Conducts a "stochastic contribution analysis" by calculating the change in the variance of the Monte Carlo output when each of the simulated input variables is kept constant at its mean, i.e. its uncertainty is removed.
stochContr(prop, plot = TRUE)The relative contributions \(C_i\) for all input variables, with the variance differences \(D_i\) as attribute "varDiff".
a propagate object.
logical. If TRUE, boxplots of the original output distribution and of those with each input fixed at its mean are drawn.
Andrej-Nikolai Spiess
This function takes the Monte Carlo simulated input data from a propagate object (...$datSIM), sequentially substitutes each input \(X_i\) by its mean \(\bar{x}_i\), and re-evaluates the model. Optional boxplots compare the original output distribution with those obtained by removing the uncertainty of each \(X_i\). The contribution of \(X_i\) is the resulting decrease in output variance,
$$D_i = \mathrm{V}(Y) - \mathrm{V}(Y \mid X_i = \bar{x}_i),$$
and the relative contributions are \(C_i = D_i/\sum_j D_j\), so that \(\sum_i C_i = 1\).
Interpretation. For a linear model with independent inputs, \(D_i = c_i^2 \mathrm{Var}(X_i)\) up to Monte Carlo noise, so that \(C_i\) equals the relative variance contribution of the law of propagation of uncertainty if all inputs have Gaussian margins or the same \(\nu\). In general, the expectation of \(\mathrm{V}(Y \mid X_i)\) over \(X_i\) is \(\mathrm{V}(Y) - V_i\), with \(V_i\) the first-order Sobol' variance, so that \(D_i/\mathrm{V}(Y)\) is a one-point estimate of the first-order index \(S_i\), evaluated at \(X_i = \bar{x}_i\) instead of averaged over \(X_i\). For nonlinear models, \(D_i\) can therefore be negative, which is reported in a message. Use sobol for variance-based sensitivity indices.
Dependent inputs. Fixing one input at its mean while keeping the simulated values of the others ignores their conditional distribution. For correlated inputs (\(|r_{ij}| > 4/\sqrt{M}\) for any pair of the \(M\) simulated input vectors) and for mc.margins = "mvt", a warning is issued, and the contributions are not interpretable.
sobol, summary.propagate (relative contributions from the law of propagation of uncertainty).
a <- c(15, 1)
b <- c(100, 5)
phi <- c(0.5, 0.02)
DAT <- cbind(a, b, phi)
EXPR <- expression(a * b^sin(phi))
RES <- propagate(EXPR, DAT, nsim = 10000)
SC <- stochContr(RES)
SC
attr(SC, "varDiff")
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