Choosing a method for dependent inputs

When inputs are correlated, sensitivity is no longer one question. An input can matter because the model uses it directly, because it moves with another input, or both. Different methods separate those effects differently.

This guide helps you choose among jaxgsa's methods. For a complete worked analysis, see Correlated Inputs.

Start with the decision

Choose the method from the question you need to answer:

Question	Method
What does each input explain, including what it carries through correlation?	Kucherenko $S_1$ or VKOGA $S_{TC}$
What can only this input explain, after accounting for the others?	Kucherenko $S_T$ or VKOGA $S_{TU}$
Which model terms carry variance, and how much comes from structure or correlation?	HDMR
How can I allocate the fitted variance to one number per input?	Shapley with the HDMR backend
Does an input change the output distribution, rather than only its variance?	HSIC, PAWN, Borgonovo delta, or optimal transport

Then choose between a dedicated design and existing data:

• If you can run the model at new points, use Kucherenko for conditional-variance indices without a surrogate.
• If you already have $(X,Y)$ data, use VKOGA for conditional-variance indices, HDMR for a term-level ANCOVA split, or a distribution-based method when variance is not the quantity you care about.
• If you need one allocation per parameter, use Shapley with the HDMR backend, but read its interpretation carefully.

Why familiar indices split

Suppose $X_1$ and $X_2$ are correlated, while the model reads only $X_1$. Observing $X_2$ still tells you something about the output because it carries information about $X_1$. But once $X_1$ is known, changing $X_2$ does nothing.

That creates two useful views:

• A correlation-inclusive measure asks what an input explains through itself and through everything it moves with. This is useful when deciding what to measure more accurately.
• A correlation-exclusive measure asks what only that input can explain. This is useful when deciding what can be fixed at a nominal value.

Under independence these views coincide. Under dependence they can rank inputs very differently.

The four variance-based routes

There is no single generalisation of Sobol' indices to dependent inputs. jaxgsa provides four routes, each with a different estimand.

Route	What it estimates	What it needs
Kucherenko	$S_1=V(\mathbb{E}(Y\mid X_i))/V(Y)$ and $S_T=\mathbb{E}(V(Y\mid {\mathbf{X}}_{\sim i}))/V(Y)$ under the declared copula	A dedicated design, $N(2D+1)$ model runs, and a declared correlation matrix
VKOGA	The same conditional-variance quantities as $S_{TC}$ and $S_{TU}$, plus $S_U$, $S_C$, and $S_{IU}$	Existing $(X,Y)$ pairs and a declared correlation matrix
HDMR	For each fitted component, a structural share $S_a$ and a correlation-driven share $S_b$	Existing $(X,Y)$ pairs; dependence is inferred from $X$
Shapley with HDMR	One ANCOVA allocation per parameter, formed by splitting each term's $S_a+S_b$ among its participants	Existing $(X,Y)$ pairs

Kucherenko

Use Kucherenko when you can still evaluate the model and want the conditional-variance quantities directly. It samples conditionally on the correlation declared on the Problem.

• $S_1$ is correlation-inclusive.
• $S_T$ is correlation-exclusive.
• There is no fitted surrogate between the model and the estimator.
• The design costs $N(2D+1)$ model evaluations.

See the Kucherenko example and the broader correlated-inputs example.

VKOGA

Use VKOGA when you have existing runs or want to reuse the same fitted model under several correlation assumptions. It fits a kernel surrogate, then samples that surrogate to estimate correlated-input variance indices.

Its two main decision measures match the Kucherenko quantities:

• $S_{TC}$ is the correlation-inclusive prioritisation measure.
• $S_{TU}$ is the correlation-exclusive fixing measure.

VKOGA also reports $S_U$, $S_C$, and $S_{IU}$ to separate uncorrelated, correlated, and independent-interaction contributions. Because these values come from a surrogate, their quality depends on the fit.

See the VKOGA example.

HDMR

Use HDMR when you care about where variance appears in the fitted model. Its ANCOVA decomposition separates every component function into:

• $S_a$: the structural contribution that would remain without correlation;
• $S_b$: the contribution driven by correlation.

This split is per term, so it can describe both main effects and interactions. That is different from the conditional-variance question answered by Kucherenko and VKOGA.

HDMR's $S_T$ needs special care under dependence. It sums every structural and correlative term containing the parameter. It is not the expected residual variance after fixing that parameter, so do not use it as a fixing measure. Use Kucherenko $S_T$ or VKOGA $S_{TU}$ for that question.

See the HDMR example.

Shapley with the HDMR backend

Use shapley.analyze(backend="hdmr", include_correlative=True) when you want one allocation per parameter from the HDMR decomposition. It splits each term's $S_a+S_b$ among the parameters in that term, and the allocations sum to 1.

This is an ANCOVA-based attribution. It is not the conditional-variance Shapley effect of Song et al. (2016), and its correlative shares can be negative. It should not be interpreted as the same quantity as Kucherenko or VKOGA.

See the Shapley example.

Distribution-based alternatives

Variance may be the wrong summary when the output is skewed, heavy-tailed, multimodal, or otherwise changes shape. The following given-data methods do not assume independent inputs:

Method	Use it when
HSIC	You want a kernel dependence measure and permutation significance tests
PAWN	You want a CDF-based, moment-independent measure
Borgonovo delta	You want a density-based measure on a fixed $[0,1]$ scale
Optimal transport	You want a Wasserstein measure split into mean-shift and shape-change components

These methods are correlation-inclusive: an input can score highly because it contains information about another input that drives the output. That is a meaningful result, not evidence that the model uses the input directly.

Worked examples are available for HSIC, PAWN, Borgonovo delta, and optimal transport.

Comparisons to avoid

Keep these boundaries clear when interpreting results:

• Kucherenko and VKOGA estimate the same pair of conditional-variance quantities. They should agree up to surrogate and Monte Carlo error.
• HDMR and HDMR-backed Shapley estimate different quantities. They do not need to agree with Kucherenko or VKOGA.
• HDMR's $S_T$ under dependence is not a conditional-variance total-effect index.
• HDMR-backed Shapley is not conditional-variance Shapley.
• None of these dependent-input indices should be placed beside jaxgsa.sobol as if they had the same meaning. Sobol' analysis assumes independent inputs and refuses a problem with declared correlation.

A practical workflow

1. Declare the input distributions and their correlation on the Problem.
2. Decide whether your question is correlation-inclusive, correlation-exclusive, term-level, allocative, or distribution-based.
3. Decide whether you can run a dedicated design or must use existing data.
4. Select the method from the table above.
5. If the method uses a surrogate, check its fit before interpreting the indices.
6. Report the method and the meaning of its index, not only the numeric value.

For setup, code, and a side-by-side comparison, continue to Correlated Inputs.