<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% Results

\section{Explanatory Notes to the Results}
\subsection{Fundamental Assumption}
First of all---before the results are computed---one has to make sure that the assumption ``there is no natural seven-day cycle'' is true. Therefore it is appropriate to have a look at the ``Witterungslagen nach \citeauthor{Schueepp1979}'' (`synoptical weather classification by \citeauthor{Schueepp1979}'). This is a classification based only on dynamics. It classifies the weather conditions in 8 main-groups which then subsequently are grouped in 5 subgroups. In this Chapter, for practical reasons, only the 8 main-classes are considered, which are: 
\begin{itemize}
\item \ac{H}
\item \ac{F}
\item \ac{T}
\item \ac{W}
\item \ac{N}
\item \ac{O}
\item \ac{S}
\item \ac{M}
\end{itemize}
\ac{H}, \ac{F}, \ac{T} are called `convective conditions' and \ac{W}, \ac{N}, \ac{O}, \ac{S} are called `advective conditions'. They all refer to the situation at the 500\,hPa-layer, while \ac{M} can have either strong winds on the ground or higher up as a jetsteam.

The count of the several ``Witterungslagen'' per weekday since 1945  shows that there is no dynamical 7-day cycle (Figure~\vref{fig:mastercount_grosswetterlagen})
%(see fig.sowieso) 
so that the assumption from above is consequently correct.

\begin{figure}
\centering
\input ./images/wetterlagen/wetterlagen_001.pdftex_t
\caption[Distribution of the different weather conditions]{Distribution of the different weather conditions (since 1945) per weekday}\label{fig:mastercount_grosswetterlagen}
\end{figure}

\subsection{Stations and their Aggregation}
In the following Chapters the meteorological-data are used from: Bargen (SH), BiÃ¨re, Binningen, Col du Grand St-Bernard, Egolzwil, GenÃ¨ve, La BrÃ©vine, Lugano, MÃ¤nnlichen, Matro, Mosen, Passo del S. Gottardo, Schaffhausen, SchÃ¼pfheim, Segl-Maria, Sion, Zollikofen and ZÃ¼rich (Figure \vref{fig:ch_stationen_karte}). This 18 \acs{CLIMAP}-stations provide temperature values since 1864.
\begin{figure}
\centering
\includegraphics[width=\linewidth{}]{./images/map/schweiz_stationshoehen}
\caption[Stations with meter above sea level]{Stations with meter above sea level: Bargen (SH) (BAR), BiÃ¨re (BIR), Binningen (Basel) (BAS), Col du Grand St-Bernard (GSB), Egolzwil (EGO), GenÃ¨ve-Observatoire (GEO), Jungfraujoch (JUN), La BrÃ©vine (BRL), Lugano (LUG), MÃ¤nnlichen (MAE), Matro (MTR), Mosen (MOA), Passo del S. Gottardo (GOH), Schaffhausen (SHA), SchÃ¼pfheim (SPF), Segl-Maria (SIA), Sion (ION), Zollikofen (Bern) (BER) and ZÃ¼rich (SMA)}\label{fig:ch_stationen_karte}
\end{figure}

The \PM{10} values are derived from \ac{NABEL}-stations which are located in: Basel, Bern, Lugano and ZÃ¼rich. The measurements go back to the year 1998. For the sommer 2007 the station Jungfraujoch is also available.

The availability of the \PM{1}-data is of a shorter time series, namely since 2003. The geographical coverage is also smaller and consists of only three stations: Basel, Bern and Lugano. Hereinafter the \PM{1}-values between March 2003 and December 2006 are analysed.

Usually the data of these stations are aggregated and the mean over all stations is used for further calculations. This allows results which reflect the entire area of Switzerland to be generated. This might smooth a possible local weekly course. To be on the safe side---regarding this ``aggregation problem''---each of the 18 stations is always considered and also separately tested.

\subsection{Faults in the Data and Data Handling}\label{results:subsec:faults_in_data}
All meteorological data are derived from the \ac{CLIMAP}. Some of them can not be true. E.\,g. the temperature series of Matro contains some very high daily temperature values (exceeding 35Â°C). They are based on wrong 10 minutes values with a temperature of 3108.2Â°C (Figure~\vref{results:fig:data_problems}). Therefore all extreme values in precipitation, daily mean temperature, daily minimum and maximum temperature had had to be manually checked for plausibility check: all values that exceed the Swiss all-time record have been dropped and those which are in the range of it have been carefully analysed. Another method to detect wrong values is to look for the outliers in the \RM{}.
\begin{figure} % falls problem Ã¼berprÃ¼fen ob auch ein Â° vor dem C in der pdftex_t datei steht.
\centering
\subfigure[Daily values]{
 \input ./images/data_problems/matro_data_problem_001.pdftex_t
}\hfill
\subfigure[10 minutes values]{
 \input ./images/data_problems/matro_data_problem_002.pdftex_t
}
\caption[Temperature in August 2003, recorded at Matro]{Temperature 2\,m above ground in August 2003, recorded at Matro.}\label{results:fig:data_problems}
\end{figure}

Even by removing the sparse existing incorrect data, the results do not change significantly. Only in the precipitation results, it does observably diminish the errorbars.

When a station contains more than 20\% unavailable data in a considered period, the station drops out automatically for this period.

As long as not stated otherwise, the analysed time series refer to the deviation from the 31-day running mean. This allows, e.\,g., to make a correlation test between the stations without testing the seasonal cycle. The resulting smaller standard deviation leads to narrower errorbars and the statistical tests reveal better results (due to the filtered time series).

\section{Particulate Matter}\label{results:sec:pm}
\citet{baeumer2007feb} hypothesised that the indirect aerosol effect is responsible for weekly cycles in meteorological data. Therefore \PM{10} and \PM{1} are analysed first.

\subsection{General Aspects of Particulate Matter Data}
Since there are no \PM{10} values available before 1998, a comparison with earlier periods is not possible. Nevertheless---thanks to the correlation between suspended matter and \PM{10}---there are facilities to convert one into the other. Based on the suspended matter measurements the \ac{BAFU} computed the \PM{10} values back until 1988. In this longer time series one can recognise a clear decreasing trend over the last 20 years, especially in the cities. The \ac{BAFU} reckons also that the maximum \PM{10}-pollution was around 1970. For this reason the further analysis of longer meteorological time series (in the following Sections) often consider the period between 1960 and 1990 \cite{buwal2005}. %, or 1950 and 2000 respectively.

One assumes that the small particles of the \PM{10}-fraction have more influence on clouds than the bigger ones (Chapter \ref{chap:theory} \nameref{chap:theory} on page \pageref{chap:theory:ccn_size}). %\cite{gong2006}.

\subsection{\acs{PM$_{10}$}}\label{results:sec:pm10}
Figure~\vref{fig:pm10_since1998} shows the weekly course of \PM{10}. Data since January 1998  have been evaluated from the five \ac{NABEL}-stations mentioned before. The range between the maximum on Wednesday and the minimum on Sunday is almost 7\,$\mu$g\,m$^{-3}$.
\begin{figure}
\centering
\input ./images/PM10/plots_AoS_RM_1998-01-01_bis_2006-12-31_005.pdftex_t
\caption[Weekly cycle of \acs{PM$_{10}$} anomaly for the period 1998-01-01--2006-12-31]{Weekly cycle of \acs{PM$_{10}$} anomaly averaged over the following stations: Basel, Bern, Lugano, ZÃ¼rich. Period: 1998-01-01 to 2006-12-31. Errorbars: $\pm1$ standard error.
}\label{fig:pm10_since1998}
\end{figure}

