Sobol Indices

Sobol indices are a way to measure how much sensitive the output of a model is for different inputs. In other words, it quantifies how much each uncertain input parameter contributes to the output variance. They are the main tool for global sensitivity analysis and come in two types:

First-Order Sobol Index ($\text{Si}$)

It measures the direct effect of a single input variable $X_i$ to the output variance. It tells you how much the output would change, on average, if you only varied that one input and kept the others at their average values.

$$S_i=\frac{D_i}{V[M(X)]}$$

Where:

• $D_i$ is the partial variance caused by the input factor $X_i$ alone. These terms are derived from an ANOVA-like decomposition of the model function.
• $V[M(X)]$ is the total variance of the model output.

Total Effect Index (${\text{S}}_{Ti}$)

It measures the total impact of an input variable $X_i$, both its direct effect and all its interaction with other inputs. A high total effect index indicates that a variable is influential, either by itself or through its combined effects with other variables.

$$\text{Total Impact of Input Variable}=\frac{\text{sum of all variance components}}{\text{total variance}}$$

Mathematically,

$$S_{Ti}=\frac{D_i+{\sum }_{j\ne i}{D}_{ij}+{\sum }_{j\ne i,k\ne i,j<k}{D}_{ijk}+\dots }{V[M(X)]}$$

Intuition

Solo Effect How will changing the quantity of the Sugar affect how sweet the cake is if we keep the rest of the ingredients more or less the same portions each time. Interaction effect How will changing the Sugar quantity affect how sweet the cake is give that we have different quantities of cocoa. Example: Maybe with a lot of cocoa, we need to use lots of sugar!

1. First Order Sobol→Tracks the solo effect.
2. Total Effect→ Tracks both Solo and Interaction Effect.