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\title{Neural network based estimates of the climate impact on mortality in Germany %
- application to storyline climate simulations}

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\author[1,*]{R. Schachtschneider}
\author[1]{J. Saynisch-Wagner}
\author[2]{A. S{\'a}nchez-Benítez}
\author[1,3]{M. Thomas}

%-------------------------------------------------------------------------------
\affil[1]{Helmholtz Centre Potsdam GFZ German Research Centre for %
Geosciences, Telegrafenberg, 14473 Potsdam, Germany}
\affil[2]{Alfred Wegener Institute, Helmholtz Centre for Polar and Marine %
Research, Am Handelshafen 12, 27570 Bremerhaven, Germany}
\affil[3]{%
%Freie Universit\"{a}t Berlin, %
Free University Berlin, %
Kaiserswerther Str. 16 - 18, %
14195 Berlin, Germany}

% Corresponding author mailing address and e-mail address:

\affil[*]{reyko.schachtschneider@gfz-potsdam.de}


\keywords{Climate change, machine learning, mortality prediction, human health, %
storyline simulations}

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\begin{abstract}
The aim of this work is the prediction of heat-related mortality for Germany 
under future, i.e. hotter, climate conditions. The prediction is made based on
2m temperature data from climate storyline simulations using machine learning
techniques. We use an echo state network for linking the outputs of storyline 
climate simulations to the target data. The target data are all-cause mortality 
rates of Germany for all ages. The network is trained with present day climate 
model outputs. Model outputs of future, i.e. 2K and 4K warmer, storylines are 
used to predict mortality rates under such climatic conditions.
We find that we can train an echo state network with recent temperature data and 
mortality and make plausible predictions about expected developments of 
mortality in Germany based on future climate storylines.
The trained network can successfully predict mortality rates for future
climate conditions. We find increased mortality during the summer months which 
is attributed to the presence of more severe heat waves. The mortality decrease 
found during winter can be explained milder winters leading to fewer deaths caused 
by respiratory diseases. However, mortality in winter is largely influenced by
other factors such as influenza waves or vaccination rate and explainability due 
to temperature is limited.
\end{abstract}

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%%% Suggested section heads:
% \section{Introduction}
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%\input{intro}
\section{Introduction}
\label{sec:intro}

%-------------------------------------------------------------------------------
% State of the art: ML and application examples

Machine learning (ML) has become a helpful tool in many areas of science and 
medicine \cite{Deo2015machine,Rajkomar2019machine}. It comprises algorithms
for revealing non-linear relationships between input and output variables and 
can be helpful in many classification tasks or regression problems. 

In medicine, ML can teach researchers new ways of studying diseases, producing 
medicines or treating patients and can be applied for early 
disease detection where severe conditions are likely to manifest at a later time
\cite{May2021eight,Rajkomar2019machine}. \cite{Than2019machine} 
used a gradient boosting model to predict the likelihood of acute myocardial 
infarction for patients. \cite{Esteva2017dermatologist} trained a 
deep neural network for classification of skin lesions into skin cancer risk 
categories.

Mortality prediction has been conducted in various fields of medicine and for
different diseases such as Chronic Obstructive Pulmonary Disease 
\cite{Moll2020machine}, COVID-19 \cite{Gao2020machine}, neonatal mortality 
\cite{Mangold2021machine}, or general mortality risk in populations with 
coronary artery disease \cite{Motwani2017machine} or general elderly populations
\cite{Rose2013mortality}. \cite{Taylor2016prediction} predict mortality of 
hospitalized patients with sepsis in a data-driven ML-approach.
The impact of climate change on mortality due to various diseases have been studied 
with the help of ML, e.g. for Malaria \cite{Nkiruka2021prediction}, 
myocardial infarctions\cite{Marien2022machine}, or chronic obstructive pulmonary 
disease \cite{Moll2020machine}.

