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Distributed Lag Neural Additive Models

Calle Helmersson, Shivang Pandey, Leonardo Olivetti, Elena Raffetti

arXiv:2609.07381v1stat.MLcs.AIcs.LGstat.APstat.ME

TL;DR

Distributed-lag models need to represent nonlinear responses over both exposure and elapsed time, but prescribed spline bases constrain representation choices. DLNAMs use learned smooth neural components within an additive model, retaining familiar distributed-lag summaries. Simulations report stronger recovery on demanding surfaces and competitive interval calibration, while the paper identifies settings where simpler established estimators remain preferable.

  • Problem

    Spline cross-bases require choices such as basis family, dimension, and knot placement, while smoothness and local adaptivity remain in tension with learned representation.

    Source: §1 · p. 2

  • Method

    DLNAMs replace prescribed spline representations with learned smooth exposure–lag components in an additive predictor, using ExU inputs, smooth activations, and learned subnetwork mixtures.

    Source: §1 · p. 2

  • Results

    On demanding response surfaces, DLNAM has lower error and bias than the comparators, and is closest to nominal cumulative coverage in 11 of 12 principal-simulation DGP-by-region cells.

    Source: §3 · p. 10

  • Takeaways & Limitations

    DLNAMs are most suited to demanding response structures, multiple lagged predictors, or analyses where repeated cross-basis specification is substantial; simpler estimators remain attractive when adequate and computational limits matter.

    Source: §3 · p. 12

    For a single exposure with a simple surface adequately represented by a low-dimensional cross-basis, DLNM is simpler, competitive, and cheaper to fit in this benchmark.

    Source: §3 · p. 11

Abstract

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We introduce Distributed Lag Neural Additive Models (DLNAMs), neural-additive analogues of Distributed Lag Non-linear Models (DLNMs) for learning nonlinear effects distributed over lags. DLNAMs replace a prespecified spline cross-basis with neural components that learn exposure--lag response surfaces, avoiding choices of basis family, dimension, and knot placement while preserving additive interpretability and familiar distributed-lag summaries. Exp-centered input layers, smooth activations, and learned subnetwork mixtures produce smooth, locally adaptive representations; pointwise uncertainty combines a conditional last-layer Laplace approximation with between-member ensemble variation. In simulations, DLNAMs generally outperformed DLNM comparators, including penalized and treed variants, in recovering known response functions, with lower bias, stronger boundary recovery, and better-calibrated cumulative intervals; gains were largest for more demanding functions. The architecture performed consistently across sample sizes, outcome families, lag horizons, and jointly fitted multi-exposure settings, retaining recovery performance as exposures were added; fit-specific changes were largely confined to optimization, and applications recovered established empirical patterns.

1 Introduction

DLNAMs replace prescribed spline cross-bases with learned smooth neural components while retaining additive interpretability and familiar distributed-lag summaries. Simulations show strongest recovery gains on demanding response surfaces, with calibration and recovery also assessed in joint-exposure fits and empirical applications.

1 Introduction

  • DLNAM targets nonlinear responses over exposure and elapsed time, replacing prescribed spline representations with learned smooth components that preserve additive reporting and familiar summaries.Its architecture combines multivariate ExU inputs, smooth activations, learned subnetwork mixtures, and uncertainty from conditional last-layer Laplace approximations plus ensemble variation.Source: §1 · pp. 1–2

2.3 Calibration

  • DLNAM is closest to nominal cumulative-interval coverage in 11 of 12 DGP-by-region cells, while coverage comparisons must be read alongside interval width.On the full surface, every estimator with higher coverage than DLNAM also has wider intervals.Source: §2.3 · pp. 3–4, Figure 1 · p. 4

2.1 Recovery

  • Across DGPs 2–4, DLNAM has the lowest error across the whole exposure grid, interior, and boundary; full-surface recovery shows a similar pattern, with T-DLNM competitive at the boundary.Source: §2.1 · p. 3, Figure 1 · p. 4

