An analytic method for sensitivity analysis of complex systems

Sensitivity analysis is concerned with understanding how the model output depends on uncertainties (variances) in inputs and then identifies which inputs are important in contributing to the prediction imprecision. Uncertainty determination in output is the most crucial step in sensitivity analysis. In the present paper, an analytic expression, which can exactly evaluate the uncertainty in output as a function of the output's derivatives and inputs' central moments, is firstly deduced for general multivariate models with given relationship between output and inputs in terms of Taylor series expansion. A $\gamma$-order relative uncertainty for output, denoted by $\mathrm{R^{\gamma}_v}$, is introduced to quantify the contributions of input uncertainty of different orders. On this basis, it is shown that the widely used approximation considering the first order contribution from the variance of input variable can satisfactorily express the output uncertainty only when the input variance is very small or the input-output function is almost linear. Two applications of the analytic formula are performed to the power grid and economic systems where the sensitivity of both actual power output and Economic Order Quantity models are analyzed. The importance of each input variable in response to the model outputs is quantified by the analytic formula.