Is Inv 17's Sobol ranking sampling-error-limited?
Inv 17 ran Saltelli 2010 / Jansen 1999 on 4,096 model evaluations and reported plume cross-section as the dominant variance driver. Is that ordering stable — or are we at the resolution floor where MC sampling error could re-rank drivers?
Polynomial chaos expansion (PCE) answers this with an algebraic alternative. Given N collocation samples we fit a Legendre-polynomial surrogate to the L3 physics model and read off Sobol indices from the coefficients directly — no MC estimator required.
From LHS to sparse PC
One fit, two orders of magnitude fewer evaluations
Order-3 PCE with 84 terms reaches 0.997 R^2 on 200 held-out samples, RMSE 0.99 µg/m³. PCE mean is 33.51 µg/m³ vs MC mean 32.74. PCE stdev is 17.14 vs MC stdev 16.67 — moments agree within 3%.
Do they rank the same drivers?
| Input | S1 PCE | S1 MC | ST PCE | ST MC | Delta ST |
|---|---|---|---|---|---|
| Wind speed (m/s) | 0.203 | 0.148 | 0.239 | 0.244 | -0.005 |
| Fuel moisture | 0.090 | 0.071 | 0.108 | 0.102 | +0.005 |
| EF chaparral | 0.005 | 0.007 | 0.006 | 0.007 | -0.001 |
| EF conifer | 0.049 | 0.100 | 0.061 | 0.067 | -0.006 |
| PBL height (m) | 0.189 | 0.159 | 0.226 | 0.203 | +0.022 |
| Plume cross-section (km) | 0.389 | 0.353 | 0.439 | 0.413 | +0.026 |
Disclosure: PCE Sobol indices are algebraic moments of the fitted surrogate, not direct estimates of the physics model. Agreement with Saltelli MC within 0.026 ST units is the cross-check. PCE truncation error (order 3, 84 terms) contributes a non-identifiable bias that the MC estimator does not have. If the surrogate’s R² = 0.997 on held-out data is representative, the ranking is stable at 34× less model work. For decision-critical rankings, keep the MC result as the primary artifact and treat PCE as a confirmation + differentiable follow-on tool.
Wind-speed sweep, zero extra model calls
A PCE surrogate lets us answer "what happens at wind speed X?" without running the physics model again. The chart shows episode-mean PM2.5 predicted by the order-3 PCE over the full wind range while holding all other inputs at nominal.
PCE is differentiable, so gradient-based optimization and calibration are free once the coefficients are fit.