Impacts of climate change and increasing numbers of death from heat disorder 
have been studied, e.g. by \cite{McGeehin2001potential,Kim2016projection,Hajat2023current,Huang2011projecting}. 
\cite{Lee2023heat} investigated the influence of heat exposure on mental health 
and connected implications following climate change. 
This includes mortality due to suicides, mental disorders, and violence among others.
\cite{Hirano2021machine} developed a ML-based model for mortality prediction 
for heat-related illnesses on the base of data from %2393 
hospitalized patients in Japan. \cite{Kim2022explainable} used a random forest 
model for heat-related mortality prediction in a detailed area within a city 
with various climatic, demographic and socio-economic sectors.
\cite{Winklmayr2022heat} employ a generalized additive model for the estimation
of heat-related deaths in Germany for the period 1992--2021.
They can characterize 
long-term trends and quantify the effect of heat on mortality over the years.
\cite{Mistry2022comparison} compare results of mortality models based on 
temperature measurement data at a number of weather stations to models based on 
global reanalyses of weather observations and climate models by the European Centre 
for Medium-Range Weather Forecasts (ECMWF).
They find that reanalysis data are a valid alternative source in temperature-related 
health risk studies.

In this study we predict the mortality rate in Germany for two different climate 
storylines computed in the project SCENIC using an echo state network (ESN). We 
show that the network can be trained on five years of monthly maximum temperature 
input data and monthly mortality rates for Germany as target data. With the trained 
network we can reliably predict mortality rates in Germany for the two given future 
climate storylines.

% Methods
%\input{method}
%===============================================================================
\section{Method}
\label{sec:method}


%-------------------------------------------------------------------------------
\subsection{Background}
\label{sec:method:bg}

% General machine learning / artificial neural networks
Artificial neural networks (ANN) are inspired by biological examples 
\cite{Lee2018machine}. Neurons in
the brain are interconnected and capable of fulfilling enormous cognitive tasks
with utmost precision. An ANN consists of a large number of artificial neurons
that mimic the behavior of connected neurons in the brain. The artificial 
neurons have an activation threshold above which they pass on information and 
weights that encode the importance of the neuron connection. Several neurons are 
organized in a layer and many layers make up a neural network. The input layers 
receives information from the outside world, a number of hidden layers further
processes the information from the input layer, and the output layer transforms
the information from the network into a result. The shape of the result depends 
on the purpose the network is designed for. There are numerous kinds of layered 
networks for purposes of , e.g., classification, regression, clustering or 
object detection.

% Training of ANNs
In the training phase the weights of an ANN are adjusted. In supervised 
learning this is 
achieved by providing input samples with the corresponding known output. 
Through iterative back-propagation techniques the weights of the ANN nodes
are adjusted to minimize the misfit between the network output and the known
result. The challenge is to avoid over-fitting and obtain an ANN that performs
well on unknown data. Unsupervised learning uses only input data to learn
the distribution of the input variables. It is useful for clustering, 
compression, or feature extraction \cite{Lee2018machine}. 


% Recurrent neural networks
A recurrent neural network (RNN) is a network with cyclic connections in which 
the neurons can send feedback signals to each other 
\cite{Grossberg2013recurrent}. They are more powerful than acyclic feed forward
neural networks (FNNs) in that they can create and process memories of arbitrary 
sequences of input patterns \cite{Schmidhuber2015deep}. 
Therefore, while FNNs represent functions, RNNs represent dynamical systems which
may develop a self-sustained temporal activation even in the absence of 
input\cite{Lukovsevivcius2009reservoir}.
Supervised Long Short-Term Memory (LSTM) RNNs can discover and memorize events 
that happened thousands of time steps ago \cite{Schmidhuber2015deep}.
Training of RNNs, however, is challenging. The well-known feed-forward 
back-propagation algorithm cannot be directly used because of the cyclic nature 
of the reservoir. One simple solution to overcome this difficulty is to neglect 
all indirect pathway (i.e. such containing cycles). However, this is a rather 
coarse approximation of the optimization problem \cite{Doya2003recurrent}. There
exist two basic ways for calculating the exact gradient fo the output with 
respect to the weights: forward and backward methods \cite{Doya2003recurrent}.
The forward method solves a linear dynamical equation system for the effect of 
a small change in a weight on the network state trajectory. While this can be 
calculated concurrently with the network dynamics the computational costs are
quite high. In the backward method the causes of the output error are computed 
backward in time. In case of discrete time steps this can be achieved by 
setting up a multi-layer feed-forward network for unrolling the multi-step 
evolution of the network.
More comprehensive overviews over RNNs and training algorithms are given by 
\cite{Tyagi2022recurrent, Doya2003recurrent, Jaeger2002tutorial, Lukovsevivcius2009reservoir}.

