Sensitivity analysis


Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system can be divided and allocated to different sources of uncertainty in its inputs. 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.
The process of recalculating outcomes under alternative assumptions to determine the impact of a variable under sensitivity analysis can be useful for a range of purposes, including:
A mathematical model can be highly complex, and as a result, its relationships between inputs and outputs may be poorly 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, absence of information and poor or partial understanding of the driving forces and mechanisms. This uncertainty imposes a limit on our confidence in the response or output of the model. Further, models may have to cope with the natural intrinsic variability of the system, such as the occurrence of stochastic events.
Good modeling practice requires that the modeler provide an evaluation of the confidence in the model. This requires, first, a quantification of the uncertainty in any model results ; and second, an evaluation of how much each input is contributing to the output uncertainty. Sensitivity analysis addresses the second of these issues, performing the role of ordering by importance the strength and relevance of the inputs in determining the variation in the output.
In models involving many input variables, sensitivity analysis is an essential ingredient of model building and quality assurance. National and international agencies involved in impact assessment studies have included sections devoted to sensitivity analysis in their guidelines. Examples are the European Commission, the White House Office of Management and Budget, the Intergovernmental Panel on Climate Change and US Environmental Protection Agency's modeling guidelines. In a comment published in 2020 in the journal Nature 22 scholars take COVID-19 as the occasion for suggesting five ways to make models serve society better. One of the five recommendations, under the heading of 'Mind the assumptions' is to 'perform global uncertainty and sensitivity analyses allowing all that is uncertain — variables, mathematical relationships and boundary conditions — to vary simultaneously as runs of the model produce its range of predictions.'

Settings, constraints, and related issues

Settings and constraints

The choice of method of sensitivity analysis is typically dictated by a number of problem constraints or settings. Some of the most common are
In uncertainty and sensitivity analysis there is a crucial trade off between how scrupulous an analyst is in exploring the input :wikt:assumption|assumptions and how wide the resulting inference may be. The point is well illustrated by the econometrician Edward E. Leamer:
I have proposed a form of organized sensitivity analysis that I call 'global sensitivity analysis' in which a neighborhood of alternative assumptions is selected and the corresponding interval of inferences is identified. Conclusions are judged to be sturdy only if the neighborhood of assumptions is wide enough to be credible and the corresponding interval of inferences is narrow enough to be useful.

Note Leamer's emphasis is on the need for 'credibility' in the selection of assumptions. The easiest way to invalidate a model is to demonstrate that it is fragile with respect to the uncertainty in the assumptions or to show that its assumptions have not been taken 'wide enough'. The same concept is expressed by Jerome R. Ravetz, for whom bad modeling is when uncertainties in inputs must be suppressed lest outputs become indeterminate.

Pitfalls and difficulties

Some common difficulties in sensitivity analysis include
There are a large number of approaches to performing a sensitivity analysis, many of which have been developed to address one or more of the constraints discussed above. They are also distinguished by the type of sensitivity measure, be it based on variance decompositions, partial derivatives or elementary effects. In general, however, most procedures adhere to the following outline:
  1. Quantify the uncertainty in each input. Note that this can be difficult and many methods exist to elicit uncertainty distributions from subjective data.
  2. Identify the model output to be analysed.
  3. Run the model a number of times using some design of experiments, dictated by the method of choice and the input uncertainty.
  4. Using the resulting model outputs, calculate the sensitivity measures of interest.
In some cases this procedure will be repeated, for example in high-dimensional problems where the user has to screen out unimportant variables before performing a full sensitivity analysis.
The various types of "core methods" are distinguished by the various sensitivity measures which are calculated. These categories can somehow overlap. Alternative ways of obtaining these measures, under the constraints of the problem, can be given.

One-at-a-time (OAT)

One of the simplest and most common approaches is that of changing one-factor-at-a-time, to see what effect this produces on the output. OAT customarily involves
Sensitivity may then be measured by monitoring changes in the output, e.g. by partial derivatives or linear regression. This appears a logical approach as any change observed in the output will unambiguously be due to the single variable changed. Furthermore, by changing one variable at a time, one can keep all other variables fixed to their central or baseline values. This increases the comparability of the results and minimizes the chances of computer programme crashes, more likely when several input factors are changed simultaneously.
OAT is frequently preferred by modellers because of practical reasons. In case of model failure under OAT analysis the modeller immediately knows which is the input factor responsible for the failure.
Despite its simplicity however, this approach does not fully explore the input space, since it does not take into account the simultaneous variation of input variables. This means that the OAT approach cannot detect the presence of interactions between input variables.

