Sensitivity analysis

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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.

1 Overview
2 Applications
3 Methodology
4 Bibliography
5 Business Context
6 See also
7 External links

[edit] 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.

[edit] Applications

Sensitivity Analysis can be used to determine:

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

Sensitivity Analysis is popular in financial applications, risk analysis, signal processing, neural networks and any area where models are developed. Sensitivity analysis can also be used in model-based policy assessment studies see e.g. [1].

[edit] 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. Other methods are based on the decomposition of the variance of the model output and are model independent (see references).

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:

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

[edit] Bibliography

Sobol’, I. M. Mathematical Modelling & Computational Experiment (Engl. Transl.) 1993, 1, 407.
Saltelli, A., M. Ratto, S. Tarantola and F. Campolongo (2005) Sensitivity Analysis for Chemical Models, Chemical Reviews, 105(7) pp 2811 – 2828.
Santner, T. J.; Williams, B. J.; Notz, W.I. In Design and Analysis of Computer Experiments; Springer-Verlag, 2003.
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), 307-323.
Cacuci, Dan G. In Sensitivity & Uncertainty Analysis, Volume 1: Theory; Chapman & Hall, 2003.
Cacuci, Dan G., Mihaela Ionescu-Bujor, Michael Navon, 2005, Sensitivity And Uncertainty Analysis: Applications to Large-Scale Systems (Volume II), Chapman & Hall
Saltelli A. Tarantola S., Campolongo, F. and Ratto, M., 2004, Sensitivity Analysis in Practice. A Guide to Assessing Scientific Models, John Wiley & Sons publishers.
A Special Issue on Sensitivity Analysis has been published in the Journal Reliability Engineering and System Safety (Volume 91, 2006). See [2]

[edit] Business Context

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.

Sensitivity analysis can help in a variety of other circumstances which can be handled by the settings illustrated below: to identify critical assumptions or compare alternative model structures, guide future data collections, detect important criteria, optimize the tolerance of manufactured parts in terms of the uncertainty in the parameters, optimize resources allocation, model simplification, model lumping and so on.