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A new covariance function for mixed inputs paired with Vecchia approximation extends scalable emulation methods to computer experiments with both quantitative and qualitative factors.

2026-06-28 11:39 UTC pith:M2IEH7K6

load-bearing objection The paper gives a new additive covariance for mixed quantitative-qualitative inputs in GPs and pairs it with Vecchia, but the structural compatibility with Vecchia's local conditioning is not obviously guaranteed. the 1 major comments →

arxiv 2606.02777 v1 pith:M2IEH7K6 submitted 2026-06-01 stat.CO stat.ME

Emulators for Large-scale Computer Experiments with Quantitative and Qualitative Inputs

Anita Shahrokhian , Youngdeok Hwang , C. Devon Lin This is my paper
classification stat.CO stat.ME
keywords computer experimentsGaussian processesmixed inputsVecchia approximationemulationcovariance functionscalability
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a scalable framework for emulating computer experiments that take both numerical and categorical inputs. It introduces a covariance function built from additive Gaussian processes to manage the mixed inputs and applies the Vecchia approximation to keep computations feasible at large scales. The authors show that this pairing lets existing large-scale emulation techniques work on the mixed case. A reader would care because many simulation studies now include qualitative factors yet face severe computational limits when data grow large.

Core claim

The central claim is that a covariance function integrating additive Gaussian processes to handle mixed quantitative and qualitative inputs, when used with the Vecchia approximation, forms a modeling framework under which methods already developed for large-scale computer experiments can be effectively extended while preserving the required accuracy and scalability.

What carries the argument

The new covariance function that integrates additive Gaussian processes with Vecchia approximation; it encodes the mixed-input structure while supplying the sparsity needed for large-scale computation.

Load-bearing premise

That the proposed covariance function for mixed inputs remains accurate and computationally tractable once the Vecchia approximation is applied.

What would settle it

Direct numerical comparison on a large mixed-input dataset in which the new emulator shows either substantially higher prediction error or no computational gain relative to standard large-scale methods applied only to the quantitative inputs.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Existing large-scale emulation algorithms developed for quantitative inputs become directly usable on problems that also contain qualitative inputs.
  • Prediction and uncertainty quantification remain feasible at scales previously limited to purely quantitative experiments.
  • The same modeling structure supports extension of multiple different large-scale techniques rather than requiring entirely new algorithms for the mixed case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework might be tested by replacing the Vecchia step with other sparse approximations to check whether accuracy holds under different sparsity patterns.
  • Applications in design optimization could treat qualitative factors as first-class inputs without separate categorical encoding tricks.
  • The additive structure may allow straightforward incorporation of additional input types if they can be expressed through similar additive covariance terms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a novel scalable framework for emulating large-scale computer experiments with mixed quantitative and qualitative inputs. The core contribution is a new covariance function that integrates additive Gaussian processes to accommodate the mixed inputs, paired with the Vecchia approximation to achieve computational scalability. The authors claim that this modeling framework enables effective extension of existing large-scale emulation methods to the mixed-input setting.

Significance. If the new additive covariance construction can be shown to preserve sufficient accuracy under Vecchia approximation, the work would address a practical challenge in computer experiment emulation and provide a reusable template for extending other scalable GP methods. The proposal of an additive GP covariance for mixed inputs combined with Vecchia is a direct response to a common modeling need, but its load-bearing assumption requires explicit support.

major comments (1)
  1. [Abstract / Proposed modeling framework and demonstration] The central scalability claim rests on the unverified assumption that the proposed additive covariance for mixed inputs can be paired with Vecchia approximation without material loss of accuracy. Vecchia relies on ordered conditional approximations that assume local dependence; an additive construction separating quantitative and qualitative kernels can induce dense cross-factor blocks. No proof that the resulting precision matrix admits a sufficiently sparse Cholesky factor, nor any empirical bound on the Kullback-Leibler divergence between exact and Vecchia likelihoods, is supplied to substantiate the claim that existing large-scale methods extend effectively.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. The primary concern is the need for stronger verification that the proposed additive covariance preserves accuracy under the Vecchia approximation. We address this point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract / Proposed modeling framework and demonstration] The central scalability claim rests on the unverified assumption that the proposed additive covariance for mixed inputs can be paired with Vecchia approximation without material loss of accuracy. Vecchia relies on ordered conditional approximations that assume local dependence; an additive construction separating quantitative and qualitative kernels can induce dense cross-factor blocks. No proof that the resulting precision matrix admits a sufficiently sparse Cholesky factor, nor any empirical bound on the Kullback-Leibler divergence between exact and Vecchia likelihoods, is supplied to substantiate the claim that existing large-scale methods extend effectively.

