% Discussion

\section{Our Motivation and the Long Road to the Results}
The main motivation surely was the results from Germany published by \citet{baeumer2007feb}. We were quite confident that one could find similar results for Switzerland too. Furthermore we wanted to provide a detailed look at the geographical differences and probable changes over time. Initially we hoped to find a spatial difference---at least between north and south of the Alps---and detect an increasing amplitude in weekly cycles over the course of time.

The fact that even to date many mechanisms in cloud physics still reveal a lot of open questions which have to be investigated made this working field additionally more interesting. As it was impossible to start a big field campaign and measure all human made effects on weather and climate within five months, we focused on achievable goals and started to analyse meteorological data from the archive.

\subsection{Assumptions and Procedures}
It is hard to overlook that---especially in the assumptions and the pro\-ce\-dure---we rely heavily on the work of \citet{baeumer2007feb}:

\begin{itemize}
\item Considering 15 year time steps: \citet{baeumer2007feb} had a good reason to do so. We did it mainly to be able to compare our results with theirs. By also analysing 30 year periods we tried to handle this coincidental fact.
\item The 31-day running mean: it is quite hazardous to take 31 days for the moving average. Therefore we calculated all \RM{} with a 7 and a 31-day running mean and the result is nearly identical. However, the 7-day running mean culls the dynamically induced weather conditions and flattens/minimises generally the \RM{}. This can be seen very well in plots with temperature data.
\end{itemize}

\subsection{Statistical Tests}\label{discussion:subsec:stattest}
Another point that we initially adopted from  \citet{baeumer2007feb} was the statistical test procedure. We advisedly did not take all available \ac{CLIMAP}-data for our computation, to prevent getting a false statistical significance. Without further analysis of the independence among the stations we applied their method.%run through the computations.
Surprisingly the weekly amplitude in temperature anomaly remains nearly constant during the last 143 years, only the warmest and the coldest day in the week changes coincidentally.
The crucial factor to reconsider our statistical methods was the fact that the one sided t-test of all  stations always reveals a significance between the maximum and the minimum while for each single station it never did. The presumption suggests itself that the stations are not independent. This can be confirmed with a correlation test in \ac{R} on the \RM{}: it shows
% As described in chapter \nameref{chap:results} section \vref{results:sec:station_aggregation} a test on the \RM{} proves 
that the stations can indeed not be considered as independent.

This result led us to the idea to make a control by performing a \WDT{} and with the help of simple simulations.

During the analysis of the precipitation data (with the t-test) we realised that many considered periods suddenly revealed a significance. The \WDT{} and the simulations showed this too. As this was not the case in the analysis of the temperature time series, we assume that this sudden significance in the real data is due to higher scatter in the precipitation \RM{} (chapter \nameref{chap:results} section \vref{results:precipitation_anomaly}).
By having a closer look at this \RM{} we detected that the precipitation data are not distributed normally. Applying a Wilcoxon test instead of the t-test would have solved this issue. But the problem is more fundamental: we compare several groups---7 in a regular week---against each other. Every one has its own distribution. However, the t-test and the Wilcoxon test are appropriate for the comparison of two groups. The following gedanken experiment should show, why we have the opinion that in this case neither the t-test nor the Wilcoxon test is advantageous.

\begin{quote}
\subsubsection{A little Gedanken Experiment}\label{discussion:subsec:gedanken_experiment}
Ten persons play with a six-sided dice. They successively roll the dice 100 times and count their points together. Now, the player with the highest score compares his dice distribution with that of the person who had the lowest score. As they all love statistics they immediately perform a one sided Wilcoxon test on these two distributions. By computing the $p$-values they want to figure out, whether this difference is significant or not.
They are so fascinated by this game, that they play it all night long, all in all 1,000 times. But it was worthwhile as the result is surprising: in more than 800 games (exceeding 80\%) the test revealed a significance at the 5\% level\footnote{For the end of the gedanken experiment see appendix~\vref{appendix:gedanken_experiment}.}.
\end{quote}

It is obvious that the dice is not unfair and they just applied a wrong statistical method to figure out whether their scores differ significantly from each other.

The same problem occurs by testing the weekdays against each other. Another analogy to the weekday analysis is, that the person who throws the highest score and the one who throws the lowest one changes arbitrarily. So it is in our case: the weekday with the highest/lowest temperature- and precipitation anomaly changes during the last 143 years quite haphazardly.

In our case there is still a problem left with the autocorrelation in the \RM{}. Especially for temperature this correlation within the time series is very high and this in turn might influence the \KW{} test negatively. The application of this test on our various simulations underlines this suspicion.

