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arxiv: 2601.05841 · v2 · pith:QWRGCQ36new · submitted 2026-01-09 · ⚛️ physics.ao-ph

Non-stationary time series attribution for heatwaves over Europe

Pascal Meurer , Sebastian Buschow , Svenja Szemkus , Petra Friederichs This is my paper

Pith reviewed 2026-05-25 07:34 UTC · model grok-4.3

classification ⚛️ physics.ao-ph
keywords heatwave attributionnon-stationary Markov processbivariate extreme value theoryextremal pattern indexCMIP6 simulationsERA5 reanalysisanthropogenic forcingtemporal dependence
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The pith

A non-stationary Markov model attributes observed European heatwave time series to anthropogenic forcing with high confidence.

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

The paper develops an attribution method that evaluates complete sequences of extreme heat days rather than isolated events. It fits non-stationary Markov chains informed by bivariate extreme value theory to capture how extremes cluster in time, condenses the spatial fields into an extremal pattern index, and computes likelihood ratios between ERA5 observations and two sets of CMIP6 runs. The ratios strongly favor the simulations that include anthropogenic greenhouse gases, with the signal already detectable in central and southern Europe by the 1970s and robust even after the overall mean temperature rise is removed. The work matters because most prior attribution studies overlook temporal dependence, so this approach tests whether forcing alters not only the average but also the sequencing and clustering of extremes.

Core claim

The likelihood ratio of the ERA5 observational time series, given the distributions estimated from CMIP6 historical natural-only and natural-plus-anthropogenic simulations via a non-stationary Markov process and the extremal pattern index, supplies very strong evidence for the anthropogenic scenario over Europe since the beginning of the 21st century; for central and southern Europe the influence of anthropogenic greenhouse gas emissions on heatwaves is already detectable in the 1970s, while no reliable signal appears beyond the general temperature increase in either the temporal dependence of extreme heat days or the shape of the extreme value distribution.

What carries the argument

The extremal pattern index (EPI) condenses spatial fields of extremes into a single compact time series; this series is then modeled by a non-stationary Markov process built on bivariate extreme value theory, allowing direct computation of the likelihood ratio between the two CMIP6 forcing ensembles.

If this is right

  • Attribution remains strong even after the mean warming trend is removed from the data.
  • For central and southern Europe the anthropogenic greenhouse-gas signal on heatwaves is already present in the 1970s.
  • No additional signal beyond the overall temperature rise is found in the temporal clustering of extremes or in the shape of their distribution.
  • The same time-series likelihood framework can be applied to other extreme-event types that exhibit clustering.

Where Pith is reading between the lines

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

  • The method could be tested on other mid-latitude regions where heatwave clustering is pronounced to see whether the same early detection holds.
  • If the CMIP6 natural-only runs systematically underestimate the variance of real-world natural variability, the reported likelihood ratios would overstate the strength of the anthropogenic signal.
  • The absence of change in the shape of the extreme-value distribution suggests that future attribution work may need to focus on frequency and duration rather than on shifts in tail heaviness.

Load-bearing premise

The probability distributions fitted to the CMIP6 simulations accurately represent the true distributions of the extremal pattern index under natural-only and natural-plus-anthropogenic conditions.

What would settle it

A direct calculation in which the likelihood of the ERA5 time series under the natural-only CMIP6 distributions equals or exceeds its likelihood under the anthropogenic distributions would falsify the attribution result.

Figures

Figures reproduced from arXiv: 2601.05841 by Pascal Meurer, Petra Friederichs, Sebastian Buschow, Svenja Szemkus.