The \KW{} test over the seven weekdays (on the average over all stations) reveals a significance at very high level with a $p$-value of $6\cdot 10^{-29}$. This means that there is a weekly cycle which can hardly be coincidence. In fact every single station
% except for the Jungfraujoch
shows a highly significant ($\alpha=0.3\%$) weekly cycle (Table \vref{table:kw_test_pm10}).
\begin{table}
\centering
\caption{\acs{PM$_{10}$}. Period: 1998-01-01 to 2006-12-31. $p$-values of the \KW test.}
\begin{tabular}{lr@{.}l}
Station&amp;	\multicolumn{2}{l}{$p$-value}\\
\hline
Basel&amp;		2&amp;30$\cdot 10^{-10}$\\ % 2&amp;297e-10\\
Bern&amp;		1&amp;09$\cdot 10^{-66}$\\ % 1&amp;087e-66\\ % &lt; 2.2e-16
% Jungfraujoch&amp;	0&amp;90\\ % 0&amp;903\\
Lugano&amp;		3&amp;98$\cdot 10^{-11}$\\ % 3&amp;982e-11\\
ZÃ¼rich&amp;		1&amp;77$\cdot 10^{-19}$\\ % 1&amp;771e-19\\ % &lt; 2.2e-16
Average&amp;	6&amp;00$\cdot 10^{-29}$ % 6&amp;003e-29   % &lt; 2.2e-16
\label{table:kw_test_pm10}
\end{tabular} 
\end{table}

The  Fourier Analysis on the time series of the \PM{10} deviation from the running mean provides a periodogram which yields a clear peak at $\frac{1}{7}$\,d$^{-1}$ and the multiples of it (vertical black lines in Figure \vref{fig:pm10_periodogram_since1998}) that in turn marks the 7-day cycle.
\begin{figure}
\centering
\input ./images/PM10/plots_spektren_RM_1998-01-01_bis_2006-12-31_004.pdftex_t
\caption[Periodogram of \acs{PM$_{10}$}]{Smoothed periodogram of \acs{PM$_{10}$}. Bandwidth=0.00243. Period: 1998-01-01 to 2006-12-31. % smoothed spans=c(20,20)
\protect\\
The first peak is at $\frac{1}{31}$\,d$^{-1}$ because smaller frequencies are filtered out through a 31-day running mean.
\protect\\
Peaks at $\frac{1}{7}$\,d$^{-1}$ and the multiples of it point to a 7-day cycle in \acs{PM$_{10}$}.
\protect\\
``The cross emblem in the upper right corner of the plot represents the bandwidth of the smoother (cross-piece) and the upper and lower bounds of a pointwise 95\% confidence interval for the spectral density about the plotted curve (vertical line of the cross)'' \protect\cite[page 49]{smith1999}.}\label{fig:pm10_periodogram_since1998}
\end{figure}

Assuming a 6 or an 8-day-week the amplitudes become at least 3 times smaller. The range for an 8-day-week is almost 1.9\,$\mu$g\,m$^{-3}$ while for a 6-day-week it is only around 1.1\,$\mu$g\,m$^{-3}$ (Figure \vref{fig:pm10_6_and_8_day_week_since1998}).
\begin{figure}
\centering
\subfigure[6-Day-Week]{
 \input ./images/PM10/plots_6-Day-Week_AoS_RM_1998-01-01_bis_2006-12-31_005.pdftex_t
}\hfill
\subfigure[8-Day-Week]{
 \input ./images/PM10/plots_8-Day-Week_AoS_RM_1998-01-01_bis_2006-12-31_005.pdftex_t
}
\caption[Weekly cycle of \acs{PM$_{10}$} anomaly]{Weekly cycle of \acs{PM$_{10}$} anomaly averaged over the following stations: Basel, Bern, Lugano, ZÃ¼rich. Period: 1998-01-01 to 2006-12-31. Errorbars: $\pm1$ standard error. Note that the range of the ordinate varies.
}\label{fig:pm10_6_and_8_day_week_since1998}
\end{figure}

\subsection{\acs{PM$_{1}$}}
The weekly course of \PM{1} has an amplitude of about 2.9 \,$\mu$g\,m$^{-3}$. Applying the \KW{} test on it, reveals a high significance ($\alpha=0.3\%$) with a $p$-value of 2.165e-06. The minimum falls on Sunday. From then on there is a constant ascent to the maximum on Friday (Figure \vref{fig:pm1_since2003}).

The  Fourier Analysis on the \PM{1}-values (Figure \vref{fig:pm1_periodogram}) provides a periodogram which yields a clear peak at $\frac{1}{7}$\,d$^{-1}$ and the multiples of it. This underlines the result from the \KW{} test, which show that the \PM{1} weekly course is statistically significant.
\begin{figure}
\centering
\subfigure[Weekly cycle of \acs{PM$_{1}$} anomaly. Errorbars $\pm1$ standard error.
]{
 \input ./images/PM1/plots_AoS_RM_2003-01-01_bis_2006-12-31_005.pdftex_t
 \label{fig:pm1_since2003}
}\hfill
\subfigure[Smoothed Periodogram with bandwidth=0.00176.% smoothed spans=c(7,7)
]{
 \input ./images/PM1/plots_spektren_RM_2003-01-01_bis_2006-12-31_003.pdftex_t
 \label{fig:pm1_periodogram}
}
\caption[\acs{PM$_{1}$}: Weekly cycle and Periodogram]{\acs{PM$_{1}$} for following stations: Basel, Bern, Lugano. Period: 2003-01-01 to 2006-12-31
}\label{fig:pm1_weekly_cycle_periodogram}
\end{figure}

The analysis of this result, by assuming a 6 and an 8-day-week (and by doing preceding procedure again), reveals apparent smaller amplitudes as for the regular week. For a 6-day-week it is about 1.1\,$\mu$g\,m$^{-3}$ and for the 8-day-week it is 0.7\,$\mu$g\,m$^{-3}$. In both cases the \KW{} test reveals no significance: neither for the shortened week ($p$-value is 0.64) nor for the extended week ($p$-value is 0.92) 

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\section{Temperature}\label{results:temperature}
\subsection{Temperature Anomaly for the Time Period between 1992 and 2007}
The analysis of the temperature anomaly for the 15 years from 1992 to 2007 reveals no significance: even if there is a small amplitude of 0.15\,K between the maximum on Wednesday and the minimum on Sunday (Figure~\vref{fig:temperature_weekly_cycle_1992_2006}), the \KW{} test reveals a $p$-value of 0.805 which means that this weekly course is not significant. Even when applying this test to each of the 18 stations, none provides a significant weekly cycle in temperature---not even at the 10\% level, all $p$-values are greater then 0.1. This result can be confirmed by applying the \WDT{} and with several simulations (Section~\vref{results:sec:simulations}).
\begin{figure}
\centering
\input ./images/temperature/plots_AoS_RM_1992-01-01_bis_2006-12-31_005.pdftex_t
\caption[Weekly cycle of temperature anomaly]{Weekly cycle of temperature (2\,m above ground) anomaly averaged over all stations. Period: 1992-01-01 to 2006-12-31. Errorbars: $\pm1$ standard error.}\label{fig:temperature_weekly_cycle_1992_2006}
\end{figure}

Examining the periodogram of the temperature deviation (Figure~\vref{fig:temp_periodogram_1992_2006}) no peak can be detected at $\frac{1}{7}$\,d$^{-1}$ and its multiples. Therefore it seems to be quite coincidental to assume a 7-day-cycle in temperature. 
\begin{figure}
\centering
\input ./images/temperature/plots_spektren_RM_1992-01-01_bis_2006-12-31_004.pdftex_t
\caption[Periodogram of temperature]{Smoothed periodogram of temperature (2\,m above ground). Bandwidth=0.00146. Period: 1992-01-01 to 2006-12-31 % smoothed spans=c(20,20)
\protect\\
The high peak at $\frac{1}{31}$\,d$^{-1}$ is because smaller frequencies are filtered out through a 31-day running mean. The very first peak at $\frac{1}{365}\,\text{d}^{-1}\approx0.003$\,d$^{-1}$ reflects the not perfectly filtered seasonal cycle.
\protect\\
No peak can be detected at $\frac{1}{7}$\,d$^{-1}$ and its multiples.
\protect\\
``The cross emblem in the upper right corner of the plot represents the bandwidth of the smoother (cross-piece) and the upper and lower bounds of a pointwise 95\% confidence interval for the spectral density about the plotted curve (vertical line of the cross)'' \protect\cite[page 49]{smith1999}.
}\label{fig:temp_periodogram_1992_2006}
\end{figure}