2.4 Architecture Ablation

  • Replacing ExU inputs with linear inputs causes the largest error increase and sharply reduces coverage; removing mixtures or smooth activations also degrades recovery in distinct ways.The ablations are one-at-a-time, not factorial.Source: §2.4 · p. 5, Figure 2 · p. 5

2.5 Joint-Exposure Recovery

  • In joint fits, DLNAM has the lowest cumulative error for every active component across the whole grid, interior, and boundary, with whole-grid degradation ratios near 1 versus single-exposure fits.It is closest to nominal coverage in all 12 active component-by-region cells; its null-exposure RMSE is 0.0124, versus 0.0123 for DLNM (QBIC).Source: §2.5 · p. 6, Figure 3 · p. 7

2.6 Chicago NMMAPS Temperature and Mortality

  • In Chicago’s 30-day temperature–mortality analysis, all five methods recover the broad pattern of risk rising toward both temperature extremes, but differ in lag allocation and estimated minimum.DLNAM spreads a lower heat-response elevation across more early lags; the application is observational and in-sample, not a test of causal or predictive superiority.Source: Figure 4 · p. 8, §2.6 · p. 8

2.7 Multi-Country DHS/MIS Malaria

  • Across 26 sub-Saharan African countries, the malaria analyses broadly agree between DLNAM and DLNM, with the largest response-shape divergence for specific humidity.The differing adjustment representations and survey-hierarchy treatments preclude interpreting that divergence as evidence favoring either model.Source: §2.7 · pp. 8–9, Figure 5 · p. 9

2.8 Computational Scaling

  • At a fixed 2500-epoch budget, DLNAM runtime rises 4.3-fold from one to four exposures, versus 6.2-fold for T-DLNM, 12.5-fold for DLNM (QBIC), 23.7-fold for DLNM (QAIC), and 33.5-fold for P-DLNM.DLNAM is slower in absolute time for one exposure; the benchmark compares implemented procedures, and timings depend on hardware and implementation.Source: §2.8 · p. 9, Figure 6 · p. 10

3 Discussion

DLNAMs recover demanding exposure–lag response surfaces while preserving additive interpretation, with their clearest advantages in boundary recovery and cumulative-interval calibration. Their practical fit depends on response complexity and computation, and the reported evidence does not establish universal tuning insensitivity or a universal threshold for preferring neural representations.

3 Discussion

  • On demanding surfaces, comparator error rises chiefly through squared bias, while DLNAM’s remaining cumulative error is predominantly variance, with lower boundary error and flatter learned components.The boundary advantage is where support is weakest; it does not imply spline estimators are generically unstable there.Source: §3 · p. 10
  • DLNAM is closest to nominal cumulative coverage in 11 of 12 principal DGP-by-region cells and all 12 active component-by-region cells in the joint experiment.The comparison is calibration relative to interval width, not coverage in isolation; intervals are neither simultaneous bands nor prediction intervals and omit some learned-representation uncertainty.Source: §3 · p. 10
  • The architecture remained effective across two outcome families, three lag horizons, and one to five simultaneous exposures, while fit-specific changes were concentrated primarily in optimization.This does not establish universal tuning insensitivity; training budget, learning rate, or batching changed when the fitting problem changed materially.Source: §3 · p. 11
  • Recovery depends on jointly representing exposure and lag through multivariate ExU inputs, jointly trained subnetworks, and smooth hidden activations, rather than merely replacing a cross-basis with a larger network.Removing these elements degrades recovery with distinct error or coverage signatures; complete fits are ensembled across initializations to temper fit-to-fit instability.Source: §3 · p. 11
  • The evidence is bounded by finite simulation replicates, one joint-exposure correlation structure, and applications aimed at response-structure recovery rather than prediction; proposed multitask and hierarchical extensions remain unevaluated.The reported architecture was not exhaustively optimized, and module changes can materially alter recovery.Source: §3 · pp. 11–12
  • DLNAMs are most relevant when response structure is demanding, several lagged predictors are modeled together, or repeated cross-basis specification is substantial; simpler smooths can remain competitive and cheaper.Abrupt critical-window structure can favor treed representations, and reported runtime comparisons depend on training budget, implementation, and whether fitting is fully joint.Source: §3 · pp. 11–12

