Euradwaste '08 - EU Bookshop - Europa
Euradwaste '08 - EU Bookshop - Europa 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
duce plots close to the diagonal, since large and small output values can be equally found in any of their regions. Additionally, since the values represented on the x axis are independent of the input parameter, many such curves may be plotted in the same graphic corresponding to one output variable and many input parameters. Additionally, a test has been developed to check if deviations from the diagonal are statistically significant or come from pure statistical randomness, see Bolado et al. (2008). Figure 1 is an example of a CSM plot for one output and nine inputs (model used in step 2 of the benchmark on SA techniques). Only inputs W and V (1) show statistically significant deviations from the diagonal. Small values of W are related to large values of the output while the largest values of the output are related to intermediate values of V (1) . For all the other inputs deviations from the diagonal are not statistically significant (large and small values of the output may be obtained in any region of those inputs). Figure 1: Contribution to the sample mean plot for the dose due to 129 I at 9·10 4 y (output variable) and all the input parameters in the model used in the second step of the benchmark on SA techniques. Other graphic tools, as for example the radar plots or the tornado bars, may be used to visualize, in a comparative manner, the value of a given sensitivity index (for example the Pearson correlation coefficient) for a given output and all or a fraction of the input parameters. But these are completely different tools for they show sensitivity indices instead of a representation of the sampled values. 393
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duce plots close to the diagonal, since large and small output values can be equally found in any of<br />
their regions. Additionally, since the values represented on the x axis are independent of the input<br />
parameter, many such curves may be plotted in the same graphic corresponding to one output variable<br />
and many input parameters. Additionally, a test has been developed to check if deviations from<br />
the diagonal are statistically significant or come from pure statistical randomness, see Bolado et al.<br />
(2008). Figure 1 is an example of a CSM plot for one output and nine inputs (model used in step 2<br />
of the benchmark on SA techniques). Only inputs W and V (1) show statistically significant deviations<br />
from the diagonal. Small values of W are related to large values of the output while the largest<br />
values of the output are related to intermediate values of V (1) . For all the other inputs deviations<br />
from the diagonal are not statistically significant (large and small values of the output may be obtained<br />
in any region of those inputs).<br />
Figure 1: Contribution to the sample mean plot for the dose due to 129 I at 9·10 4 y (output variable)<br />
and all the input parameters in the model used in the second step of the benchmark on SA techniques.<br />
Other graphic tools, as for example the radar plots or the tornado bars, may be used to visualize, in<br />
a comparative manner, the value of a given sensitivity index (for example the Pearson correlation<br />
coefficient) for a given output and all or a fraction of the input parameters. But these are completely<br />
different tools for they show sensitivity indices instead of a representation of the sampled values.<br />
393
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