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Local Polynomial Estimation for Sensitivity Analysis on Models With Correlated Inputs
Sébastien Da Veiga, François Wahl, Fabrice Gamboa
TL;DR
Correlated inputs complicate the interpretation of first-order sensitivity indices, while existing approaches can involve high-dimensional calculations. The paper estimates conditional moments with local polynomial methods and reports good precision at relatively low computation time.
Problem
With correlated inputs, first-order indices can count shared sensitivity repeatedly, complicating interpretation, while high-dimensional conditional-density estimation and integration can be difficult.
Source: §1 · p. 2, §1.3 · p. 4Method
The paper estimates conditional moments using local polynomial tools, then computes sensitivity indices from a separate sample of input factors without further model runs.
Source: §2 · p. 4Results
The estimators show good precision at relatively low computation time, and the two methods produce comparable results in a simple analytical case.
Source: §3.1 · p. 8, §4 · p. 9Takeaways & Limitations
The estimators require only smooth marginals and can be used as a data-driven black-box routine that avoids the Bayesian approach’s high-dimensional numerical integration issue.
Source: §4 · p. 9The variance of the proposed estimator is not derived because it requires third and fourth moments of the local linear estimator, a problem left beyond the paper’s scope.
Source: §2.3 · p. 6
Abstract
from arXiv · showhide
Sensitivity indices when the inputs of a model are not independent are estimated by local polynomial techniques. Two original estimators based on local polynomial smoothers are proposed. Both have good theoretical properties which are exhibited and also illustrated through analytical examples. They are used to carry out a sensitivity analysis on a real case of a kinetic model with correlated parameters.
1. MODELS WITH CORRELATED INPUTS
With correlated inputs, classical variance decomposition no longer holds, and first-order indices can count shared sensitivity more than once, complicating interpretation. The section reviews three approaches that address this setting, each with limitations.
1. MODELS WITH CORRELATED INPUTS
- Correlated inputs break the variance-decomposition property, can make first-order indices count shared sensitivity repeatedly, and render FAST and Sobol unusable as designed.First-order indices nevertheless remain useful for identifying the factor whose fixing would most reduce output variance.Source: §1 · p. 2
1.1 Multidimensional Sensitivity Analysis
- Jacques et al. group inputs into mutually independent vectors and estimate first-order multidimensional indices, but fully correlated inputs prevent useful grouping and group scores cannot isolate a driver.The indices can be estimated by Monte Carlo.Source: §1.1 · pp. 2–3
1.2 Correlation-Ratios With Known Conditional Density Functions
- Ratto et al. estimate first-order indices from samples of the conditional output distribution, using Monte Carlo or replicated Latin hypercube sampling, but good precision requires many model evaluations.The approach is therefore suited only to cases where model runs have very low computational cost.Source: §1.2 · p. 3
1.3 Bayesian Sensitivity Analysis
- Oakley and O’Hagan model the response as a Gaussian random field and compute sensitivity indices from its posterior, allowing correlated inputs and reducing Monte Carlo standard errors when model runs are fixed.Their approach requires multidimensional integrals, which may need numerical estimation when analytic evaluation is unavailable.Source: §1.3 · p. 3
- With unknown conditional densities, high-dimensional approximation faces the curse of dimensionality, while the required integrals can reach dimension 2d − 1 and require many draws for reasonable accuracy.The cited public-domain software does not yet support dependent inputs.Source: §1.3 · p. 4
2. NEW ESTIMATION METHODOLOGY
The paper proposes local-polynomial estimators of sensitivity indices using conditional moments. The approach estimates those moments from model outputs, then uses a second input-only sample to compute indices without additional model runs.
2. NEW ESTIMATION METHODOLOGY
- Local-polynomial estimates of conditional moments provide the basis for two sensitivity-index estimators, approximating conditional expectations marginally rather than approximating the full model function.The authors state that the marginal approach avoids the multidimensional integration problem of the Bayesian approach.Source: §2 · p. 4
2.1 Formulation of the Estimators
- The estimators use model-output data to estimate conditional moments and a second input-only sample to estimate their empirical moments.The second sample requires no further model runs; the model is run only for the first sample.Source: §2 · p. 4, §2.1 · p. 5, §2.4 · p. 6
2.2 Bandwidth and Orders Selection
- The authors use EBBS for bandwidth selection and focus on local linear regression for the conditional mean and p = 1, q = 1 for the conditional variance.They describe EBBS as estimating MSE empirically and report that p = 1, q = 1 was adequate in their simulations.Source: §2.2 · p. 5
2.3 Theoretical Properties of the Estimators
- Under their stated assumptions, T1 is asymptotically unbiased and T2 is consistent.The paper notes that calculating T1's variance is beyond its scope; bootstrap methods can estimate it in practical cases.Source: §2.3 · pp. 5–6
- The variance of T1 is not derived because doing so requires third and fourth moments of the local linear estimator; bootstrap methods can estimate it in practical cases.The authors leave the variance calculation for future research.Source: §2.3 · p. 6
2.4 Application to Sensitivity Analysis
- Assuming the joint input distribution can be generated at low computational cost, the authors describe a procedure for estimating first-order sensitivity indices; simulations show n of order 100 and n′ around 2000 suffice for accurate estimates.The last situation leads to less accurate approximations of first-order sensitivity indices.Source: §2.4 · pp. 6–7
3. EXAMPLES
The examples assess the two local-polynomial estimators in analytical comparisons and a correlated-parameter kinetic model. They show that accounting for input correlations matters for interpreting sensitivity, and that parameter importance in the chemical model varies with feed and operating conditions.
