Derivative-Based Global Sensitivity Measures and Poincaré Bounds

Why Average Derivatives Can Help

A global sensitivity analysis asks how uncertainty in model inputs contributes to variation in an output over specified input distributions. Variance-based indices answer this question by decomposing output variance. Derivative-based global sensitivity measures instead summarize how rapidly the output changes as each input varies across its distribution. They can be attractive when model derivatives are available, including through sensitivity equations or automatic differentiation, and can sometimes be estimated with fewer model evaluations than direct variance-based indices.

A derivative measure is not itself a variance contribution. Its value depends on the units and scale of the input, as well as the output. Under suitable input-distribution assumptions, a Poincaré inequality supplies a rigorous upper bound connecting an expected squared derivative to a variance-based total-effect index. The distinction matters: a bound may be useful for screening, but it is not generally an estimate of the index.

Definitions and Assumptions

Let $X=(X_1,\ldots,X_d)$ be model inputs with a specified joint distribution, let $Y=f(X)$ be a scalar output, and write $X_{-i}$ for all inputs except $X_i$. Assume initially that the inputs are mutually independent, that $0<\operatorname{Var}(Y)<\infty$, and that $f$ is differentiable in each input with square-integrable partial derivatives. These assumptions let us define the measures below and apply the distribution-specific inequality.

Definition: The derivative-based global sensitivity measure (DGSM) for input $X_i$ is $$ \nu_i=\mathbb{E}\left[\left(\frac{\partial f(X)}{\partial X_i}\right)^2\right]. $$ It averages squared local derivatives over the specified input distribution. The variance-based total-effect index is $$ S_{T_i}=\frac{\mathbb{E}\left[\operatorname{Var}(Y\mid X_{-i})\right]}{\operatorname{Var}(Y)} =\frac{\mathbb{E}\left[\left(Y-\mathbb{E}[Y\mid X_{-i}]\right)^2\right]}{\operatorname{Var}(Y)}. $$ It measures the fraction of output variance attributable to input $X_i$, including its contributions through interactions with other inputs.

The derivative measure $\nu_i$ has units of output squared divided by input squared. Consequently, comparing raw DGSM values across parameters with different units or scales can be misleading. The total-effect index is dimensionless. A Poincaré constant provides the scale adjustment needed to turn a derivative measure into a bound.

Definition: A probability distribution for a scalar variable $X_i$ satisfies a Poincaré inequality with constant $C_i$ if, for every suitable differentiable function $g$, $$ \operatorname{Var}(g(X_i))\leq C_i\,\mathbb{E}\left[(g'(X_i))^2\right]. $$ For $X_i\sim U(a_i,b_i)$, a sharp constant is $C_i=(b_i-a_i)^2/\pi^2$. For $X_i\sim N(\mu_i,\sigma_i^2)$, the constant is $C_i=\sigma_i^2$.

The stated total-effect bound uses independent inputs, so that the conditional distribution of $X_i$ given $X_{-i}$ remains its marginal distribution. If inputs are dependent, marginal Poincaré constants alone do not generally justify this bound; suitable inequalities for the conditional distributions would be needed instead. This is important when interpreting sensitivity results for correlated parameters.

Deriving the Poincaré Bound

Fix a value $x_{-i}$ of the other inputs and consider the one-variable function $g(t)=f(t,x_{-i})$. Independence means that, conditional on $X_{-i}=x_{-i}$, the variable $X_i$ still has its specified marginal distribution. Applying its Poincaré inequality to $g$ gives a bound on the conditional variance. Averaging that inequality over $X_{-i}$ yields the result.

\begin{aligned} \operatorname{Var}(Y\mid X_{-i}=x_{-i}) &\leq C_i\,\mathbb{E}\left[\left(\frac{\partial f(X)}{\partial X_i}\right)^2\middle|X_{-i}=x_{-i}\right],\\ \mathbb{E}\left[\operatorname{Var}(Y\mid X_{-i})\right] &\leq C_i\,\mathbb{E}\left[\left(\frac{\partial f(X)}{\partial X_i}\right)^2\right] =C_i\nu_i. \end{aligned}

Dividing by the positive output variance gives the total-effect bound. Since a variance-based total-effect index is between zero and one, the bound can be sharpened by taking its minimum with one.