% Echo state networks
\subsection{The echo state network}
\label{sec:esn}
The ESN is a special form of a recurrent neural network that was introduced 
independently by \cite{Jaeger2001echo} and \cite{Maass2002real}. Originally, 
it incorporates an algorithm for supervised learning in which only the 
output layer is modified \cite{Doya2003recurrent}. 
A schematic of an ESN is shown in \fig~\ref{fig:esn_scheme}.
%\textcolor{red}{ToDo: Elaborate!}


\begin{figure}
    \centering
    \includegraphics[width=\oneFigureWidth]{Images/figure1}
    \caption{Schematic of an echo state network. The input layer is connected to 
      a reservoir. In the reservoir there are connections between some of the 
      nodes, including recurrent connections. The output layer is connected to 
      the reservoir and to the input layer. After adaption of the neuron activations
      in the reservoir, the output weights are determined by a regression that 
      minimizes the misfit, $E$, between prediction and target.}
    \label{fig:esn_scheme}
\end{figure}


In our network, the input weights are initialized randomly from a uniform 
distribution over the interval $[-0.5, 0.5]$. The reservoir weights are initialized
by the same procedure and in addition a fraction $(1-c)$ of the weights, where 
$c \in [0, 1]$ is the connectivity of the reservoir, is set to zero. During the training 
process, the reservoir weights remain 
unchanged. Only the neuron activations are updated based on training input. The 
neuron activation decides whether a neuron is important and will be activated or not. 
The activation values are updated using
%
\begin{equation}
    \label{eq:neur_act}
    \vec{x}_{n} = (1-\alpha) \vec{x}_{n-1} 
      + \alpha \tanh(\mat{W}_{\mathrm{in}} + \mat{W}\vec{x}_{n-1})
\end{equation}
%
where $\vec{x}_{n}$ are the neuron activations at step $n$, $\mat{W}_{\mathrm{in}}$ 
is the input weight matrix, $\mat{W}$ is the recurrent weight matrix of the 
reservoir, and $\alpha \in (0, 1]$ is the leakage parameter. The leakage parameter
determines the proportion of the neuron activation update due to new input data at 
step $n$. Finally, the output weight matrix is determined by a ridge regression that
minimizes the misfit between outputs and targets in a least squares sense.
The ESN is described in more detail by \cite{Lukovsevicius2012practical}. The ESN 
parameters used in this study are given in Table~\ref{tab:esn_params}.

\begin{table}
    \centering
    \caption{Parameters of the echo state network used in this study.
      $\alpha$ is the leakage parameter, $r_{\text{s}}$ is the spectral radius of
      the reservoir weights matrix, $c$ is the connectivity of the reservoir, and 
      $\lambda$ is the regression coefficient in
      the ridge regression of the output weights matrix.}
    \begin{tabular}{l|lllll}
        Parameter & Reservoir size & $\alpha$ & $r_{\text{s}}$ & $c$ & $\lambda$\\
        \hline
        Value & 9,000 & 0.5 & 1.25 & 0.5 & 1e-8 
    \end{tabular}
    \label{tab:esn_params}
\end{table}

