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arxiv: 2509.24087 · v3 · pith:BRVR4NLXnew · submitted 2025-09-28 · 📊 stat.AP

A penalized distributed lag non-linear Lee-Carter framework for regional weekly mortality forecasting

Jens Robben , Karim Barigou This is my paper

Pith reviewed 2026-05-22 13:03 UTC · model grok-4.3

classification 📊 stat.AP
keywords mortality forecastingLee-Carter modeldistributed lag non-linearregional analysistemperature effectsinfluenzaweekly mortalitypenalized models
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The pith

An extended Lee-Carter model with penalized distributed lag terms for heat, cold and influenza improves accuracy in regional weekly mortality forecasts.

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

The paper develops a forecasting framework that extends the Lee-Carter model to include age- and region-specific seasonal effects along with penalized distributed lag non-linear components that capture the delayed and non-linear impacts of temperature extremes and influenza on mortality. It accommodates overdispersed counts through a negative binomial distribution, models the latent factors with SARIMA processes, and links regions with copulas to reflect shared influences. Tests on French regional data from 1990 to 2019 show the resulting forecasts are well-calibrated and more accurate than benchmark models, while exposing clear differences in how these factors affect mortality risks across ages and places. Accurate short-term mortality predictions support better planning for public health responses and insurance risk management during extreme weather or outbreaks.

Core claim

By adding penalized distributed lag non-linear components to the Lee-Carter model together with age- and region-specific seasonal effects, the framework captures the delayed and non-linear effects of heat, cold, and influenza on mortality. Temporal dynamics of the latent factors are modeled with SARIMA processes and cross-regional dependencies are captured through copulas, while a negative binomial distribution handles overdispersion. Applied to regional French mortality data from 1990-2019, the approach produces well-calibrated forecast distributions, improves predictive accuracy relative to benchmark models, and reveals substantial heterogeneity in temperature- and influenza-related risks.

What carries the argument

Penalized distributed lag non-linear components that recover the delayed and non-linear effects of heat, cold, and influenza on mortality, integrated into the Lee-Carter structure for age- and region-specific modeling.

Load-bearing premise

The chosen penalty tuning and lag structure in the distributed lag non-linear components correctly recover the true delayed and non-linear effects of heat, cold, and influenza without bias from post-hoc selection or overfitting to the 1990-2019 French data.

What would settle it

Generating forecasts for a later period such as 2020-2023 and checking whether the predicted probability distributions match the observed weekly death counts, or whether the estimated temperature-mortality relative risk curves align with independent epidemiological evidence, would test the calibration and the non-linear component.

Figures

Figures reproduced from arXiv: 2509.24087 by Jens Robben, Karim Barigou.

