Additive Nonparametric Regression with Spatial and Network Objects

1 Introduction

• Existing image-on-image methods do not account for ROI-level spatial predictors paired with network predictors defined over nested subnetworks.The resulting spatial and network predictors are misaligned because ROIs are nested within subnetworks.Source: §1 · p. 3
• AMO-GP is a two-stage semi-parametric framework for image-on-image regression with spatially varying functions and a network predictor.The first stage models each network edge as a bilinear effect of its nodes’ latent vectors; the second models additive predictor effects.Source: §1 · p. 3
• Spatially indexed functional predictors have spatially varying coefficient functions, while network-node effects use a nonlinear function of estimated latent effects.Gaussian-process priors are assigned to the coefficient functions and the function capturing node-specific latent effects.Source: §1 · p. 3
• The additive structure is presented as interpretable and regularizing, reducing model complexity and avoiding joint covariance kernels across predictors.The framework is intended to integrate diverse image data while quantifying uncertainty for inference.Source: §1 · p. 3

2 Multi-modal Neuroimaging Data Application

• The application models working-memory task activation as a function of cortical morphological features and resting-state connectivity.The motivation is that individual task-based fMRI activation maps may have low reliability, while morphological and resting-state features are considered more stable.Source: §2 · pp. 4–5
• The ABCD analysis uses baseline imaging and cognitive measures from a random subsample of 80 children aged 9–10 years at one study site.Sample size is restricted by computational constraints.Source: §2 · p. 5
• Task fMRI supplies the response, while structural MRI supplies cortical morphology and resting-state fMRI supplies connectivity predictors.The task measure is the emotional N-back 2-back versus 0-back contrast for 148 Destrieux-atlas ROIs; morphology includes cortical thickness and sulcal depth.Source: §2 · p. 5
• Resting-state connectivity is represented by a symmetric subnetwork adjacency matrix whose entries are Fisher Z-transformed average pairwise ROI correlations.The network is based on 333 cortical-surface ROIs grouped into 13 Gordon subnetworks.Source: §2 · pp. 5–6
• Centroid-distance matching nests Destrieux ROIs within Gordon subnetworks, aligning the two atlases for cross-modality and subject comparisons.The alignment retained 145 Destrieux ROIs within 11 Gordon subnetworks and excluded salience and sensorimotor-mouth networks for sparse coverage.Source: §2 · p. 6
• The authors identify atlas selection and alignment as sensitivity considerations for this analysis.The matched ROI-to-subnetwork assignment had a median distance of 6.6 mm, with a range of 1.4–17.2 mm.Source: §2 · p. 6

3.1 Multi-object image characterization

• AMO-GP jointly represents spatial images as ROI-indexed functions and connectivity as a network object linked through shared topology.Source: §3.1 · p. 6

3.2 Model Framework

• AMO-GP combines network manifold regression with nonparametric functional-predictor modeling in two stages, with no noticeable loss in predictive uncertainty reported for ABCD t-fMRI prediction.Source: §3.2 · p. 7

3.2.1 Stage 1: Latent scale representation of brain networks

• Stage 1 models network edges with subject intercepts and node latent vectors whose inner products encode connectivity, reducing parameters from P(P −1)/2 to PR.Source: §3.2.1 · p. 7

3.2.2 Stage 2: Additive Semi-parametric Regression Framework

• Stage 2 adds nonlinear node and functional-predictor effects under independent GP priors, enabling flexible relationships and information sharing across ROIs and subjects.Source: §3.2.2 · pp. 8–9
• The w predictor is linear with ROI-varying coefficients, and posterior consistency of AMO-GP is studied in the Supplementary Section A.Source: §3.2.2 · pp. 8–9

4 Posterior Computation: Parallel Computation Over Subnetworks

• For the empirical studies, the network model uses an identity link and fixes intrinsic dimensionality at R = 6.Source: §4 · p. 9
• A Gibbs sampler runs for 15,000 iterations, with the first 10,000 discarded as burn-in; estimates for bui,p are derived from the remaining 5,000 iterations.Source: §4 · p. 9
• Parallel updates across subnetworks reduce computation, whose dominant costs scale approximately as max_p V^3 for varying coefficients and n^3 for node effects.Source: §4 · p. 10
• A moderate sample size was used for feasible computation, with larger datasets identified as a challenge for future work.Source: §4 · p. 10

5.1 Simulated Data Generation

• The simulations use 50 subnetworks with 12 ROIs each and 100 training and 100 test samples, across four scenarios and 10 independent repetitions.Scenarios 1 and 3 are well-specified; Scenarios 2 and 4 are misspecified.Source: §5.1 · p. 10

5.2 Competitors & Metrics of Comparison

• Competitors comprise nested models using predictor subsets and non-nested models using all predictors without AMO-GP’s additive multi-object structure.The non-nested competitors do not account for network topology, spatial associations in functional predictors, or their structural relationships.Source: §5.2 · p. 10

5.2.1 Competing Nested Models

• The nested Network model predicts the response from estimated network features; Function & Network adds one functional predictor to that regression.Both nested models share AMO-GP’s first stage and modify its second stage.Source: §5.2.1 · p. 11

