Adding structure to generalized additive models, with applications in ecology

David L Miller; Ken Newman; Thomas Cornulier

REVIEW 2 major objections 5 minor 26 references

Adding structure to generalized additive models, with applications in ecology

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper shows that varying-coefficient, scalar-on-function, and distributed-lag models are all generalized additive models, so vector-valued environmental covariates can enter directly instead of as hand-chosen summaries.

desk verdict Useful, clearly-written tutorial on fitting three existing structured-covariate GAM extensions in mgcv, but the flagship SoFR comparison is reported ambiguously and needs correcting before the paper's adoption argument can be trusted. read the letter →

arxiv 2508.07915 v2 pith:N2DO4OXB submitted 2025-08-11 stat.ME q-bio.QM

David L Miller , Ken Newman , Thomas Cornulier This is my paper

classification stat.MEq-bio.QM MSC 62G0862P12

keywords generalizedadditivemodelsvarying-coefficientscalar-on-functionregressiondistributedlagfunctionaldataanalysistensorproductssummationconventionecologicalmodelling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Many ecological covariates are inherently non-scalar: a single breeding-success observation comes with a full year of daily temperatures or monthly prey counts. The usual workaround is to reduce such a series to one number by averaging or picking a window, which can discard the signal of interest and hide which time periods matter. This paper claims that three existing model classes—varying-coefficient, scalar-on-function, and distributed-lag models—can all be expressed as generalized additive models, so the data-driven weighting happens inside the model rather than before it. Fitted this way, the models estimate smooth weight functions over the covariate index (depth, time, lag), making the relevant aggregation a result rather than an assumption. If the claim is right, ecologists can adopt these structured analyses with the same penalized-spline machinery they already use, without bespoke software.

What carries the argument

The argument runs on three pieces of machinery: (1) tensor products, which build multivariate smooths from univariate marginal bases so covariates on different scales can interact; (2) the summation convention, which treats matrix-valued covariates by evaluating the smooth at each column and summing over columns, approximating an integral; and (3) linear functionals ("by" variables), which multiply a smooth by a scalar, factor, or vector before it enters the linear predictor. Depending on the type of "by" variable, this yields varying-coefficient terms ($s(z)\cdot x$), scalar-on-function terms ($\sum_j s(v_j)x_i(v_j)$), or, with a tensor product, distributed-lag terms ($\sum_t s(x_{it}, t)$)

What would settle it

Refit the kittiwake model of Section 4 on the same data with both the full scalar-on-function terms and the Frederiksen-style averaged covariates, and compare predictive performance on years or sites held out during fitting; if the averaged model matches or beats the functional model, the central practical claim that data-driven weighting avoids information loss is contradicted.

Watch

Extended reading notes

Core claim

The central claim is that moving from scalar covariates to vector-valued covariates does not require leaving the generalized additive model class. The paper shows that a smooth term $s_j$ in the linear predictor can be pre-multiplied by a linear functional $L_{ij}$—the "by" variable—and that when the covariates are matrices, the summation convention lets the model sum the smooth over columns. That single mechanism, combined with tensor products for anisotropic interactions, yields three model families: varying-coefficient models $s(z)\cdot x$, scalar-on-function regression $\sum_j s(v_j)x_i(v_j)$, and distributed-lag models $\sum_t s(x_{it}, t)$. The paper demonstrates on soil carbon, kittiw

Load-bearing premise

The practical case for these models rests on the illustrative analyses being valid, and Section 4.2 is internally ambiguous about whether the scalar-on-function model or the simple averaged-covariate model actually had the lower AIC, so if the averaged models truly win, the paper's argument for adoption weakens.