All in all one could say that---for this issue---the \KW{} test seems to be the most eligible significance test that we know. But it is highly recommended to check the results carefully with other tests.

Since statistical tests are not always unproblematic and since we do not feel very comfortable basing all our results and statements only on one statistical test, we decided to do a few additional examinations.

\subsection{Simulation}
The simulations however are a good method of controlling. They are far too simple to call them a model. The simulation of the reality is not very accurate either, but nevertheless they provide a check whether a result is plausible or not. Applying the t-test on a random series divided by the number of weekdays, and testing the maximum against the minimum, was then also how we figured out that the initially pursued path had to be wrong.
% hier kommt vermutlich noch der Text von Thommy
%discussion wilcoxon test unbrauchbart kruskal wallis ist gut da uniform und wir uns in 5 % der fälle irren was wir uns auch tun sollten auf einem siginifikants niveau von $\alpha=5%$.

\subsection{``6-8-Day-Test''}
For the \WDT{} one can say that, it is likely improbable that a 6, 7 and 8-day-cycle exists at the same time. Even if this were the case, it would for sure not be human made. Insofar it is a simple plausibility control to answer the question as to whether there is a human made weekly cycle or not.

\subsection{What else did we do}
At this point we have to admit that we really tried to do almost everything conceivable to find a weekly cycle in climatology. Besides the division in the four seasons we tried to find a cycle at least in one of the 8 different weather conditions. We thought that maybe Milano, Genova and Torino together could cause a 7-day precipitation cycle during a longer \acl{S}. We even looked only at periods, in which \acl{H} and \acl{F} last longer than 14 day and tried to find a weekly cycle, but without success.

Another basic approach to detect a 7 day cycle is by means of a Fourier-Model-fitting-curve in \ac{MATLAB}. As we could not find anything in the \RM{} for temperature data, we tried a different approach. This time the seasonal cycle is eliminated by subtracting the Fourier-Model-fitting-curve from the daily measured values. This results in residuals. The latter can be considered as a new \RM{}. Now, by performing a second fitting curve on this time series, it is possible to compute the periodicity of this fitting curve. No 7-day cycle could be found either.
%In the calculation with the running mean, we called that \RM{}

\section{Discussion of the Results}
As we could not find any weekly periodicities in climatology, there is little left to discuss about weekly cycles. As a matter of fact the discussion leads mainly to the question of the right method of detecting a 7-day cycle. Depending on that, one finds a weekly cycle or not.

Nevertheless there are a few aspects concerning the ``climate results'' which might be interesting to discuss.

\subsection{Anomalies since 1865}
Considering the plots since 1865 (chapter~\nameref{chap:results} on page~\pageref{chap:results}ff) one expects that for the considered time period of 15 years the weekly cycles are arbitrary.

The anomaly ranges of all parameters do not increase over time and remain more or less ``constant''. This can be considered as an indication, that the human made \ac{PM} pollution is not responsible for a potential weekly cycle. The arbitrarily changing maximums and minimums in the last 50 years imply that there might be no human impact.

\subsection{Summer 2007 (Case Study)}
In longer time series the \PM{10} usually reveals a clear weekly cycle with a minimum on Sundays and an enhancement during the midweekdays (\nameref{chap:results} Chapter, Section~\vref{results:sec:pm10}ff). For the summer 2007 the \PM{10} maximum occurs on Monday and the minimum on Tuesday (\nameref{chap:results} Chapter, Section~\vref{results:sec:summer2007}ff). This unusual 
%anomalous
result can be explained with the precipitation anomaly and the `weather conditions'---in fact they strongly relate to each other:
\begin{itemize}
\item On weekends there is hardly any precipitation while Tuesday is the day with the second highest rainfall during the week and the latter is responsible for the washout of the \PM{10} (Figure~\vref{fig:precip_anomaly_summer2007}). %..latter washes out the \PM{10}
\item Probably more important however are the 'weather conditions'. The sunny and dry weekends (\acl{H}) make an accumulation of \PM{10} possible while the high number of \acl{W} on Tuesday cut back and dilute the aerosols (Figure~\vref{fig:count_weather_conditions_summer2007}).
\end{itemize}

\subsection{Simulation}
Assuming that the weekly cycles are absolutely random, a good simulation should reveal approximately the same weekly amplitudes as in reality. If this is not the case, there must be a deficiency in the simulation. This deficiency can result simply because an important feature has been forgotten, or else the assumption of no weekly cycle is wrong.