4 Methods

This section describes DLNAMs’ neural architecture, fitting procedure, and pointwise uncertainty approach, alongside simulation designs and evaluation targets for exposure–lag recovery. It also specifies benchmark and comparator choices, joint-exposure recovery, observational applications, and limits on likelihood-matched and estimator-only comparisons.

4.1 Distributed Lag Neural Additive Model

  • DLNAM learns neural exposure–lag components instead of prescribed cross-bases; Mixture DLNAM combines jointly trained subnetworks, while ensembles average complete fits to reduce initialization variability.Components are expressed as contrasts from reference values; subnetworks within a fit may reinforce or partially offset one another.Source: §4.1 · pp. 13–14

4.2 Neural Component Architecture

  • Coordinate-wise ExU units learn local sensitivity around input locations, while smooth Mish activations propagate that sensitivity; reported fits use Mish, and SiLU was not separately evaluated.The architecture assigns ExU to sensitivity location and scale and the activation to smooth propagation through the component.Source: §4.2 · pp. 14–15

4.3 Optimization

  • Fits minimize average negative log-likelihood with AdamW, cosine annealing, gradient clipping, and usually full-batch updates; Chicago additionally uses a roughness penalty for training-budget stability.Dropout, subnetwork dropout, and an output penalty were examined preliminarily but did not consistently improve recovery and were inactive in reported analyses.Source: §4.3 · p. 15

4.4 Last-Layer Laplace Confidence Intervals

  • Pointwise uncertainty combines a target-specific last-layer Laplace approximation conditional on the learned representation with variation between ensemble members.The reported interval is a Gaussian moment-matched summary of the member-level mixture, not full posterior uncertainty in the learned representation.Source: §4.4 · pp. 15–17

4.5 Simulation Design

  • The simulation used a single-exposure distributed-lag Poisson model with lag horizon 14, 5,000 analyzed observations, and reference exposure 20.Four smooth generating surfaces included one separable case and three nonseparable cases with differing lag patterns across exposure, keeping spline comparators within their intended function class.Source: §4.5 · pp. 17–18

4.6 Evaluation Metrics

  • Evaluation prioritizes cumulative log-relative-risk contrasts, with lag-specific surfaces as complementary diagnostics because isolated surface locations are not directly observed by the likelihood.Targets are assessed over the full grid, exposure-boundary regions, and their interior; interval coverage is pointwise for the generating function, not simultaneous over the grid.Source: §4.6 · pp. 18–19

4.7 Model Specifications and Fitting

  • The benchmark used three subnetworks per DLNAM component and three independently initialized fits; comparators followed literature-grounded configurations rather than DGP-specific optimization.Surface subnetworks used a 128-unit ExU input layer, a 128-unit Mish hidden layer, and a linear output; principal simulation and Chicago fits used 5,000 configured full-batch epochs without early stopping.Source: §4.7 · pp. 19–20
  • P-DLNM used rank-10 penalized cross-bases in simulation and fixed 5×5 bases on Chicago, while T-DLNM used a Gaussian log(1+Y) likelihood rather than a count likelihood.P-DLNM coupled marginal cubic P-splines with second-order penalties and REML; on Chicago, fitted complexity rose with rank. T-DLNM retained the authors’ simulation settings and serves as a flexible-method benchmark, not a likelihood-matched comparator; its exponentiated contrasts are only a proxy for relative risk.Source: §4.7 · pp. 20–21