Editorial extensions

If this is right

Ecologists can include entire time series or spatial profiles as covariates instead of pre-chosen averages, with the aggregation weights estimated from data.
The estimated weight functions are directly interpretable: they show which months, depths, or lag days drive the response, and in what direction.
Because the terms are smooth and penalized, the models borrow strength across index values and avoid the multicollinearity and parameter inflation of treating each component as a separate covariate.
Covariates with different index scales (days versus months) can enter the same model because each has its own smooth over its own index.
The same fitted-value framework can be extended hierarchically, for example by sharing information between sites, within the existing GAM toolbox.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the Section 4.2 text says both "the corresponding average models had lower AICs" and "Delta AIC > 20 between our scalar-on-function regression and best average model," a reader should settle which model actually won before using the kittiwake example as evidence; the two sentences point in opposite directions.
The same "smooth times covariate, summed over an index" construction should transfer directly to spatial surfaces or trait gradients as the index domain, not only time and depth.
A testable extension is to compare held-out predictive accuracy of data-driven weighting versus hand-picked summaries across many species and sites; the paper's examples suggest but do not formally establish that the structured models pay for their extra complexity.
Because the terms enter only the linear predictor, they can be embedded in hierarchical Bayesian population models; the paper notes this direction, which would let forecasts propagate uncertainty through the estimated weight functions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, this paper is a teaching-oriented synthesis, not a new method. Its value is in one place: it shows ecologists, in a single article, how to set up varying-coefficient, scalar-on-function, and distributed-lag models using mgcv's tensor products, summation convention, and by variables. The exposition is clear, the code examples are practical, and the soil carbon application is a nice demonstration of spatial VCMs.

The weak spot is real and it is in Section 4.2. The sentence as written says the 'average models had lower AICs' with delta AIC > 20 between the SoFR and best average model. If that's literal, the flagship SoFR example is actually worse than simple averaging, which cuts against the paper's claim. If the intended reading is the opposite, the sentence is mistyped. Either way, the text as it stands cannot support the paper's adoption argument. The deviance-explained phrase is likewise ambiguous. This is not a minor typo; it's the key evidence in the paper's core example. The supplementary material may contain the right comparison, but the main text needs to state it plainly and, ideally, add out-of-sample validation or cross-validation to show the structured models are not just in-sample winners.

The aphid DLM is more convincing, though based on only 141 observations from three adjacent east-coast traps, so the generalizability is limited. The paper acknowledges this. The soil example is fine. The references are appropriate; the self-citation to Miller (2025) is for background, not load-bearing.

On balance, the paper deserves a serious referee. The methods are standard, so the refereeing burden is mostly on clarity and evidence, not mathematical novelty. I'd recommend the editor send it out, with a request that the authors resolve the AIC contradiction, clarify the deviance comparison, and either show cross-validation or temper the usefulness claims. With those fixes, it would be a genuinely useful practitioner resource.

Referee Report

2 major / 5 minor

Summary. The paper is a methods tutorial aimed at ecologists. It presents three extensions of generalized additive models for vector-valued covariates—varying-coefficient models, scalar-on-function (signal) regression, and distributed lag models—and shows how each can be implemented in the R package mgcv using tensor products, the summation convention, and 'by' variables. The mathematical exposition is cast as an extension of the standard GAM additive predictor, and the three model classes are illustrated with ecological applications: soil organic carbon as a varying-coefficient model, kittiwake breeding success as a scalar-on-function regression, and aphid arrival dates as a distributed lag model. The paper claims that these approaches avoid arbitrary aggregation of structured covariates, reduce information loss, and can be fitted with standard software. The central contribution is didactic rather than novel statistical theory.

Significance. If the illustrative analyses are reported correctly, the paper is a useful practical bridge between functional-data-analysis methods and routine GAM software. The tensor-product construction (Appendix A) and the cross-basis derivation for distributed lags (Appendix B) are standard but clearly presented, and the paper gives concrete formula and data-layout guidance that ecologists can follow. The use of posterior sampling for uncertainty and the attention to model structure are good practice. However, the paper's adoption argument depends on the examples demonstrating that the proposed models outperform or at least interpretably augment simple averaging approaches, and one of those comparisons is currently reported in a self-contradictory way. The central derivations are sound, but the evidence base needs correction before the claims can be relied upon.

