Sensitivity analysis

Sensitivity analysis is the study of how the variation in the output of a model (numerical or otherwise) can be apportioned, qualitatively or quantitatively, to different sources of variation.

Overview

A mathematical model is defined by a series of equations, input factors, parameters, and variables aimed to characterize the process being investigated. Input is subject to many sources of uncertainty including errors of measurement, absence of information and poor or partial understanding of the driving forces and mechanisms.

This 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 provides an evaluation of the confidence in the model, possibly assessing the uncertainties associated with the modeling process and with the outcome of the model itself. Uncertainty and Sensitivity Analysis offer valid tools for characterizing the uncertainty associated with a model.

Applications

Sensitivity Analysis can be used to determine:

1. The model resemblance with the process under study
2. The quality of model definition
3. Factors that mostly contribute to the output variability
4. The region in the space of input factors for which the model variation is maximum
5. Optimal - or instability - regions within the space of factors for use in a subsequent calibration study
6. Interactions between factors

Sensitivity Analysis is popular in financial applications, risk analysis, signal processing, neural networks and any area where models are developed.

Methodology

There are several possible procedures to perform uncertainty (UA) and sensitivity analysis (SA). The most common sensitivity analysis is sampling-based. A sampling-based sensitivity is one in which the model is executed repeatedly for combinations of values sampled from the distribution (assumed known) of the input factors.

In general, UA and SA are performed jointly by executing the model repeatedly for combination of factor values sampled with some probability distribution. The following steps can be listed:

1. Specify the target function and select the input of interest
2. Assign a distribution function to the selected factors
3. Generate a matrix of inputs with that distribution(s) through an appropriate design
4. Evaluate the model and compute the distribution of the target function
5. Select a method for assessing the influence or relative importance of each input factor on the target function.

Bibliography

1. Sobol’, I. M. Mathematical Modelling & Computational Experiment (Engl. Transl.) 1993, 1, 407.
2. Saltelli, A., M. Ratto, S. Tarantola and F. Campolongo (2005) Sensitivity Analysis for Chemical Models, Chemical Reviews, 105(7) pp 2811 – 2828.
3. Santner, T. J.; Williams, B. J.; Notz, W.I. In Design and Analysis of Computer Experiments; Springer-Verlag, 2003.
4. Saisana M., Saltelli A., Tarantola S., 2005, Uncertainty and Sensitivity analysis techniques as tools for the quality assessment of composite indicators, Journal Royal Statistical Society A, 168 (2), 1-17.
5. Cacuci, D. G. In Sensitivity & Uncertainty Analysis, Volume 1: Theory; Chapman & Hall, 2003.
6. Saltelli A. Tarantola S., Campolongo, F. and Ratto, M., 2004, Sensitivity Analysis in Practice. A Guide to Assessing Scientific Models, John Wiley & Sons publishers.

Business Context

Sensitivity analysis can provide information to managers about which elements of the business require more concentration. For example if sales, variable costs, fixed costs or output were to increase or decrease by 10% which would have the most effect on profit?

See also

• What-if analysis

External links

• A forum on sensitivity analysis (main source)
• The SIMLAB software for sensitivity analysis: download it for free
• Sensitivity Analysis Index