4.8 Joint-Exposure Simulation

  • The joint study tested four active additive surfaces and a null exposure across five correlated seasonal series in 50 datasets of n=5000.Comparators used a coordinate-wise DLNM sweep, joint REML for P-DLNM, and separate target-exposure T-DLNM fits; recovery was scored on active cumulative effects and null-exposure leakage, with joint-to-single RMSE ratios measuring degradation. The five-component DLNAM used a reduced learning rate and 5000 epochs after baseline training showed occasional high-loss convergence and under-training.Source: §4.8 · pp. 21–22

4.9 Chicago Temperature and Mortality Analysis

  • The Chicago analysis modeled daily mortality against temperature over lags 0–30, centered at 10.3 °C, using 4638 complete days and shared adjustment variables.The shared controls were date, day of week, lag 0–1 mean dew point temperature, ozone, and PM10. DLNAM used a Poisson working likelihood, whereas T-DLNM retained its Gaussian log(1+Y) model; the in-sample observational analysis assesses empirical plausibility, not causal effects or forecasting.Source: §4.9 · p. 22

4.10 Multi-Exposure Malaria Analysis

  • The malaria application modeled 348,565 childhood outcomes with a Bernoulli-logit DLNAM across five environmental exposures and six monthly lags before testing.Each target-specific fit used the source study’s adjustment set, with adjusting exposures retaining full lag surfaces; the DLNM used the same environmental controls but different functional forms and survey-hierarchy treatment. The comparison is between complete observational specifications and assesses feasibility and interpretation, not comparative recovery.Source: §4.10 · pp. 22–23

S1 Notation

Table S1 collects recurring notation used in the main and supplementary methods; symbols local to individual derivations are defined where they occur.

S1 Notation

  • Table S1 summarizes recurring notation across the main and supplementary methods.Source: Table S1 · p. 27
  • Symbols specific to individual derivations are defined where they occur.Source: Table S1 · p. 27

S2 Last-Layer Laplace Confidence Intervals

This section develops conditional last-layer Laplace uncertainty for DLNAM targets, combining Gaussian approximations conditional on learned representations and mixing weights with variation across ensemble members. It clarifies that the resulting ensemble intervals are approximate, pointwise confidence intervals for response functions—not full-posterior or outcome-prediction intervals.

S2.1 Setup, Partition, and Scope

  • Conditioning on the learned representation makes each selected component affine in its final-layer parameters, enabling lower-dimensional generalized linear model curvature calculations while fixing mixture weights.This avoids the scale non-identifiability that would arise if mixture weights and corresponding output layers varied together, but omits within-member mixing-weight uncertainty; the construction adapts the last-layer approach of LA-NAM [17].Source: S2.1 Setup, Partition, and Scope · p. 28

S2.2 Prior and Laplace Approximation

  • A post-hoc Gaussian prior on penalized final-layer parameters yields a conditional Laplace approximation, with the trained expansion center retained rather than re-optimized.The intercept has a flat, improper prior, and the prior precision is selected after network training; for working-likelihood fits the result is a Gaussian working analogue, not a literal posterior.Source: S2.2 Prior and Laplace Approximation · pp. 29–30

S2.3 Last-Layer Curvature

  • Because the predictor is affine in the last-layer parameters, no predictor-curvature term enters the Hessian, so no Gauss–Newton approximation is required; W/φ is expected linear-predictor information pulled back to the last-layer parameterization.The lower bound φ̂ ≥ 1 prevents apparent underdispersion from understating uncertainty, and the pragmatic large-sample approximation is ν = n − 1. The effective last-layer dimension γ quantifies penalized last-layer directions supported by the data, not residual degrees of freedom of the complete trained network, and is not substituted for ν.Source: S2.3 Last-Layer Curvature · pp. 30–32

S2.4 Evidence-Style Prior Precision

  • The evidence-style precision update profiles out the unpenalized intercept and uses an effective count of data-constrained penalized directions.For ensembles, effective dimensions and squared penalized weights are pooled across members before the shared fixed-point update; this is not a product of independent marginal likelihood contributions.Source: S2.4 Evidence-Style Prior Precision · pp. 33–34

