Add an Implementation of Linear Function from Saltelli et al. (2000)

[damar-wicaksono]

The linear function is an M-dimensional test function for sensitivity analysis introduced in ¹ and later employed in ².
The function reads:

$$ \mathcal{M}(\boldsymbol{x}) = \sum_{i = 1}^M x_i, $$

where $x_i$ is uniformly distributed in $[x_{o, i} - \sigma_{o, i}, x_{o, i} + \sigma_{o, i}]$, where $x_{o, i} = 3^{i - 1}$ and $\sigma_{o, i} = 0.5 x_{o, i}$.

The function does not contain any interaction between the inputs, but the larger the index of the input variable is, the larger its variance (or uncertainty) becomes.
An input becomes more important the larger its index.

Footnotes

1. A. Saltelli, K. Chan, and E. Scott, 2000. Sensitivity Analysis: Wiley Series in Probability and Statistics. Wiley, New York. ↩

2. X. Sun, B. Croke, A. Jakeman, S. Roberts, "Benchmarking Active Subspace methods of global sensitivity analysis against variance-based Sobol’ and Morris methods with established test functions," Environmental Modelling & Software, vol. 149, p. 105310, 2022. DOI: 10.1016/j.envsoft.2022.105310 ↩