\subsection{Temperature Anomaly since 1865}
Considering the plots since 1865 (Figure~\vref{fig:temp_steps_since_1865}) one can not recognise a clear weekly trend which lasts for more than 30 years. The amplitudes however are---during the considered 140 years---more or less around 0.12\,K.

\begin{figure}
\centering
\subfigure[1872-01-01--1886-12-31]{
 \input ./images/changeovertime/temperature/AoS_RM_temp_ch_1872-01-01_bis_1886-12-31_005.pdftex_t
}\hfill
\subfigure[1887-01-01--1901-12-31]{
 \input ./images/changeovertime/temperature/AoS_RM_temp_ch_1887-01-01_bis_1901-12-31_005.pdftex_t
}\hfill
\subfigure[1902-01-01--1916-12-31]{
 \input ./images/changeovertime/temperature/AoS_RM_temp_ch_1902-01-01_bis_1916-12-31_005.pdftex_t
}\vspace{1em}\\
\subfigure[1917-01-01--1931-12-31]{
 \input ./images/changeovertime/temperature/AoS_RM_temp_ch_1917-01-01_bis_1931-12-31_005.pdftex_t
}\hfill
\subfigure[1932-01-01--1946-12-31]{
 \input ./images/changeovertime/temperature/AoS_RM_temp_ch_1932-01-01_bis_1946-12-31_005.pdftex_t
}\hfill
\subfigure[1947-01-01--1961-12-31]{
 \input ./images/changeovertime/temperature/AoS_RM_temp_ch_1947-01-01_bis_1961-12-31_005.pdftex_t
}\vspace{1em}\\
\subfigure[1962-01-01--1976-12-31]{
 \input ./images/changeovertime/temperature/AoS_RM_temp_ch_1962-01-01_bis_1976-12-31_005.pdftex_t
}\hfill
\subfigure[1977-01-01--1991-12-31]{
 \input ./images/changeovertime/temperature/AoS_RM_temp_ch_1977-01-01_bis_1991-12-31_005.pdftex_t
}\hfill
\subfigure[1992-01-01--2006-12-31]{
 \input ./images/changeovertime/temperature/AoS_RM_temp_ch_1992-01-01_bis_2006-12-31_005.pdftex_t
}
\caption[Weekly cycles of temperature anomaly since 1865]{Weekly cycles of temperature anomaly since 1865 in 15 year time steps. Average over all stations. Errorbars: $\pm1$ standard error. Note that the range of the ordinate varies from period to period.}\label{fig:temp_steps_since_1865}
\end{figure}


\subsection{Temperature Anomaly for a 30 Year Time Period starting at 1960}
Considering the time period with the highest \PM{10} air pollution (Section~\vref{results:sec:pm}) the weekly course reveals a maximum on Monday and a minimum on Sunday. The amplitude between the two is 0.07\,K. It is remarkable that, by doubling the considered time period, the amplitude almost bisects (Table~\vref{tab:temp_amplitudes_15vs30_year}).
\begin{table}
\centering
\caption{Amplitudes of weekly temperature anomalies. 15 vs. 30 year periods}\label{tab:temp_amplitudes_15vs30_year}
\begin{tabular}{ll|ll}
15 years&amp;amplitude&amp;amplitude&amp;30 years\\
\hline
1872--1886&amp;0.132&amp;0.12&amp;1870--1899\\ % 0.132
1887--1901&amp;0.218&amp;&amp;\\ % 0.218
1902--1916&amp;0.113&amp;0.02&amp;1900--1929\\ % 0.113
1917--1931&amp;0.098&amp;&amp;\\ % 0.098
1932--1946&amp;0.103&amp;0.07&amp;1930--1959\\ % 0.103
1947--1961&amp;0.102&amp;&amp;\\ % 0.102
1962--1976&amp;0.141&amp;0.07&amp;1960--1989\\ % 0.141
1977--1991&amp;0.055&amp;&amp;\\ % 0.055
1992--2006&amp;0.139&amp;&amp;\\ % 0.139
\hline
average&amp;0.122&amp;0.072&amp;average
\end{tabular} 
\end{table}

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\section{Precipitation}\label{results:precipitation_anomaly}
The weekly course in the mean precipitation anomaly reveals a minimum on Wednesday and a maximum on Saturday which leads to an amplitude of more than 4\,mm (Figure \vref{fig:precipitation_weekly_cycle_1992_2006}).
\begin{figure}
\centering
\input ./images/precipitation/plots_AoS_RM_1992-01-01_bis_2006-12-31_005.pdftex_t
\caption[Weekly cycle of precipitation]{Weekly cycle of daily precipitation (5:40 to 5:40 UTC) anomaly averaged over all stations. Period: 1992-01-01 to 2006-12-31. Errorbars: $\pm1$ standard error.}\label{fig:precipitation_weekly_cycle_1992_2006}
\end{figure}
According to the errorbars one could assume that this cycle could be significant: the errorbars of Wednesday and those of Sunday do not overlap. But the \KW{} test reveals another result: the aberrations from the medians are not significant.

The second test---with the assumed 6 and 8-day-weeks---reveals similar amplitudes as the regular week and the errorbars of the maximum and the minimum do not overlap either (Figure~\vref{fig:precipitation_6_8_weekly_cycle}).
\begin{figure}
\centering
\subfigure[6-Day-Week
]{
 \input ./images/precipitation/plots_6-Day-Week_AoS_RM_1992-01-01_bis_2006-12-31_005.pdftex_t
 \label{fig:precipitation_6-day_weekly_cycle}
}\hfill
\subfigure[8-Day-Week
]{
 \input ./images/precipitation/plots_8-Day-Week_AoS_RM_1992-01-01_bis_2006-12-31_005.pdftex_t
 \label{fig:precipitation_8-day_weekly_cycle}
}
\caption[Weekly cycles of precipitation for different lengths of weeks]{Weekly cycles of daily precipitation (5:40 to 5:40 UTC) anomaly averaged over all stations for different lengths of weeks. Period: 1992-01-01 to 2006-12-31. Errorbars: $\pm1$ standard error.
Note that the range of the ordinate varies.}\label{fig:precipitation_6_8_weekly_cycle}
\end{figure}

Moreover with an appropriate model one can point out that such a weekly course---with an amplitude in the same range---can simply be simulated (Section~\vref{results:sec:simulations}).

\subsection{Precipitation in the Past for 15 Year Time Series}
Going back into the past---in 15 year time steps---the maximum and the minimum seem to vary arbitrarily. The weekly anomaly range in precipitation between 1992 and 2006 is almost 0.4\,mm. Compared to the past this is remarkably high. Only in the period between 1872 and 1886 does the amplitude yield a similar range (Figure~\vref{fig:rain_steps_since_1865}).
\begin{figure}
\centering
\subfigure[1872-01-01--1886-12-31]{
 \input ./images/changeovertime/precipitation/AoS_RM_rain_ch_1872-01-01_bis_1886-12-31_005.pdftex_t
}\hfill
\subfigure[1887-01-01--1901-12-31]{
 \input ./images/changeovertime/precipitation/AoS_RM_rain_ch_1887-01-01_bis_1901-12-31_005.pdftex_t
}\hfill
\subfigure[1902-01-01--1916-12-31]{
 \input ./images/changeovertime/precipitation/AoS_RM_rain_ch_1902-01-01_bis_1916-12-31_005.pdftex_t
}\vspace{1em}\\
\subfigure[1917-01-01--1931-12-31]{
 \input ./images/changeovertime/precipitation/AoS_RM_rain_ch_1917-01-01_bis_1931-12-31_005.pdftex_t
}\hfill
\subfigure[1932-01-01--1946-12-31]{
 \input ./images/changeovertime/precipitation/AoS_RM_rain_ch_1932-01-01_bis_1946-12-31_005.pdftex_t
}\hfill
\subfigure[1947-01-01--1961-12-31]{
 \input ./images/changeovertime/precipitation/AoS_RM_rain_ch_1947-01-01_bis_1961-12-31_005.pdftex_t
}\vspace{1em}\\
\subfigure[1962-01-01--1976-12-31]{
 \input ./images/changeovertime/precipitation/AoS_RM_rain_ch_1962-01-01_bis_1976-12-31_005.pdftex_t
}\hfill
\subfigure[1977-01-01--1991-12-31]{
 \input ./images/changeovertime/precipitation/AoS_RM_rain_ch_1977-01-01_bis_1991-12-31_005.pdftex_t
}\hfill
\subfigure[1992-01-01--2006-12-31]{
 \input ./images/changeovertime/precipitation/AoS_RM_rain_ch_1992-01-01_bis_2006-12-31_005.pdftex_t
}
\caption[Weekly cycles of precipitation anomaly since 1872]{Weekly cycles of precipitation anomaly averaged over all stations since 1872 in 15 year time steps. Errorbars: $\pm1$ standard error. Note that the range of the ordinate varies from period to period.}\label{fig:rain_steps_since_1865}
\end{figure}

