Bibliography

Annotated references for every method implemented in salib, organized by family. Each entry links back to the method page that uses it.

Variance-based (Sobol’)

Sobol’ (1993) Sobol’, I. M. “Sensitivity estimates for nonlinear mathematical models.” Mathematical Modelling and Computational Experiments 1(4), 407–414. The foundational paper. Introduces the ANOVA-like functional decomposition of a square-integrable function into terms of increasing dimensionality, and defines the variance-based sensitivity indices S_i and S_{Ti} that all subsequent estimators target.

Saltelli et al. (2010) → method page Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M., & Tarantola, S. “Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index.” Computer Physics Communications 181(2), 259–270. The improved estimator for both S_i and S_{Ti} that superseded Sobol’ (2001). Uses the f(B)(f(A_B^{(i)}) - f(A)) formulation for first-order and the squared-difference formulation for total-effect. The default choice in most SA software.

Jansen (1999) → method page Jansen, M. J. W. “Analysis of variance designs for model output.” Computer Physics Communications 117(1–2), 35–43. Alternative total-effect estimator that compares f(B) to f(A_B^{(i)}) rather than f(A) to f(A_B^{(i)}). Empirically better convergence for small N and models with high-order interactions.

Janon et al. (2014) → method page Janon, A., Klein, T., Lagnoux, A., Nodet, M., & Prieur, C. “Asymptotic normality and efficiency of two Sobol index estimators.” ESAIM: Probability and Statistics 18, 342–364. Proves that their estimator achieves the semiparametric efficiency bound for S_i — no other estimator from the same Saltelli design can have lower asymptotic variance.

Owen (2013) → method page Owen, A. B. “Better estimation of small Sobol’ sensitivity indices.” ACM Transactions on Modeling and Computer Simulation 23(2), Article 11. Three-matrix design for improved second-order interaction estimates S_{ij}. Introduces a third base matrix to separate first-order and interaction effects more cleanly.

Plischke et al. (2013) → method page Plischke, E., Borgonovo, E., & Smith, C. L. “Global sensitivity measures from given data.” European Journal of Operational Research 226(3), 536–550. Estimate first-order Sobol’ indices from observational data by binning inputs and computing between-bin variance of conditional means. No designed experiment required.

Homma & Saltelli (1996) Homma, T., & Saltelli, A. “Importance measures in global sensitivity analysis of nonlinear models.” Reliability Engineering & System Safety 52(1), 1–17. Introduces the total-effect index S_{Ti} as a complement to first-order indices. Shows that S_{Ti} = 0 is necessary and sufficient for X_i to be non-influential.

Elementary effects

Morris (1991) → method page Morris, M. D. “Factorial sampling plans for preliminary computational experiments.” Technometrics 33(2), 161–174. The original OAT screening design. Defines elementary effects on a p-level grid and proposes r random trajectories to estimate \mu_i and \sigma_i.

Campolongo et al. (2007) → method page Campolongo, F., Cariboni, J., & Saltelli, A. “An effective screening design for sensitivity analysis of large models.” Environmental Modelling & Software 22(10), 1509–1518. Introduces \mu_i^* (mean of absolute elementary effects) to fix the cancellation problem in Morris’s signed mean. Also extends Morris to grouped factors.

Frequency-based

Cukier et al. (1973) → method page Cukier, R. I., Fortuin, C. M., Shuler, K. E., Petschek, A. G., & Schaibly, J. H. “Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I. Theory.” Journal of Chemical Physics 59(8), 3873–3878. The original FAST method. Maps each input to a characteristic frequency and uses Fourier decomposition to extract first-order sensitivity from the model output’s power spectrum.

Saltelli et al. (1999) → method page Saltelli, A., Tarantola, S., & Chan, K. P.-S. “A quantitative model-independent method for global sensitivity analysis of model output.” Technometrics 41(1), 39–56. Extended FAST (eFAST): adds total-effect indices by computing the complementary variance at all non-\omega_i frequencies.

Tarantola et al. (2006) → method page Tarantola, S., Gatelli, D., & Mara, T. A. “Random balance designs for the estimation of first order global sensitivity indices.” Reliability Engineering & System Safety 91(6), 717–727. RBD-FAST: uses a single random sample for all factors by permuting columns independently, reducing cost from d \times N to N.

Plischke (2010) Plischke, E. “An effective algorithm for computing global sensitivity indices (EASI).” Reliability Engineering & System Safety 95(4), 354–360. Bias correction for RBD-FAST via the \lambda = 2M/N bandwidth factor.

Distribution-based

Borgonovo (2007) → method page Borgonovo, E. “A new uncertainty importance measure.” Reliability Engineering & System Safety 92(6), 771–784. The \delta moment-independent importance measure: the expected L^1 distance between conditional and unconditional output densities. Captures distributional shifts that variance-based indices miss.

Pianosi & Wagener (2015) → method page Pianosi, F., & Wagener, T. “A simple and efficient method for global sensitivity analysis based on cumulative distribution functions.” Environmental Modelling & Software 67, 1–11. PAWN: CDF-based sensitivity using Kolmogorov–Smirnov distance between conditional and unconditional output CDFs. Simpler and more robust than kernel-density-based measures.

