Sensitivity analysis

Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in its inputs. This involves estimating sensitivity indices that quantify the influence of an input or group of inputs on the output. A related practice is uncertainty analysis, which has a greater focus on uncertainty quantification and propagation of uncertainty; ideally, uncertainty and sensitivity analysis should be run in tandem.

A mathematical model (for example in biology, climate change, economics, renewable energy, agronomy...) can be highly complex, and as a result, its relationships between inputs and outputs may be faultily understood. In such cases, the model can be viewed as a black box, i.e. the output is an "opaque" function of its inputs. Quite often, some or all of the model inputs are subject to sources of uncertainty, including errors of measurement, errors in input data, parameter estimation and approximation procedure, absence of information and poor or partial understanding of the driving forces and mechanisms, choice of underlying hypothesis of model, and so on. This uncertainty limits our confidence in the reliability of the model's response or output. Further, models may have to cope with the natural intrinsic variability of the system (aleatory), such as the occurrence of stochastic events.

In models involving many input variables, sensitivity analysis is an essential ingredient of model building and quality assurance and can be useful to determine the impact of a uncertain variable for a range of purposes, including:

The object of study for sensitivity analysis is a function ${\displaystyle f}$, (called "mathematical model" or "programming code"), viewed as a black box, with the ${\displaystyle p}$-dimensional input vector ${\displaystyle X=(X_{1},...,X_{p})}$ and the output ${\displaystyle Y}$, presented as following:

$${\displaystyle Y=f(X).}$$

The variability in input parameters ${\displaystyle X_{i},i=1,\ldots ,p}$ have an impact on the output ${\displaystyle Y}$. While uncertainty analysis aims to describe the distribution of the output ${\displaystyle Y}$ (providing its statistics, moments, pdf, cdf,...), sensitivity analysis aims to measure and quantify the impact of each input ${\displaystyle X_{i}}$ or a group of inputs on the variability of the output ${\displaystyle Y}$ (by calculating the corresponding sensitivity indices). Figure 1 provides a schematic representation of this statement.

Taking into account uncertainty arising from different sources, whether in the context of uncertainty analysis or sensitivity analysis (for calculating sensitivity indices), requires multiple samples of the uncertain parameters and, consequently, running the model (evaluating the ${\displaystyle f}$-function) multiple times. Depending on the complexity of the model there are many challenges that may be encountered during model evaluation. Therefore, the choice of method of sensitivity analysis is typically dictated by a number of problem constraints, settings or challenges. Some of the most common are:

To address the various constraints and challenges, a number of methods for sensitivity analysis have been proposed in the literature, which we will examine in the next section.