Journal cover Journal topic
Earth System Dynamics An interactive open-access journal of the European Geosciences Union
Journal topic
Earth Syst. Dynam., 10, 31–43, 2019
https://doi.org/10.5194/esd-10-31-2019
Earth Syst. Dynam., 10, 31–43, 2019
https://doi.org/10.5194/esd-10-31-2019

Research article 07 Jan 2019

Research article | 07 Jan 2019

# The effect of univariate bias adjustment on multivariate hazard estimates

The effect of univariate bias adjustment on multivariate hazard estimates
Jakob Zscheischler1,2,3, Erich M. Fischer1, and Stefan Lange4 Jakob Zscheischler et al.
• 1Institute for Atmospheric and Climate Science, ETH Zurich, Universitaetstrasse 16, 8092 Zurich, Switzerland
• 2Climate and Environmental Physics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
• 3Oeschger Centre for Climate Change Research, University of Bern, 3012 Bern, Switzerland
• 4Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, P.O. Box 60 12 03, 14412 Potsdam, Germany

Correspondence: Jakob Zscheischler (jakob.zscheischler@climate.unibe.ch)

Abstract

1 Introduction

With ongoing climate change, climate impact modeling has become an important pillar of climate research, informing decision makers and risk managers in many sectors that are affected by climate variability. Impact models such as hydrological models, crop models and epidemiological models usually rely on absolute thresholds in their driving climate variables such as temperature, precipitation, wind speed and humidity ; and thus require unbiased climate information as input. However, biases continue to persist in global and regional climate models , rendering bias adjustment an undesired but often unavoidable data processing step for climate impact modeling .

The need for easily available information on climate impacts has led to a sometimes overly uncritical use of bias adjustment in large impact modeling projects, even though its current usage might, in some cases, result in ill-informed adaptation decisions . It is often assumed that biases are stationary, and hence that bias adjustment can be developed in the current climate and applied in a warmer world. It is not clear, however, for which variable and to what extent this assumption is met. Moreover, while uncertainties related to the choice of climate model and impact model as well as future socioeconomic scenarios are frequently carried through the modeling chain , uncertainties associated with the choice of the applied bias-adjustment method are rarely reported , though there are exceptions .

Assessing the appropriateness of bias adjustment for impact modeling is not an easy task . While the quality of bias adjustment on climate variables can be evaluated against observed quantities in the climate domain, the implications for modeled impacts are much harder to determine. On the one hand, limited length and a lack of homogeneity in observed climate impacts render an evaluation of modeled impacts very challenging . On the other hand, many impacts rely on the complex interaction of multiple climate variables across time and space , preventing an evaluation in the climate domain. For instance, temperature and precipitation variability affect crop yields and other ecosystem services such as net carbon uptake . Flood occurrence, its intensity and associated damages depend on the temporal and spatial characteristics of precipitation, soil moisture, river flow and surge . Drought-related impacts depend on precipitation, evapotranspiration, and temperature; their spatial distribution; and their interaction with human activities . Fire occurrence, and strength, not only requires available fuel and an ignition source, but is also strongly dependent on relative humidity and temperature . Infrastructural damage is particularly large when strong winds and extreme precipitation occur jointly . Finally, climate-related human-health impacts are linked to a suite of climatic drivers . It is currently largely unknown how widely used statistical bias-adjustment approaches affect modeled impacts that depend on multiple drivers such as the ones mentioned above.

In this paper we study the limitations of univariate bias-adjustment methods for multivariate impacts. We focus on global-scale climate outputs from global circulation models (GCMs), which are frequently used for global impact studies . However, our analyses are of a rather conceptual nature and thus also apply to other climate model outputs, for instance from regional climate models. First, we investigate whether traditional univariate bias-adjustment methods generally reduce biases in impacts that depend on multiple dependent drivers. Bias adjustment may also amplify uncertainties inherent to the observations . Hence, using a model environment with multiple models and multiple runs for a single model, we further estimate how uncertainties related to internal variability may be propagated and amplified through incomplete bias adjustment. We use multivariate hazard indicators as proxies for actual impacts to evaluate the effect of bias adjustment on impacts and to overcome the challenge of missing impact data. It can be assumed that “real” impacts are in many cases more complex and also depend on more driving variables. Therefore, our analysis provides a rather conservative estimate of the potential effects of univariate bias adjustment on modeled impacts.

