A very efficient method within the screening methods in identifying important factors with few simulations is the elementary effects method (EE method). It is very simple, easy to implement and the results are clear to be interpreted. It was introduced in Morris (1991) and has been refined by Campolongo and co-workers in Campolongo et al (2007). Because of the ABS structure’s complexity it is computationally expensive and EE method is very well suited for the sensitivity analysis of the ABS model’s output.
The elementary effect (EE) of a specific input factor is the difference in the model output when this particular input factor is changed, while the rest of the input factors are kept constant. The method is thus based on one-at-a-time sensitivity analysis. However, in the EE method the one-at-a-time analysis is done many times for each input, each time under different settings of the other input factors, and the sensitivity measures are calculated from the empirical distribution of the elementary effects.
Let us assume that there are k uncertain input parameters X1, X2, . . . , Xk (assumed to be
independent) in our model. Examples of input parameters are the mean and standard deviation of the default distribution.
To each input factor we assign a range and a distribution. For example, we could assume that X1 is the mean of the default distribution and that it takes values in the range [5%,30%]
uniformly, that is, each of the values in the range is equally likely to be chosen. We could of course use non-uniform distributions as well, for example, an empirical distribution.
These input parameters and their ranges create an input space of all possible combinations of values for the input parameters. To apply the EE method we map each of the ranges to the unit interval [0,1] such that the input space is completely described by a k-dimensional unit cube.
The original method by Morris provides two sensitivity measures for each input factor i= 1,2, . . . k:
1
We have been using the ’sobolset’ class (with the ’MatousekAffineOwen’ scramble algorithm) and ’RandStream’
class (with the ’mrg32k3a’ generator algorithm) in MATLABr for generating Sobol sequences and pseudo
7.5. Sensitivity Analysis - Elementary Effects 69
• µi used to detect input factors with an important overall influence on the output;
• σi used to detect input factors involved in interaction with other factors or whose effect is
not linear.
In order to estimate the sensitivity measures, a number of elementary effects must be calculated for each input factor. Morris suggested an efficient design that builds r trajectories in order to compute r elementary effects. Each trajectory is composed by (k+ 1) points in the input space such that each input factor changes value only once. A characteristic of this design is that the points on the same trajectory are not independent and in fact two consecutive points differ only in one component. Points belonging to different trajectories are independent since the starting points of the trajectories are independent.
Once a trajectory has been generated, the model is evaluated at each point of the trajectory and one elementary effect for each input factor can be computed. The EE of input factor i is either:
EEi(X(l)) =
Y X(l+1)−Y X(l)
∆ (7.3)
if the ithcomponent ofX(l) has been increased by ∆ or
EEi(X(l)) =
Y X(l)−Y X(l+1)
∆ (7.4)
if the ith component of X(l) has been decreased by ∆, whereY X(l) is the model output of interest calculated in the point X(l) on the trajectory.
By randomly sampling r trajectories, r elementary effects can be estimated for each input. Usually the number of trajectories (r) depends on the number of factors, on the computational cost of the model, and on the number of levels (p) that each input can vary across. It has been proven that the best choice is to letpbe an even integer and ∆ to be equal to 2(pp−1) (see Saltelli
et al (2004) and Saltelli et al (2008)).
The sensitivity measures are defined as the mean and the standard deviation of the distri- bution for the elementary effects of each input:
µi = Pr j=1EEij r (7.5) and σi= s Pr j=1(EE j i −µi)2 r−1 . (7.6)
70 Chapter 7 - Global sensitivity analysis for ABS
When considering elementary effects with opposite signs related to theith factor, the effects may cancel each other out generating a low µi value. To overcome this problem Morris recommends
to consider both µi and σi simultaneously in order to be able to draw some conclusions on the
factor importance.
In Campolongo et al (2007), two improvements of the original EE method are proposed. Firstly, the sampling strategy to generate the trajectories is constructed such that optimized trajectories are generated. A large number of different trajectories (e.g. 1000) is constructed and then r of them are selected in order to get the maximum spread in the input space. (See Campolongo et al (2007) for the all details about the design that builds the r trajectories of (k+ 1) points in the input space.) The second improvement is the introduction of a new sensitivity measure based on the absolute values of the elementary effects:
µ∗
i =
Pr
j=1|EEij|
r . (7.7)
This new sensitivity measure overcomes the cancelation effect mentioned earlier and can alone be used to assess the importance of each factor in the model.
Section 7.6 presents the results obtained by applying this methodology to the ABS model.