The trained network can then be used to either forecast new values of the time
series of input data from the known inputs in the past or to compute the response
to new inputs. Here, we use the latter functionality since the goal is to predict 
socio-economic parameters for new given inputs (i.e. in warmer world scenarios).
The output is then determined by
%
\begin{equation}
    \label{eq:esn_output}
    \vec{y}_{n} = \mat{W}_{\mathrm{out}}[1;\vec{u}_{n};\vec{x}_{n}]
\end{equation}
%
where $\mat{W}_{\mathrm{out}}$ is the matrix of output weights, 
$[\cdot;\cdot;\cdot]$ denotes the vertical concatenation of vectors, 
$\vec{u}_{n}$ is the input vector, and $\vec{x}_{n}$ is the vector of neuron
activations. Thus, the output at a certain time step is determined by both, 
the state of the reservoir and directly by the input data at that time. 
As indicated in \fig~\ref{fig:esn_scheme}, an error measure is calculated from the 
predictions and the target values. We choose the root-mean-square (RMS) error.

As the network is initialized with random values, the network state after training
can be different for each realization. Therefore, we compute an ensemble of trained 
ESNs and predictions and calculate the arithmetic mean from the ensemble of
predictions. The ensemble size is 25. The statistics of the ensemble are used to
estimate the robustness of the approach.


% Data
%\input{data}
%===============================================================================
\section{Data}
\label{sec:data}

As input data for training the ESN we used outputs of global climate simulations.
The used simulations are so-called climate storyline simulations. They recreate
extreme weather events in climate models by keeping some boundary conditions (in 
this case winds in the higher troposphere) close to states of the atmosphere 
observed in the past during those events. This process is called nudging. 
When different climate conditions 
(e.g. elevated CO2 concentrations) are applied to the climate model with the same 
nudging, it is possible to study how the extreme event would enfold under those 
new conditions.
The storyline simulations are computed using a model developed at the Alfred 
Wegener Institute, Helmholtz Centre for Polar and Marine Research (AWI). 
The model is called AWI CM1 and a description can be found in %the paper by 
\cite{Sidorenko2015towards}.

For the current time scenario the tropospheric winds in the climate model are 
nudged to ECMWF weather reanalysis data (ERA5). The two additional future climate 
conditions considered are elevated concentrations of CO2 that result in increased 
global mean temperatures of +2K and +4K scenarios w.r.t. pre-industrial times,
respectively. The method for computing climate storylines is described in detail 
in
%by 
\cite{Sanchez-Benitez2022thejuly}.

The climate storyline simulations that constitute our training data are available 
in a period from 2015 to 2021. The warmer world storylines that are used for the 
mortality rate predictions are available in a period from 2017 to 2021. In order to 
exclude excess mortality due to the COVID-19 pandemic we restrict the training 
and the prediction to before 2020. From the AWI CM1 model we used hourly 2m 
temperature fields from which we compute monthly maximum values. The data have 
a $1^{\circ}\times 1^{\circ}$ spatial resolution. We consider a region within the 
longitudinal range from $10^{\circ}$W to $40^{\circ}$E and the latitudinal range 
from $30^{\circ}$N to $65^{\circ}$N.
This covers a region large enough to include all influences on people in Germany 
but reduces the computational effort compared to using the global data set.
The temperatures are normalized such that the interval [-13$^{\circ}$C, 47$^{\circ}$C] 
maps to [0,1]. The temperature data used as inputs in this study have been published by 
\cite{Sanchez-Benitez2023global}. An example of a temperature field 
before normalization used as input is given in \fig~\ref{fig:temp_in}. 

\begin{figure}
    \centering
    \includegraphics[width=\oneFigureWidth]{Images/figure2.png}
    \caption{Example of temperature input data. Shown are the monthly 
      maximal values at each grid point for July 2017. The input data
      consist of one 2D distribution of the 2m temperature for each input 
      month. For the analysis, temperatures are normalized such that all 
      values lie in the interval [0,1] over the whole investigation period. 
      The red diamonds denote the locations of temperature time series shown in 
      \fig~\ref{fig:temp_vs_time}.}
    \label{fig:temp_in}
\end{figure}