In case the simulated amplitudes differ from reality, one can distinguish between two circumstances:
\begin{description}
\item[Overestimation] The simulation has a higher amplitude, which can be reduced to a deficiency in the simulation.
\item[Underestimation] The opposite is the case. The simulated amplitude is smaller than in reality. The simulation ignores a forcing that could be an anthropogenically induced 7-day forcing. But it could be just as well an other forcing, like e.\,g., a dynamical one.
\end{description}

% a monte carlo simulation---as described in chapter \ref{chap:methods} section \vref{methods:simulation}---should give approximately the same amplitude as the real amplitude.
% because the observed amplitude is for one measured value more or less the same in all periods.
%In the analysis of the temperature time series 
%In a simulation with a higher amplitude there is a lack of an important feature in the simulation. If the opposite is the case---the amplitude is less in the simulation---there is a lack of an important feature. This important feature can than be the 7-day anthropogenic forcing or an other forcing.
As shown in \nameref{chap:results} Chapter, Section~\vref{results:subsec:autocorrelation}, the autocorrelation of the time series is an important feature. Nevertheless the inference that an underestimation of the simulated amplitude---that exhibits the same autocorrelation as the original time series---must be due to an anthropogenic forcing, is wrong. The amplitude could not be as high in the 6 and 8-day-week, if this was the case.

Generally one can say that simulations with a high autocorrelation usually underestimate the weekly amplitude while those without autocorrelation normally overestimate it. At this point one has to emphasise, that the so called ``real autocorrelation'' only looks at the preceding day, or the following day respectively. It therefore depends on the parameter whether this ``real autocorrelation'' reflects the original \RM{} or not. The temperature e.\,g. displays a rather high autocorrelation ($\varrho=0.81$). Various fluctuations underlie the real \RM{}---despite the subtraction of the 31 running mean---e.\,g. dynamical ones, which last usually over several days. The dynamic cycle again force the \RM{} literally towards a defined direction. It is very unlikely that a random simulation with an implemented autocorrelation of 0.8 would do so. The simulation with the ``real autocorrelation'' of precipitation ($\varrho=0.36$) on the contrary reflects the reality quite well.

In all simulations the dynamical forcing is excluded, even though it is a very important factor for the weather conditions in the middle latitudes. As shown in chapter \nameref{chap:results} in section \vref{results:sec:summer2007} it can be the most important thing for a cycle---in this case weekly one.

\subsection{Number of Significant Tests}
Performing the three statistical tests (t-test, Wilcoxon test and \KW{} test) on the precipitation data and counting the number of significant outputs reveals a similar result to the gedanken experiment (Section~\vref{discussion:subsec:gedanken_experiment}): most of the t-tests---namely around 80 of 140---between the maximum and the minimum get significant (Figure~\vref{fig:sig_tests}). Regardless whether this test is applied on a 6, 7 or 8-day-week, it provides approximately the same picture. This result underlines our assumption, that this kind of statistical test is not appropriate for our study.

The Wilcoxon test reveals about half as many significant cases than the t-test does. Nevertheless the comparison with the 6 and 8-day-week provides even more significant cases. Insofar a 6-day cycle or an 8-day cycle would be more likely than a 7-day one. This might be true, but it underlines our results: there is no significant 7-day-cycle.
\begin{figure}
\centering
\input ./images/anzahl_signifikante_tests/plot_sig.pdftex_t
\caption[Number of siginificant tests on precipitation]{Number of siginificant tests (for $\alpha=5\%$) on precipitation data in 15 years period steps between 1872-01-01 and 2006-12-31. The total amount of tests in every single bar is 140.}\label{fig:sig_tests}
\end{figure}

\section{Faults in the Data and Data Handling}
Series of measurements often do contain wrong values, this is bothersome but practically unavoidable. The question is, how to detect and eradicate them. The best would be to check the time series with the highest available resolution for wrong values. Due to the lack of a direct access to this data this was not possible for us.

The difference between a corrected series and the original series (contain wrong values) is normally not recognisable in the results (\nameref{chap:results} Chapter, Section  \vref{results:subsec:faults_in_data}). In the weekly mean values of precipitation between 1992-01-01 and 2006-12-31 one mean is slightly higher and the standard deviation is much bigger in the uncorrected series. This is caused by the fact that one value in the precipitation series of Bargen (SH) is extremely high (1637.0\,mm at 1998-11-29). There is no other fanciful value in this series. Normally there is more than one wrong value in a series, and they can be distributed evenly over the weekdays.

