Regression-Based Methods

Sensitivity indices from ordinary least squares — standardized coefficients and partial correlations on raw and rank-transformed data.

When to use: Your model is approximately linear (SRC, PCC) or monotonic (SRRC, PRCC). You have an existing sample set from any sampler. Very cheap — just OLS on the data. Good first pass before committing to expensive Sobol’. Always check the R^2 diagnostic before trusting the indices.

Theory

SRC — Standardized Regression Coefficients

Saltelli & Marivoet (1990) Comp. Stat. Data Anal. 9(1), 55–64. [bib]

Fit an OLS regression Y \approx \beta_0 + \boldsymbol{\beta} \cdot \mathbf{X} and standardize each coefficient by the input/output standard deviations:

\text{SRC}_i = \beta_i \cdot \frac{\sigma_{X_i}}{\sigma_Y}

For a truly linear model with independent inputs, \text{SRC}_i^2 \approx S_i — the squared standardized coefficient recovers the first-order Sobol’ index. The R^2 of the linear fit is the load-bearing diagnostic: if R^2_{\text{linear}} > 0.7, SRC indices are trustworthy; below that, the model is too nonlinear for linear regression to capture.

SRRC — Standardized Rank Regression Coefficients

Replace both \mathbf{X} and Y with their ordinal ranks, then compute SRC on the rank-transformed data (Spearman regression). SRRC captures monotonic nonlinear relationships — the rank transform linearizes any monotonic function. Trust if R^2_{\text{rank}} > 0.7.

PCC — Partial Correlation Coefficients

For each factor X_i, regress X_i on all other factors and Y on all other factors. The Pearson correlation of the residuals is the partial correlation:

\text{PCC}_i = \operatorname{corr}\!\big(X_i - \hat{X}_i^{(\sim i)},\; Y - \hat{Y}^{(\sim i)}\big)

PCC isolates X_i’s unique linear contribution after removing the linear effects of every other factor. In a correlated-input design, PCC separates individual influence from collinearity; with independent inputs, PCC and SRC agree in sign and rank ordering but differ in magnitude (PCC normalizes by residual variance, SRC by total variance).

PRCC — Partial Rank Correlation Coefficients

PCC computed on rank-transformed data. Captures monotonic partial contribution. The standard tool for screening in Monte Carlo uncertainty analyses — Marino et al. (2008) J. Theor. Biol. cite PRCC as the default for biological models.

Code

estimate_regression_indices returns all four indices plus both R^2 diagnostics in a single call. Sampler-agnostic — works on any (X, Y) dataset.

use salib::estimators::estimate_regression_indices;
use ndarray::Array2;

// x: (N, d) input matrix, y: N-element output vector
let indices = estimate_regression_indices(x.view(), &y).unwrap();

println!("{indices}");
// Regression indices (d=3)
//   R²(linear) = 0.9987  R²(rank) = 0.9992
//
//   Factor      SRC      SRRC       PCC      PRCC
//   ------   ------    ------    ------    ------
//        0   0.8942    0.8951    0.9988    0.9992
//        1   0.4472    0.4476    0.9962    0.9976
//        2  -0.0012   -0.0018   -0.0084   -0.0121

Verify on linear fixture Y = 2 X_0 + X_1 (N = 1024, seed [0u8; 32]):

Diagnostic	Value
R^2_{\text{linear}}	> 0.99
\text{SRC}_0 / \text{SRC}_1	\approx 2.0 (matches coefficient ratio)
\text{SRC}_2	\approx 0 (absent factor)

Verify on Ishigami (N = 4096, seed [0u8; 32]):

Diagnostic	Value	Interpretation
R^2_{\text{linear}}	0.19	Well below 0.7 — SRC untrustworthy
R^2_{\text{rank}}	0.19	SRRC also untrustworthy
\text{SRRC}_1	\approx 0	Correctly detects \sin^2(x_2) non-monotonicity

When to use each variant

All four are sampler-agnostic and cost O(N d^2 + d^3) — negligible compared to any model-evaluation budget. None recover Sobol’ indices unless the model is linear (SRC) or monotonic (SRRC/PRCC). The R^2 diagnostic is the load-bearing trust signal.

Workflow

1. Run estimate_regression_indices on your existing (X, Y) data.
2. Check R^2_{\text{linear}} and R^2_{\text{rank}}.
3. If R^2 > 0.7: trust the corresponding indices. Identify dominant factors by |\text{SRC}| or |\text{PRCC}| rank ordering.
4. If R^2 < 0.7: the model is too nonlinear for regression-based analysis. Proceed to variance-based (Sobol’) or distribution-based (Borgonovo, PAWN) methods.