major comments (2)

[Section 4.2, paragraph beginning 'We compared model (6)...'] The AIC comparison is internally contradictory. The text states that 'the corresponding average models had lower AICs' and then gives '∆AIC >20 between our scalar-on-function regression and best average model'; the second clause implies the scalar-on-function model is worse by more than 20 AIC units. If that is the correct reading, the flagship SoFR example actually shows the average model winning, which undercuts the claim that SoFR reduces information loss. If the intended reading is that SoFR has the lower AIC, then the sentence is mistyped. Either way, the reported comparison is not usable as evidence. Please state clearly which model has lower AIC, report actual AIC (and deviance-explained) values for both the SoFR and each average model, and, if the average models are better, re-examine or re-frame the example. This is load-bearing because the paper's usefulness claim rests on the
[Section 5.2, final paragraph and Supplementary Material C] The distributed lag model is advocated as an improvement over the standard January–February average-temperature model and a weekly-average model, but the quantitative comparison is only in Supplementary Material C. The main text says 'Overall we find that the distributed lag approach gives significant, interpretable insight', but no main-text statistics are provided. Given that the tutorial's practical value is demonstrated through these examples, the comparison should be reported in the main text, or the claim should be softened to 'interpretable insight' without asserting relative predictive improvement. Please also clarify whether 'significant' is used in the statistical sense or colloquially.

minor comments (5)

[Section 4.2, Figure 3 caption and text] The text says 'The right four plots relate to the calanus effect, the left four to SST', but the figure caption and the next paragraph say the left four plots are calanus and the right four are SST. These are contradictory; please correct.
[Section 4.2, model comparison sentence] The phrase 'deviances explained (> 35% difference in both cases)' is vague. Report the deviance-explained values for each model, not just a percentage difference.
[Equations (6) and (7), index conventions] The summation ranges (18 months; days 15 to 120; 59 lags) are motivated by the applied context, but the manuscript should state more explicitly that these windows are fixed a priori and that only the weights within the windows are estimated. As written, 'effectively perform model selection during fitting' may overstate the data-driven aspect.
[Section 5.2, final paragraph] The sentence beginning 'These are reasonable when compared to the parameters required...' is grammatically incomplete and should be rephrased.
[Throughout] A few typographical issues: 'anholocylic' should be 'anholocyclic' (Section 5.2), and Figure 4 is referenced as 'Figure 4)' with a stray parenthesis (Section 4.2).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the paper maps existing model classes (VCM, SoFR, DLM) into mgcv machinery via self-contained derivations; self-citations are ancillary pointers, and the Section 4.2 AIC ambiguity is a consistency issue, not a circular one.

full rationale

This is a methods tutorial showing how three pre-existing model classes (varying-coefficient, scalar-on-function, and distributed lag models) can be expressed in the mgcv GAM framework using tensor products, the summation convention, and 'by' variables. The derivation chain is self-contained, not circular. The SoFR construction is an explicit algebraic equivalence: the functional integral (4) is approximated by quadrature in (5), and Section 2.3 shows this is exactly a vector-valued linear functional with the summation convention; the smooth s(v) is estimated from data, not defined in terms of the response. The DLM is derived from first principles in Appendix B as a tensor product of marginal smooths in x and lag, followed by the summation convention over lags, matching the standard cross-basis construction with external citations (Gasparrini 2011; Armstrong 2006). The VCM is simply the numeric 'by' variable form in Section 2.3. No fitted parameter is renamed as a prediction; the paper makes no out-of-sample predictive claims, and the example analyses are illustrative fits. The paper cites its own prior work (Miller 2025 for Bayesian interpretation and posterior sampling; Pedersen et al. 2019 for hierarchical GAMs), but these are ancillary pointers: the central implementability claim rests on the paper's own equations and on external references (Wood 2017, Hastie and Tibshirani 1993), not on the self-citations. They are therefore not load-bearing. The one notable issue is internal consistency in Section 4.2: the text says 'the corresponding average models had lower AICs (ΔAIC >20 between our scalar-on-function regression and best average model)', which literally implies the averaged Frederiksen model outperforms the SoFR model by >20 AIC, undercutting the paper's promotion of SoFR; if the intended claim is the opposite, the sentence is mistyped. This weakens the illustrative evidence for the usefulness claim but is not a circularity: it does not make any derivation equivalent to its inputs. Overall, no significant circularity; score 2 reflects only the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper's central claim is methodological and relies on standard GAM theory and mgcv's implementation. No new entities are introduced. The main analyst-chosen inputs are lag windows and basis dimensions, listed above.