Nevertheless the \KW{} test does not reveal a significance for the average over all stations nor for a single one. This applies to all analysed time periods (Table~\vref{tab:precip_pvalues}).
\begin{sidewaystable}
\centering
\caption{$p$-values of the \KW{} test on precipitation anomaly. The values are separated by time periods and stations}\label{tab:precip_pvalues}
\begin{tabular}{llllllllll}
Periods &amp; 1872-- &amp; 1887-- &amp; 1902-- &amp; 1917-- &amp; 1932-- &amp; 1947-- &amp; 1962-- &amp; 1977-- &amp; 1992--\\
Stations &amp; 1886 &amp; 1901 &amp; 1916 &amp; 1931 &amp; 1946 &amp; 1961 &amp; 1976 &amp; 1991 &amp; 2006\\
\hline
Bargen (SH)&amp;0.64&amp;0.41&amp;0.98&amp;0.55&amp;0.92&amp;0.69&amp;0.9&amp;0.41&amp;0.26\\
Binningen&amp;0.45&amp;0.12&amp;0.78&amp;0.49&amp;0.83&amp;0.85&amp;1&amp;0.99&amp;0.82\\
Col du Grand St-Bernard&amp;0.54&amp;0.32&amp;0.56&amp;0.49&amp;0.43&amp;0.83&amp;0.87&amp;0.55&amp;0.82\\
Egolzwil&amp;0.93&amp;0.34&amp;0.78&amp;0.99&amp;0.91&amp;0.91&amp;0.53&amp;0.5&amp;0.93\\
GenÃ¨ve&amp;0.61&amp;0.22&amp;0.41&amp;0.4&amp;0.84&amp;0.56&amp;&amp;&amp;\\
La BrÃ©vine&amp;0.89&amp;0.79&amp;0.86&amp;0.91&amp;0.95&amp;0.67&amp;0.85&amp;0.68&amp;0.44\\
Lugano&amp;0.97&amp;0.22&amp;0.88&amp;0.14&amp;0.77&amp;0.65&amp;0.97&amp;0.79&amp;0.7\\
Matro&amp;&amp;0.72&amp;0.25&amp;0.56&amp;0.76&amp;0.73&amp;0.57&amp;0.8&amp;\\
Mosen&amp;&amp;0.21&amp;0.82&amp;0.73&amp;0.68&amp;0.94&amp;0.52&amp;0.99&amp;0.25\\
MÃ¤nnlichen&amp;0.99&amp;0.65&amp;0.9&amp;0.87&amp;0.83&amp;0.68&amp;0.3&amp;0.74&amp;0.45\\
Passo del S. Gottardo&amp;&amp;&amp;0.35&amp;0.54&amp;0.56&amp;0.81&amp;&amp;&amp;\\
Schaffhausen&amp;&amp;&amp;0.79&amp;0.67&amp;0.89&amp;0.63&amp;0.97&amp;1&amp;0.5\\
SchÃ¼pfheim&amp;0.78&amp;0.97&amp;0.96&amp;0.99&amp;0.71&amp;0.88&amp;0.32&amp;0.84&amp;0.65\\
Segl-Maria&amp;0.35&amp;0.65&amp;0.91&amp;0.72&amp;0.89&amp;0.67&amp;0.69&amp;0.79&amp;0.68\\
Zollikofen&amp;0.18&amp;0.37&amp;0.95&amp;0.97&amp;0.17&amp;0.97&amp;1&amp;0.93&amp;0.76\\
ZÃ¼rich&amp;0.14&amp;0.54&amp;0.91&amp;0.41&amp;0.91&amp;0.44&amp;0.7&amp;0.99&amp;0.58\\
Average over stations&amp;0.37&amp;0.41&amp;0.4&amp;0.83&amp;0.74&amp;0.8&amp;0.99&amp;0.81&amp;0.34\\
\end{tabular}
\end{sidewaystable}

\subsection{Precipitation for Time Series of 30 Years}
The analysis of 30 year time series does not reveal large differences compared to those of 15 years. Every timestep provides a graph with a divers look. The one between 1960 and 1990---which corresponds to the time period with the highest \PM{10} pollution---has an amplidude of a little bit more than 1\,mm (Figure~\vref{fig:precip_1961_1990}). All errorbars overlap with each other and the \KW{} test does not reveal a significance. Compared to the 6 and 8-day-week the amplitudes are all in the same range---around 1\,mm: the amplitude of the 8-day-week exceeds the regular one while the 6-day-week does not.
\begin{figure}
\centering
 \input ./images/precipitation/30years/plots_AoS_RM_1960-01-01_bis_1989-12-31_005.pdftex_t
\caption[Weekly cycle of precipitation over 30 years]{Weekly cycle of daily precipitation (5:40 to 5:40 UTC) anomaly over 30 years, averaged over all stations. Period: 1960-01-01 to 1989-12-31. Errorbars: $\pm1$ standard error.}\label{fig:precip_1961_1990}
\end{figure}

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\section{Daily Temperature Range}

\subsection{15 Year Steps}
The weekly cycle of the temperature range ($T_{\text{max}}-T_{\text{min}}$) between 1992-01-01 and 2006-12-31 reveals an amplitude of 0.094\,K. The maximum is on Friday and the minimum on Sunday. Table~\vref{table:amplitudes_of_temprangeanom} shows the $p$-values of the \KW{} test and compares the amplitudes between the different time periods. The latter seem to get smaller with time. But due to the fact that none of the periods analysed provide a statistically significant weekly course, this change over time can just be arbitrary as well.
\begin{table}
\centering
\caption[Temperature range anomalies. Amplitudes and $p$-values]{Daily temperature range ($T_{\text{max}}-T_{\text{min}}$) anomalies. Amplitudes and $p$-values of the \KW{} test in 15 year time steps.}\label{table:amplitudes_of_temprangeanom}
\begin{tabular}{l|ll}
Period &amp; Amplitude [K] &amp; $p$-value\\
\hline
1932-1946&amp;	0.144	&amp;0.389\\ %0.3889677\\
1947-1961&amp;	0.167	&amp;0.127\\ %0.1265178\\
1962-1976&amp;	0.134	&amp;0.832\\ %0.8321202\\
1977-1991&amp;	0.132	&amp;0.621\\ %0.6208974\\
1992-2006&amp;	0.094	&amp;0.933\\ %0.9334676\\
\end{tabular} 
\end{table}
\begin{figure}
\centering
\input ./images/range/plots_AoS_RM_1992-01-01_bis_2006-12-31_005.pdftex_t
\caption[Weekly cycle of temperature range]{Weekly cycle of daily temperature range ($T_{\text{max}}-T_{\text{min}}$) anomaly averaged over all stations. Period: 1992-01-01 to 2006-12-31. Errorbars: $\pm1$ standard error.}\label{fig:range_weekly_cycle}
\end{figure}

The test with the 6 and the 8-day-week unveil vaguely the same amplitudes as the regular week.