Derivative-based local methods

Local derivative-based methods involve taking the partial derivative of the output Y with respect to an input factor Xi:
where the subscript X0 indicates that the derivative is taken at some fixed point in the space of the input. Adjoint modelling and Automated Differentiation are methods in this class. Similar to OAT, local methods do not attempt to fully explore the input space, since they examine small perturbations, typically one variable at a time.

Regression analysis

, in the context of sensitivity analysis, involves fitting a linear regression to the model response and using standardized regression coefficients as direct measures of sensitivity. The regression is required to be linear with respect to the data because otherwise it is difficult to interpret the standardised coefficients. This method is therefore most suitable when the model response is in fact linear; linearity can be confirmed, for instance, if the coefficient of determination is large. The advantages of regression analysis are that it is simple and has a low computational cost.

Variance-based methods

Variance-based methods are a class of probabilistic approaches which quantify the input and output uncertainties as probability distributions, and decompose the output variance into parts attributable to input variables and combinations of variables. The sensitivity of the output to an input variable is therefore measured by the amount of variance in the output caused by that input. These can be expressed as conditional expectations, i.e., considering a model Y = f for X =, a measure of sensitivity of the ith variable Xi is given as,
where "Var" and "E" denote the variance and expected value operators respectively, and X~i denotes the set of all input variables except Xi. This expression essentially measures the contribution Xi alone to the uncertainty in Y, and is known as the first-order sensitivity index or main effect index. Importantly, it does not measure the uncertainty caused by interactions with other variables. A further measure, known as the total effect index, gives the total variance in Y caused by Xi and its interactions with any of the other input variables. Both quantities are typically standardised by dividing by Var.
Variance-based methods allow full exploration of the input space, accounting for interactions, and nonlinear responses. For these reasons they are widely used when it is feasible to calculate them. Typically this calculation involves the use of Monte Carlo methods, but since this can involve many thousands of model runs, other methods can be used to reduce computational expense when necessary. Note that full variance decompositions are only meaningful when the input factors are independent from one another.

Variogram analysis of response surfaces (''VARS'')

One of the major shortcomings of the previous sensitivity analysis methods is that none of them considers the spatially ordered structure of the response surface/output of the model Y=f in the parameter space. By utilizing the concepts of directional variograms and covariograms, variogram analysis of response surfaces addresses this weakness through recognizing a spatially continuous correlation structure to the values of Y, and hence also to the values of.
Basically, the higher the variability the more heterogeneous is the response surface along a particular direction/parameter, at a specific perturbation scale. Accordingly, in the VARS framework, the values of directional variograms for a given perturbation scale can be considered as a comprehensive illustration of sensitivity information, through linking variogram analysis to both direction and perturbation scale concepts. As a result, the VARS framework accounts for the fact that sensitivity is a scale-dependent concept, and thus overcomes the scale issue of traditional sensitivity analysis methods. More importantly, VARS is able to provide relatively stable and statistically robust estimates of parameter sensitivity with much lower computational cost than other strategies. Noteworthy, it has been shown that there is a theoretical link between the VARS framework and the variance-based and derivative-based approaches.

Screening

Screening is a particular instance of a sampling-based method. The objective here is rather to identify which input variables are contributing significantly to the output uncertainty in high-dimensionality models, rather than exactly quantifying sensitivity. Screening tends to have a relatively low computational cost when compared to other approaches, and can be used in a preliminary analysis to weed out uninfluential variables before applying a more informative analysis to the remaining set. One of the most commonly used screening method is the elementary effect method.

Scatter plots

A simple but useful tool is to plot scatter plots of the output variable against individual input variables, after sampling the model over its input distributions. The advantage of this approach is that it can also deal with "given data", i.e., a set of arbitrarily-placed data points, and gives a direct visual indication of sensitivity. Quantitative measures can also be drawn, for example by measuring the correlation between Y and Xi, or even by estimating variance-based measures by nonlinear regression.

Alternative methods

A number of methods have been developed to overcome some of the constraints discussed above, which would otherwise make the estimation of sensitivity measures infeasible. Generally, these methods focus on efficiently calculating variance-based measures of sensitivity.