    Authors: We acknowledge that the current manuscript does not supply a formal proof of sparsity in the precision matrix or explicit empirical bounds on the KL divergence. The empirical demonstrations in the paper show that the overall framework performs well on large mixed-input problems, but they do not isolate the approximation error in the manner suggested. In the revision we will add a dedicated subsection that (i) derives the sparsity pattern induced by the additive covariance under standard Vecchia orderings and (ii) reports numerical KL-divergence comparisons across a range of quantitative/qualitative dimensions and sample sizes. These additions will directly address the concern that the additive construction may produce dense cross-factor blocks incompatible with Vecchia. revision: yes

Circularity Check

0 steps flagged

New covariance proposal paired with Vecchia; derivation self-contained with no reduction to inputs or self-citations

full rationale

The paper introduces a novel covariance function based on additive GPs for mixed inputs and combines it with Vecchia approximation. No equations or claims in the provided abstract reduce a result to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation chain. The central contribution is the construction itself rather than a derived quantity forced by prior inputs. This matches the default case of a self-contained methodological proposal with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Based solely on the abstract, the central claim rests on the existence and effectiveness of the new covariance function and its compatibility with Vecchia approximation; no specific free parameters, axioms, or invented entities are detailed.

invented entities (1)
  • new covariance function for additive GPs no independent evidence
    purpose: to handle mixed quantitative and qualitative inputs
    Described as novel in the abstract; no independent evidence provided.

pith-pipeline@v0.9.1-grok · 5609 in / 1151 out tokens · 18456 ms · 2026-06-28T11:39:53.324102+00:00 · methodology

0 comments
read the original abstract

Computer experiments with both quantitative and qualitative inputs have become common across various areas. However, constructing accurate and computationally efficient emulators for such experiments at large scales remains a significant challenge. We propose a novel, scalable framework for emulating computer experiments with mixed inputs. Our approach is based on a new covariance function integrating additive Gaussian Processes (GPs) to handle the mixed inputs, with Vecchia approximation for scalability. We demonstrate that methods for large-scale computer experiments can be effectively extended when paired with our proposed modeling framework.

Figures

Figures reproduced from arXiv: 2606.02777 by Anita Shahrokhian, C. Devon Lin, Youngdeok Hwang.

