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The Projection Pursuit |
An Investigation of Methods for Visualising Highly Multivariate Datasets## 2.3 InterpretationHaving obtained an optimal projection, it is essential that this can be
easily interpreted. Since the projection is a linear mapping, interpretation
is fairly straighforward. Having optimised in terms of b, then a unit change in the _{j}jth original variable
causes a change (a) in the projection space.
Since the projection is linear, this statement is independent of the values of
other variables. Also due to linearity, a change by an amount _{j}, b_{j}k in the
jth variable leads to a change (ka)
in the projection space. Using this fact, one can plot the ´change
vectors' for a given point in the plot in projection space when each of the
initial variables changes by one standard deviation. This is illustrated in
figure 4._{j}, kb_{j}Figure 4: Minimised MNND projection of census data - Interpretation Plot This gives some clues as to the variables causing the ´spur' in the projection shown in figure 3. Although a number of possible variable combinations could cause this, figure 4 suggests that very high unemployment levels or low crowding could cause this, perhaps with low levels of the other variables. ## 2.4 Choice of IAt this point, some further discussion about the choice of the index
function, The result of this approach is shown in figure 5, together with an interpretation plot in figure 6. The spur feature has now completely disappeared, and the projected points now form a more symmetrical pattern, but a number of outliers are visible in many directions around the outside of the cloud. The interpretation plot should help in identifying the nature of the outliers as in the previous example. A further possibility is to consider means of highlighting geographical
trends. In this case, the idea of ´projection' takes a different form.
Here, a one-dimensional projection is used, where i and j are
neighbours, and zero if they are not. Neighbourhood may be defined in a number
of ways. Typically zones are neighbours if they share a common boundary, or if
their centroids are less than some distance apart. If the z values are
standardised to have variance one and mean zero, then the above expression
simplifies to
If we are attempting to maximise or minimise this expression, the denominator
may be ignored, since it is a positive constant. Thus, for projection pursuit
designed to high geographical relationships, a suitable
Thus, here the projection pursuit problem can be stated as
since in this case
Interpretation of the single-dimensional projection is probably best done by
tabulating the elements of Applying the method to the census data gives the maps in Appendix B, which show indices for maximum and minimum spatial autocorrelation respectively. The coefficients of projection (adjusted for scale) are given in table 1. Here it can be seen that the maximising map mostly picks up an urban/rural trend, whereas more subtle differences are picked out in the minimising map. In particular it highlights the way some nearby rural areas differ. The strongest contributing variables in the maximising case are CROWD, LLTI and UNEMP. It is suggested that this linear combination of variables is perhaps a useful indicator of 'urbanness' in the sense that high values tend to coincide with inner cities and low values with rural areas. On the other hand, the coefficients for the minimising case give a very different index. This index is useful for differentiating between nearby places, and is more strongly influenced by variables that are more spatially variable. A good example is SPF which has a much greater weighting in the minimising index. Although there is no strong geographical trend in the proportion of single parent families, it can be used as a means of differentiating between nearby places. Another variable that does this is CROWD which is possibly a differentiator between affluent and poor rural communities. ## 2.5 Geographically Weighted RegressionAt this point, another trend-based method of analysis should be considered
briefly. This is the method of
Note that this differs from the projection pursuit using Moran's-I in two major
ways. First, whereas the projection pursuit method produces just one map, (or
two if Moran's I is both minimised and maximised), GWR produces a map for each
regression coefficient, plus one for the intercept coefficient. Secondly,
project pursuit treats all variables identically, whereas GWR requires that one
variable has to be ´singled out' as the dependent variable. A
comprehensive example of the technique is given in Brunsdon |

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