\subsection{30 Year Steps}
Looking backwards---in 30 year steps---reveals that there has never been a significant weekly cycle in temperature range in Switzerland since 1887. This applies both to the average over all stations and to every single station except Bargen (between 1947 and 1976). The latter reveals---in the indicated time period---a $p$-value of 0.030, which is indeed significant ($\alpha=5\%$) but not highly significant ($\alpha=0.03\%$).
% total sind 59 KW tests gemacht worden

The amplitude comparison between the 6, 7 and 8-day-weeks shows that the regular week does not yield higher amplitudes than the 6 and 8-day-week---this holds in the past as well as to date (Table~\vref{table:temprange_amplitudes_30y}).
\begin{table}
\centering
\caption[Temperature range anomalies. Amplitudes for for a 6, 7 and 8-day-week]{Daily temperature range ($T_{\text{max}}-T_{\text{min}}$) anomalies. Amplitudes for a 6, 7 and 8-day-week in 30 year time steps.}\label{table:temprange_amplitudes_30y}
\begin{tabular}{l|lll}
&amp;\multicolumn{3}{l}{Amplitude [K]}\\
Period&amp;6-day-week &amp;7-day-week &amp;8-day-week \\
\hline
1887--1916&amp;0.076&amp;0.089&amp;0.033\\
1917--1946&amp;0.066&amp;0.086&amp;0.114\\
1947--1976&amp;0.051&amp;0.152&amp;0.074\\
1977--2006&amp;0.075&amp;0.065&amp;0.181
\end{tabular} 
\end{table}

\section{Sunshine Duration}
Considering the period between 1992 and 2007, Friday was the day with the highest sunshine duration while Thursday had the lowest. The mean difference is 16.1\,minutes. (Figure~\vref{fig:sunshine_weekly_cycle_1992_2006})
\begin{figure}
\centering
\input ./images/sunshine/plots_AoS_RM_1992-01-01_bis_2006-12-31_005.pdftex_t
\caption[Weekly cycle of sunshine duration]{Weekly cycle of daily sunshine duration anomaly averaged over all stations. Period: 1992-01-01 to 2006-12-31. Errorbars: $\pm1$ standard error.}\label{fig:sunshine_weekly_cycle_1992_2006}
\end{figure}

The \KW{} test implies no significance for this cycle. The same is true for the test with the shortened and extended weeks: the amplitudes are of the same range as for the normal week (Table~\vref{table:sunshine_amplitude_kw}).
\begin{table}
\centering
\caption[Sunshine duration anomalies. Amplitudes and $p$-values for a 6, 7 and 8-day-week.]{Daily sunshine duration anomalies. Amplitudes and $p$-values of the \KW{} test for a 6, 7 and 8-day-week. Period: 1992-01-01 to 2006-12-31} \label{table:sunshine_amplitude_kw}
\begin{tabular}{l|lll}
parameter&amp;6-day-week&amp;7-day-week&amp;8-day-week\\
\hline
Amplitude [minute]&amp;10.65&amp;16.1&amp;20.66\\
$p$-value&amp;0.85&amp;0.49&amp;0.29
\end{tabular} 
\end{table}
A look into the past and applying \WDT{} shows that an amplitude of this range is absolutely normal and therefore could easily be random.

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\section{Pressure}
The weekly pressure course has an amplitude of 0.134\,hPa (Figure~\vref{fig:pressure_cycle_1992_2006}). As Figure~\vref{fig:mastercount_grosswetterlagen} at the beginning of this Chapter permits to assume, the pressure cycle is not significant. The \KW{} test results in a $p$-value of 0.99 and the 6 and 8-day-week cycles deliver 5 times higher amplitudes than the regular week, namely almost 0.7\,hPa. In this respect it would be more justified to assume a 6 or 8-day periodicity than a 7-day cycle.
\begin{figure}
\centering
\input ./images/pressure/plots_AoS_RM_1992-01-01_bis_2006-12-31_005.pdftex_t
\caption[Weekly cycle of pressure anomaly]{Weekly cycle of pressure anomaly averaged over all stations. Period: 1992-01-01 to 2006-12-31. Errorbars: $\pm1$ standard error.}\label{fig:pressure_cycle_1992_2006}
\end{figure}

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\section{Four Seasons}
Considering the weekly temperature and precipitation anomalies for each season (Spring, Summer, Autumn, Winter), no significant cycle can be detected either. This can be verified by the application of a \KW{} test ($p$-values for precipitation in Table~\vref{tab:seasons_rain_KW}).
\begin{table}
\centering
\caption[Precipitation anomalies. $p$-values since 1870 in 30 year steps by season.]{Daily precipitation anomalies. $p$-values of the \KW{} test since 1870 in 30 year steps by season.}
\begin{tabular}{l|r@{.}lr@{.}lr@{.}lr@{.}l}
 &amp; \multicolumn{2}{l}{$p$-values}\\
Periods &amp; \multicolumn{2}{l}{Spring} &amp; \multicolumn{2}{l}{Summer} &amp; \multicolumn{2}{l}{Autumn} &amp; \multicolumn{2}{l}{Winter}\\
\hline
1870-01-01--1899-12-31 &amp; 0&amp;127 &amp; 0&amp;164 &amp; 0&amp;992 &amp; 0&amp;243\\
1900-01-01--1929-12-31 &amp; 0&amp;900 &amp; 0&amp;351 &amp; 0&amp;501 &amp; 0&amp;836\\
1930-01-01--1959-12-31 &amp; 0&amp;625 &amp; 0&amp;990 &amp; 0&amp;514 &amp; 0&amp;773\\
1960-01-01--1989-12-31 &amp; 0&amp;961 &amp; 0&amp;831 &amp; 0&amp;992 &amp; 0&amp;505\\
\label{tab:seasons_rain_KW}
\end{tabular} 
\end{table}

Figure~\vref{fig:seasons_rain} shows the precipitation anomaly per season, for the period between 1960 and 1990.
The comparison of the plots visualises, that the maximum and minimum change rather coincidentally  among the seasons while the weekly range remains approximately the same. The weekly range varies is between 0.27 an 0.34\,mm which is distinctly higher than the average over all seasons together. However the dataset is four times smaller. Compared to the amplitudes of the 15 year periods they are nothing particular.
\begin{figure}
\centering
\subfigure[Spring]{
 \input ./images/seasons/precipitation/plots_AoS_RM_spring_1960-01-01_bis_1989-12-31_005.pdftex_t
 \label{fig:seasons_rain_spring}
}\hfill
\subfigure[Summer]{
 \input ./images/seasons/precipitation/plots_AoS_RM_summer_1960-01-01_bis_1989-12-31_005.pdftex_t
 \label{fig:seasons_rain_summer}
}
\subfigure[Autumn]{
 \input ./images/seasons/precipitation/plots_AoS_RM_autumn_1960-01-01_bis_1989-12-31_005.pdftex_t
 \label{fig:seasons_rain_autumn}
}\hfill
\subfigure[Winter]{
 \input ./images/seasons/precipitation/plots_AoS_RM_winter_1960-01-01_bis_1989-12-31_005.pdftex_t
 \label{fig:seasons_rain_winter}
}
\caption[Weekly cycles of precipitation by season]{Weekly cycles of daily precipitation (5:40 to 5:40 UTC) anomaly by season, averaged over all stations. Period: 1960-01-01 to 1989-12-31. Errorbars: $\pm1$ standard error. Note that the range of the ordinate varies from period to period.}\label{fig:seasons_rain}
\end{figure}