5.2.2 Non-nested Competitors

• Non-nested competitors are BART, a feedforward neural network, and BIRD-GP, each using all three predictors without AMO-GP’s additive multi-object structure.BIRD-GP uses deep kernel learning within a Gaussian process framework.Source: §5.2.2 · p. 11

5.2.3 Metrics of Comparison

• Nested-model comparisons use posterior predictive loss; evaluation also reports training fit and complexity, test MSPE and R2, and 95% interval coverage and length.NN does not provide model complexity or predictive intervals; coefficient-estimation accuracy is assessed separately.Source: §5.2.3 · p. 12

5.3 Simulation Results

• BART, NN, and BIRD-GP substantially underperform AMO-GP overall (NN fares worst), while Scenario 4 marginally favors BIRD-GP; coefficient recovery declines under misspecification, especially with heavy-tailed noise and interactions.The authors report similar performance between BIRD-GP and AMO-GP in Scenario 4.Source: §5.3 · pp. 12, 15
• Across Scenarios 1–3, AMO-GP consistently outperforms the nested models in accuracy and complexity, with substantially lower G(M), P(M), and D(M).The table reports average training and test performance over 10 repetitions.Source: §5.3 · p. 12, Table 2 · p. 13
• Training intervals under-cover most for AMO-GP, whose intervals are very narrow; on test data, its coverage is at least 90% across scenarios.Figure 2 summarizes 95% interval coverage and length; NN intervals are unavailable.Source: §5.3 · pp. 12–13, Figure 2 · p. 14

6 ABCD Data Analysis

• The analysis uses 80 randomly selected ABCD subjects, split equally into training and test sets, because AMO-GP is computationally demanding for large samples.The predictors are cortical thickness, sulcal depth, and resting-state connectivity; the outcome is task-based fMRI activation.Source: §6 · p. 15

6.1 Analysis & Results

• AMO-GP outperforms BART and BIRD-GP out of sample and modestly improves on nested competitors, whose overall performance is comparable.The comparison suggests limited additional contribution from cortical thickness and sulcal depth after accounting for the network predictor.Source: §6.1 · p. 15
• AMO-GP has the highest out-of-sample coverage at 88.34%, with wider but more plausible intervals than BIRD-GP and BART.BIRD-GP coverage is 44.24% and BART coverage is 50.38%; their lower coverage is described as indicating underestimated predictive uncertainty.Source: §6.1 · p. 16
• Across subnetworks, in-sample R2 ranges from approximately 0.32 to 0.87, with standard deviations of 0.04 to 0.11 across subsamples.The reported variability suggests stable performance across the random training-set splits.Source: §6 · p. 15, §6.1 · p. 16
• Cortical thickness is generally negatively associated with working-memory task response, particularly in parietal regions, while the right temporal association is positive.These are conditional effects controlling for resting-state connectivity and sulcal depth.Source: §6.1 · pp. 16–17
• Sulcal-depth associations with task activation are mixed: positive in frontal and insula regions and negative in parietal, temporal, and occipital regions.Figure 3 maps coefficient functions by Destrieux region, with shading indicating coefficient values.Source: §6.1 · p. 17, Figure 3 · p. 17

7 Conclusion and Future Work

• AMO-GP models additive nonlinear effects with Gaussian process priors on spatial coefficients for functional predictors and node effects for network predictors.It targets uncertainty-quantified prediction of brain activation maps from cortical thickness, sulcal depth, and connectivity networks.Source: §7 · pp. 17–18
• The framework provides interpretable conditional effects and assesses the relative contributions of network and structural images to task-based responses.The article describes it as one of the first image-on-image regression models to incorporate both functional and network predictors.Source: §7 · p. 18
• Computational constraints limited sample size, requiring cautious interpretation of the findings.The conclusion identifies this as a limitation of the ABCD analysis.Source: §7 · p. 18
• Atlas selection and alignment are important because structural regions spanning multiple subnetworks may affect prediction accuracy, although inference and interpretation remain clear without ideal nesting.Reprocessing the data to a common atlas and repeating the analysis is described as beyond this work’s scope.Source: §7 · p. 18
• Gaussian process priors offer modeling flexibility but present computational challenges with large datasets.Efficient distributed Gaussian process computation is proposed as a future direction.Source: §7 · p. 18

ABCD Data Acknowledgment

• The ABCD Study is designed to recruit more than 10,000 children aged 9–10 and follow them for 10 years into early adulthood.It is a multisite, longitudinal study.Source: ABCD Data Acknowledgment · p. 18
• The ABCD Study is supported by the National Institutes of Health and additional federal partners.The passage lists award numbers for the support.Source: ABCD Data Acknowledgment · p. 18
• ABCD consortium investigators designed and implemented the study and/or provided data, but did not necessarily participate in the analysis or writing.The manuscript notes that its views may not reflect those of NIH or ABCD consortium investigators.Source: ABCD Data Acknowledgment · p. 19

