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Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be apportioned to different sources of uncertainty in its inputs.^{[1]}^{[2]} 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 variable under analysis Sensitivity analysis can be useful for a range of purposes,^{[3]} including
Taking an example from economics, in any budgeting process there are always variables that are uncertain. Future tax rates, interest rates, inflation rates, headcount, operating expenses and other variables may not be known with great precision. Sensitivity analysis answers the question, "if these variables deviate from expectations, what will the effect be (on the business, model, system, or whatever is being analyzed), and which variables are causing the largest deviations?"
A mathematical model is defined by a series of equations, input variables and parameters aimed at characterizing some process under investigation. Some examples might be a climate model, an economic model, or a finite element model in engineering. Increasingly, such models are highly complex, and as a result their input/output relationships 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 (aleatory), such as the occurrence of stochastic events.^{[5]}
Good modeling practice requires that the modeler provides an evaluation of the confidence in the model. This requires, first, a quantification of the uncertainty in any model results (uncertainty analysis); and second, an evaluation of how much each input is contributing to the output uncertainty. Sensitivity analysis addresses the second of these issues (although uncertainty analysis is usually a necessary precursor), performing the role of ordering by importance the strength and relevance of the inputs in determining the variation in the output.^{[2]}
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 (see e.g. the guidelines for impact assessment), the White House Office of Management and Budget, the Intergovernmental Panel on Climate Change and US Environmental Protection Agency's modelling guidelines.
The choice of method of sensitivity analysis is typically dictated by a number of problem constraints or settings. Some of the most common are
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.^{[2]} They are also distinguished by the type of sensitivity measure, be it based on (for example) variance decompositions, partial derivatives or elementary effects. In general, however, most procedures adhere to the following outline:
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.
This section discusses various types of "core methods", distinguished by the various sensitivity measures that are calculated (note that some of these categories "overlap" somewhat). The following section focuses on alternative ways of obtaining these measures, under the constraints of the problem.
One of the simplest and most common approaches is that of changing one-factor-at-a-time (OFAT or OAT), to see what effect this produces on the output.^{[13]} ^{[14]} ^{[15]} 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 (all ‘effects’ are computed with reference to the same central point in space) 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.^{[9]}
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.^{[16]}
Local methods involve taking the partial derivative of the output Y with respect to an input factor X_{i}:
where the subscript X^{0} indicates that the derivative is taken at some fixed point in the space of the input (hence the 'local' in the name of the class). Adjoint modelling^{[17]}^{[18]} and Automated Differentiation^{[19]} are methods in this class. Similar to OAT/OFAT, local methods do not attempt to fully explore the input space, since they examine small perturbations, typically one variable at a time.
A simple but useful tool is to plot scatter plots of the output variable against individual input variables, after (randomly) 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 X_{i}, or even by estimating variance-based measures by nonlinear regression.^{[10]}
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 (i.e. a hyperplane, hence with no quadratic terms, etc., as regressors) 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^{[20]}^{[21]}^{[22]} 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(X) for X={X_{1}, X_{2}, ... X_{k}}, a measure of sensitivity of the ith variable X_{i} 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 X_{i}. This expression essentially measures the contribution X_{i} alone to the uncertainty (variance) in Y (averaged over variations in other variables), 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 X_{i} and its interactions with any of the other input variables. Both quantities are typically standardised by dividing by Var(Y).
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 (such as emulators) 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.^{[23]}
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 (i.e. in terms of variance). 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.^{[24]}^{[25]}
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 (most often due to computational expense). Generally, these methods focus on efficiently calculating variance-based measures of sensitivity.
Emulators (also known as metamodels, surrogate models or response surfaces) are data-modelling/machine learning approaches that involve building a relatively simple mathematical function, known as an emulator, that approximates the input/output behaviour of the model itself.^{[26]} In other words, it is the concept of "modelling a model" (hence the name "metamodel"). 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(X). By running the model at a number of points in the input space, it may be possible to fit a much simpler emulator η(X), such that η(X)≈f(X) to within an acceptable margin of error. Then, sensitivity measures can be calculated from the emulator (either with Monte Carlo or analytically), 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.^{[27]}
Clearly the crux of an emulator approach is to find an η (emulator) that is a sufficiently close approximation to the model f. This requires the following steps,
Sampling the model can often be done with low-discrepancy sequences, such as the Sobol sequence 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.
A high-dimensional model representation (HDMR)^{[32]}^{[33]} (the term is due to H. Rabitz^{[34]}) 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 (second or third-order and above). The terms in the truncated series can then each be approximated by e.g. polynomials or splines (REFS) 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 being that they are able to emulate models with higher dimensionality than full-order emulators.
The Fourier Amplitude Sensitivity Test (FAST) uses the Fourier series to represent a multivariate function (the model) in the frequency domain, using a single frequency variable. Therefore, the integrals required to calculate sensitivity indices become univariate, resulting in computational savings.
Methods based on Monte Carlo filtering.^{[35]}^{[36]} These are also sampling-based and the objective here is to identify regions in the space of the input factors corresponding to particular values (e.g. high or low) of the output.
In uncertainty and sensitivity analysis there is a crucial trade off between how scrupulous an analyst is in exploring the input assumptions and how wide the resulting inference may be. The point is well illustrated by the econometrician Edward E. Leamer (1990):^{[37]}
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.^{[38]}
Some common difficulties in sensitivity analysis include
Some examples of sensitivity analyses performed in various disciplines follow here.