\PM{10} reveals a significant weekly cycle for each season (Table~\vref{tab:seasons_PM10_KW}).
\begin{table}
\centering
\caption[\acs{PM$_{10}$}. $p$-values by season.]{\acs{PM$_{10}$}. $p$-values of the \KW{} test by season. Period: 1998-01-01 to 2006-12-31}
\begin{tabular}{l|r@{.}l}
Season &amp; \multicolumn{2}{l}{$p$-value}\\
\hline
Spring &amp; 2&amp;41$\cdot 10^{-12}$\\ % 2&amp;411625e-12\\
Summer &amp; 1&amp;51$\cdot 10^{-06}$\\ % 1&amp;509175e-06\\
Autumn &amp; 8&amp;32$\cdot 10^{-07}$\\ % 8&amp;321645e-07\\
Winter &amp; 8&amp;05$\cdot 10^{-04}$ % 0&amp;8049048e-03
\label{tab:seasons_PM10_KW}
\end{tabular} 
\end{table}

The amplitudes however vary sizeably between 4.78\,$\mu$g\,m$^{-3}$ for summers and 9.20\,$\mu$g\,m$^{-3}$ for springs (Figure~\vref{fig:seasons_PM10}). It is remarkable that in summer, while \PM{10} reveals its smallest amplitude, precipitation reveals its highest one.
\begin{figure}
\centering
\subfigure[Spring]{
 \input ./images/seasons/PM10/plots_AoS_RM_spring_1998-01-01_bis_2006-12-31_005.pdftex_t
 \label{fig:seasons_PM10_spring}
}\hfill
\subfigure[Summer]{
 \input ./images/seasons/PM10/plots_AoS_RM_summer_1998-01-01_bis_2006-12-31_005.pdftex_t
 \label{fig:seasons_PM10_summer}
}
\subfigure[Autumn]{
 \input ./images/seasons/PM10/plots_AoS_RM_autumn_1998-01-01_bis_2006-12-31_005.pdftex_t
 \label{fig:seasons_PM10_autumn}
}\hfill
\subfigure[Winter]{
 \input ./images/seasons/PM10/plots_AoS_RM_winter_1998-01-01_bis_2006-12-31_005.pdftex_t
 \label{fig:seasons_PM10_winter}
}
\caption[Weekly cycles of \acs{PM$_{10}$} by season]{Weekly cycles of \acs{PM$_{10}$} anomaly by season, averaged over the following stations: Basel, Bern, Lugano, ZÃ¼rich. Period: 1998-01-01 to 2006-12-31. Errorbars: $\pm1$ standard error. Note that the range of the ordinate varies from period to period.}\label{fig:seasons_PM10}
\end{figure}

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\section{Summer 2007: a Case Study}\label{results:sec:summer2007}
The summer 2007---here, only the time period between June 1\st{} and September the 30\kth{} is considered---had particularly often sunny and warm weekends while the midweekdays often were colder, cloudier and with more precipitation. In the following temperature, sunshine duration, \PM{10}, atmospheric pressure and precipitation anomaly are analysed  as well as the number of the sundry `weather conditions' on each weekday. Furthermore we analyse these anomalies with the help of a spectral analysis and try to find a periodicity by dint of a Fourier fitting curve.

The temperature anomaly reveals a rather clear weekly cycle with an amplitude of more than 2.5Â°C (Figure~\vref{fig:temp_anomaly_summer2007}). The temperature minimum is on Thursday while the maximum falls on Sunday. Compared to the results for a longer time series, e.\,g., those in Section~\vref{results:temperature} and those by \citet{baeumer2007feb}, who found an amplitude around 0.2Â°, this is more than 10 times larger.

The temperature range anomaly reveals a weekly course with a minimum on Tuesday and a maximum on Sunday (Figure~\vref{fig:t_range_anomaly_summer2007}). Contemplating the amplitude one gets a range of more than two degrees.
\begin{figure}
\centering
\subfigure[Mean temperature anomaly.
]{
 \input ./images/sommer2007/temperature/plots_AoS_RM_2007-06-01_bis_2007-09-30_005.pdftex_t
 \label{fig:temp_anomaly_summer2007}
}\hfill
\subfigure[Temperature range ($T_{\text{max}}-T_{\text{min}}$) anomaly
]{
 \input ./images/sommer2007/range/plots_AoS_RM_2007-06-01_bis_2007-09-30_005.pdftex_t
 \label{fig:t_range_anomaly_summer2007}
}
\caption[Weekly cycles for summer 2007. Temperature and temperature range.]{Weekly cycles of anomalies for summer 2007, averaged over the following stations: Binningen, Col du Grand St-Bernard, La BrÃ©vine, Lugano, Matro, Schaffhausen, Segl-Maria, Zollikofen, ZÃ¼rich. Period: 2007-06-01 to 2007-09-30. Errorbars $\pm1$ standard error.}
\end{figure}

The anomaly of the sunshine duration (Figure~\vref{fig:summer_sunshine}) looks similar to the one with the temperature. The sunshine-maximum is on Saturday while the minimum falls on Tuesday. The amplitude is almost 200 minutes, so that generally %(in the mean) 
there is exceeding %(over)
three hours more sunshine on Sundays than on Tuesdays---at least during the summer 2007.
\begin{figure}
\centering
\subfigure[Sunshine duration anomaly, averaged over the following stations: Binningen, Col du Grand St-Bernard, Lugano, Schaffhausen, Zollikofen, ZÃ¼rich.]{
 \input ./images/sommer2007/sunshine/plots_AoS_RM_2007-06-01_bis_2007-09-30_005.pdftex_t
 \label{fig:summer_sunshine}
}\hfill
\subfigure[Daily precipitation anomaly, averaged over the following stations: Binningen, Col du Grand St-Bernard, La BrÃ©vine, Matro, Schaffhausen, Segl-Maria, Zollikofen, ZÃ¼rich.]{
 \input ./images/sommer2007/precipitation/plots_AoS_RM_2007-06-01_bis_2007-09-30_005.pdftex_t
 \label{fig:precip_anomaly_summer2007}
}
\caption[Weekly cycles for summer 2007. Sunshine duration and precipitation.]{Weekly cycles of anomalies for summer 2007. Period: 2007-06-01 to 2007-09-30. Errorbars $\pm1$ standard error.}
\end{figure}

The anomaly in precipitation has its maximum on Wednesday and its minimum on Saturday (Figure~\vref{fig:precip_anomaly_summer2007}). The anomaly range exceeds 8\,mm and is hence more than an order of magnitude higher than the amplitudes in a 15 year time series (Section~\vref{results:precipitation_anomaly}).

Saturday is the day with the highest mean in atmospheric pressure, the minimum falls on Monday. Compared to the longer time series, the range of about 4.5\,hPa is extremely high (Figure~\vref{fig:press_anomaly_summer2007}).
\begin{figure}
\centering
\subfigure[Boxplot]{
 \input ./images/sommer2007/pressure/plots_AoS_RM_2007-06-01_bis_2007-09-30_002.pdftex_t
 \label{fig:summer_pressure_boxplot}
}\hfill
\subfigure[Errorbars: $\pm1$ standard error]{
 \input ./images/sommer2007/pressure/plots_AoS_RM_2007-06-01_bis_2007-09-30_005.pdftex_t
 \label{fig:summer_pressure_cycle}
}
\caption[Weekly cycle of pressure for summer 2007]{Weekly cycle of air pressure at sea level (\acs{QFF}) for summer 2007, averaged over the following stations: Binningen, Lugano, Schaffhausen, Zollikofen, ZÃ¼rich. Period: 2007-06-01 to 2007-09-30. Errorbars $\pm1$ standard error.}\label{fig:press_anomaly_summer2007}
\end{figure}
%gehÃ¶rt evtl auch in den diskussionsteil
The weekly cycle in the atmospheric pressure indicates the reason for the sunny and warm weekends during the summer 2007 and a look at the distribution of the `weather conditions' per weekday (Figure~\vref{fig:count_weather_conditions_summer2007}) verifies the hypothesis %suspicion 
that it has to be due to dynamical reasons.

\begin{figure}
\centering
\input ./images/wetterlagen/sommer_wetterlagen_001.pdftex_t
\caption[Distribution of the different weather conditions per 
weekday during the summer]{Distribution of the different weather conditions per weekday during the summer. Period: 2007-06-01 to 2007-09-30}\label{fig:count_weather_conditions_summer2007}
\end{figure}

