Derivative-Based Methods

Sensitivity measures derived from model gradients — upper bounds on total-effect indices without the full Saltelli design.

When to use: Gradient information is available or cheap (analytical, adjoint, or finite-difference). You want provable upper bounds on each factor’s total contribution to output variance. Fewer model evaluations than Sobol’ — N \cdot (d + 1) for forward FD vs N \cdot (d + 2) for Saltelli, but each evaluation includes a gradient.

DGSM

Sobol’ & Kucherenko (2009) Math. Comp. Sim. 79(10), 3009–3017. [bib]

Theory

For a square-integrable model f(\mathbf{x}) with independent inputs, the Derivative-based Global Sensitivity Measure for factor X_i is the expected squared partial derivative:

\nu_i = \mathbb{E}\!\left[\left(\frac{\partial f}{\partial x_i}\right)^2\right]

The Poincare inequality links \nu_i to the total-effect Sobol’ index. For an input X_i with distribution \mu_i and Poincare constant C_P(\mu_i):

S_{Ti} \leq \frac{C_P(\mu_i) \cdot \nu_i}{\operatorname{Var}(Y)}

This bound is provable, not empirical. A factor with \nu_i \cdot C_P / \operatorname{Var}(Y) < \delta contributes provably less than \delta to total variance. The bound can be loose — it is one-sided and depends on the spectral gap of the input distribution — but it is sufficient for screening: factors below the threshold are safely fixed.

Poincare constants

Per Roustant, Barthe & Iooss (2017), the closed-form constants for common distributions are:

Use poincare_constant to derive C_P from a Distribution. Other distributions (Beta, Gamma, truncated families) require numerical Kummer-function solvers not yet implemented — supply the constant directly.

Gradient computation

estimate_dgsm takes a pre-computed gradient matrix. The caller chooses the gradient source:

• Analytical — closed-form derivative of the model. Zero extra model evaluations.
• Adjoint — same interface as analytical; distinguished by the solver backend.
• Finite-difference — via finite_difference_gradients:
  • FdKind::Forward: (f(x + \varepsilon e_i) - f(x)) / \varepsilon, O(\varepsilon) error, N \cdot d extra evaluations.
  • FdKind::Central: (f(x + \varepsilon e_i) - f(x - \varepsilon e_i)) / (2\varepsilon), O(\varepsilon^2) error, 2 N d extra evaluations.

Typical step sizes: \varepsilon = 10^{-6} for forward, 10^{-4} to 10^{-5} for central (balancing truncation vs round-off).

Code

use salib::estimators::{
    estimate_dgsm, finite_difference_gradients,
    poincare_constant, FdKind,
};
use salib::{Distribution, tree_var};
use salib::samplers::{LhsSampler, Sampler};
use ndarray::Array2;

// Sample inputs (N = 4096, d = 3, Uniform[-pi, pi])
let dist = Distribution::Uniform { lo: -std::f64::consts::PI, hi: std::f64::consts::PI };
let cp = poincare_constant(&dist).unwrap();   // (2*pi)^2 / pi^2 = 4.0

// Compute gradients via central finite-difference
let gradients = finite_difference_gradients(
    x.view(), 1e-5, FdKind::Central, |xi| model(xi),
);

let var_y = tree_var(&y);
let indices = estimate_dgsm(
    gradients.view(),
    &[cp, cp, cp],   // one Poincare constant per factor
    var_y,
).unwrap();

println!("{indices}");
// Factor   nu     ST_upper
//      0  7.7721    2.1820
//      1 24.5001    6.8770
//      2 10.9184    3.0650

Verify against Ishigami closed-form (N = 4096, seed [0u8; 32], central FD with \varepsilon = 10^{-5}):

Factor	\hat{\nu}_i	Analytic \nu_i	\hat{S}_{Ti}^{\text{upper}}	Analytic S_{Ti}
x_1	7.7721	7.72	2.1820	0.5576
x_2	24.5001	24.50	6.8770	0.4424
x_3	10.9184	10.99	3.0650	0.2437

All S_{Ti}^{\text{upper}} \geq S_{Ti} — the Poincare bound holds. The bound is loose because \operatorname{Uniform}[-\pi, \pi] has C_P = 4; the screening value is in the provable direction, not tight estimation.

Choosing DGSM vs Sobol’

DGSM is a screening instrument. If the upper bound for a factor is below your tolerance, the factor is provably unimportant — fix it and reduce d before running a full Sobol’ analysis. If the bound is above the tolerance, DGSM cannot distinguish “genuinely important” from “loose bound” — proceed to variance-based methods.