be the sum of these individual forces. When
the puck is in equilibrium there are no resultant forces acting on it and this
sum will be zero. Denoting the position vectors of S to
_{1}S by _{m}S_{1} to S_{m}, and
putting u = (_{i}u)_{i},
v_{i}^{T} we havewhich may be solved for
Thus, for each case R nonlinear. ^{2}
Viewing the projection in this explicit form allows several of its properties
to be deduced. First, assuming that the u_{i} lies within the convex hull of
the points S_{1} to S_{m},. Due to the regular
spacing of these points, this convex hull will be an m-sided regular
polygon. Note that if some of the x values are negative
this property need not hold, but that often each variable is re-scaled to avoid
negative values. Two typical methods of doing this are the local metric
(L-metric) rescaling, in which the minimum and maximum values of
_{ij}x for each _{ij}j are respectively mapped onto zero and
one respectively:and the global metric (G-metric), in which the rescaling is applied to the data set as a whole, rather than on a variable by variable basis:
in each case, the rescaled
The weighted centroid interpretation of the projection also allows some other
properties to become apparent. If, for a given u_{i} will be the
zero vector. This is a rather strange property, since it implies that
observations in which all variables take on a very high constant value (once
re-scaled) will be projected onto the same point as observations in which all
variables take on a very low constant value. More generally, it suggested that
observations which take on very similar values for re-scaled data will be
mapped into regions close to the origin.A RADVIZ projection for the census data is shown in figure 8. The result is superficially similar to the maximised MNND projection pursuit, showing a circular cluster of points and identifying outliers around this. Figure 8: A RADVIZ Projection of the Census Data
For general data sets this property could lead to difficulties in interpreting
the plots, but it is particularly useful when considering
In the case x) for a given case, more than
one composition can project onto the same _{ij}u_{i}, as
discussed above.
It is also interesting to note that for a given set of variables, there are
several possible RADVIZ projections, since the $m$ initial
One way of deciding which of these should be used is to use an index, in a
similar manner to projection pursuit in the previous section. In fact, the
same indices could be used - for example maximising the variance of the
m is very large. |

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