3. EXAMPLES
- The examples use ˆS(1) and ˆS(2), estimating conditional expectations by local linear regression and conditional variances with p = 1 and q = 1.Bandwidths are selected using the estimated-bias method.Source: §3 · p. 7
3.1 Analytical Examples
- With σ = 1.2 and ρ = −0.8, ignoring correlation suggests fixing X3 rather than X1 for the greatest variance reduction, contrary to the correlated-input calculations.The example warns that neglecting correlation can produce misleading sensitivity interpretations.Source: §3.1 · p. 7
- The multidimensional index for (X2, X3) equals that for X1, yet its high value comes from X2; the proposed method estimates each first-order index.The example illustrates that the grouped index alone does not distinguish which factor accounts for the sensitivity.Source: §3.1 · p. 7
- Across simulations, ˆT2’s variance was at least comparable to ˆT1’s and sometimes lower, but computing ˆT2 is more difficult.The authors do not recommend one estimator over the other theoretically because computing the variance of ˆT1 is mathematically complex.Source: §3.1 · p. 7
- In the second analytical example, the local-polynomial and Oakley–O’Hagan results are comparable, though analytic multidimensional integration may not be possible with dependent inputs.If numerical integration is feasible instead, it may yield less accurate approximations.Source: §3.1 · p. 8
3.2 Practical Example from Chemical Field : Isomerization of the Normal Butane
- For c1, parameters 1 and 6 are most important for C3+C5; for c2, parameters 4 and 2 are most important, matching the shift from reaction (1) to reaction (3).The two feed conditions use the same operating conditions but differ in feed composition; the estimates from the second estimator are similar.Source: §3.2 · p. 9
- The study of other outputs and conditions found that influential parameters depend on feed and operating conditions, while every parameter affects at least one output in at least one condition.Chemical-engineer input is needed to identify critical outputs and which parameters merit attention.Source: §3.2 · p. 9
4. DISCUSSION AND CONCLUSION
The estimators provide first-order sensitivity indices for correlated inputs, with theoretical support and practical precision at relatively low computation time. They assume smooth marginals, and the authors recommend using them alongside Oakley and O’Hagan’s approach.
4. DISCUSSION AND CONCLUSION
- Local polynomial estimators provide first-order sensitivity indices for dependent inputs, with theoretical properties and practical precision at relatively low computation time.Higher precision requires more calculation time; users can adjust hyperparameters such as polynomial order.Source: §4 · p. 9
- The estimators assume only smooth marginals and require fewer model runs than classical sampling methods.The authors describe their method and Oakley and O’Hagan’s as complementary approaches and suggest combining them for more accurate results.Source: §4 · p. 9
- Future work will develop multi-output sensitivity indices using multivariate nonparametric regression.Source: §4 · p. 9
A.1 Assumptions
The proofs use regularity conditions on kernels, input densities, and conditional moments. The compact-support assumption can be relaxed with truncation, while bounded inputs commonly have compactly supported densities.
A.1 Assumptions
- The assumptions require positive bandwidths and regularity of the kernels, marginal densities, and conditional moments.The conditions include positive density near the evaluation point and smoothness or continuity requirements for relevant functions.Source: §A.1 · p. 10
- Kernel conditions include symmetry and boundedness; bounded support is required in one set of assumptions but is described as technical, with Gaussian kernels also usable.The text identifies the A0/B0 conditions as standard and gives n^-1/5 as the order of theoretically optimal constant bandwidths under classical MSE or MISE considerations.Source: §A.1 · p. 10
- The positive-density condition ensures the experimental design is sufficiently rich, while bounded second derivatives and Lipschitz regularity provide local smoothness.Source: §A.1 · p. 10
- The compact-support condition on the marginal density is made for presentation and can be relaxed using conventional truncation techniques.The authors note that input factors in sensitivity analysis are bounded in practice and therefore have densities with compact support.Source: §A.1 · p. 10
A.2 Proof of Theorem 1
The proof of Theorem 1 relies on the asymptotic bias and variance behavior of local linear regression. It derives the estimator’s conditional properties and then evaluates them under the stated assumptions.
A.2 Proof of Theorem 1
- Theorem 1 follows directly from the asymptotic bias and variance behavior of local linear regression.Source: §A.2 · p. 10
- The proof invokes Fan et al.’s established result under assumptions (A0)–(A4), then conditions on predictors so the sampled conditional observations are i.i.d.Source: §A.2 · p. 10
- It obtains the variance of estimator T1 using the classical empirical variance formula and evaluates its behavior under the stated assumptions.The derivation uses independence of the auxiliary sample from X and Y and proceeds analogously to standard MISE evaluation.Source: §A.2 · p. 10
A.3 Proof of Theorem 2
The proof of Theorem 2 builds on established asymptotic results for residual-based conditional-variance estimation. It derives conditional bias and variance for estimator T2 under the stated assumptions.
A.3 Proof of Theorem 2
- Theorem 2’s proof begins from Fan and Yao’s asymptotic results for the residual-based conditional-variance estimator under assumptions (B0)–(B4).Source: §A.3 · p. 11
- The proof expresses T2 using estimated conditional variances evaluated at the auxiliary sample points, then derives its conditional bias and variance.Source: §A.3 · p. 11
- The derivation uses independence of the auxiliary sample from X and Y and applies assumptions (B4) and (C0) to obtain the stated asymptotic expressions.Source: §A.3 · p. 11