Environmental computer models are increasingly used in a wide variety of studies and applications. For example, global climate models are used for both short-term weather forecasts and long-term climate change. Moreover, computer models are increasingly used for environmental decision-making at a local scale, for example for assessing the impact of a waste water treatment plant on a river flow, or for assessing the behavior and life-length of bio-filters for contaminated waste water.
In both cases sensitivity analysis may help to understand the contribution of the various sources of uncertainty to the model output uncertainty and the system performance in general. In these cases, depending on model complexity, different sampling strategies may be advisable and traditional sensitivity indices have to be generalized to cover multiple model outputs,^{[39]} heteroskedastic effects and correlated inputs.^{[8]}
In a decision problem, the analyst may want to identify cost drivers as well as other quantities for which we need to acquire better knowledge in order to make an informed decision. On the other hand, some quantities have no influence on the predictions, so that we can save resources at no loss in accuracy by relaxing some of the conditions. See Corporate finance: Quantifying uncertainty. Additionally to the general motivations listed above, sensitivity analysis can help in a variety of other circumstances specific to business:
However there are also some problems associated with sensitivity analysis in the business context:
Examples from research-led sensitivity analyses can be found on gender wage gap in Chile^{[40]} and water sector interventions in Nigeria.
In modern econometrics the use of sensitivity analysis to anticipate criticism is the subject of one of the ten commandments of applied econometrics (from Kennedy, 2007^{[41]} ):
Thou shall confess in the presence of sensitivity. Corollary: Thou shall anticipate criticism [•••] When reporting a sensitivity analysis, researchers should explain fully their specification search so that the readers can judge for themselves how the results may have been affected. This is basically an ‘honesty is the best policy’ approach, advocated by Leamer, (1978^{[42]}).
Sensitivity analysis can also be used in model-based policy assessment studies.^{[43]} Sensitivity analysis can be used to assess the robustness of composite indicators,^{[44]} also known as indices, such as the Environmental Performance Index.
Sensitivity Analysis is common in many areas of physics and chemistry.^{[45]}
With the accumulation of knowledge about kinetic mechanisms under investigation and with the advance of power of modern computing technologies, detailed complex kinetic models are increasingly used as predictive tools and as aids for understanding the underlying phenomena. A kinetic model is usually described by a set of differential equations representing the concentration-time relationship. Sensitivity analysis has been proven to be a powerful tool to investigate a complex kinetic model.^{[46]}^{[47]}^{[48]}
Kinetic parameters are frequently determined from experimental data via nonlinear estimation. Sensitivity analysis can be used for optimal experimental design, e.g. determining initial conditions, measurement positions, and sampling time, to generate informative data which are critical to estimation accuracy. A great number of parameters in a complex model can be candidates for estimation but not all are estimable.^{[48]} Sensitivity analysis can be used to identify the influential parameters which can be determined from available data while screening out the unimportant ones. Sensitivity analysis can also be used to identify the redundant species and reactions allowing model reduction.
Modern engineering design makes extensive use of computer models to test designs before they are manufactured. Sensitivity analysis allows designers to assess the effects and sources of uncertainties, in the interest of building robust models. Sensitivity analyses have for example been performed in biomechanical models,^{[49]} tunneling risk models,^{[50]} amongst others.
In a meta analysis, a sensitivity analysis tests if the results are sensitive to restrictions on the data included. Common examples are large trials only, higher quality trials only, and more recent trials only. If results are consistent it provides stronger evidence of an effect and of generalizability.^{[51]}
Sometimes a sensitivity analysis may reveal surprising insights about the subject of interest. For instance, the field of multi-criteria decision making (MCDM) studies (among other topics) the problem of how to select the best alternative among a number of competing alternatives. This is an important task in decision making. In such a setting each alternative is described in terms of a set of evaluative criteria. These criteria are associated with weights of importance. Intuitively, one may think that the larger the weight for a criterion is, the more critical that criterion should be. However, this may not be the case. It is important to distinguish here the notion of criticality with that of importance. By critical, we mean that a criterion with small change (as a percentage) in its weight, may cause a significant change of the final solution. It is possible criteria with rather small weights of importance (i.e., ones that are not so important in that respect) to be much more critical in a given situation than ones with larger weights.^{[52]}^{[53]} That is, a sensitivity analysis may shed light into issues not anticipated at the beginning of a study. This, in turn, may dramatically improve the effectiveness of the initial study and assist in the successful implementation of the final solution.
Producing time-critical accurate knowledge about the state of a system (effect) under computational and data acquisition (cause) constraints is a major challenge, especially if the knowledge required is critical to the system operation where the safety of operators or integrity of costly equipment is at stake, e.g., during manufacturing or during environment substrate drilling. Understanding and interpreting, a chain of interrelated events, predicted or unpredicted, that may or may not result in a specific state of the system, is the core challenge of this research. Sensitivity analysis may be used to identify which set of input data signals has a significant impact on the set of system state information (i.e. output). Through a cause-effect analysis technique, sensitivity can be used to support the filtering of unsolicited data to reduce the communication and computational capabilities of a standard supervisory control and data acquisition system.^{[8]}
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 SA - 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 (e.g. which group needs to be protected) all the way to technical (e.g. which variable can be treated as a constant).
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,^{[54]} 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.^{[55]} 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 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 (the 'treatment') on some objects (the 'experimental units'). 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.
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