Pianosi & Wagener (2018) Pianosi, F., & Wagener, T. “Distribution-based sensitivity analysis from a generic input-output sample.” Environmental Modelling & Software 108, 197–207. Refinements to the PAWN methodology with improved conditioning strategies.

Maume-Deschamps & Niang (2018) Maume-Deschamps, V., & Niang, I. “Estimation of quantile oriented sensitivity indices.” Statistics & Probability Letters 134, 122–127. Partition-based estimator for QOSA with convergence analysis.

Fort et al. (2016) → method page Fort, J.-C., Klein, T., & Rachdi, N. “New sensitivity analysis subordinated to a contrast.” Communications in Statistics — Theory and Methods 45(15), 4349–4364. QOSA: quantile-oriented sensitivity analysis. Measures factor importance at a specific quantile level, capturing tail sensitivity that variance-based methods average away.

Derivative-based

Sobol’ & Kucherenko (2009) → method page Sobol’, I. M., & Kucherenko, S. “Derivative based global sensitivity measures and their link with global sensitivity indices.” Mathematics and Computers in Simulation 79(10), 3009–3017. DGSM: defines \nu_i = \mathbb{E}[(\partial f / \partial x_i)^2] and proves upper bounds linking \nu_i to total-effect indices S_{Ti}. Cheap when gradients are available.

Regression

Saltelli & Marivoet (1990) → method page Saltelli, A., & Marivoet, J. “Non-parametric statistics in sensitivity analysis for model output: A comparison of selected techniques.” Computational Statistics & Data Analysis 9(1), 55–64. SRC, SRRC, PCC, PRCC: regression-based sensitivity measures. SRC² approximates S_i for linear models; rank-transformed versions (SRRC, PRCC) capture monotonic nonlinearity.

Surrogate

Xiu & Karniadakis (2002) → method page Xiu, D., & Karniadakis, G. E. “The Wiener-Askey polynomial chaos for stochastic differential equations.” SIAM Journal on Scientific Computing 24(2), 619–644. Generalized polynomial chaos: match orthogonal polynomial families to input distributions (Legendre for uniform, Hermite for Gaussian). Sobol’ indices are analytic from the expansion coefficients.

Blatman & Sudret (2011) → method page Blatman, G., & Sudret, B. “Adaptive sparse polynomial chaos expansion based on least angle regression.” Journal of Computational Physics 230(6), 2345–2367. Sparse PCE via LARS: automatic basis selection when the full polynomial basis has more terms than data points. Makes PCE practical in moderate-to-high dimensions.

Li et al. (2001) → method page Li, G., Rosenthal, C., & Rabitz, H. “High dimensional model representations.” Journal of Physical Chemistry A 105(33), 7765–7777. HDMR: decompose f into a hierarchy of component functions (constant, univariate, bivariate, …). Cut-HDMR evaluates components along cuts through a reference point.

Constantine (2015) → method page Constantine, P. G. Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies. SIAM, Philadelphia. The monograph. Eigendecomposition of C = \mathbb{E}[\nabla f \nabla f^T] reveals the directions in input space where the function varies most. Practical algorithms for gradient estimation, subspace identification, and dimension reduction.

Game-theoretic

Song, Nelson & Staum (2016) → method page Song, E., Nelson, B. L., & Staum, J. “Shapley effects for global sensitivity analysis: Theory and computation.” SIAM/ASA Journal on Uncertainty Quantification 4(1), 1060–1083. Shapley values applied to sensitivity analysis. Unlike Sobol’ indices, Shapley effects always sum to \operatorname{Var}(Y) and handle correlated inputs without ambiguity.

Experimental design

Fisher (1925) → method page Fisher, R. A. Statistical Methods for Research Workers. Oliver & Boyd, Edinburgh. The origin of analysis of variance. Two-way and higher-order ANOVA partition total variation into main effects, interactions, and error.

Brennan (2001) → method page Brennan, R. L. Generalizability Theory. Springer, New York. The standard reference for G-theory. Extends classical reliability theory to multiple sources of measurement error (facets). D-studies optimize measurement designs for target reliability.

Hickernell (1998) → method page Hickernell, F. J. “A generalized discrepancy and quadrature error bound.” Mathematics of Computation 67(221), 299–322. L2-star discrepancy and its connection to quasi-Monte Carlo integration error via the Koksma–Hlawka inequality.

Box, Hunter & Hunter (1978) → method page Box, G. E. P., Hunter, W. G., & Hunter, J. S. Statistics for Experimenters. Wiley, New York. The textbook for designed experiments. Fractional factorial designs, confounding, resolution, and the analysis of main effects and interactions.

General references

Saltelli et al. (2008) Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., & Tarantola, S. Global Sensitivity Analysis: The Primer. Wiley, Chichester. The comprehensive textbook. Covers all variance-based, screening, and moment-independent methods with worked examples and practical guidance. The standard reference for the field.

Iooss & Lemaître (2015) Iooss, B., & Lemaître, P. “A review on global sensitivity analysis methods.” In Uncertainty Management in Simulation-Optimization of Complex Systems, Operations Research/Computer Science Interfaces Series 59, 101–122. Springer. Survey of the full landscape: variance-based, screening, moment-independent, metamodel-based, and derivative-based methods with a decision flowchart for method selection.