2 Data and methods

## 2.1 Data

Observations. We use daily temperature and relative humidity from the ERA-Interim reanalysis as main reference dataset. We further use daily temperature and relative humidity from the observational dataset EWEMBI (Lange2016, 2018), which has been used in the Inter-Sectoral Impact Model Intercomparison Project phase 2b (ISIMIP2b; ).

Model simulations. We use daily model output from the historical runs of the Coupled Model Intercomparison Project Phase 5 (CMIP5; ). A main objective of the CMIP5 model experiments is to improve our understanding of the climate system, and to provide estimates of future climate change. All 29 model simulations used in this study are listed in Table A1.

All data have been bilinearly interpolated to a 2.5 by 2.5 regular latitude–longitude grid prior to analysis. The studied time period is 1981–1995. Our analysis is based on daily values during the hottest month of the year at each grid point, which results in about 450 samples. We focus on the hottest month to avoid dealing with seasonality and because, arguably, fire risk and heat stress are most relevant during this time period. The hottest month was identified based on the climatology of ERA-Interim monthly temperature data. Although the effectiveness of bias adjustment is typically evaluated outside the calibration period (Maraun2013), here we focus our analysis completely on the selected 15-year period. This allows for a separate assessment of the effect of bias adjustment on univariate versus multivariate impacts independently from other effects such as cross-validation error. A good performance in the calibration period is a necessary requirement for an effective bias-adjustment approach. Furthermore, cross-validation might not help to diagnose whether a bias-adjustment approach is effective .

## 2.2 Methods

Hazard indicators. We use the Wet Bulb Globe Temperature (WBGT; ) as an indicator for heat stress and the Chandler Burning Index (CBI; ) as an indicator for fire risk. Both are relatively simple indicators that can be computed solely from daily temperature and relative humidity. WBGT and its variants have been extensively used to assess projections of heat stress under climate change . CBI is one of many fire risk indicators , mainly chosen here for its simplicity. The hazard intensities of WBGT and CBI vary along different gradients in the temperature–humidity domain . WBGT increases with hotter and more humid conditions and is equally dependent on temperature and humidity (Fig. 1). In contrast, CBI increases with hotter and drier conditions and its variability is mostly driven by humidity and much less by temperature. Using two hazard indicators that depend differently on the same climatic drivers enables us to study how the relationship between hazard direction and driver distribution changes the way bias adjustment affects modeled hazards . While certainly more sophisticated indicators exist, both for fire risk and heat stress , our goal here is rather to provide an illustrative example of the issues associated with bias adjustment and impact modeling than to provide the most reliable hazard projections. We estimate WBGT following the approach outlined in the supplement of . CBI can be computed as

$\begin{array}{ll}& \mathrm{CBI}=\\ \text{(1)}& & \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(\left(\left(\mathrm{110}-\mathrm{1.373}\phantom{\rule{0.125em}{0ex}}\mathrm{RH}\right)-\mathrm{0.54}\left(\mathrm{10.2}-T\right)\right)\mathrm{124}\cdot {\mathrm{10}}^{-\mathrm{0.0142}\phantom{\rule{0.125em}{0ex}}\mathrm{RH}}\right)/\mathrm{60},\end{array}$

where RH is relative humidity in % and T is temperature in degrees Celsius.

Figure 1Distribution of daily temperature versus relative humidity during July (1981–1995) for a grid point in central Africa. ERA-Interim (a); model simulations from the model IPSL-CM5A-MR (b); bias-adjusted with UQM (c); bias-adjusted with MBCn (d). Pink lines depict levels of equal heat stress (WBGT). Violet lines depict levels of equal fire risk (CBI). Red points denote values for which WBGT exceeds its 95th percentile.

Bias adjustment. We employ two different bias-adjustment methods, the widely used univariate empirical quantile mapping (UQM; ) and the multivariate bias adjustment in n dimensions (MBCn) developed by . UQM applies separate corrections to a fixed number of quantiles to adjust a modeled empirical cumulative distribution towards the observed empirical cumulative distribution. Hence, if Xo and Xm are the observed and modeled values, respectively, then

$\begin{array}{}\text{(2)}& {\stackrel{\mathrm{^}}{X}}_{\mathrm{m}}={F}_{\mathrm{o}}^{-\mathrm{1}}\left({F}_{\mathrm{m}}\left({X}_{\mathrm{m}}\right)\right),\end{array}$

where Fm is the empirical cumulative distribution function of Xm and ${F}_{\mathrm{o}}^{-\mathrm{1}}$ is the inverse empirical distribution function (or quantile function) corresponding to Xo. Values between the predefined quantiles are approximated using linear interpolation. We apply UQM with the R package qmap using 100 quantiles.