Figure 1
Figure 1. Figure 1: AR6 regions (Iturbide et al., 2020) of (a) northern, (b) central, and (c) southern Europe as used in this study with North Africa excluded and restricted to land points only. A grid cell is considered as land point when at least 50% of the cell is occupied by land. 2.2 Extremal pattern index (EPI) Our compact representation of a spatially extended multivariate weather event uses the EPI as suggested in Sze… view at source ↗
Figure 2
Figure 2. Figure 2: Mean T2max anomalies in ERA5 between (a) 07 June to 5 July 2025, (b) 15 July to 01 August 2025, and (c) 07 August to 19 August 2025. Only grid points exceeding the 99%-quantile are shown. In (d) the EPI for T2max from June to August 2025 for northern (blue), central (orange) and southern (green) European regions from [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Regions of the censored likelihood model according to Eq. (12), whereby the likelihood contribution depends on which variable is exceeding the threshold or not. Often, l ⋆ is modelled using a parametric family. A common choice is the logistic model with l ⋆ (v1,v2) =  v 1/α 1 + v 1/α 2 α , vj ≥ 0. (11) The dependence parameter 0 < α ≤ 1 represents independence if α = 1 and complete dependence if α → 0. I… view at source ↗
Figure 4
Figure 4. Figure 4: Estimated coefficients for the model with constant threshold (Sect. 4.1) in the southern European region. The coefficients (x-axis) are shown for (a) the threshold exceedance ϕ, (b) the scale σ, (c) the shape ξ, and (d) the dependence α. The estimates for the different climate models are represented as box-whiskers, those for ERA5 as green crosses. The whiskers and fliers cover the whole range of data. and… view at source ↗
Figure 5
Figure 5. Figure 5: Temporal evolution of (a) threshold exceedance probability, (b) scale, (c) shape, and (d) dependence parameters for the model with constant threshold in the southern European region. Mean (solid line) and standard deviation (shading) for the different climate models is plotted based on the estimates in [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Same as [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Logarithmic likelihood ratio including ± standard deviation (y-axis left) for the northern European region, estimated according to Sect. 4.6. Green crosses indicate the number of threshold exceedances of the ERA5 EPI over the 95%-quantile in the respective summer (y-axis right). for a single northern European summer time series. All events are documented as extreme heat events. The likelihood ratio of the … view at source ↗
Figure 9
Figure 9. Figure 9: EPI of ERA5 and the logarithmic likelihood ratio for the summers 2022 and 2025 in the northern European region and the constant threshold model. The estimate shown in [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Same as [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Same as [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Accumulative likelihood ratio for all summers since 1940 up to a given year n (x-axis). This Figure is based on Eq. (7), when assuming that the prior probability of both scenarios are equal. 6 Discussion and conclusion The Markov process model, which is based on bivariate extreme value theory, has been shown to be suitable for modelling the temporal dependence of the EPI, and thus deriving likelihood of t… view at source ↗
read the original abstract

The increasing occurrence of extreme weather events since the beginning of the 21st century has led to the development of new methods to attribute extreme events to anthropogenic climate change. The way in which the extreme event is defined has a major influence on the attribution result. A frequently disregarded or overlooked aspect concerns the temporal dependence and the clustering of extremes. This study presents an approach for attributing complete time series during extreme events to anthropogenic forcing. The approach is based on a non-stationary Markov process using bivariate extreme value theory to model the temporal dependence of the time series. We calculate the likelihood ratio of an observational time series from ERA5 given the distributions as estimated from CMIP6 simulations with historical natural-only and natural and anthropogenic forcing scenarios. The spatial fields are condensed by the extremal pattern index (EPI) as a compact description of spatial extremes. In addition, the study examines the extent to which attribution statements about the occurrence of extreme heat events change when the effect of the mean warming is eliminated. The resulting attribution statement provides very strong evidence for the scenario with anthropogenic drivers over Europe, especially since the beginning of the 21st century. For central and southern Europe, the influence of anthropogenic greenhouse gas emissions on heatwaves could already have been proven in the 1970s using today's methods. There is no reliable signal apart from a general increase in temperature, neither in terms of the temporal dependence of extreme heat days nor in terms of the shape of the extreme value distribution.

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 / 2 minor

Summary. The paper presents a non-stationary Markov process based on bivariate extreme value theory to attribute complete time series of heatwaves (via the extremal pattern index EPI) over Europe. It computes the likelihood ratio of the ERA5 observational series against distributions fitted from CMIP6 historical natural-only versus natural-plus-anthropogenic ensembles, concluding very strong evidence for anthropogenic forcing (detectable since the 1970s in central/southern Europe) with no residual signal beyond mean warming in temporal dependence or extreme-value shape.