Figure 1
Figure 1. Figure 1: The 12 administrative regions in Metropolitan France. In this paper, we examine the 30-year period from 1990 to 2019, and focus on five-year age groups beginning at age 50 and including an open-ended 95+ age category. Our analysis covers 1The database is available at https://www.insee.fr/fr/information/4769950. 2The matching table is available at https://www.insee.fr/fr/information/7766585 [PITH_FULL_IMAG… view at source ↗
Figure 2
Figure 2. Figure 2: Stacked female death counts (left panel) and exposures (right panel) per week for the twelve French administrative regions over the years 1990-2019. The weekly death counts and expo￾sures per region are aggregated across the different age groups. Next, we analyze the overdispersion in our data set of weekly death counts across different regions and age groups. Compared to annual country-level death statist… view at source ↗
Figure 3
Figure 3. Figure 3: Left panel: the logarithm of the weekly death rates for females in the administrative region of Ile-de-France for 5 different age groups 50-54, 60-64, 70-74, 80-84, and 90-94, across the years 1990-2019. Right panel: the average of the log death rates per week across the years 1990-2019, centered around zero for comparability reasons. 2010). Henceforth, we extract daily average temperatures across a spatia… view at source ↗
Figure 4
Figure 4. Figure 4: Left panel: the daily average temperature on a spatial grid covering metropolitan France at July 20, 2003. Right panel: the weekly average of the daily average temperature values in the 29th ISO week of the year 2003 per administrative region. Since the mortality data in Section 2.2 is defined at the level of administrative regions, we aggregate the daily temperature data into population-weighted regional … view at source ↗
Figure 5
Figure 5. Figure 5: The weekly incidence rates of influenza-like illness in three administrative regions in France: Hauts-de-France (32), Bretagne (53), and Auvergne-Rhˆone-Alpes (84) from 1990-2019. 3 Model specification We develop a weekly, region, and age-specific mortality model designed to (1) capture seasonal variations in weekly mortality rates and (2) quantify the effects of heat waves, cold spells, and influenza outb… view at source ↗
Figure 6
Figure 6. Figure 6: Estimated parameters from the model: ˆαx,r (A), βˆ x (B), ˆκt,r (C), ˆγx (D), λˆw,r (E), ˆδx (F), and ˆϵx (G). Female data, French administrative regions, 5-year age groups 50−54, 55−59, ..., 95+, and years 1990 − 2019. Confidence intervals are based on the Hessian of the penalized negative binomial log-likelihood. The age-specific parameters, with the exception of ˆαx,r, do not vary by region r and are sh… view at source ↗
Figure 7
Figure 7. Figure 7: Estimated female weekly death rates from 1990-2019 for the age groups 70-74 (top) and 90-94 (bottom) in Hauts-de-France (left), Bretagne (middle), and Auvergne-Rhˆone-Alpes (right). The gray line represents the logarithm of the observed death rates, while the blue and red line reflect, respectively, the logarithm of the death rates as estimated by the baseline model and the full model including the DLNM co… view at source ↗
Figure 8
Figure 8. Figure 8: Heat map of the squared Pearson residuals ρ 2 x,t,w,r across age and time for Hauts-de-France (left), Bretagne (middle), and Auvergne-Rhˆone-Alpes (right). Green cells indicate areas with a good fit, while red and black cells correspond to areas with a poor fit (ρ 2 x,t,w,r > χ2 1 ). analysis under the assumption that the weekly death counts follow a Poisson distribution, i.e., without accounting for overd… view at source ↗
Figure 9
Figure 9. Figure 9: 95% uncertainty bounds around the logarithm of the estimated female weekly death rates from 1990-2019 for the age groups 50-54, 70-74 and 90-94 in Hauts-de-France (left), Bretagne (middle), and Auvergne-Rhˆone-Alpes (right). d’Azur. These regions represent a region in the north, west, and south of France, respectively. We provide the plots for the remaining regions in the Supplementary Material. For each r… view at source ↗
Figure 10
Figure 10. Figure 10: Estimated overall relative risk for temperature (top row) and influenza (bottom row) in Hauts-de-France (left), Bretagne (middle), and Provence-Alpes-Cˆote d’Azur (right) for the age groups 50–54, 70–74, and 90–94. Shaded areas represent 95% point-wise confidence intervals [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Estimated relative risk per lag at the 99.5th percentile of the region-specific weekly average temperatures (top row) and ILI exceedance rates (bottom row) in Hauts-de-France (left), Bretagne (middle), and Auverge-Rhˆone-Alpes (right) for the age groups 50–54, 70–74, and 90–94. Shaded areas represent 95% point-wise confidence intervals. mortality risk), and green areas to regions where RR < 1 (decreased m… view at source ↗
Figure 12
Figure 12. Figure 12: Estimated relative risk (RR) surfaces of the lag-specific temperature– and ILI–mortality as￾sociations in Hauts-de-France (left), Bretagne (middle), and Auvergne-Rhˆone-Alpes (right) for the age group 90–94. Red areas indicate an increased RR, while green areas indicate a decreased RR. Proposed model vs. benchmarks. We compare the mortality forecasts with those from several benchmark models, each assuming… view at source ↗
Figure 13
Figure 13. Figure 13: Forecasted female weekly death rates on log-scale according to Model 4 in Hauts-de-France (left), Bretagne (middle), and Auvergne-Rhˆone-Alpes (right) over the period 2015–2019 using simulated exogenous factors. We show the 95% prediction intervals based on 10 000 simulated trajectories of temperature, ILI incidence rates, and the latent mortality index for the age groups 50-54, 70-74, and 90-94. The obse… view at source ↗
Figure 14
Figure 14. Figure 14: Forecasted weekly death rates for women in Hauts-de-France (left), Bretagne (middle), and Auvergne-Rhˆone-Alpes (right) over the period 2020–2024 using the realized exogenous factors. We show the 95% prediction intervals based on 10 000 simulated trajectories of the latent mortality index for the age groups 50-54, 70-74, and 90-94. The observed death rates from 2020-2024 are visualized in black with a ’+’… view at source ↗
read the original abstract

Accurate forecasts of weekly mortality are essential for public health and the insurance industry. We develop a forecasting framework that extends the Lee-Carter model with age- and region-specific seasonal effects and penalized distributed lag non-linear components that capture the delayed and non-linear effects of heat, cold, and influenza on mortality. The model accommodates overdispersed mortality rates via a negative binomial distribution. We model the temporal dynamics of the latent factors in the model using SARIMA processes and capture cross-regional dependencies through a copula-based approach. Using regional French mortality data (1990-2019), we demonstrate that the proposed framework yields well-calibrated forecast distributions and improves predictive accuracy relative to benchmark models. The results further show substantial heterogeneity in temperature- and influenza-related relative risks between ages and regions. These findings underscore the importance of incorporating exogenous drivers and dependence structures into a weekly mortality forecasting framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a penalized distributed lag non-linear extension of the Lee-Carter model for regional weekly mortality forecasting. It adds age- and region-specific seasonal effects, penalized DLNM terms to capture delayed and non-linear impacts of heat, cold, and influenza, a negative binomial likelihood for overdispersion, SARIMA dynamics on the latent factors, and a copula to model cross-regional dependence. Applied to French regional mortality data 1990–2019, the authors claim that the framework produces well-calibrated forecast distributions, improves predictive accuracy relative to benchmark models, and reveals substantial heterogeneity in temperature- and influenza-related relative risks across ages and regions.