MBCn is a bias-adjustment method in which both the marginal distribution of each individual variable and the multivariate dependence structure are corrected at the same time (Cannon2018). This is achieved by an iterative approach, which first applies a random rotation R[j] to the multivariate observed and modeled data distribution

$\begin{array}{}\text{(3)}& & {\stackrel{\mathrm{̃}}{X}}_{\mathrm{m}}^{\left[j\right]}={X}_{\mathrm{m}}^{\left[j\right]}{R}^{\left[j\right]},\text{(4)}& & {\stackrel{\mathrm{̃}}{X}}_{\mathrm{o}}^{\left[j\right]}={X}_{\mathrm{o}}^{\left[j\right]}{R}^{\left[j\right]}.\end{array}$

Subsequently, quantile mapping (Eq. 2) is applied to the rotated data ${\stackrel{\mathrm{̃}}{X}}_{\mathrm{m}}^{\left[j\right]}$ with ${\stackrel{\mathrm{̃}}{X}}_{\mathrm{o}}^{\left[j\right]}$ as a reference, yielding ${\stackrel{\mathrm{^}}{X}}_{\mathrm{m}}^{\left[j\right]}$. Then the inverse rotation is applied

$\begin{array}{}\text{(5)}& {X}_{\mathrm{m}}^{\left[j+\mathrm{1}\right]}={\stackrel{\mathrm{^}}{X}}_{\mathrm{m}}^{\left[j\right]}{{R}^{\left[j\right]}}^{-\mathrm{1}}.\end{array}$

The observed data are carried forward to the next iteration unchanged ${X}_{\mathrm{o}}^{\left[j+\mathrm{1}\right]}={X}_{\mathrm{o}}^{\left[j\right]}$. These steps are repeated until the modeled data distribution has converged to the observed distribution. One may interpret the algorithm such that the random rotations allow an information exchange between the different dimensions.

Finally, we use the bias-adjustment method (Lange2017) applied in ISIMIP2b . This method adjusts relative humidity with parametric quantile mapping using beta distributions to model simulated and observed daily values , and temperature with an additive correction of monthly mean temperature and a multiplicative correction of daily anomalies from the monthly mean temperature .

We adjust daily values of temperature and relative humidity in CMIP5 during the hottest month at each grid point and evaluate the change in bias for WBGT and CBI.

Figure 2Quantile–quantile plot of observed versus modeled values for temperature (a), relative humidity (b), heat stress (WBGT, c) and fire risk (CBI, d) for a grid point in central Africa; original model output (from IPSL-CM5A-MR) in red, model output corrected with UQM in blue, model output corrected with MBCn in green. The vertical dashed line in (c) shows the 90th percentile of observed WBGT. The same data as in Fig. 1 were used. Note that in panels (a, b, d) the green circles cover most of the blue circles, as in these cases both bias adjustments yield virtually identical results.

Perfect model approach. Internal variability can lead to uncertainties in the bias adjustment, which may be amplified through an inadequate choice of bias adjustment. Since fully coupled ocean–atmosphere models provide different realizations of unforced internal variability, observations and models as well as different simulations of the same model are not expected to agree on a year-by-year basis. Even when using a time period of 2–3 decades, this constitutes a substantial source of uncertainty . Using the multimodel environment of CMIP5, we study whether UQM increases uncertainties related to internal variability for multivariate hazard indicators. We conduct a perfect model approach to separate uncertainties associated with internal variability and the choice of bias adjustment . To this end we use the five available initial-condition members of the model CanESM to estimate the influence of internal variability as a source of uncertainty. First, we bias-adjust CanESM runs against each other to estimate the range of the uncertainty due to internal variability (“noise”). Bias adjusting all other model runs against all five CanESM runs then provides an estimate for the full uncertainty range (“full range”). Comparing the range of the noise with the full range, we estimate whether univariate bias adjustment amplifies uncertainties related to internal variability.