\section[Comparison of our Work with Bäumer and Vogel (2007)]{Comparison of our Work with \protect{\citet{baeumer2007feb}}}
To make sure that our results, that differ from those of \citet{baeumer2007feb} are not based on a lack of understanding on our part, we verified our computation-procedure by using the data from Germany (Figure  \vref{fig:baeumer_vergleich}). It is possible to reproduce their results by applying the same statistical proceeding.% and get a signifigant weekly cycle too.
We could reproduce the weekly anomalies for each single station except for Hohenpeissenberg. Here our anomalies differ conspicuously from those of  \citet{baeumer2007feb}.% (Appendix ......).
It is quite opaque, how this can come into being. Nevertheless this verification with the German WMO-data illustrates that our results are not a question of a wrong series (like e.\,g. Hohenpeissenberg), % error-corrected time series or not,
and it neither seems to be a question of region. It solely depends on, whether their method is correct or not. And assuming their method is correct would imply that there has to be---beside the 7-day cycle---also a 6 and 8-day cycle (Figure \vref{fig:baeumer_vergleich_weeks}). This is rather unprobable.

\begin{figure}
\centering
\subfigure[Original Figure from \protect\citet{baeumer2007feb}
]{
 \includegraphics[page=2, width=0.49\linewidth{}, height=56.5mm, viewport=87 615 271 733, clip]{../literatur/papers/Baeumer_Vogel_2006}
 \label{fig:german_orig}
}\hfill
\subfigure[Our reproduction of their result
]{
 \vspace{-5cm}
 \input ./images/germany/plots_RM_1991-01-01_bis_2005-12-31__run_time_2008_02_07_1829_005.pdftex_t
 \label{fig:german_rm}
}
\caption[Verifying of our calculation procedure with the German stations]{Verifying of our calculation procedure with the German stations: Aachen, Berlin-Tempelhof, Düsseldorf, Frankfurt/Main Flughafen, Helgoland, Hohenpeissenberg, Kahler Asten, Karlsruhe, Konstanz, Rostock-Warnemünde, Stuttgard-Echterdingen and Zugspitze. 
\protect\\
The \RM{} of all stations is grouped directly by weekday (assumption that the stations are absolutely selfcontained), no average over the stations.
\protect\\
Period: 1991-01-01 to 2005-12-31. Errorbars $\pm1$ standard error.
}\label{fig:baeumer_vergleich}
\end{figure}

\begin{figure}
\centering
\subfigure[6-Day-Week. Amplitude is 0.17. % 0.167509
]{
 \input ./images/germany/plots_6-Day-Week_RM_1991-01-01_bis_2005-12-31_005.pdftex_t
 \label{fig:german_6-day-week_rm}
}\hfill
\subfigure[7-Day-Week. Amplitude is 0.17. % 0.16909
]{
 \input ./images/germany/plots_RM_1991-01-01_bis_2005-12-31_005.pdftex_t
 \label{fig:german_7-day-week_rm}
}\hfill
\subfigure[8-Day-Week. Amplitude is 0.30 % 0.295929
]{
 \input ./images/germany/plots_8-Day-Week_RM_1991-01-01_bis_2005-12-31_005.pdftex_t
 \label{fig:german_8-day-week_rm}
}\\\vspace{2ex}
\subfigure[6-Day-Week. Average over stations. % 0.167509
]{
 \input ./images/germany/plots_6-Day-Week_AoS_RM_1991-01-01_bis_2005-12-31_005.pdftex_t
 \label{fig:german_6-day-week_AoS_rm}
}\hfill
\subfigure[7-Day-Week. Average over stations. % 0.16909
]{
 \input ./images/germany/plots_AoS_RM_1991-01-01_bis_2005-12-31_005.pdftex_t
 \label{fig:german_7-day-week_AoS_rm}
}\hfill
\subfigure[8-Day-Week. Average over stations. % 0.295929
]{
 \input ./images/germany/plots_8-Day-Week_AoS_RM_1991-01-01_bis_2005-12-31_005.pdftex_t
 \label{fig:german_8-day-week_AoS_rm}
}
\caption[Weekly cycles of temperature anomaly for a 6, 7 and 8-day-week with German stations]{Weekly cycles of temperature anomaly for a 6, 7 and 8-day-week with the following German stations: Aachen, Berlin-Tempelhof, Düsseldorf, Frankfurt/Main Flughafen, Helgoland, Hohenpeissenberg, Kahler Asten, Karlsruhe, Konstanz, Rostock-Warnemünde, Stuttgard-Echterdingen and Zugspitze.
\protect\\
In the first row the \RM{} of all stations is grouped directly by weekday (assumption that the stations are absolutely selfcontained), no average over the stations. In the second row a single time series is first computed with mean values for every single day.
\protect\\
Errorbars $\pm1$ standard error. The range of the ordinate is from week to week different. Period 1991-01-01 to 2005-12-31. Errorbars $\pm1$ standard error.
}\label{fig:baeumer_vergleich_weeks}
\end{figure}
