Elementary effects method

The elementary effects (EE) 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. The elementary effects (EE) 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. To exemplify the EE method, let us assume to consider a mathematical model with k {displaystyle k} input factors. Let Y {displaystyle Y} 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 p {displaystyle p} selected levels in the space of the input factors. The region of experimentation Ω {displaystyle Omega } is thus a k {displaystyle k} -dimensional p {displaystyle p} -level grid. Each trajectory is composed of ( k + 1 ) {displaystyle (k+1)} points since input factors move one by one of a step Δ {displaystyle Delta } in { 0 , 1 / ( p − 1 ) , 2 / ( p − 1 ) , . . . , 1 } {displaystyle {0,1/(p-1),2/(p-1),...,1}} while all the others remain fixed. Along each trajectory the so-called elementary effect for each input factor is defined as: where X = ( X 1 , X 2 , . . . X k ) {displaystyle mathbf {X} =(X_{1},X_{2},...X_{k})} is any selected value in Ω {displaystyle Omega } such that the transformed point is still in Ω {displaystyle Omega } for each index i = 1 , … , k . {displaystyle i=1,ldots ,k.} r {displaystyle r} elementary effects are estimated for each input d i ( X ( 1 ) ) , d i ( X ( 2 ) ) , … , d i ( X ( r ) ) {displaystyle d_{i}left(X^{(1)} ight),d_{i}left(X^{(2)} ight),ldots ,d_{i}left(X^{(r)} ight)} by randomly sampling r {displaystyle r} points X ( 1 ) , X ( 2 ) , … , X ( r ) {displaystyle X^{(1)},X^{(2)},ldots ,X^{(r)}} . Usually r {displaystyle r} ~ 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 p {displaystyle p} , 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 p {displaystyle p} and Δ {displaystyle Delta } is p {displaystyle p} even and Δ {displaystyle Delta } equal to p / [ 2 ( p − 1 ) ] {displaystyle p/} , as this ensures equal probability of sampling in the input space.