Figure 3Fraction of grid points for which UQM increases bias (a) or does not result in a reduction of more than 50 % in biases (b) of modeled hazards. Box plots represent the CMIP5 multimodel ensemble after UQM and highlight the median (horizontal line), interquartile range (box), 1.5 times the interquartile range (whiskers) and outliers (points). Colored dots represent the results for the ISIMIP2b bias adjustment applied to the four GCMs used in ISIMIP2b. Shown are the metrics RMSE between empirical cumulative distribution functions and absolute differences in the 90th (Δq90) and 95th percentile (Δq95) between hazard indicators computed from (bias-adjusted) model outputs and observations (ERA-Interim for CMIP5 and EWEMBI for ISIMIP2b).

3 Results

Statistical bias adjustment that is separately applied to each marginal distribution of a multivariate distribution such as UQM ensures that the bias-adjusted modeled marginal distributions are well aligned with the observed marginal distributions. We illustrate the effect of bias adjustment for one selected grid point in central Africa. UQM “squeezes” and “stretches” a modeled multivariate distribution along the marginal axes to match the observations of the marginals (Fig. 1c). For hazards that are a function of multiple drivers and that vary along a diagonal gradient such as WBGT, UQM may not be able to reduce biases. In fact, for many percentiles, it can even increase biases as illustrated in Fig. 2c (blue dots). CBI seems to be less affected, likely because its variability is mostly driven by a single variable, namely relative humidity (Figs. 12).

Bias adjustment can also affect the relationship between a multivariate hazard indicator and its individual contributing climate variables. For instance, looking at the raw model output of IPSL-CM5A-MR in Fig. 1b, one would infer that high heat stress (WBGT values exceeding their 95th percentile, highlighted in red) is reached at rather low relative humidity. Instead, for ERA-Interim data for the same grid point, high heat stress is associated with high relative humidity (Fig. 1a). UQM improves this mismatch to some extent but not entirely (Fig. 1c). Using MBCn for bias adjustment we would infer the correct contribution of individual variables. This is an important aspect, which needs to be taken into account when applying bias adjustment because it may lead to incorrect conclusions about which climate drivers are most relevant for extreme hazards and impacts. Consequently, we might also focus on the wrong aspects to improve in numerical climate models.

Figure 4(a) Fraction of grid points for which biases in WBGT are larger than 1 K before or after the application of bias adjustment. (b–c) As in Fig. 3 only for WBGT, based on the subset of grid points identified by (a).

Figure 5Fraction of models for which UQM increases biases (a) or does not decrease biases by more than 50 % in the 90th percentile of WBGT (b).

Regions for which biases only slightly increase or decrease include locations where biases are small to start with. To study for how many grid points this is the case, we compute the fraction of grid points for which the bias in WBGT is larger than 1 K either before or after bias adjustment. This is the case for 50 %–90 % of the grid points, depending on the model and the metric (Fig. 4a). Recomputing Fig. 3 based on this subset reduces the fraction of locations where bias adjustment does not achieve the two benchmarks by about half (Fig. 4b, c). Because these numbers strongly depend on the size of the accepted bias (1 K in our example), we continue the analysis with all grid points.

In Australia, the Sahel, and some parts in sub-Saharan Africa and South America, UQM increases biases in the 90th percentile of WBGT for a large fraction of models (Fig. 5a). In those regions, but also some other areas in the world, UQM reduces WBGT biases in the 90th percentile by less than 50 % in the majority of all models (Fig. 5b). Overall, in nearly 11 % of the land area, for more than 50 % of the models, UQM does not reduce the bias of the 90th percentile of WBGT by more than 50 %. MBCn, on the other hand, is able to fit the hazard estimates well where UQM fails (not shown for all grid points but Figs. 1d and 2c illustrate the worst case).

To improve the usage of bias-adjustment methods it would be important to know whether we can make any a priori statements as to whether UQM adjustment will lead to a reduction in biases of modeled multivariate hazards or impacts. Generally, regions where UQM fails to improve biases in WBGT coincide with regions where the observed correlations between temperature and relative humidity is outside the CMIP5 range (Fig. 6a). UQM fails when (univariate) mean biases in temperature are small (Fig. 6b), and when the correlation between the driving variables (in this case temperature and relative humidity) are not well captured (Fig. 6d). The mean bias in relative humidity looks similar for both cases (Fig. 6c). Overall this means that if mean biases in climate drivers are large, any bias adjustment will lead to a substantial reduction in biases of hazard or impact indicators (consistent with Fig. 4).

