Tools › Statistics › Uncertainty and Monte Carlo › Sensitivity Analysis › Example
Saltelli's Ishigami function: first-order and total Sobol' indices against the analytic values
| Tool | Sensitivity Analysis |
| Source | Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M., and Tarantola, S. (2010). "Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index." Computer Physics Communications, Vol. 181, No. 2, pp. 259–270. |
| Level | Graduate |
Problem
For the Ishigami function y = sin x₁ + 7 sin² x₂ + 0.1 x₃⁴ sin x₁ with x₁, x₂, x₃ uniform on (−π, π), estimate the first-order and total Sobol' sensitivity indices by the Saltelli sampling design and compare them with the analytic values; show why a one-at-a-time sweep over the same ranges misses the sensitivities.
Inputs to Enter
- Model y = sin(x1) + 7*sin(x2)^2 + 0.1*x3^4*sin(x1) (loaded by the Ishigami example)
- x1, x2, x3 base 1, range −π to π, uniform; Monte Carlo N = 20,000 (the example's setting; the test runs 100,000)
Answers
- Analytic: V = 13.8446, S₁ = 0.3139, S₂ = 0.4424, S₃ = 0, S_T1 = 0.5576, S_T2 = 0.4424, S_T3 = 0.2437; mean 3.5
- Tool at N = 100,000 within 0.012 of every analytic index; numpy at N = 200,000: S = 0.3157, 0.4436, −0.0011; S_T = 0.5615, 0.4422, 0.2448
- One-at-a-time swings are all zero (sin, sin² and x⁴ take the same value at both ends of the range) and the spider curves are non-monotone; the linear fit has R² well below 0.4
Notes
Choose the Ishigami example in the selector. The Sobol' estimates carry Monte Carlo error of order 0.01 at the example's N; the test asserts the tool within 0.012.