S2.5 Inferential Targets and Propagation

  • For centered exposure–lag and covariate contrasts, Gaussian covariance propagation is exact conditional on the learned representation, and interval endpoints are transformed for reported estimands.Exponentiation maps log-link contrasts to relative risks and logit-link contrasts to odds ratios; identity-link estimates require no transformation.Source: S2.5 Inferential Targets and Propagation · p. 35

S2.6 Ensembles

  • Ensemble intervals combine conditional last-layer uncertainty with between-member variation as a Gaussian moment-matched summary of the member-level mixture.This constructed uncertainty measure is neither sampling variance of the ensemble mean nor full posterior uncertainty in the learned representation; its between-member term captures disagreement among independently initialized fits.Source: S2.6 Ensembles · pp. 35–36

S2.7 Regularity Conditions and Scope

  • The procedure is approximate, relying on a trained last-layer estimate treated as a posterior or working-mode analogue and on curvature and ensemble-Gaussian assumptions.Conditional on the trained network parameters, the affine predictor makes its Jacobian and inferential-target propagation exact, while canonical-link likelihood curvature is also exact.Source: S2.7 Regularity Conditions and Scope · p. 36
  • Ensemble intervals combine conditional member-level Laplace uncertainty with between-member variation, capturing fit instability only insofar as independently initialized fits disagree.Member approximations condition on fitted network parameters, including mixture weights, and do not represent a posterior over the complete parameter vector.Source: S2.7 Regularity Conditions and Scope · p. 36
  • The reported intervals are pointwise confidence intervals for f_m, F_m, and h_k, not simultaneous bands or predictions of future outcomes.Predictive quantiles require a fully specified outcome distribution and integration over uncertainty in the fitted linear predictor; a mean–variance specification alone is insufficient.Source: S2.7 Regularity Conditions and Scope · pp. 36–37

S3 Data-Generating Surfaces

The four data-generating processes span separable and structurally demanding exposure–lag response surfaces; cumulative relative-risk curves are the primary simulation estimand.

S3 Data-Generating Surfaces

  • Generating surfaces are reported and scored after centering at x⋆= 20.Cumulative curves are the primary simulation estimand.Source: S3 Data-Generating Surfaces · p. 37
  • DGP 2 combines opposing-sign contributions at intermediate lags with distinct high- and low-exposure responses at early and later lags.Figure S1 shows the resulting exposure–lag surface and cumulative curve.Source: S3 Data-Generating Surfaces · p. 37
  • DGP 4 places sharp responses at both exposure extremes, with different dominant lag timing.Figure S1 shows the resulting exposure–lag surface and cumulative curve.Source: S3 Data-Generating Surfaces · p. 37
  • DGP 3 combines a narrow high-exposure response with a broader low-exposure arm on a slower lag kernel.Figure S1 shows the resulting exposure–lag surface and cumulative curve.Source: S3 Data-Generating Surfaces · p. 37
  • DGP 1 is separable: a smooth exposure response is multiplied by a single exponentially decaying lag kernel.Figure S1 shows the resulting exposure–lag surface and cumulative curve.Source: S3 Data-Generating Surfaces · p. 37

S4 Monte Carlo Uncertainty

Monte Carlo uncertainty calculations account for replicate-level dependence across evaluation points and use jackknife or delta-method procedures for reported summaries.

S4 Monte Carlo Uncertainty

  • The plotted 95% Monte Carlo confidence interval is bS ± 1.96 MCSE(bS), and delta-method propagation through the square-root transformation estimates uncertainty for RMSE.The RMSE calculation starts from replicate-wise mean squared error.Source: S4 Monte Carlo Uncertainty · pp. 37–39
  • Coverage uncertainty is estimated from replicate-level summaries, preserving dependence among evaluation points within each fitted replicate.The same replicate-level approach accounts for dependence when calculating Monte Carlo uncertainty for averaged replicate-wise quantities.Source: S4 Monte Carlo Uncertainty · pp. 37–38
  • The joint-to-single degradation ratio uses first-order propagation with the covariance between its two Monte Carlo RMSE estimates set to zero.No covariance term is included in this ratio approximation.Source: S4 Monte Carlo Uncertainty · p. 39
  • Monte Carlo uncertainty for regional squared bias and variance uses delete-one-replicate jackknifing because these summaries depend on the complete replicate collection.The calculations apply separately to cumulative and full-surface targets and each evaluation region.Source: S4 Monte Carlo Uncertainty · p. 39

