Elementary effects method
Encyclopedia
The elementary effects method is the most used 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.
input factors. Let 
be the output of interest (a scalar for simplicity):
The original EE method of Morris provides two sensitivity measures for each input factor:
These two measures are obtained through a design based on the construction of a series of trajectories in the space of the inputs, where inputs are randomly moved One-At-a-Time (OAT).
In this design, each model input is assumed to vary across
selected levels in the space of the input factors. The region of experimentation 
is thus a 
-dimensional 
-level grid.
Each trajectory is composed of
points since input factors move one by one of a step 
in 
while all the others remain fixed.
Along each trajectory the so called elementary effect for each input factor is defined as:
where
is any selected value in 
such that the transformed point is still in 
for each index 
elementary effects are estimated for each input 
by randomly sampling 
points 
.
Usually
~ 4-10, depending on the number of input factors, on the computational cost of the model and on the choice of the number of levels 
, since a high number of levels to be explored needs to be balanced by a high number of trajectories, in order to obtain an exploratory sample.
It is demonstrated that a convenient choice for the parameters
and 
is 
even and 
equal to 
, 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
equals the step taken by the inputs in the space of the quantiles.
The two measures
and 
are defined as the mean and the standard deviation of the distribution of the elementary effects of each input:
These two measures need to be read together (e.g. on a two-dimensional graph) 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
and 
correspond to a non-influent input.
An improvement of this method was developed by Campolongo et al. who proposed a revised measure
, 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:
The use of
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 
.
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..
Sensitivity analysis
Sensitivity analysis is the study of how the variation in the output of a statistical model can be attributed to different variations in the inputs of the model. Put another way, it is a technique for systematically changing variables in a model to determine the effects of such changes.In any...
. It is applied to identify non-influential inputs for a computationally costly mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
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 input factors. Let be the output of interest (a scalar for simplicity):
The original EE method of Morris provides two sensitivity measures for each input factor:
- the measure
, assessing the overall importance of an input factor on the model output; - the measure
, describing non-linear effects and interactions.
These two measures are obtained through a design based on the construction of a series of trajectories in the space of the inputs, where inputs are randomly moved One-At-a-Time (OAT).
In this design, each model input is assumed to vary across
selected levels in the space of the input factors. The region of experimentation is thus a -dimensional -level grid.Each trajectory is composed of
points since input factors move one by one of a step in while all the others remain fixed.Along each trajectory the so called elementary effect for each input factor is defined as:
-
,
where
is any selected value in such that the transformed point is still in for each index elementary effects are estimated for each input by randomly sampling points .Usually
~ 4-10, depending on the number of input factors, on the computational cost of the model and on the choice of the number of levels , since a high number of levels to be explored needs to be balanced by a high number of trajectories, in order to obtain an exploratory sample.It is demonstrated that a convenient choice for the parameters
and is even and equal to , 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
equals the step taken by the inputs in the space of the quantiles.The two measures
and are defined as the mean and the standard deviation of the distribution of the elementary effects of each input:-
, -
.
These two measures need to be read together (e.g. on a two-dimensional graph) 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
and correspond to a non-influent input.An improvement of this method was developed by Campolongo et al. who proposed a revised measure
, 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:-
.
The use of
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 .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..