Sensitivity

Heijungs, Reinout

doi:10.1007/978-3-031-49317-1_9

Reinout Heijungs^2,3

116 Accesses

Abstract

This chapter presents the basic elements of sensitivity analysis (SA), with an emphasis on their use in life cycle assessment. We discuss topics such as local and global SA, one-at-a-time and all-at-a-time SA, uncertainty apportioning, and the use of scenarios for addressing sensitivity.

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Notes

1.
The name has an extra ‘r’.
2.
OAT is also called OFAT, for one-factor-at-a-time (Borgonovo 2017, Montgomery 2017, Wei et al. 2015). Saltelli and Annoni (2010) use the abbreviation ‘OAT’ for both one-at-a-time and one-factor-at-a-time. Clemen and Reilly (2014) use the term “one-way sensitivity analysis”, which should not be confused with one-way ANOVA (Sect. 5.10.2). Saltelli (1999) uses the term ‘elementary OAT’ (EOAT) besides OAT.
3.
We choose to add quotation marks to the word ‘explain’, because it is not an interpretative explanation, but only a mathematical one, similar to the coefficient of determination \(R^2\) (Sect. 5.9.4) that tells one how much of the variance is ‘explained’ by the ‘explanatory’ variables.
4.
We observe that Mahmood and Gheewala 2023 consider the study of sensitivity for choices as a local sensitivity analysis.
5.
We ignore the issue that in forming these derivatives, we assume that the variables \(x_1\), \(x_2\) and y are continuous, while they here represent integer-valued outcomes of a die.
6.
The term ‘screening’ in sensitivity analysis has nothing to do with ‘screening LCA studies’, which were briefly mentioned in Footnote 2 on Sect. 6.1. Screening LCAs in general will not consider uncertainty and sensitivity.
7.
The term Jacobian is also used as an abbreviation of the Jacobian determinant, which is \(\det (\textbf{J})\). This determinant is seldom of interest to sensitivity analysis, because it requires that \(m=k\), which is rarely the case. Nevertheless, Clifford (1973) uses it extensively in a chapter on “error propagation from two observables to two parameters”.
8.
In LCA, this form appears in Hong et al. (2010), who use the term ‘relative sensitivity’. The only difference is that they write an extra \(\overline{x}\): “\(\dfrac{\partial \ln y}{\partial \ln x_i}=\dfrac{\partial y/y}{\partial x_i/x_i}\overline{x}\)”, which is weird anyhow. Probably it is a typo.
9.
The elasticity is called a ‘multiplier’ in that source, and the local sensitivity a ‘perturbation analysis’.
10.
This is somewhat comparable to the use of r and \(r^2\) for measuring correlation; see Sect. 4.6.2.5.
11.
We remark that the bar is traditionally used to indicate a sample mean (Sect. 4.2.1). Probably these authors do not imply that the mean is by definition the nominal value.
12.
Indeed, in the LCA literature we often find such definitions, for instance Eq. (3) of Gaudreault et al. (2011), Eq. (3) of Van Stappen et al. (2018) and Eq. (16) of Shimako et al. (2018).
13.
Silva et al. (2018) use a unique way to assess the sensitivity, involving a “probability of coincidence”, illustrated in their Fig. 9.1. This indicator resembles our overlap metrics (Sect. 8.8.14) that were used to discern competing products. As such, we are not convinced in its use to measure sensitivity for choices.
14.
Indeed, ISO 14044 (ISO 2006b) stipulates that “systems shall be compared using ... equivalent methodological considerations, such as performance, system boundary, data quality, allocation procedures” and that “allocation procedures shall be uniformly applied to similar inputs and outputs of the system under consideration”.
15.
Substances in group 2B are “possibly carcinogenic to humans” (IARC 2006).
16.
We also allow for open intervals or semi-open intervals, such as \(\left( x_{j,\text {min}}\;,\;x_{j,\text {max}}\right) \), but for ease of notation we will always write closed intervals.
17.
See, e.g., Verghese et al. (2010) and Martínez-Rocamora et al. (2016); also Cerdas et al. (2017) and Hollberg et al. (2021) mention them in their overviews of visualizations in LCA, but in a different way. We discussed such alleged ‘spider plots’ (or ‘spider diagrams’, or ‘spider charts’) under the name of ‘radar plots’ in Sect. 6.8.
18.
Tornado diagrams can also be used to display correlations in uncertainty apportioning (Sect. 9.7; see Cooke and Van Noortwijk 2000).
19.
Morgan and Henrion (1990) speak of “discretizing a continuous distribution”.
20.
Given our previous remarks on the contradicting definitions of global and local, the reader should not be surprised to find Morris’ method here under the heading of GSA, while, for instance, Morio (2011) lists it as LSA.
21.
Another conventional choice is \(p=5\) because it yields values corresponding to the quartiles of a distribution.
22.
This set-up is due to Campolongo et al. (2007). The original proposal by Morris (1991) uses \(\mu =\dfrac{1}{r}\sum \limits _{l=1}^r EE_{j,l}\).
23.
Also here, Morris (1991) uses \(\sigma _j\), without absolute values. Saltelli et al. (2008) recommend to report both \(\mu \), \(\mu ^*\), \(\sigma \) and \(\sigma ^*\), “so as to extract the maximum amount of sensitivity information”. In the present example, this would not add information, because the dependence is monotonous throughout.
24.
The details for this latter paper are described in the Supplementary Information, which is unfortunately no longer available.
25.
See https://cran.r-project.org/web/packages/sensitivity/index.html.
26.
We observe that these authors define the standardized regression coefficients with \(\dfrac{s_Y}{s_{X_j}}\) instead of \(\dfrac{s_{X_j}}{s_Y}\). Probably this is a typo, and not a variation.
27.
These latter authors refer to Geisler et al. (2005), De Koning et al. (2010) and Mutel et al. (2013) as precursors, but those papers use Spearman correlation coefficients or the FAST method; see below.
28.
These authors use it for the “standard regression coefficient”.
29.
Some authors, for instance Chan et al. (2000), use the notation \(X_{\sim j}\) for \(X_{-j}\).
30.
Another, very early, contribution is by McRae et al. (1982).
31.
Kleijnen (2008) uses the term “design and analysis of simulation experiments”, abbreviated as DASE.
32.
Probably it is even longer than printed, because the missing closing parenthesis in the original suggests that some more symbols are forgotten. Their Table 8.6 suggests that their equation contains 20 terms, excluding the constant.
33.
This is not an official acronym that codes for a term, but it is just “derived from the authors names”.
34.
See Footnote 37 on Sect. 7.10.4 for the ‘big-oh’ notation.
35.
Golub and Van Loan (1983) refer to this result as the ‘Sherman–Morrison–Woodbury formula’.

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Reinout Heijungs
Institute of Environmental Sciences, Leiden University, Leiden, The Netherlands
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Heijungs, R. (2024). Sensitivity. In: Probability, Statistics and Life Cycle Assessment. Springer, Cham. https://doi.org/10.1007/978-3-031-49317-1_9

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