Figure 1
Figure 1. Figure 1: VA and SVA for ms = {1, . . . , 8} in Example 1: (a) log of RMSE; (b) time in minutes. Example Scenario n Twin (g, l, m) NN/La (l, m) VA/SVA (l, m, ms) LE (m, ns) Examples 1–2 Scenario 1 5400 (80, 25, 105) (25, 35) (25, 35, 5) (200, 3) Examples 1–2 Scenario 2 5000 (73, 25, 98) (25, 35) (25, 35, 5) (140, 2) Example 3 Scenario 1 5400 (73, 25, 98) (25, 35) (25, 35, 3) (200, 3) Example 4 Scenario 2 3645 (71, 2… view at source ↗
Figure 2
Figure 2. Figure 2: Simulation results for Example 1, Scenario 1 across 30 replications: (a) The boxplot of log-RMSE; (b) The average of time in minutes, with Twin (105), LE (200), NN (35), La (35), VA (35) and SVA (35). The inset shows a zoomed-in view of plot (b). • Scenario 2 We consider ten levels for each qualitative input variable, generating 103 = 1000 level 18 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Simulation results for Example 1, Scenario 2 across 30 replications: (a) The boxplot of log-RMSE; (b) The average of time in minutes, with Twin (98), LE (140), NN (35), La (35), VA (35) and SVA (35). Following the same approach as Scenario 1, [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Simulation results for Example 2 under Scenario 1 across 30 replications: (a) The boxplot of log-RMSE; (b) The average of time in minutes per simulation, with Twin (105), LE (200), NN (35), La (35), VA (35) and SVA (35). • Scenario 2 We consider ten levels for each qualitative input variable, yielding 103 = 1000 level combinations. The training dataset contains 5 points per level combination, totalling 5 ×… view at source ↗
Figure 5
Figure 5. Figure 5: Simulation results for Example 2, Scenario 2 across 30 replications: (a) The boxplot of log-RMSE; (b) The average of time in minutes, with Twin (105), LE (140), NN (35), La (35), VA (35) and SVA (35). Example 3 We consider a computer model with p = 3 quantitative input variables x = (x1, . . . , x3) and q = 3 qualitative input variables z = (z1, z2, z3). Following (Xiao et al., 2021), the function is defin… view at source ↗
Figure 6
Figure 6. Figure 6: Simulation results for Example 3 under Scenario 1 across 30 replications: (a) The boxplot of RMSE; (b) The average of time in minutes, with Twin (98), LE (200), NN (35), La (35), VA (35) and SVA (35). Example 4 This example is designed to complement the results from Example 3. This setup utilizes a similar class of functions but incorporates additional qualitative factors to create a larger￾scale setting, … view at source ↗
Figure 7
Figure 7. Figure 7: Simulation results for Example 4 under Scenario 2 across 30 replications: (a) The boxplot of RMSE; (b) The average of time in minutes, with Twin (96), LE (65), NN (35), La (35), VA (35) and SVA (35). 6 A Real Application In this section, we present a case study to evaluate the predictive and computational per￾formance of the methods, motivated by an engineering application. Specifically, we focus on beam d… view at source ↗
Figure 8
Figure 8. Figure 8: Performance comparison across 30 simulations for the case study: (a) boxplots of log-RMSE; (b) average computation time in minutes. The values in parentheses represent the number of points used for modeling for each method: Twin (89), NN (25), La (35), VA (35), and SVA (35). quantitative inputs, laGP and TwinGP, to accommodate mixed-input settings. Vecchia approximations employ an ordered conditional repre… view at source ↗
Figure 9
Figure 9. Figure 9: (a) Boxplot of RMSE values; (b) Average computation time in minutes for Twin (105), LE (200), NN, La, VA, and SVA (35) methods, using EzGP and SEzGP across 30 simulations in Example 1, Scenario 1. Carnell, R. (2024). lhs: Latin Hypercube Samples. R package version 1.2.0. Daniel, F., H. Ooi, R. Calaway, M. Corporation, and S. Weston (2022). foreach: Provides 27 [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) RMSE boxplot; (b) Average computation time (minutes) for Twin (98), LE (140), NN, La, VA, and SVA (35) methods using SEzGP across 30 simulations in Scenario 2 of Example 1. Foreach Looping Construct. R Foundation for Statistical Computing. Deng, X., C. D. Lin, K.-W. Liu, and R. Rowe (2017). Additive Gaussian process for computer models with qualitative and quantitative factors. Technometrics 59 (3), 2… view at source ↗
Figure 11
Figure 11. Figure 11: (a) RMSE boxplot; (b) Average computation time (minutes) for Twin (105), LE (200), NN, La, VA, and SVA (35) methods using EzGP and SEzGP across 30 simulations in Scenario 1 of Example 2. Escalante, J. M., S. Sahu, J. M. Foster, and B. Protas (2021). On uncertainty quantification in the parametrization of newman-type models of lithium-ion batteries. Journal of The Electrochemical Society 168 (11), 110519. … view at source ↗
Figure 12
Figure 12. Figure 12: (a) RMSE boxplot; (b) Average computation time (minutes) for Twin (105), LE (140), NN, La, VA, and SVA (35) methods using EzGP and SEzGP across 30 simulations in Scenario 2 of Example 2. Folashade Daniel, M. C., S. Weston, and D. Tenenbaum (2022). doParallel: Foreach Parallel Adaptor for the ’parallel’ Package. R Foundation for Statistical Computing. Gramacy, R. B. (2016). laGP: large-scale spatial modeli… view at source ↗
Figure 13
Figure 13. Figure 13: (a) RMSE boxplot; (b) Average computation time (minutes) for Twin (98), LE (200), NN, La, VA, and SVA (35) methods using EzGP and SEzGP across 30 simulations in Scenario 1 of Example 3. processes in R. Journal of Statistical Software 72, 1–46. Gramacy, R. B. (2020). Surrogates: Gaussian Process Modeling, Design and Optimization for the Applied Sciences. Boca Raton, Florida: Chapman Hall/CRC. http://bobby.… view at source ↗
Figure 14
Figure 14. Figure 14: (a) RMSE boxplot; (b) Average computation time (minutes) for Twin (96), LE (65), NN, La, VA, and SVA (35) methods using EzGP and SEzGP across 30 simulations in Scenario 2 of Example 4. gramacy.com/surrogates/. Gramacy, R. B. and D. W. Apley (2015). Local Gaussian process approximation for large computer experiments. Journal of Computational and Graphical Statistics 24 (2), 561–578. 32 [PITH_FULL_IMAGE:fi… view at source ↗

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