Theorem: For mutually independent inputs, if the marginal distribution of $X_i$ satisfies a Poincaré inequality with constant $C_i$, and the stated integrability and differentiability conditions hold, then $$ S_{T_i}\leq \frac{C_i\nu_i}{\operatorname{Var}(Y)}. $$ Thus $S_{T_i}\leq \min\left(1,C_i\nu_i/\operatorname{Var}(Y)\right)$. The inequality is an upper bound, not an equality in general.

Worked Calculations

Worked Example: Uniform Inputs and a Linear Output

Suppose a hypothetical normalized QSP output is $Y=2X_1+X_2$, where $X_1\sim U(0,1)$ and $X_2\sim U(0,2)$ are independent. The output is measured in arbitrary response units and both inputs are dimensionless. The model is differentiable, its derivatives are finite, and both uniform distributions have finite Poincaré constants, so the bound applies.

The derivatives and DGSM values are $$ \frac{\partial f}{\partial X_1}=2,\qquad \frac{\partial f}{\partial X_2}=1,\qquad \nu_1=2^2=4,\qquad \nu_2=1^2=1. $$ For a uniform distribution, $\operatorname{Var}(X)=(b-a)^2/12$. Independence therefore gives $$ \operatorname{Var}(Y)=4\left(\frac{1}{12}\right)+\frac{4}{12} =\frac{1}{3}+\frac{1}{3}=\frac{2}{3}. $$

The Poincaré constants are $C_1=1/\pi^2$ and $C_2=4/\pi^2$. Hence $$ S_{T_1}\leq\frac{(1/\pi^2)4}{2/3} =\frac{6}{\pi^2}\approx 0.6079,\qquad S_{T_2}\leq\frac{(4/\pi^2)1}{2/3} =\frac{6}{\pi^2}\approx 0.6079. $$ In this additive model, the exact total-effect indices are each $1/2$, since each input contributes $1/3$ of variance out of $2/3$. The bounds are valid but not tight.

Worked Example: A Valid Bound Larger Than One

Let $X\sim U(-1,1)$ and $Y=X^2$, a hypothetical scalar response. The function is differentiable throughout the input range and has finite variance and squared derivative, so the Poincaré inequality applies. There is only one input, and therefore its exact total-effect index is one.

The output moments are $$ \mathbb{E}[Y]=\mathbb{E}[X^2]=\frac{1}{3},\qquad \mathbb{E}[Y^2]=\mathbb{E}[X^4]=\frac{1}{5}, $$ so $$ \operatorname{Var}(Y)=\frac{1}{5}-\left(\frac{1}{3}\right)^2 =\frac{4}{45}\approx 0.08889. $$ Since $\partial f/\partial X=2X$, $$ \nu=\mathbb{E}[4X^2]=\frac{4}{3}. $$ The uniform Poincaré constant on $[-1,1]$ is $C=4/\pi^2$. The resulting bound is $$ S_T\leq \frac{(4/\pi^2)(4/3)}{4/45} =\frac{60}{\pi^2}\approx 6.079. $$ This is mathematically valid but uninformative until capped at one. It does not imply that a sensitivity index can exceed one; it demonstrates that the inequality can be loose.

Worked Example: Normal Inputs and Distribution-Specific Constants

Consider the hypothetical model $Y=3X_1-X_2$, with independent inputs $X_1\sim N(0,4)$ and $X_2\sim N(0,1)$. The second argument in each normal distribution is the variance. The model is smooth, its output variance is finite, and normal inputs have Poincaré constants equal to their variances.

The partial derivatives are 3 and $-1$, so $\nu_1=9$ and $\nu_2=1$. Independence gives $$ \operatorname{Var}(Y)=3^2(4)+(-1)^2(1)=36+1=37. $$ The constants are $C_1=4$ and $C_2=1$. The bounds are therefore $$ S_{T_1}\leq\frac{4(9)}{37}=\frac{36}{37}\approx0.9730,\qquad S_{T_2}\leq\frac{1(1)}{37}=\frac{1}{37}\approx0.02703. $$ For this additive linear model, the exact indices equal these values because each input's contribution to output variance is exactly its derivative squared times its input variance. Equality in this example should not be assumed for nonlinear models.