Supplementary Materials

• Supplementary materials cover posterior consistency, computation, simulations, and additional results from the real-data analysis.Source: Supplementary Materials · p. 19
• An R implementation of AMO-GP is available.Source: Supplementary Materials · p. 19

• The ABCD data repository grows and changes over time.Source: Data Availability · p. 19

B.1 Conditional Posterior Distributions in Stage 1 of AMO-GP

• The supplement provides full conditional distributions for MCMC estimation, with Stage 1 notation aligned to Section 3.2.1 of the main manuscript.Source: Supplementary Materials for Additive… · p. 23, §B.1 · p. 30

Supplementary Materials for Additive Nonparametric Regression with Spatial and

• The supplementary file has four sections covering posterior consistency, MCMC conditionals, simulation settings, and additional ABCD analysis results.Source: Supplementary Materials for Additive… · p. 23

A Posterior Consistency of AMO-GP

• Under assumptions (A.1)–(A.4), Pn,f0[Π(f: ||f−f0||n > ε | O1:n)] converges to 0 as n→∞.Source: A Posterior Consistency of AMO-GP · p. 24

A.1 Proof of Theorem 1

• The proof decomposes posterior error over sieve shells and bounds test-error contributions before taking n to infinity.Source: §A.1 · pp. 28–29

B.1 Conditional Posterior Distributions in Stage 1 of AMO-GP

• Stage 1 supplementary derivations give conditional posterior distributions, including for μ_i under a flat prior and for the latent network variables.The passages identify conditional posterior derivations for μ_i and ũ_i,p.Source: §B.1 · pp. 30–31

B.2 Conditional Posterior Distributions in Stage 2 of AMO-GP

• Stage 2 specifies the subnetwork response model and conditional posterior calculations, uses Metropolis–Hastings for log variance parameters, and describes test-set functional-effect prediction.Subnetwork ROIs have coordinates; the model combines f_p and coefficient-weighted predictors with noise, and uses exponential covariance kernels.Source: §B.2 · pp. 32–34

C.1 Network Model for Stage 1

• The simulations use 50 subnetworks, 12 ROIs per subnetwork, and 100 training and test subjects; functional predictors are independently generated as standard normal variables.Network locations are sampled uniformly over [0, 10]^3 and shared across subjects within each subnetwork; each scenario is repeated for 10 independent replicates.Source: §C.1 · p. 35, §C.2 · p. 35, §C.5 · p. 36

C.3 Gaussian Process Specifications

• The simulation’s network and functional effects use independent, zero-mean Gaussian processes with exponential covariance kernels and parameters fixed across subnetworks.GPs specify the network effect fN_p and functional coefficient functions βp,g(v) and βp,w(v).Source: §C.3 · p. 35

C.4 Scenario-Specific Data Generation

• Four simulation scenarios cover an additive GP model, nonlinear network effects, heavy-tailed noise, and non-additive interactions, with interaction strength set by ρ = 0.3.Each scenario uses response noise variance 0.01; Scenario 3 is generated as in Scenario 1 but with heavy-tailed noise.Source: §C.4 · p. 36

C.6 Coefficient Recovery Evaluation

• AMO-GP coefficient recovery was evaluated using relative error across spatial locations and subnetworks, but only AMO-GP was assessed because competing methods lacked comparable coefficient estimates.Recovery was accurate under correct specification and mild deviations, but degraded with stronger misspecification.Source: §C.6 · pp. 36–37
• Coefficient-estimation error was lowest in Scenario 1, increased moderately in Scenario 2 and variably in Scenario 3, and was largest in Scenario 4.Scenario 2 includes nonlinear network-effect misspecification, Scenario 3 heavy-tailed noise, and Scenario 4 non-additive interactions.Source: Figure S1 · p. 37, §C.6 · p. 37

D.1 Traceplots indicating Stage 2 convergence of AMO-GP

• For a representative ABCD subsample of 40 participants, Stage 2 parameter traceplots showed satisfactory MCMC convergence across subnetworks.Figure S2 displays 15,000 iterations for all 11 subnetworks, with seven parameters per subnetwork.Source: §D.1 · p. 38, Figure S2 · p. 40

D.2 Analysis of in-sample R2 across multiple subsamples

• Retrosplenial temporal (0.87±0.06), cingulo-parietal (0.78±0.04), and sensorimotor hand (0.69±0.11) showed consistently high in-sample R2.These subnetworks had relatively low dispersion across subsamples.Source: §D.2 · p. 41
• None (0.32 ± 0.08) and fronto-parietal (0.35 ± 0.08) had lower mean R2 with comparable variability, indicating weaker and less stable fits.Source: §D.2 · p. 41
• Across functional subnetworks, mean in-sample R2 values ranged from approximately 0.32 (none) to 0.87 (retrosplenial temporal), with Figure S3 showing boxplots from 10 independent subsamples of size 40.Source: §D.2 · p. 41, Figure S3 · p. 41

D.3 Mean squared error (in-sample and predictive) for each subnetwork

• Table S1 reports in-sample MSE and out-of-sample MSPE for competing models across functional subnetworks.The supplied passages do not state table values or identify a best-performing model.Source: §D.3 · p. 42, Table S1 · p. 42