\Fig~\ref{fig:temp_vs_time} shows examples of temperature time series at
four locations in Germany. Even though the global mean temperature is only 
2K or 4K higher than in pre-industrial times, the local maximum 
temperature difference can be much higher.

\begin{figure}
    \centering
    \includegraphics[width=\oneFigureWidth]{Images/figure3.png}
    \caption{Examples of monthly maximum temperature development over time 
    for all considered scenarios. The locations of the profiles are shown in 
    \fig~\ref{fig:temp_in}. The temperature differences between current and 
    future scenarios can be much larger than the average difference.}
    \label{fig:temp_vs_time}
\end{figure}

The target data are monthly mortality values published by the German Federal
Statistical Office. They are available under the code \emph{12613-03-01-4-B} at
https://www.regionalstatistik.de \cite{regionalstatistik}. The data are 
all-cause mortality values over all ages. The mortality was 
scaled by the population of each federal state to yield the mortality per 
100,000 inhabitants. The population statistics are available at the same website
under the code \emph{12411-05-01-4-B}. The population data have annual values 
so the same value was used for each month of the year to scale the monthly 
mortality.


% Results
%\input{results}
%===============================================================================
\section{Results}
\label{sec:results}

With the described ESN we are able to predict mortality rates in Germany from 
monthly maps of maximum 2m temperatures in Europe. 
\Fig~\ref{fig:esn_training} shows the time 
series of the target values used for training the network (left part) and the 
subsequent test predictions (right part, red circles). In the test months, the 
prediction of mortality is very accurate given the short training period. 
The RMS error is 1.7, i.e. $\approx$~1.9\% in the test period.

\begin{figure}
    \centering
    \includegraphics[width=\oneFigureWidth]{Images/figure4.png}
    \caption{Training of the echo state network. This panel shows the five years 
    of target data used for training and testing. The test period is seven 
    months from June to December 2019 in the right part of the panel. The RMS 
    error in the test period is 1.7, i.e. $\approx$~1.9\%.}
    \label{fig:esn_training}
\end{figure}

In \fig~\ref{fig:mean_pred}, we show the mortality rate prediction for the two 
climate storylines investigated in this study. We calculate ensembles of trained
ESNs and predictions to test the robustness of the approach since the initialization 
of the ESN's input layer and reservoir are performed with random numbers. The ensemble 
size is 25. The blue 
and red shaded areas indicate the ensemble spread, i.e. the mean $\pm$ two standard 
deviation (SD), while the circles represent the mean values for each month. The SD 
for the +2K runs ranges between 0.42 and 1.0, for the +4K runs between 0.64 and 1.32, 
respectively. This is in the range of 0.5 to 1\% of the target values, showing that 
the results are robust.


\begin{figure}
    \centering
    \includegraphics[width=\oneFigureWidth]%
      {Images/figure5.png}
    \caption{Mortality rate predictions for warmer climate storyline scenarios. 
    Here, we show the prediction
      of the mortality rate for the warmer world scenarios: +2K in red and 
      +4K in blue, along with the target values of the recent climate scenario 
      on which the network was trained (black). The shaded areas represent one 
      standard deviation from the ensemble mean values.
      The periods highlighted with orange
      background are the summer months (June--August). The warmer world 
      scenarios are available only from 2017.}
    \label{fig:mean_pred}
\end{figure}


The difference
between the predictions for the +2K and +4K scenarios and the target values 
representing the current mortality rates is shown in \fig~\ref{fig:diff_pred}. 
During the summer months, the values are elevated while in winter the values are
lower than in the reference period. The increase in mortality rate in the summer
months is negligible for +2K storyline and 4-5 per 100,000
people for the +4K storyline. The latter corresponds to about
3,200--4,000 additional fatalities in Germany per month.