Using the Measures in Practice

A practical analysis usually evaluates $\nu_i$ by averaging squared derivatives over samples from the specified input distribution. Analytic derivatives, sensitivity equations, or automatic differentiation can provide the gradients. If finite differences are used, the step size must be checked: steps that are too large introduce truncation error, while steps that are too small can amplify solver tolerances and floating-point error. Repeating the calculation with alternative step sizes, and comparing selected derivatives against an independent method, helps identify problems.

The Poincaré constant must match the distribution actually used in the analysis. If a parameter is sampled uniformly over a range, use the uniform constant for that range; if it is sampled normally, use its variance. Reparameterization also changes the derivative. For example, differentiating with respect to $\log X_i$ measures response change per unit change in log-parameter, not per unit change in $X_i$. The distribution and its Poincaré constant must be expressed in that same coordinate before applying a bound.

A small bound can be useful for screening: it guarantees that the total-effect index is no larger than that bound under the assumptions. A large bound does not establish that the input is influential; the bound may simply be loose. A very small output variance also makes the ratio unstable or large, so inspect the output variance and the unnormalized quantity $C_i\nu_i$ alongside the ratio.

Approach	What it summarizes	Useful distinction
DGSM with a Poincaré bound	Average squared local derivative and an upper bound on total effect	Uses gradients; the bound may be conservative
Variance-based total-effect index	Variance attributable to an input, including interactions	Directly targets a variance fraction but requires its own estimation strategy
Morris elementary effects	Finite changes over selected input perturbations	Useful for screening without requiring derivatives, but is not the same expectation of squared derivatives
Shapley effects	Allocation of explained output variance among inputs, including in correlated-input settings	Addresses attribution under dependence; the independent-input Poincaré bound does not automatically replace it

Diagnostics and Common Pitfalls

Before reporting a derivative-based bound, verify that the output and its gradients are finite over the region sampled, that the input distributions match the intended uncertainty analysis, and that the independence assumption is defensible. Discontinuities, event-driven outputs, threshold definitions, or solver artifacts can make derivatives unstable or undefined. In such cases, a derivative average may fail to represent meaningful output changes, even if a numerical routine returns a value.

Comparing raw DGSM values across differently scaled inputs. A large $\nu_i$ can reflect the units used for $X_i$. Report the parameter scale and use the appropriate $C_i$ when forming the bound.
Calling a DGSM a Sobol index. The DGSM averages local squared slopes; the total-effect index is a variance fraction. They are connected by an inequality only under its assumptions.
Treating a bound above one as a contradiction. The inequality can be loose. Since $S_{T_i}\leq1$, report the capped bound or state that the derived bound is uninformative.
Using marginal constants for correlated parameters. Dependence changes conditional input distributions. Marginal constants alone do not establish the displayed total-effect bound.
Ignoring numerical-gradient diagnostics. A gradient dominated by solver error can distort $\nu_i$; check step-size or solver-tolerance sensitivity.

Key takeaway: For independent inputs with suitable Poincaré inequalities, $C_i\nu_i$ bounds the unnormalized total-effect variance. The resulting index bound is rigorous under the stated assumptions, but it can be conservative and should not be mistaken for a direct variance-based estimate.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

For independent $X_1\sim U(0,1)$, $X_2\sim U(0,1)$, and $Y=X_1+X_2$, calculate $\nu_1$, $\operatorname{Var}(Y)$, and the Poincaré bound for $S_{T_1}$.
What assumptions allow the marginal Poincaré constant for $X_i$ to be used in the total-effect bound?
In the $Y=X^2$, $X\sim U(-1,1)$ example, why does a bound above one not invalidate the result?
Why is it generally inappropriate to rank parameters using raw DGSM values when their units or ranges differ?

Solutions. (1) Since $\partial Y/\partial X_1=1$, $\nu_1=1$. Each input has variance $1/12$, so $\operatorname{Var}(Y)=1/6$. Here $C_1=1/\pi^2$, giving $S_{T_1}\leq(1/\pi^2)/(1/6)=6/\pi^2\approx0.6079$. The exact index is $1/2$. (2) The inputs are independent, and the output and derivatives satisfy the required integrability and differentiability conditions; the marginal distribution of $X_i$ is then also its conditional distribution given the other inputs. (3) A Poincaré bound is an upper bound and may be loose. The actual index remains at most one; taking the minimum of one and the derived bound gives the useful universal cap. (4) A derivative's magnitude depends on the coordinate scale and units. Use the appropriate input distribution and Poincaré constant, and interpret any ranking in the context of the chosen parameterization.