S5 Mixture Subnetwork Decomposition

The mixture decomposition illustrates how weighted subnetworks share functional complexity, while the ablation indicates recovery degrades without the mixture construction.

S5 Mixture Subnetwork Decomposition

  • The decomposition indicates that weighted subnetworks distribute functional complexity, combining different response features through reinforcement or partial cancellation on the log-relative-risk scale.The ablation shows recovery degrades when the mixture construction is removed, but does not reveal how subnetworks combine.Source: S5 Mixture Subnetwork Decomposition · pp. 39–40
  • The decomposition is illustrative, not representative, and does not establish that particular subnetwork roles are identifiable or stable across fits.Source: S5 Mixture Subnetwork Decomposition · p. 40
  • Figure S2 decomposes cumulative relative-risk functions into the fitted Mixture DLNAM and its three weighted subnetwork contributions.The illustration uses one ensemble member fitted to one replicate of each DGP, without ensemble averaging.Source: S5 Mixture Subnetwork Decomposition · p. 39, Figure S2 · p. 40

S6 Multivariate ExU Input Layers

The section compares three multivariate ExU input layers for exposure–lag surfaces, with the reported coordinate-wise concatenation performing best across the tested DGPs and targets.

S6 Multivariate ExU Input Layers

  • The scalar ExU handles one input coordinate; exposure–lag surfaces use two, requiring a multivariate extension.The compared alternatives are coordinate-wise concatenation and unified layers that allow coordinate mixing.Source: S6 Multivariate ExU Input Layers · p. 40
  • Concatenation applies independent scalar ExUs to each coordinate, leaving exposure–lag interaction to subsequent hidden layers.For a surface, its width is split between exposure and lag units.Source: S6 Multivariate ExU Input Layers · p. 40
  • Unified local-bias and shared-bias layers let each first-layer unit depend on all coordinates; local-bias retains unit-specific localization, unlike shared-bias.The local-bias layer reduces to the scalar ExU when p = 1; shared-bias uses one location per coordinate across units.Source: S6 Multivariate ExU Input Layers · pp. 40–41
  • The layers are matched by output width, not parameter count, and otherwise use the reported input scaling and initialization settings.Their input-layer parameter counts are 2d for concatenation, 2pd for unified local-bias, and pd + p for unified shared-bias.Source: S6 Multivariate ExU Input Layers · p. 41
  • Across DGPs 1–4, concatenation has the lowest error for both cumulative and full-surface targets, while shared-bias performs worst.The comparison supports concatenation over the two unified alternatives, not universal optimality.Source: S6 Multivariate ExU Input Layers · p. 41

S7 Prospective Extensions

The prospective extensions use neural components for multitask and hierarchical distributed-lag models. They share representations across outcomes or pool group deviations toward global components, with corresponding limits on flexibility and uncertainty propagation.