Figure 6Reasons why UQM may fail. (a) Regions in which the correlation between temperature and relative humidity in ERA-Interim is outside the CMIP5 model range; (b–d) conditions where UQM does not lead to a substantial reduction in biases in WBGT (measured as difference in empirical cumulative distribution function RMSE between model output and ERA-Interim before and after bias adjustment). Shown is the mean bias in temperature (T, b) and relative humidity (RH, c) as well as the mean difference in correlation over all grid cells between temperature and relative humidity (d). Blue (red) represents the distribution across models for cases in which UQM does (does not) lead to a reduction in biases of WBGT by at least 50 %.

Figure 7Illustration of the perfect model approach to study the effect of internal variability and UQM on the uncertainty range across models. Shown is the grid point of Austin, Texas, US. UQM was separately applied to temperature (T) and relative humidity (RH), using multiple runs (five) of CanESM as observations. CanESM runs bias-adjusted against themselves represent the noise associated with internal variability (blue). The full range represents the range of all model simulations bias-adjusted against all CanESM runs (red).

Internal variability can impact the effectiveness of bias adjustment because it introduces uncertainties that may be amplified by incomplete bias adjustment. For temperature and relative humidity individually, the perfect model approach reveals little difference between the range of the noise (uncertainty associated with internal variability) and the uncertainty range of all models (Fig. 7 illustrates the approach for the grid point of Austin, Texas, US). This is to be expected, as these driver variables can be perfectly adjusted. In the case of WBGT and CBI, however, UQM leads to much larger uncertainties for the full range than for the noise for some percentiles (Fig. 7). Overall, the full range of bias-adjusted model simulations for the 90th percentile of WBGT is a factor of 10 larger than the range expected from internal variability alone in many regions, including the Amazon, eastern North America and Indonesia (Fig. 8a). The regions with large increases in uncertainty roughly coincide with regions where the between-model variability in the correlation between temperature and relative humidity is very high compared to the variability within the CanESM runs (Fig. 8b). This is consistent with Fig. 6d, which shows that univariate bias adjustment is not very successful when the correlation structure between models and reference is not well matched. For a fixed percentile threshold, the fraction of grid points for which the full range is at least 2 times the range of the noise varies between 10 % and 40 % for CBI, and between 30 % and 70 % for WBGT (Fig. 9). If we use MBCn to adjust biases, the uncertainty associated with internal variability is largely indistinguishable from the full uncertainty range (Fig. 9, dashed lines).

Figure 8(a) Ratio between full range and noise (uncertainty associated with internal variability) at the 90th percentile of heat stress (WBGT) based on a perfect model approach using UQM to adjust biases; (b) ratio between the range of all models and the range of all five CanESM runs of correlation between temperature and relative humidity.

Figure 9Fraction of grid points for which the full range is at least twice the range of the noise (uncertainty associated with internal variability) in the perfect model approach. Solid lines represent cases computed with UQM; dashed lines represent cases computed with MBCn.

4 Discussion

We find that UQM of temperature and relative humidity often does not lead to a substantial reduction in WBGT biases. In a sizable number of cases UQM even leads to an increase in the original biases, particularly for high percentiles that are potentially most impact relevant. The fire risk indicator CBI is less affected by these issues because, although temperature is required for calculating it, it is largely dominated by variations in relative humidity. Our findings on the limited effectiveness of univariate bias adjustment are admittedly based on one bias-adjustment approach and two hazard indicators. Nevertheless, we expect that our results also hold for other multivariate hazards or impacts whose drivers are adjusted with bias-adjustment approaches that do not explicitly correct for dependencies between variables.

We show that for nearly 40 % of the land area, UQM fails to reduce biases in the 90th percentile of WBGT by at least 50 % in more than half of the models. The challenges in correcting high percentiles of WBGT suggest that a direct application of UQM to a warmer climate may lead to large errors. Multivariate bias adjustment such as MBCn offer remedies, though for a generic impact modeling project such as ISIMIP, all variables and dependencies would need to be corrected at once, requiring large amounts of good climate observations to fill the high-dimensional data space (Cannon2018). Furthermore, observational datasets would need to be carefully tested as to whether they represent the desired dependencies correctly .