The weekly course of \PM{10} is---compared with those of a longer time series (Section~\vref{results:sec:pm10})---rather unusual: the \PM{10} maximum falls on Monday and the minimum on Tuesday while the range is a little less than 4.5\,$\mu$g\,m$^{-3}$ (Figure~\vref{fig:summer_PM10}).
\begin{figure}
\centering
 \input ./images/sommer2007/PM10/plots_AoS_RM_2007-06-01_bis_2007-09-30_005.pdftex_t
\caption[Weekly cycle of \acs{PM$_{10}$} anomaly for summer 2007]{Weekly cycle of \acs{PM$_{10}$} anomaly for summer, averaged over the following stations: Basel, Bern, Jungfrau, Lugano, ZÃ¼rich. Period: 2007-06-01 to 2007-09-30. Errorbars $\pm1$ standard error.}\label{fig:summer_PM10}
\end{figure}

According to the \KW{} test all these cycles except the \PM{10} are significant on a 10\% level ($p$-values are in Table~\vref{table:kw_test_summer2007}).
\begin{table}
\centering
\caption[$p$-values for summer 2007]{$p$-values of the \KW{} test for summer 2007. Period: 2007-06-01 to 2007-09-30.}
\begin{tabular}{lr@{.}l}
Parameter&amp;	\multicolumn{2}{l}{$p$-value}\\
\hline
\PM{10}	&amp;	0&amp;964\\
pressure&amp;	6&amp;09$\cdot 10^{-3}$\\
precipitation &amp;	4&amp;19$\cdot 10^{-2}$\\
sunshine&amp;	9&amp;66$\cdot 10^{-2}$\\
temperature&amp;	8&amp;45$\cdot 10^{-2}$\\
temperature range &amp;	3&amp;33$\cdot 10^{-2}$
\label{table:kw_test_summer2007}
\end{tabular} 
\end{table}

A further analysis of the anomalies has been done in \ac{MATLAB}, by generating a fitting curve with help of the ``general model Fourier 1'' of the form:
$$f(x) =  a_0 + a_1 \cdot \cos{(x \cdot w)} + b_1 \cdot \sin{(x \cdot w)}$$
For the temperature and the pressure anomaly the fitting curve reveals a periodicity of 7 days within the 95\% confidence interval (Figure~\vref{fig:matlab_summer_pressure_fitting} and Table~\vref{tab:summer_matlab}).
\begin{figure}
\centering
\input ./images/sommer2007/pressure/matlab_pressure_fit.pdftex_t
% \includegraphics[viewport=85 50 710 507, clip, width=\linewidth{}]{./images/sommer2007/pressure/matlab_sommer07_pressure_fit.pdf}
\caption[Fitting curve for air pressure]{Fitting curve for air pressure at sea level (\acs{QFF}) during the summer 2007. Period: 2007-06-01 to 2007-09-30.}\label{fig:matlab_summer_pressure_fitting}
\end{figure}
\begin{table}
\centering
\caption[Periodicities of the fitting curves for temperature and pressure]{Periodicities of the fitting curves for temperature and pressure anomalies during the summer 2007. Period: 2007-06-01 to 2007-09-30. In parentheses are the 95\% confidence bounds}\label{tab:summer_matlab}
\begin{tabular}{l|ll@{--}l}
Parameter &amp;\multicolumn{3}{l}{Periodicities [d]}\\
\hline
Temperature &amp; 7.012 &amp; (7.094 &amp; 6.931)\\ % Temperature &amp; 7.0117 &amp; (7.09403 &amp; 6.93126)\\
Pressure &amp; 6.991 &amp; (7.080 &amp; 6.903) % Pressure &amp; 6.99064 &amp; (7.08044 &amp; 6.90308)
\end{tabular}
\end{table}

% Fit Curve: pressure  vs time
% General model Fourier1:
%        f(x) =  a0 + a1*cos(x*w) + b1*sin(x*w)
% Coefficients (with 95\% confidence bounds):
%        a0 =    0.001345  (-0.5314, 0.5341)
%        a1 =      0.4609  (-1.026, 1.948)
%        b1 =      -1.854  (-2.67, -1.038)
%        w =      0.8988  (0.8874, 0.9102)
% Periodicity:  6.99064 (7.08044, 6.90308)
%
% Goodness of fit:
%   SSE: 1006
%   R-square: 0.1786
%   Adjusted R-square: 0.1573
%   RMSE: 2.945

% Fit Curve: temperature vs time
%
% General model Fourier1:
%        f(x) =  a0 + a1*cos(x*w) + b1*sin(x*w)
% Coefficients (with 95\% confidence bounds):
%        a0 =    0.008192  (-0.3549, 0.3713)
%        a1 =       1.381  (0.8248, 1.937)
%        b1 =     -0.3369  (-1.338, 0.6647)
%        w =      0.8961  (0.8857, 0.9065)
% Periodicity: 7.0117 day (7.09403, 6.93126)
%
% Goodness of fit:
%   SSE: 467.6
%   R-square: 0.2053
%   Adjusted R-square: 0.1847
% RMSE: 2.008

For all other parameters (anomalies of: precipitation, \PM{10}, sunshine duration) the fitting curve reveals no 7-day periodicity.

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\section{Simulations}\label{results:sec:simulations}
Simulations can be conducted with the measured values or the \RM{}. Likewise it is feasible to differentiate between 6, 7 and 8-day in the simulations.

For all simulations the \RM{} is used and a 7-day-week.

\subsection{Random Series}\label{results:subsec:randomseries}
The two simulations described in the \nameref{chap:methods} Chapter, Section~\vref{methods:simulation_random_series}, provide roughly % oder approximately
the same results.
In both cases the simulated mean amplitude is higher as the real one. Figure~\vref{fig:simulation_1_precipitation} shows the histogram of the simulated amplitudes for precipitation.
\begin{figure}
\centering
\subfigure[The \RM{} in random order (sample)]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_001.pdftex_t
 \label{fig:simulation_precipitation_sample}
}\hfill
\subfigure[Series with the same standard deviation and length as \RM{}
]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_011.pdftex_t
 \label{fig:simulation_precipitation_distribution}
}
\caption[Histograms of 1,000 (random) simulated amplitudes for precipitation]{Histograms of 1,000 simulated amplitudes for daily precipitation (5:40 to 5:40 UTC). The amplitude of the original time series (0.144) is at the blue line. 95\% of the simulated values are within the marked vertical grey lines. Period: 1960-01-01 to 1989-12-31.
}\label{fig:simulation_1_precipitation}
\end{figure}

A histogram of the $p$-values reveals that in more than 70\% of the sim\-u\-la\-tions---the one sided Wilcoxon test---are significant ($\alpha=5\%$). The distribution of the $p$-values is not uniform (Figure~\vref{fig:simulation_1_p-values_wilcoxon}). In contrast to this the distribution of the $p$-values of the \KW{} test is uniform and around 5\% of the simulations are significant (Figure~\vref{fig:simulation_1_p-values_kw}).
\begin{figure}
\centering
\subfigure[One sided Wilcoxon test]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_015.pdftex_t
 \label{fig:simulation_1_p-values_wilcoxon}
}\hfill
\subfigure[\KW{} test
]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_014.pdftex_t
 \label{fig:simulation_1_p-values_kw}
}
\caption[Histograms of $p$-values for 1,000 simulated precipitation time series]{Histograms of $p$-values for 1,000 simulated daily precipitation (5:40 to 5:40 UTC) time series. Latter has the length and same standard deviation as \RM{} for precipitation. Values below 5\% are grey shaded. Period: 1960-01-01 to 1989-12-31.
}\label{fig:simulation_1_p-values}
\end{figure}