Emulators

Emulators are data-modeling/machine learning approaches that involve building a relatively simple mathematical function, known as an emulator, that approximates the input/output behavior of the model itself. In other words, it is the concept of "modeling a model". The idea is that, although computer models may be a very complex series of equations that can take a long time to solve, they can always be regarded as a function of their inputs Y = f. By running the model at a number of points in the input space, it may be possible to fit a much simpler emulator η, such that η ≈ f to within an acceptable margin of error. Then, sensitivity measures can be calculated from the emulator, which will have a negligible additional computational cost. Importantly, the number of model runs required to fit the emulator can be orders of magnitude less than the number of runs required to directly estimate the sensitivity measures from the model.
Clearly, the crux of an emulator approach is to find an η that is a sufficiently close approximation to the model f. This requires the following steps,
  1. Sampling the model at a number of points in its input space. This requires a sample design.
  2. Selecting a type of emulator to use.
  3. "Training" the emulator using the sample data from the model – this generally involves adjusting the emulator parameters until the emulator mimics the true model as well as possible.
Sampling the model can often be done with low-discrepancy sequences, such as the Sobol sequence – due to mathematician Ilya M. Sobol or Latin hypercube sampling, although random designs can also be used, at the loss of some efficiency. The selection of the emulator type and the training are intrinsically linked since the training method will be dependent on the class of emulator. Some types of emulators that have been used successfully for sensitivity analysis include,
The use of an emulator introduces a machine learning problem, which can be difficult if the response of the model is highly nonlinear. In all cases, it is useful to check the accuracy of the emulator, for example using cross-validation.

High-dimensional model representations (HDMR)

A high-dimensional model representation is essentially an emulator approach, which involves decomposing the function output into a linear combination of input terms and interactions of increasing dimensionality. The HDMR approach exploits the fact that the model can usually be well-approximated by neglecting higher-order interactions. The terms in the truncated series can then each be approximated by e.g. polynomials or splines and the response expressed as the sum of the main effects and interactions up to the truncation order. From this perspective, HDMRs can be seen as emulators which neglect high-order interactions; the advantage is that they are able to emulate models with higher dimensionality than full-order emulators.

Fourier amplitude sensitivity test (FAST)

The Fourier amplitude sensitivity test uses the Fourier series to represent a multivariate function in the frequency domain, using a single frequency variable. Therefore, the integrals required to calculate sensitivity indices become univariate, resulting in computational savings.

Other

Methods based on Monte Carlo filtering. These are also sampling-based and the objective here is to identify regions in the space of the input factors corresponding to particular values of the output.

Applications

Examples of sensitivity analyses can be found in various area of application, such as:
It may happen that a sensitivity analysis of a model-based study is meant to underpin an inference, and to certify its robustness, in a context where the inference feeds into a policy or decision making process. In these cases the framing of the analysis itself, its institutional context, and the motivations of its author may become a matter of great importance, and a pure sensitivity analysis – with its emphasis on parametric uncertainty – may be seen as insufficient. The emphasis on the framing may derive inter-alia from the relevance of the policy study to different constituencies that are characterized by different norms and values, and hence by a different story about 'what the problem is' and foremost about 'who is telling the story'. Most often the framing includes more or less implicit assumptions, which could be political all the way to technical.
In order to take these concerns into due consideration the instruments of SA have been extended to provide an assessment of the entire knowledge and model generating process. This approach has been called 'sensitivity auditing'. It takes inspiration from NUSAP, a method used to qualify the worth of quantitative information with the generation of `Pedigrees' of numbers. Likewise, sensitivity auditing has been developed to provide pedigrees of models and model-based inferences. Sensitivity auditing has been especially designed for an adversarial context, where not only the nature of the evidence, but also the degree of certainty and uncertainty associated to the evidence, will be the subject of partisan interests. Sensitivity auditing is recommended in the European Commission guidelines for impact assessment, as well as in the report Science Advice for Policy by European Academies.

Related concepts

Sensitivity analysis is closely related with uncertainty analysis; while the latter studies the overall uncertainty in the conclusions of the study, sensitivity analysis tries to identify what source of uncertainty weighs more on the study's conclusions.
The problem setting in sensitivity analysis also has strong similarities with the field of design of experiments. In a design of experiments, one studies the effect of some process or intervention on some objects. In sensitivity analysis one looks at the effect of varying the inputs of a mathematical model on the output of the model itself. In both disciplines one strives to obtain information from the system with a minimum of physical or numerical experiments.