Euradwaste '08 - EU Bookshop - Europa

teria to divide a sample in two or more parts are set on the input space and the test is performed using
the corresponding points in the output space.
2.2.2 Variance based methods
The variance, or equivalently the standard deviation, and the entropy, are the main measures of uncertainty
in the theory of Probability. The larger the variance of a random variable is, the less accurate
our knowledge about it is. Decreasing the variance of a given output variable is quite an attractive
target, that may be achieved sometimes by decreasing the variance of input parameters (this is
not always true, remember the possibility of risk dilution). This is what makes so attractive methods
that try to find out what fraction of the output uncertainty (variance) may be attributed to the uncertainty
(variance) in each input.
Variance based methods find their theoretical support in Sobol’s decomposition of any integrable
function in the unit reference hypercube into 2 k orthogonal summands of different dimension: the
mean value of the function, k functions which depend each one only on one input parameter, k(k-
1)/2 functions that depend only on two input parameters, k(k-1)(k-2)/6 that depend only on three
input parameters and so on. Replacing any output variable of the system model (our function) by its
Sobol’s decomposition in the integral used to compute its variance produces in a straightforward
manner the decomposition of the variance in its components. The quotient between each component
of the variance and the total variance provides the fraction of the variance attributed to each single
input parameter (main effects), each combination of only two input parameters (second order interactions)
and so on. These are called Sobol’s sensitivity indices; see Sobol (1993). It is important to
remark that Sobol’s decomposition is equivalent to the classical Analysis of Variance (ANOVA)
used in Statistics.
Several algorithms have been proposed to compute Sobol’s indices, the first one by himself. The
main problem is related to the efficiency of the method. It needs one specific sample to compute
each sensitivity index. Since its development, huge efforts have been done to improve the strategies
(algorithms) to compute Sobol’s indices, see for example Saltelli (2002) and Tarantola et al. (2006).
It remains as a powerful but expensive method (in terms of computational cost).
Independently, and quite before the development of Sobol’s decomposition and Sobol’s indices, a
method had been developed to compute first order sensitivity indices (equivalent to first order
Sobol’s sensitivity indices): the Fourier Amplitude Sensitivity Test (FAST), see Cukier et al.
(1973), Schaibly and Shuler (1973) and Cukier et al. (1975). In order to compute sensitivity indices,
these authors create a search curve that covers reasonably well the input space. Each input parameter
is assigned an integer frequency. Varying simultaneously all input parameters according to that
set of frequencies generates the search curve. Equally spaced points are sampled from the search
curve and used to perform a Fourier analysis. The coefficients corresponding to the frequency (and
its harmonics) assigned to each input parameter are used to compute the corresponding sensitivity
index. Saltelli et al. (1999) did further improvements of the method, among them the possibility of
computing total sensitivity indices for a given input parameter (the fraction of the variance due to it
and all its interactions of any order). FAST remains unable to compute sensitivity indices for interactions.
Correlation Ratios are an alternative to Sobol’s method and FAST to compute first order sensitivity
indices using a normal sample (SRS, LHS, etc.). So, though a method used to compute variance
based sensitivity indices, it could also be considered Monte Carlo based.
391