Euradwaste '08 - EU Bookshop - Europa

2.2.3 Graphical methods
Let us call X=(X1,X2,…,XK) the vector of input parameters and Y to a given scalar output variable.
For a given input Xi, the scatter-plot is the projection of the sample points (X,Y) on the (Xi,Y)
plane. This representation allows the examination of the dependence between Y and Xi. Scatterplots
are very helpful to identify linear relations, monotonic relations and the existence of thresholds
among other potential trends. A wise use of transformations (e.g. logarithmic, ranks, etc.) may
also provide a lot of information about input/output relations. They may be used as supporting material
to explain the results obtained by means of numeric sensitivity techniques, but also to prevent
the use of inadequate techniques.
Three-dimensional scatter-plots or XYZ plots show the projection of the sample points (X,Y) on the
(Xi,Xj,Y) space. The information they are able to provide is also valuable. The extraction of such
information is limited, though challenging, due to obvious interpretation problems when a 3-D figure
is shown on a 2-D display. Software packages that allow changing the angle of the view may
enhance and broaden their applicability.
Extensions of scatter-plots are matrices of scatter-plots and overlay scatter-plots. Matrices of scatter-plots
show simultaneously, under a matrix format, the scatter-plots of different pairs of input
parameters/output variables. They allow identifying quite quickly the pairs with most remarkable
relations, but they are also affected by the loss of accuracy due to including several plots in a reduced
space, typically a fraction of a page. Overlay scatter-plots allow showing on the same plot the
scatter-plot of one output and several inputs. In order to distinguish the points corresponding to different
inputs, different symbols (dots, circles, crosses, diamonds, etc.) and different colours are
used. Frequently only a few inputs may be represented due to either the different scales used in the
plot or to the difficulties to interpret correctly so many overlapped different symbols.
Cobweb plots have been designed to show multidimensional samples in a two-dimensional graph,
see Cooke and Van Noortwijk (1999). Vertical parallel lines separated by equal distances are used
to represent the sampled values of a given number of inputs/outputs, usually not more than ten or
twelve, in order to keep the plot sufficiently clear. Each vertical line is used for a different input/output
and either the raw values or the ranks may be represented (either raw values or ranks in
all lines, never mixed). Sampled values are marked in each vertical line and jagged lines connect
the values corresponding to the same run. Coloured lines can be used to display the different regions
of any input parameter or output variable. Moreover, flexible conditioning capabilities enable
an extensive insight into particular regions of the mapping. The cobweb plots are usually provided
with ‘cross densities’ showing the density of line crossings midway between the vertical axes.
Therefore, an informed and careful analysis of cobweb plots enables the characterisation of dependence
and conditional dependence.
The Contribution to the sample mean plot (CSM plot) represents the fraction of a given output variable
mean that is due to any given fraction of smallest values of any input. This is obtained by putting
on the x axis the values of the Empirical Cumulative Distribution Function (ECDF) of any input
parameter (this values are 1/n, 2/n,…, 1 for any sample of size n of any continuous random
variable), and putting on the y axis the fraction of the output variable sample mean corresponding to
the smallest value of the input parameter, to the two smallest values, and so on. This way, a monotonic
non-decreasing curve is obtained. Plotting the ECDF in the x axis means that equal lengths
represent approximately regions of equal probability of the input parameter. The more the plot deviates
from the diagonal in a given region, the more (or the less) that region of the input variable
contributes to the sample mean or the sample variance. In fact, non-important input variables pro-
392