The perfect model approach demonstrates that uncertainties related to internal variability and the use of an incomplete bias adjustment can lead to substantial uncertainty in multivariate hazards, particularly for more extreme percentiles. The large increase in uncertainty illustrated by the full range stems from a combination of initial uncertainty related to internal variability and the inability of UQM to adjust dependencies adequately. Therefore, uncertainties related to internal variability or other types of uncertainties may be strongly increased by incomplete bias adjustment. These uncertainties are likely sensitive to the length of the time period, as longer time periods will reduce the noise component related to internal variability due to a better sampling of the tails. These types of uncertainty are typically not communicated and accounted for in impact modeling chains. Hence, to be fully transparent, impact modeling should account for uncertainties associated with the chosen bias-adjustment approach in addition to uncertainties related to the choice of climate model and impact model.

While efforts to improve climate models and reduce their biases will continue , impact assessments need to become more transparent. In particular, uncertainties need to be well communicated to aid adaptation planning . Many artifacts of currently widely used bias-adjustment methods are probably unknown because of the way bias adjustment is evaluated . Evaluation of multivariate relationships and extremes need to become standard in the evaluation of climate and impact models as well as an evaluation of the appropriateness of the chosen bias-adjustment approach . Overall, multivariate bias-adjustment methods should be favored in impact modeling to ensure that multivariate impacts are captured more realistically.

As long as major biases in climate models persist, some form of bias adjustment is unavoidable in model climate impacts. In the fortunate case where large multimodel ensembles are available and a clear target is identified, ensemble members can be selected according to how well they match certain criteria associated with the target . Process-based observational constraints are one way forward to select the most promising subset of models . Yet, in many cases, this information is not available and large model ensembles may be too expensive to obtain. To achieve the most reliable outcome with respect to modeled impacts, bias adjustment preferably takes into account the known limitations of the relevant climate model as well as characteristics of the target system for which the bias adjustment is applied . In large-scale impact modeling projects such as ISIMIP, global flood modeling , global crop modeling , and generic modeling of high-impact events , achieving these standards is extremely challenging, if not unfeasible. In these cases, typically, a single bias-adjustment method is applied to a set of climate variables that then serves as input for a variety of impact models across regions and sectors. The usefulness of such one-size-fits-all approaches may be debated , yet decision makers urgently require robust information on potential impacts of climate change. Impact modeling frameworks such as ISIMIP and many others sample the uncertainty related to the chosen climate model and impact model but do not take into account uncertainties related to the applied bias-adjustment method. To be transparent towards potential users, however, the scientific community should provide information on all uncertainties associated with modeled impacts.

5 Conclusions

Climate impact modeling is crucial to translate information from climate projections into potential impacts to aid decision-making and planning. Due to persistent biases in current climate models, bias adjustment is an integral part of most impact modeling activities. Our results demonstrate that univariate bias adjustment can increase biases of simulated hazards that depend on multiple correlated climatic drivers. Univariate bias adjustment can furthermore lead to a large increase in the uncertainty of such modeled hazards and impacts. Both aspects are particularly severe when studying extremes. More importantly, if univariate bias adjustment does not adequately adjust biases in hazards in the present-day climate, future projections of such hazards have to be interpreted very carefully. Our findings highlight that impact modeling chains need to incorporate uncertainties associated with the choice of bias adjustment into their uncertainty assessment to be transparent for decision makers.

Code and data availability
Code and data availability.

All datasets and the code to compute the different bias adjustments are freely available from the sources mentioned in Sect. 2 (Data and methods).

Appendix A

Table A1The 29 CMIP5 models used in this study. We use daily data from the historical simulation during the period 1981–1995.

Author contributions
Author contributions.

JZ and EMF conceived the study. SL computed the bias adjustment following the ISIMIP2b scheme. JZ analyzed all data and produced all figures. JZ wrote the first draft, all authors commented on the draft and all revisions.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

We thank Alex Cannon for helpful discussions related to the application of the MBCn approach. Jakob Zscheischler acknowledges financial support from the SNSF (Ambizione grant PZ00P2_179876). Stefan Lange acknowledges funding from the European Union's Horizon 2020 research and innovation program under grant agreement no. 641816 (CRESCENDO).

Edited by: Ben Kravitz
Reviewed by: Stefan Hagemann and one anonymous referee

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