S7.1 Multitask DLNAM

  • Multitask DLNAMs share subnetwork parameters across outcomes while allowing outcome-specific intercepts and mixing weights.The shared representation induces cross-outcome regularization but does not specify residual dependence.Source: S7.1 Multitask DLNAM · p. 41
  • Figure S3 compares cumulative ExU recovery across four DGPs using estimated and generating curves, regional RMSE, and pointwise interval coverage.It reports Monte Carlo means across fitted replicates, with boundary and interior regions defined by exposure percentiles.Source: Figure S3 · p. 42

S7.2 Hierarchical DLNAM

  • Hierarchical DLNAMs combine global components with group-specific neural deviations, whose mixing weights are shrunk toward zero to pool groups toward the global components.As group-level variances approach zero, the model reduces to a common Mixture DLNAM; nested or crossed groupings can add deviation terms and pooling distributions.Source: S7.2 Hierarchical DLNAM · pp. 42–43
  • Unlike the multitask construction, hierarchical deviation subnetworks are not shared across groups, and full uncertainty propagation for deviations and pooling parameters requires extending the last-layer construction.That construction conditions on fitted mixing weights and learned representations.Source: S7.1 Multitask DLNAM · p. 41, S7.2 Hierarchical DLNAM · p. 43

S8 Surface-Target Comparisons

This section repeats estimator comparisons and architecture ablations for the full exposure–lag surface, as a complementary target to the cumulative summary.

S8 Surface-Target Comparisons

  • The surface analysis scores every exposure-grid point at every lag, providing a finer-grained view of how fitted effects are allocated across exposure and lag.It is presented as complementary, not superior, to cumulative recovery.Source: S8 Surface-Target Comparisons · p. 43
  • The surface-target model comparison uses Figure S4 and Tables S10–S12, while the architecture ablation uses Figure S5 and Table S14.The section repeats these analyses from the main simulation for the full surface.Source: S8 Surface-Target Comparisons · p. 43
  • The main text emphasizes the cumulative curve as the primary simulation estimand and applied summary.Surface scoring is not treated as a superior recovery target.Source: S8 Surface-Target Comparisons · p. 43

S9 Analysis Specifications

The supplementary specifications collect study designs, fitting and comparator settings, and further simulation comparisons; the treed comparator's nonlinear implementation and retry procedure are also documented.

S9 Analysis Specifications

  • Tables S2 and S3 summarize study design, DLNAM fitting, comparators, adjustments, software, computing, and study-specific departures.Settings shared across analyses are listed once.Source: S9 Analysis Specifications · p. 43, Table S2 · p. 47, Table S3 · p. 48
  • For criterion-selected cross-basis DLNMs, QAIC and QBIC differ only in the penalty applied to cross-basis complexity.The candidate fit is scored using its number of cross-basis coefficients and fitted dispersion.Source: S9 Analysis Specifications · p. 43
  • All 800 treed fits in the principal comparison completed; 83 needed multiple initialization attempts, with at most seven attempts.Failed initializations were retried with a new seed up to ten times; none were excluded.Source: S9 Analysis Specifications · p. 43
  • Supplementary Tables S4–S9 report simulation outcomes by DGP, method, and exposure region: cumulative log-relative-risk RMSE and pointwise 95% coverage, with bias, variance, relative RMSE, interval width, and interval decomposition.Tables S4–S7 use R = 200; S8–S9 use R = 50. Relative RMSE uses DLNAM total RMSE within each DGP as the common denominator, and interval width is on the log-relative-risk scale.Source: Table S4 · p. 49, Table S5 · p. 50, Table S6 · p. 51
  • Table S10 reports log-relative-risk RMSE and pointwise 95% interval coverage by DGP, method, and exposure region, with Monte Carlo standard errors, using R = 200.Table S11 reports squared bias and variance. Table S12 reports relative RMSE and mean pointwise 95% interval width; relative RMSE uses DLNAM total RMSE within each DGP as the common denominator.Source: Table S10 · p. 55, Table S11 · p. 56, Table S12 · p. 57
  • The surface ExU and architecture-ablation tables report log-relative-risk RMSE and pointwise 95% interval coverage by DGP, method, and exposure region, with R = 50.The ExU comparison is Table S13; the architecture ablation is Table S14.Source: Table S13 · p. 58, Table S14 · p. 59
  • The joint-model table reports cumulative log-relative-risk RMSE and pointwise 95% interval coverage by component, method, and exposure region, with R = 50..Source: Table S15 · p. 60

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