Tools › Statistics › Uncertainty and Monte Carlo › Sensitivity Analysis › Example
Sobol' g-function with a = (0, 1, 4.5, 9, 99, 99, 99, 99): analytic Sobol' indices
| Tool | Sensitivity Analysis |
| Source | Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., and Tarantola, S. (2008). Global Sensitivity Analysis: The Primer. Wiley, Chichester. Sobol' g-function. |
| Level | Graduate |
Problem
The g-function g = Π (|4x_i − 2| + a_i)/(1 + a_i) over eight uniform (0, 1) inputs with a = (0, 1, 4.5, 9, 99, 99, 99, 99) has partial variances V_i = 1/(3(1 + a_i)²). Estimate the first-order and total indices by Monte Carlo and check the ranking of the inputs against the analytic indices.
Inputs to Enter
- Model loaded by the g-function example (eight factors (abs(4*xi − 2) + a_i)/(1 + a_i))
- x1 … x8 base 0.25, range 0 to 1, uniform; N = 20,000 (the test runs 50,000)
Answers
- Analytic V = 0.46542, mean 1; S₁ = 0.716, S₂ = 0.179, S₃ = 0.0237, S₄ = 0.0072; x₅ … x₈ negligible (a = 99)
- Tool at N = 50,000 within 0.015 of S₁ and S₂, within 0.01 of S₃ and S₄; ranking x₁ > x₂ > x₃ > x₄
- numpy at N = 200,000: S = 0.7116, 0.1806, 0.0228, 0.0072; S_T = 0.7828, 0.2428, 0.0344, 0.0105
Notes
Choose the Sobol' g-function example in the selector; estimates carry Monte Carlo error at the example's N.