Difference between Sobol indices and total Sobol indices?

Given a mathematical model $Y\widetilde Y(X_i)$, where $X_i=x_i^*$ represents a particular point estimate for input variable $X_i$. In sensitivity analysis, Sobol indices explain the importance of an input factor $X_i$ on the variance of the output $Y$ such that:

$$S_i=\displaystyle \dfrac{V_{X_{x \sim i}}(E_{X_i}(Y|X_i=x_i^*))}{V(Y)},\quad 0 \leq S_i \leq 1$$

This represents the variance of the expectation of $X$ in the red box over the total variance of $Y$.

Total Sobol indices are seemingly given by this Wikipedia entry:

$$S_{Ti}=\displaystyle 1-\dfrac{V_{X_{x \sim i}}(E_{X_i}(Y|X_i=x_i^*))}{V(Y)}$$

But what do they really mean? I'm not convinced.

• May be this reference is very useful Sobol I (1993) Sensitivity estimates for non linear mathematical models. Mathematical Modelling and Computational Experiments 1:407–414. and this book Roger Ghanem,David Higdon,Houman Owhadi ," Handbook of Uncertainty Quantification ", Elsevier. Commented Nov 27, 2017 at 12:27

The reason why total Sobol' indices are interesting is interactions. Two inputs $x_1$ and $x_2$ are interacting when their joint effect on the output is different from the sum of their individual effects. Consider for instance the following model

$$f(\mathbf{x}) = x_1 . x_2$$

It is possible to measure interactions by computing higher order Sobol' indices, that is Sobol' indices for groups of variables. These can be defined in two ways, depending if one counts the interactions of the subgroups or not. The problem of this approach is that the number of Sobol' indices grow geometrically with the number of inputs so that computing them quickly become intractable. Total Sobol' indices is a viable alternative: the total index for a given input $x_i$ represent the effect of all the group of variables that contain $x_i$. Hence, the difference between the total index and first order index of $x_i$ is the amount of interactions that $x_i$ contributes to.

Note that unlike the first order indices, the sum of total indices can exceed one. The equality is obtained when there are no interactions.

For a deeper understanding, have a look at the papers from Saltelli and coworkers for interpretation in terms of variance lowering when freezing a variable and to the seminal papers from Sobol' for the interpretation in terms of the ANOVA decomposition, also named HDMR, Sobol' or Hoefding decomposition (the one from 2001 is very clear and concise but might need a few readings if you are not familiar with the domain). Both approach have their merits and complement each other for a deep understanding of the meanings of sensitivity indices.

Regarding the estimation of total Sobol' indices, a review of modern variance based estimator can be found in [1].

-- Reminder -- First order Sobol' indices are

the variance of the conditional expectation of the output given the value of an input, normalised by the total variance.

Total Sobol' indices are

the complementary of the variance of the conditional expectation given the values of all but an input, normalised by the total variance.

This is equal, thanks to the total variance theorem, to

The expectation of the conditional variance of the output given the values of all but an input, normalised by the total variance.

[1] Andrea Saltelli, Paola Annoni, Ivano Azzini, Francesca Campolongo, Marco Ratto, Stefano Tarantola, Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Computer Physics Communications, Volume 181, Issue 2, February 2010, Pages 259-270, ISSN 0010-4655, http://dx.doi.org/10.1016/j.cpc.2009.09.018. (http://www.sciencedirect.com/science/article/pii/S0010465509003087)

• Thank you for the nice clarification. I have read a few papers by Saltelli and Dimov, including **Global Sensitivity Analysis, The Primer. My question lies along the lines of calculating the $S_{Ti}$... It appears from the formula above that it is simply 1-$S_i$, which doesn't seem right. Am I missing something?
– HCAI
Commented Sep 25, 2013 at 12:52
• Incidentally, what is the meaning of the wiggle sign in $V_{xi\sim}$ for example?
– HCAI
Commented Sep 25, 2013 at 12:58
• The ~i subscript means : “all but i”. There are several formulae to compute total indices, have a look at Andrea Saltelli, Paola Annoni, Ivano Azzini, Francesca Campolongo, Marco Ratto, Stefano Tarantola, Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Computer Physics Communications, Volume 181, Issue 2, February 2010, Pages 259-270, ISSN 0010-4655, dx.doi.org/10.1016/j.cpc.2009.09.018. (sciencedirect.com/science/article/pii/S0010465509003087) for a review and recommendations. Commented Sep 25, 2013 at 13:41
• Thank you for the reference, I have read it, very nice! It looks like I have the first formula wrong. I wonder if I'm incorrectly visualising S_i. Does it mean the variance of the expectation of one variable X_i divided by the total variance of Y? Or is there a subtlety here that I'm missing? It's equation 2 in that paper... I'm confused by E_{X_{\sim i}}
– HCAI
Commented Sep 25, 2013 at 21:43
• @HCAI: I have added textual definition which may circumvent notational issues. Reading a step by step derivation from the seminal papers from Saltelli or Sobol' might help you to get a better grasp on the fundamental concepts. Commented Sep 26, 2013 at 8:19