free parameters (3)

Maximum lag T = 59 days for aphid model = 59
Chosen by hand to cover January 1 to February 28; affects the DLM's window for temperature effects.
Calanus window of 18 months and SST window of days 15 to 120 = 18 months; 106 days
Selected by the analyst based on ecological reasoning, not derived; these choices define the quadrature grid for SoFR.
Basis dimensions k for smooth terms = not specified in main text
Set by analyst to be large enough; the penalty estimates effective complexity. Not central to the methodological claim.

assumptions (5)

standard math Tensor product smooth construction
Used throughout to build bivariate smooths of different-scale variables; see Section 2.1 and Appendix A.
domain assumption Summation convention in mgcv
Software-specific convention that sums smooth terms over matrix columns; used in SoFR and DLM. It is mathematically equivalent to quadrature, but relies on mgcv's implementation.
domain assumption Smoothness penalties provide appropriate regularization
The paper relies on mgcv's penalized likelihood to avoid overfitting with flexible smooths; this is standard GAM theory.
domain assumption Response distributions are correctly specified for the examples
The examples assume normality for log carbon and aphid arrival date, and negative binomial for fledgling counts; if mis-specified, inference could be biased.
standard math Soap film smoother appropriately handles the coastline boundary
Used in the soil carbon VCM to prevent leakage across non-adjacent areas; see Wood et al. 2008.

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Cite this review

Pith. "Pith review of Adding structure to generalized additive models, with applications in ecology." pith.science (2026). https://pith.science/paper/N2DO4OXB

@misc{pith2026250807915,
  author       = {Pith},
  title        = {Pith review of: Adding structure to generalized additive models, with applications in ecology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N2DO4OXB}},
  note         = {Machine review of arXiv:2508.07915}
}

abstract

Generalized additive models (GAMs) connecting a set of scalar covariates that map 1-1 to a response are commonly employed in ecology and beyond. However, covariates are often inherently non-scalar, taking multiple values for each observation of the response. They can sometimes have a temporal structure, e.g., a time series of temperatures, or a spatial structure, e.g., multiple soil pH measurements made at nearby locations. While aggregating or selectively summarizing such covariates to yield a scalar covariate allows the use of standard GAM fitting procedures, exactly how to do so can be problematic and information is necessarily lost. Naively including all $p$ components of a vector-valued covariate as $p$ separate covariates, say, without recognizing the structure, can lead to problems of multicollinearity, data sets that are excessively wide given the sample size, and difficulty extracting the primary signal provided by the covariate. Here we introduce three useful extensions to GAMs that efficiently and effectively handle vector-valued covariates without requiring one to choose aggregations or selective summarizations. These extensions are varying-coefficient, scalar-on-function and distributed lag models. While these models have existed for some time they remain relatively underused in ecology. This article aims to show when these models can be useful and how to fit them with the popular R package \mgcv{}.

Figures

Figures reproduced from arXiv: 2508.07915 by the authors.

**Figure 2.** Term plots for our soil organic carbon model. Left: coefficient estimates, [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Data and resulting smooths from our analysis of kittiwake breeding success. The left four plots relate to the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Results from scalar-on-function regression modelling of kittiwakes. Left: kittiwake breeding success trend; [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Temperature time series and arrival dates for the [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Three plots showing different views of the distributed lag effect of temperature and lag for [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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