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specific sampling strategy, with the only exception of the simplest method (correlation ratios - CR).
Graphical methods provide complementary visual information that helps understanding the meaning
of numerical sensitivity indices and the global structure of the system model.
2.2.1 Monte Carlo based methods
Monte Carlo based methods may be divided in three types: regression/correlation based methods,
Monte Carlo Filtering and gridding methods. Regression/correlation methods assume that inputs
and outputs can have a linear relation. The simplest method is Pearson’s correlation coefficient.
This is a measure of linear relation that takes values between –1 and +1. Positive values indicate
joint linear increase or decrease, negative values indicate that when the input increases the output
decreases and vice versa. The closer the value is to +1 or to –1, the stronger the relation is, thus the
more important the input parameter is. An alternative related measure of sensitivity is the slope or
regression coefficient of the output versus the input (after standardisation of the sampled values,
subtracting the corresponding sample mean and dividing by the corresponding standard deviation,
in order to avoid scale effects in the sensitivity indices). When the importance of several inputs is
analysed at the same time, the tool used is multiple regression together with input and output standardisation.
In this case, the indices used are the Partial Correlation Coefficients (PCC) and the
Standardised Regression Coefficients (SRC). These methods become really important when the inputs
are correlated, otherwise they are respectively exactly the same as Pearson’s correlation coefficient
and regression coefficients in simple regression. The importance of PCCs and SRCs comes
from the fact that they measure respectively the correlation and the regression coefficient between
one input and one output after removing the influence of all the other inputs. Unfortunately, by default,
these indices are used to analyse only main effects, interactions are hardly ever considered in
the analysis. Additionally, hardly ever practitioners study possible transformations of inputs and
outputs to get a more appropriate regression model.
In many cases, instead of linear, the relation between inputs and outputs is monotonic, in those
cases it is convenient to transform inputs and outputs into their ranks, the largest sampled value is
transformed into n (sample size), the second largest one into n-1,…, the smallest into 1. This way,
the same tools (renamed as Partial Rank Regression Coefficients -PRCC- and Standardised Rank
Regression Coefficients –SRRC-) may be used to assess the importance of each input parameter.
The reliability of the results obtained via linear regression depends on the Coefficient of Determination
(R 2 ) of the regression model obtained. If R 2 is close to 1 the results are very reliable, if it is
close to 0, it means that this sensitivity method is not appropriate to study the system model at hand.
Monte Carlo Filtering (MCF) is based on dividing the output sample in two or more subsets according
to some criterion (achievement of a given condition, exceeding a threshold, etc.) and testing if
the inputs associated to those subsets are different or not. As an example, we could divide the output
sample in two parts, the one that exceeds a safety limit and the rest. We could wonder if points
in both subsamples are related to different regions of a given input or if they may be related to any
region of that input. In the first case knowing the value of that input parameter would be important
in order to be able to predict if the safety limit will be exceeded or not, while in the second case it
would not be. The tools used to provide adequate answers to this type of questions are a set of parametric
and non-parametric statistics and their associated tests, as for example the two-sample t
test, the two-sample F test, the two-sample Smirnov test, the k-sample Smirnov test, the Cramervon
Mises test, the Wilcoxon test (also known as Mann-Whitney test) and the Kruskal-Wallis test,
see Conover (1980) for details about each specific statistic and test. In general, non-parametric tests
should be preferred for fewer restrictions are imposed on the samples used. The idea behind Gridding
and the tests used are similar to Monte Carlo Filtering. The only real difference is that the cri-
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