The weekly cycles of the first 10 simulated time series indicate that both simulations create a time series with different characteristics than the original ones (Figure~\vref{fig:simulation_1_precipitation_weekly_cycle}). The original weekly cycle is always smoother than the simulated ones---no matter which meteorological parameter is considered. The original time series must therefore have a feature that both simulations do not include.
\begin{figure}
\centering
\input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_019.pdftex_t
\caption[Weekly cycles of the first 10 simulated time series for precipitation]{Weekly cycles of the first 10 simulated time series for daily precipitation (5:40 to 5:40 UTC). The thick black line is the weekly cycle of the original \RM{}, the dotted colour lines are simulated. Period: 1960-01-01 to 1989-12-31.}\label{fig:simulation_1_precipitation_weekly_cycle}
\end{figure}

\subsection{Autocorrelation}\label{results:subsec:autocorrelation}
Figure~\vref{fig:simulation_2_precipitation_autocorrelation} shows the histogram of the simulated amplitudes. The histogram of the time series with autocorrelation $\varrho \approx 0$ (Figure~\vref{fig:simulation_2_amp_cor0}) looks similar to the results of a random simulated series (Figure~\vref{fig:simulation_1_precipitation}). This applies also to the histogram of $p$-values (Figure~\vref{fig:simulation_2_p-values_kw_cor0}) and the weekly cycles of the first 10 time series (Figure~\vref{fig:simulation_2_wc_cor0}). This has to be the case, as it is basically the same.

In contrast, the mean amplitude of simulated series with higher autocorrelation is lower. The simulated mean amplitude of series with the same autocorrelation as the original series (for precipitation $\varrho = 0.357$) is only slightly lower than the original one (Figure~\vref{fig:simulation_2_amp_correal}). However the simulations with the autocorrelation set to 0.9 are an order of magnitude smaller.
\begin{figure}
\centering
\subfigure[Series with $\varrho \approx 0$]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_031.pdftex_t
 \label{fig:simulation_2_amp_cor0}
}\hfill
\subfigure[Series with $\varrho \approx 0.36$. This corresponds to the one of the original precipitation series.
]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_021.pdftex_t
 \label{fig:simulation_2_amp_correal}
}
\hfill
\subfigure[Series with $\varrho \approx 0.9$
]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_041.pdftex_t
 \label{fig:simulation_2_amp_cor0.9}
}
\caption[Histograms of 1,000 simulated amplitudes for precipitation]{Histograms of 1,000 simulated amplitudes for daily precipitation (5:40 to 5:40 UTC). The amplitude of the original time series (0.144) is at the blue line. 95\% of the simulated values are within the marked vertical grey lines. Period: 1960-01-01 to 1989-12-31.}\label{fig:simulation_2_precipitation_autocorrelation}
\end{figure}

% \begin{figure}
% \centering
% \subfigure[Series with autocorrelation approximately 0]{
%  %\input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_015.pdftex_t
%  \label{fig:simulation_2_p-values_wilcoxon_cor0}
% }\hfill
% \subfigure[Series with approximately the autocorrelation of the original serie
% ]{
%  %\input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_014.pdftex_t
%  \label{fig:simulation_2_p-values_wilcoxon_correal}
% }
% \hfill
% \subfigure[Series with autocorrelation approximately 0.9
% ]{
%  %\input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_014.pdftex_t
%  \label{fig:simulation_2_p-values_wilcoxon_cor0.9}
% }
% \caption{p-values of Wilcoxon one sided of 1000 simulated time series for precipitation (daily precipitation total 5:40 to 5:40 UTC from 1960-01-01 to 1989-12-31).
% }\label{fig:simulation_2_p-values_wilcoxon}
% \end{figure}

From the Figure~\vref{fig:simulation_2_p-values_kw} it follows that the number of significant ($\alpha=5\%$) simulations (grey shaded in the Figure) decrease with an increasing autocorrelation ($\varrho$). Around 60 of 1000 simulations have a $p$-value below 5\% if the autocorrelation is 0 (Figure~\vref{fig:simulation_2_p-values_kw_cor0}), slightly more then 20 if the autocorrelation is 0.357 and 0 if the autocorrelation is very high (0.9).
\begin{figure}
\centering
\subfigure[Series with $\varrho \approx 0$]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_034.pdftex_t
 \label{fig:simulation_2_p-values_kw_cor0}
}\hfill
\subfigure[Series with $\varrho \approx 0.36$. This corresponds to the one of the original series.
]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_024.pdftex_t
 \label{fig:simulation_2_p-values_kw_correal}
}
\hfill
\subfigure[Series with $\varrho \approx 0.9$
]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_044.pdftex_t
 \label{fig:simulation_2_p-values_kw_cor0.9}
}
\caption[Histograms of $p$-values for 1,000 simulated precipitation time series with different autocorrelations]{Histograms of $p$-values of the \KW{} test of 1,000 simulated daily precipitation (5:40 to 5:40 UTC) time series with different autocorrelations ($\varrho$). Period: 1960-01-01 to 1989-12-31.
}\label{fig:simulation_2_p-values_kw}
\end{figure}

A higher autocorrelation smoothes the weekly cycle as shown in Figure~\vref{fig:simulation_2_wc}.
\begin{figure}
\centering
\subfigure[Series with $\varrho \approx 0$]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_039.pdftex_t
 \label{fig:simulation_2_wc_cor0}
}\hfill
\subfigure[Series with $\varrho \approx 0.36$. This corresponds to the one of the original series.
]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_029.pdftex_t
 \label{fig:simulation_2_wc_correal}
}
\hfill
\subfigure[Series with $\varrho \approx 0.9$
]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_049.pdftex_t
 \label{fig:simulation_2_wc_cor0.9}
}
\caption[Weekly cycles of the first 10 simulated precipitation time series]{Weekly cycles of the first 10 simulated daily precipitation (5:40 to 5:40 UTC) time series. The thick black line is the weekly cycle of the original \RM{}, the dotted colour lines are simulated. Period: 1960-01-01 to 1989-12-31.
}\label{fig:simulation_2_wc}
\end{figure}

% \begin{figure}
% \centering
% \subfigure[Series with autocorrelation approximately 0]{
%  %\input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_015.pdftex_t
%  \label{fig:simulation_2_ts_cor0}
% }\hfill
% \subfigure[Series with approximately the autocorrelation of the original serie
% ]{
%  %\input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_014.pdftex_t
%  \label{fig:simulation_2_ts_correal}
% }
% \hfill
% \subfigure[Series with autocorrelation approximately 0.9
% ]{
%  %\input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_014.pdftex_t
%  \label{fig:simulation_2_ts_cor0.9}
% }
% \caption{Time series of the first 10 simulated time series for precipitation (daily precipitation total 5:40 to 5:40 UTC from 1960-01-01 to 1989-12-31). The strong black line is the original \RM{}, the dotted colour ones are the ones from the simulations.).
% }\label{fig:simulation_2_ts}
% \end{figure}

\subsection{Take Samples}
If samples of the length of a week are taken from the time series, the mean simulated amplitude is slightly higher than the real one and the weekly cycle is as smooth as the original one (Figure~\vref{fig:simulation_3}).
\begin{figure}
\centering
\subfigure[Histograms of 1,000 simulated amplitudes for daily precipitation (5:40 to 5:40 UTC). The amplitude of the original time series (0.144) is at the blue line. 95\% of the simulated values are within the marked vertical grey lines.
]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_051.pdftex_t
 \label{fig:simulation_3_amplitude}
}\hfill
\subfigure[Weekly cycle of the first 10 simulated time series for daily precipitation total (5:40 to 5:40 UTC). The thick black line is the weekly cycle of the original \RM{}, the dotted colour lines are simulated.
]{
 \input ./images/simulation/precipitation/simulation_rain_1960-01-01-1989-12-31_059.pdftex_t
 \label{fig:simulation_3_wc}
}
\caption[Simulated time series with samples out of
the original time series. Histograms of simulated amplitudes and weekly cycles]{Simulated time series with samples out of
the original time series, which then exhibits the same length as the week. Period: 1960-01-01 to 1989-12-31}\label{fig:simulation_3}
\end{figure}</pre></body></html>