Significance. If the CMIP6-fitted distributions are faithful representations of the true conditional distributions of the EPI process, the method would offer a useful extension of attribution techniques that explicitly incorporates temporal clustering and non-stationarity. The early-detection claim and the finding of no additional signal beyond mean warming would be of broad interest to the detection-and-attribution community.

major comments (2)
  1. [Likelihood ratio computation] The attribution statement rests on the premise that the probability distributions estimated from the CMIP6 hist-nat and hist ensembles accurately represent the true distributions of the EPI under each forcing (abstract; § on likelihood ratio). No validation against held-out data, cross-validation, or sensitivity tests to model bias in tail shape or Markov dependence parameters is reported; any systematic discrepancy directly biases the likelihood ratio and therefore the central claim.
  2. [Results on mean-warming removal] The additional test that removes mean warming and reports no residual signal in temporal dependence or extreme-value shape (abstract) inherits the identical requirement that the CMIP6-fitted distributions faithfully capture the conditional process; without evidence on this point the conclusion that there is 'no reliable signal apart from a general increase in temperature' cannot be assessed.
minor comments (2)
  1. [Abstract] The abstract states 'very strong evidence' but supplies no numerical likelihood-ratio values, confidence intervals, or error bars; these should be reported in the main text and abstract.
  2. [Methods] Notation for the non-stationary Markov process and the bivariate extreme-value dependence function should be defined explicitly with equation numbers in the methods section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. We address the major concerns regarding the validation of the CMIP6-fitted distributions below and outline planned revisions to strengthen the paper.

read point-by-point responses

  1. Referee: [Likelihood ratio computation] The attribution statement rests on the premise that the probability distributions estimated from the CMIP6 hist-nat and hist ensembles accurately represent the true distributions of the EPI under each forcing (abstract; § on likelihood ratio). No validation against held-out data, cross-validation, or sensitivity tests to model bias in tail shape or Markov dependence parameters is reported; any systematic discrepancy directly biases the likelihood ratio and therefore the central claim.

    Authors: We recognize that the attribution results depend on the assumption that the CMIP6 ensembles provide faithful representations of the EPI distributions under the respective forcings. The manuscript does not include explicit held-out validation or cross-validation, as the full ensembles are used to estimate the parameters. However, the use of multi-model ensembles is standard in detection and attribution studies to account for model uncertainty. To address this, we will add sensitivity analyses to variations in tail shape and Markov dependence parameters in the revised version, including tests using subsets of models. This addition will be included as a new paragraph in the methods section and discussed in the results. revision: partial

  2. Referee: [Results on mean-warming removal] The additional test that removes mean warming and reports no residual signal in temporal dependence or extreme-value shape (abstract) inherits the identical requirement that the CMIP6-fitted distributions faithfully capture the conditional process; without evidence on this point the conclusion that there is 'no reliable signal apart from a general increase in temperature' cannot be assessed.

    Authors: The mean-warming removal analysis is indeed subject to the same modeling assumptions. We will revise the manuscript to explicitly state that this conclusion is conditional on the accuracy of the fitted distributions and incorporate the sensitivity tests described in response to the first comment. This will help assess the robustness of the finding that no residual signal exists beyond mean warming. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent CMIP6 fits

full rationale

The paper estimates distributions from separate CMIP6 historical natural-only and natural-plus-anthropogenic ensembles, then forms the likelihood ratio of the ERA5 EPI series against those fitted distributions. This step does not reduce by the paper's own equations to any parameter fitted directly to the ERA5 observations themselves. No self-citation load-bearing steps, self-definitional constructions, or fitted-input-called-prediction patterns appear in the derivation chain. The central attribution result therefore remains independent of the target data and is self-contained against the external simulation benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only information limits the ledger to domain assumptions required for the likelihood comparison; no free parameters or invented entities are explicitly introduced in the provided text.

axioms (2)
  • domain assumption CMIP6 historical natural-only and natural-plus-anthropogenic ensembles furnish unbiased estimates of the true distributions of the extremal pattern index under each forcing scenario
    Invoked when the likelihood ratio of the ERA5 series is evaluated against the two sets of model-derived distributions.
  • domain assumption The extremal pattern index condenses spatial extreme fields without discarding information relevant to the attribution question
    Used to reduce the spatial fields to a univariate time series for the Markov model.

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