\begin{figure}
    \centering
    \includegraphics[width=\oneFigureWidth]%
      {Images/figure6.png}
    \caption{Differences between predictions and current mortality rates. Shown are the +2K scenario (red) and +4K scenario (blue). The intervals with the orange 
      highlighted background are the summer months (June--August).}
    \label{fig:diff_pred}
\end{figure}

% Discussion
%\input{disc}
%-------------------------------------------------------------------------------
\section{Discussion}
\label{sec:disc}

The predictions of mortality rates in warmer world scenarios show elevated 
values in summer months and decreased values in winter months. We interpret
those results as higher mortality due to higher heat stress in summer. High 
day temperatures in combination with high temperatures at night are a major 
cause of fatality\cite{Watts2019the2019}.

A decrease of mortality in winter time is plausible since warmer winters may
lead to less severe influenza waves. However, this is hard to quantify since 
the impact of a influenza wave also depends on many other factors such as 
vaccination rate, effectiveness of the vaccination, etc. Therefore, this 
study's analysis is focused on the summer months and heat-related mortality. 
Even with 
heat related mortality there are factors other than the ambient temperature. 
After the 2003 heatwave in Europe with thousands of fatalities among elderly
people as reported by \cite{DeBono2004Impacts} 
preventive measures have been taken in many 
European countries. This has lead to a reduced number of fatalities in the 
severe 2006 heatwave in comparison to the previous heatwaves without those
measures\cite{Fouillet2008has}.  However, \cite{Winklmayr2022heat} show 
that despite the evident adaption to heat, high temperature periods in summer 
still pose a significant thread to human health in Germany.


% Justification of used data set
In order to reduce computational costs we used only monthly maximum temperatures
as input data. However, the results show that our network is able to model 
the connection between temperature inputs and mortality on a very accurate level.
The spatial extend of input data is significantly larger than the investigated 
domain. Spatially extended high temperature distributions allow the 
conclusion that the temporal extend at one location in or near the center is also high since the weather conditions would not change very quickly if high 
temperatures are observed all over the domain. Therefore, the use of monthly 
maximum temperatures is a reasonable choice.

Taking those facts into consideration, this analysis gives a hint as to 
what the impact of future climate change concerning mortality could be. Society 
and politics have to take the appropriate measures not only to prevent the warming 
as far as possible but also to reduce the risk of death under such climate 
conditions should they occur.

One drawback of ESN is that techniques that attribute the outcome to specific 
(parts of) the input data such as layer-wise relevance propagation cannot be 
used due to the cyclic nature of the network. However, the advantage of yielding
robust results with short input data time series outweighs that disadvantage 
of lack of interpretability.


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% DATA SECTION and ACKNOWLEDGMENTS
%
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\section*{Availability of materials and data}
The target mortality and population data used in this study are publicly available 
on the website of the German statistical service (see reference numbers in 
\sect~\ref{sec:data} of the main text). 
The climate model outputs used as input data for the network training and mortality 
prediction are available at https://zenodo.org/record/8014199. The R code used in 
this project is available from the authors upon request. Please contact R. 
Schachtschneider (reyko.schachtschneider@gfz-potsdam.de) for further information.


\section*{Acknowledgments}
This work was supported by funding from the Helmholtz Research Field Earth \& 
Environment for the Innovation Pool Project SCENIC. Climate storyline
simulations were carried out at the supercomputing facilities of the German
Climate Computing Center (DKRZ).
%The storyline simulations
%from which our input data were taken, were prepared by Antonio S{\'a}nchez-Benítez.


\section*{Author contributions statement}
R.S. conducted the research, model development, interpretation and wrote the 
first manuscript draft.
J.S.-W. contributed to the methodology, financing of the project and project 
supervision.
A.S.-B. conducted the climate simulations and provided the input data.
M.T. contributed to project supervision. All authors were involved in revising 
the manuscript.

\section*{Competing interests}
The authors declare no competing interests.

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% REFERENCES and BIBLIOGRAPHY
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