Elementary effects method: Difference between revisions

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The '''elementary effects (EE) method''' is the most used{{cnCitation needed|date=January 2010}} screening method in [[sensitivity analysis]]. It is applied to identify non-influential inputs for a computationally costly [[mathematical model]] or for a model with a large number of inputs, where the costs of estimating other sensitivity analysis measures such as the variance-based measures is not affordable. Like all screening, the EE method provides qualitative sensitivity analysis measures, i.e. measures which allow the identification of non-influential inputs or which allow to rank the input factors in order of importance, but do not quantify exactly the relative importance of the inputs.

==Methodology==

To exemplify the EE method, let us assume to consider a mathematical model with <math> k </math> input factors. Let <math> Y </math> be the output of interest (a scalar for simplicity):<br />

: <math> Y = f(X_1, X_2, ... X_k).</math>

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Each trajectory is composed of <math>(k+1)</math> points since input factors move one by one of a step <math> \Delta </math> in <math>\{0, 1/(p-1), ... , 1-1/(p-1), 1\}</math> while all the others remain fixed. Figure 1 shows an example of trajectory in three dimensions.

Along each trajectory the so called ''elementary effect'' for each input factor is defined as:<br />

: <math> d_i(X) = \frac{Y(X_1, \ldots ,X_{i-1}, X_i + \Delta, X_{i+1}, \ldots, X_k ) - Y( \mathbf X)}{\Delta} </math>,

where <math> \mathbf{X} = (X_1, X_2, ... X_k)</math> is any selected value in <math> \Omega </math> such that the transformed point is still in <math> \Omega </math> for each index <math> i=1,\ldots, k. </math>

<math> r </math> elementary effects are estimated for each input <math> d_i\left(X^{(1)} \right), d_i\left( X^{(2)} \right), \ldots, d_i\left( X^{(r)} \right) </math> by randomly sampling <math> r </math> points <math> X^{(1)}, X^{(2)}, \ldots , X^{(r)}</math>.

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p/[2(p-1)]</math>, as this ensures equal probability of sampling in the input space.

In case input factors are not uniformly distributed, the best practice is to sample in the space of the quantiles and to obtain the inputs values using inverse cumulative distribution functions. Note that in this case <math> \Delta </math> equals the step taken by the inputs in the space of the quantiles.<br />

In case input factors are not uniformly distributed, the best practice is to sample in the space of the quantiles and to obtain the inputs values using inverse cumulative distribution functions. Note that in this case <math> \Delta </math> equals the step taken by the inputs in the space of the quantiles.<br />

The two measures <math> \mu </math> and <math> \sigma </math> are defined as the mean and the standard deviation of the distribution of the elementary effects of each input:<br />

: <math> \mu_i = \frac{1}{r} \sum_{j=1}^r d_i \left( X^{(j)} \right) </math>,<br />

: <math> \sigma_i = \sqrt{ \frac{1}{(r-1)} \sum_{j=1}^r \left( d_i \left( X^{(j)} \right) - \mu_i \right)^2} </math>.<br />

These two measures need to be read together (e.g. on a two-dimensional graph, see Figure 2) in order to rank input factors in order of importance and identify those inputs which do not influence the output variability. Low values of both <math> \mu </math> and <math> \sigma </math> correspond to a non-influent input.<br />

An improvement{{cnCitation needed|date=January 2010}} of this method was developed by Campolongo et al.<ref>Campolongo, F., J. Cariboni, and A. Saltelli (2007). An effective screening design for sensitivity analysis of large models. ''Environmental Modelling and Software'', '''22''',

1509&ndash;1518.</ref>{{Better source|date=January 2010}} who proposed a revised measure <math> \mu^* </math>, which on its own is sufficient to provide a reliable ranking of the input factors. The revised measure is the mean of the distribution of the absolute values of the elementary effects of the input factors:<br />

: <math> \mu_i^* = \frac{1}{r} \sum_{j=1}^r \left| d_i \left( X^{(j)} \right) \right| </math>.<br />

The use of <math> \mu^* </math> solves the problem of the effects of opposite signs which occurs when the model is non-monotonic and which can cancel each other out, thus resulting in a low value for <math> \mu </math> (see Figure 3 for an example).<br />

An efficient technical scheme to construct the trajectories used in the EE method is presented in the original paper by Morris while an improvement strategy aimed at better exploring the input space is proposed by Campolongo et al..