Significance. If the central claims hold after proper validation, the work would meaningfully advance applied mortality forecasting by integrating exogenous drivers and spatial dependence into a weekly, regional Lee-Carter framework. The combination of DLNM penalties with copula dependence structures addresses a recognized gap in short-term public-health forecasting.

major comments (2)
  1. [Methods (DLNM specification)] Methods section on the penalized DLNM components: the description of penalty-parameter tuning, lag order, and knot placement does not specify whether selection was performed via nested cross-validation on held-out periods or on the full 1990–2019 dataset. Without this detail the claim that the DLNM terms recover unbiased delayed and non-linear effects of temperature and influenza cannot be evaluated, directly threatening the reported forecast calibration and accuracy gains.
  2. [Results (forecast evaluation)] Results and validation: the abstract asserts well-calibrated forecast distributions and accuracy improvements, yet the manuscript provides no explicit hold-out period, rolling-window scheme, or proper scoring-rule comparison that isolates the contribution of the DLNM and copula terms from the high-dimensional regional/age structure. This omission leaves open the possibility that reported gains are inflated by overfitting.
minor comments (1)
  1. [Abstract] The abstract would benefit from a concise quantitative statement of the accuracy improvement (e.g., percentage reduction in CRPS or log-score relative to the best benchmark).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which highlight important aspects of methodological transparency and validation rigor. We address each major comment below and commit to revisions that strengthen the manuscript without altering its core contributions.

read point-by-point responses

  1. Referee: Methods section on the penalized DLNM components: the description of penalty-parameter tuning, lag order, and knot placement does not specify whether selection was performed via nested cross-validation on held-out periods or on the full 1990–2019 dataset. Without this detail the claim that the DLNM terms recover unbiased delayed and non-linear effects of temperature and influenza cannot be evaluated, directly threatening the reported forecast calibration and accuracy gains.

    Authors: We agree that explicit documentation of the hyperparameter selection process is essential for evaluating the DLNM effects. In the original analysis, penalty parameters, lag orders, and knot placements were determined via grid search and cross-validation on the full 1990–2019 dataset, following standard DLNM practice for stable effect estimation. To address the referee's concern, we will revise the Methods section to describe this procedure in detail, discuss its implications for potential optimism bias in effect recovery, and add a sensitivity analysis using time-series nested cross-validation on held-out periods. This will allow readers to better assess the unbiasedness of the reported temperature and influenza effects. revision: yes

  2. Referee: Results and validation: the abstract asserts well-calibrated forecast distributions and accuracy improvements, yet the manuscript provides no explicit hold-out period, rolling-window scheme, or proper scoring-rule comparison that isolates the contribution of the DLNM and copula terms from the high-dimensional regional/age structure. This omission leaves open the possibility that reported gains are inflated by overfitting.

    Authors: We acknowledge that the validation procedures require clearer exposition to rule out overfitting concerns. The manuscript employs a rolling-window forecasting evaluation with held-out periods in the later years of the sample, but the description was not sufficiently detailed. In the revision, we will expand the Results section to explicitly document the hold-out scheme, rolling-window implementation, and use of proper scoring rules such as the Continuous Ranked Probability Score (CRPS) and logarithmic score. We will also include ablation experiments comparing the full model to reduced versions without the DLNM components and without the copula, thereby isolating their specific contributions beyond the regional/age structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the modeling framework

full rationale

The paper constructs an extended Lee-Carter model by adding age- and region-specific seasonal effects, penalized distributed lag non-linear (DLNM) terms for temperature and influenza, negative binomial overdispersion, SARIMA dynamics on latent factors, and a copula for cross-regional dependence. These components are introduced as independent modeling choices drawn from established statistical literature rather than derived from the target forecasts or fitted parameters. The empirical demonstration on 1990-2019 French regional data then evaluates predictive accuracy and heterogeneity against benchmarks; no equation reduces a claimed prediction or uniqueness result to a quantity defined solely by the same fitted values or by a self-citation chain. The derivation chain therefore remains self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The central claim rests on several standard statistical modeling choices whose validity is assumed rather than derived from first principles within the paper.

free parameters (2)
  • penalty parameters for DLNM
    Tuning parameters that control smoothness of the distributed lag non-linear effects; chosen or cross-validated on the data.
  • lag order and knot placement
    Structural choices in the distributed lag non-linear components that affect how delayed effects are represented.
axioms (3)
  • domain assumption Mortality counts are adequately described by a negative binomial distribution
    Invoked to accommodate overdispersion in weekly regional counts.
  • domain assumption Latent temporal factors follow SARIMA processes
    Used to model the dynamics of the age- and region-specific factors.
  • domain assumption Cross-regional dependence can be captured by a copula
    Assumed to handle joint forecast distributions across regions.

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