Variance-based sensitivity analysis (often referred to as the Sobol’ method or Sobol’ indices, after Ilya M. Sobol’) is a form of global sensitivity analysis. Working within a probabilistic framework, it decomposes the variance of the output of the model or system into fractions which can be attributed to inputs or sets of inputs. For example, given a model with two inputs and one output, one might find that 70% of the output variance is caused by the variance in the first input, 20% by the variance in the second, and 10% due to interactions between the two. These percentages are directly interpreted as measures of sensitivity. Variance-based measures of sensitivity are attractive because they measure sensitivity across the whole input space (i.e. it is a global method), they can deal with nonlinear responses, and they can measure the effect of interactions in non-additive systems.
Decomposition of variance From a black box perspective, any model may be viewed as a function Y=f(X), where X is a vector of d uncertain model inputs {X1, X2, ... Xd}, and Y is a chosen univariate model output (note that this approach examines scalar model outputs, but multiple outputs can be analysed by multiple independent sensitivity analyses). Furthermore, it will be assumed that the inputs are independently and uniformly distributed within the unit hypercube, i.e. X i ∈ [ 0 , 1 ] {\displaystyle X_{i}\in [0,1]} for i = 1 , 2 , . . . , d {\displaystyle i=1,2,...,d} . This incurs no loss of generality because any input space can be transformed onto this unit hypercube. f(X) may be decomposed in the following way,
Y = f 0 + ∑ i = 1 d f i ( X i ) + ∑ i < j d f i j ( X i , X j ) + ⋯ + f 1 , 2 , … , d ( X 1 , X 2 , … , X d ) {\displaystyle Y=f_{0}+\sum _{i=1}^{d}f_{i}(X_{i})+\sum _{i<j}^{d}f_{ij}(X_{i},X_{j})+\cdots +f_{1,2,\dots ,d}(X_{1},X_{2},\dots ,X_{d})}
where f0 is a constant and fi is a function of Xi, fij a function of Xi and Xj, etc. A condition of this decomposition is that,
∫ 0 1 f i 1 i 2 … i s ( X i 1 , X i 2 , … , X i s ) d X k = 0 , for k = i 1 , . . . , i s {\displaystyle \int _{0}^{1}f_{i_{1}i_{2}\dots i_{s}}(X_{i_{1}},X_{i_{2}},\dots ,X_{i_{s}})dX_{k}=0,{\text{ for }}k=i_{1},...,i_{s}}
i.e. all the terms in the functional decomposition are orthogonal. This leads to definitions of the terms of the functional decomposition in terms of conditional expected values,
f 0 = E ( Y ) {\displaystyle f_{0}=E(Y)}
f i ( X i ) = E ( Y | X i ) − f 0 {\displaystyle f_{i}(X_{i})=E(Y|X_{i})-f_{0}}
f i j ( X i , X j ) = E ( Y | X i , X j ) − f 0 − f i − f j {\displaystyle f_{ij}(X_{i},X_{j})=E(Y|X_{i},X_{j})-f_{0}-f_{i}-f_{j}}
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