Elementary Effects

Elementary Effects method (EE) is one of the most used screening technique in sensi- tivity analysis. It develops the idea of Morris method, which in turn is based on the OAT method. Elementary Effects determine which input factors have larger effect on the objective functions, measuring their type of contribution (e.g. linear, non-linear, interactive or negligible) by few simple indicators. Originally only two measures were computed. The most important is µ, which describes the overall influence of the se- lected variable to the objective functions. Then it comes σ, estimating how the variable contributes inside the entire domain and how it interacts with other variables. Later reviews of the model introduced a new parameter µ∗. It estimates the mean contribute of the absolute value of the chosen decision variable. This last parameter allows to highlight factors whose contributes take opposite sign in the domain, realizing a null µ value and a non-zero σ one. However, this information could be already understood from the previous two parameters, but the evaluation of µ∗ is done with no extra com- putational cost. In alternative to the absolute value it could be considered the squared effects, which has been proven to be a less robust measure.

The Elementary Effect procedure develops a randomized OAT experiment, requiring the following entries: a random starting point in the domain space x = (x1, . . . , xk), a

number of step p, the grid division of the k -dimensional region of experiment in p-levels and a step size. Moreover all the input factors of the objective function are considered uniformly distributed, so any value inside the domain space is equally probable. In [5], the EE of the i -th input factor associated to the point x is defined as:

di(x) =

 y(x1, . . . , xi−1, xi+ ∆, xi+1, . . . , xk) − y(x)

∆



(5.8)

In eq. 5.8 the argument of the first function evaluation y(x1, . . .) is the vector x+ei∆ ∈

Ω ∀i = 1, . . . , k, with Ω domain space and ei the vector, having all components null

but the i -th one. ∆ is the step size and it takes value in {_p−11 , . . . , 1 − _p−11 }. The elementary effect of the i -th input factor is obtained by randomly sampling different point x in the domain and building its final distribution Fi, such as dix ∼ Fi. Suggested

values for the parameter p and ∆ are respectively a general even value for p and ∆ = _2(p−1)p . These allow the method to realize a sampling which guarantee more or less an equal-probability for each Fi.

Finally to properly obtain the distribution of each elementary effect, Morris in [42] suggested to build r trajectories each with k + 1 points. These provide k elementary effects, for a total cost of the experiment of r(k + 1) evaluation runs. Saltelli later proved that the performance of such sensitivity analysis was quite satisfying, however they are still liable to statistical error of Type II7 while quite robust against Type

7_{Typically three types of error are reported (different authors introduce others of them):}

Type I Rejecting the null hypothesis when it is true. It occurs when a non-influential factor is wrongly identified as so.

Type II Accepting the null hypothesis when it is false. This occurs when the analysis fails in highlight a factor which has considerable influence.

Type III correctly rejecting the null hypothesis for the wrong reason. Type III errors are difficult to identifies and can be translated as a researcher providing the right answer to the wrong question.

I error. It happens because with a non-monotonic objective function, negative and positive contribution can cancel out each other, thus producing a low µ value. Anyway one could still identify this factor by the associated σ value. In fact, in this case it would be rather large, if the factor is non-negligible but it cancels out itself. The introduction of µ∗ leads to a simpler treating of this question.

A further improvement of the method was presented in [5]. The article introduces a sort of modification on the trajectory, since the original one often leads to non-optimal coverage of the input set. The idea consists in generating much more trajectories than those necessary (10–50 times the parameter r ) and then chose those with the highest spread over the search space. Classification of the best trajectories is performed using a distance metric dml between two of them:

dml =        k+1 X i=1 k+1 X j=1 v u u t k X z=1 xm i (z) − xlj(z) 2 for m 6= l 0 otherwise (5.9) Here xm

i (z) identifies the z -coordinate of the i -th point of the m-th trajectory. Then

dml evaluates the geometric distance between each couple of point of the trajectories.

Obviously one needs to define a 0 in this metric, therefore a reference trajectory which allows the distances evaluation. Finally the choice of the r trajectories is done evalu- ating the distance covered by each possible combination of an r sample of trajectories over M possibilities. The total covered distance is defined as:

Dcomb =

q d2

1 + d22+ . . . + d2r

where each di is a different path dml. Literature suggests to prefer always this kind of

sampling strategy, since it provides much better performance than he Morris sampling. This happens mainly due to its improved ability in scanning the design space without further model evaluation.

The Elementary Effect has also the possibility to work with groups of variables, thanks to the parameter µ∗. This gives to the model an advantage over many others, since it can deal quite easily with problems with a great number of factors. EE can evaluate the contribution of a group of factors, despite in such frame it loses information on the single variables. The idea consists in realizing a step which involves all the factors in the same group, considering it as a single variable. However in this formulation the step size keep on being fixed, but its direction varies for each factor, e.g. the step could increase or decrease by ∆ each factor randomly. To properly measure the contribute and the effect of this different jump, only the parameter µ∗ can be used. In fact it has been observed that µ can hardly obtain correct results. The elementary effect for groups is evaluated by:

|dui(x)| =     y(_ex) − y(x) ∆     (5.10)

In eq. 5.10 subscript u identifies the vector of variables ui = (xi1, xi2, . . .), which

enumerates all the variables inside the i -th group. The _ex represents the vector having ui entries moved by a ±∆ quantity.

In conclusion, improved Morris method can gather quite good sensitivity analysis results without a too wide number of function evaluations. This is a key point of all the screening techniques. Comparing it to the Variance-based method 5.3.3, it can

67 5.3. Sensitivity analysis methods

obtain good results with a smaller number of evaluation, although it lacks in accuracy with respect to this other cited. On the other hand it can easily perform group analysis which are very useful in first testing. If the grouping is properly designed, it can retrieve results similar to those achieved in simple Morris model.