\subsection{Results timing alfaro}
\label{sec:results-timing-alfaro}

\subsection{First experiment}
\label{sec:first-experiment}

The first experiment did not show a clear pattern. It seems like all
the differences come from randomness, and the difference between the
glmfit and the nullfit is not significant at all, probably because the
time it takes for the glmfit itself without the reading is negligible
as compared to the time it takes to read the data. The results are
shown in Figure~\ref{fig:alfaro.timing.plot.1}.

\begin{figure}[!h]
  \centering
  \includegraphics[height = 6.5in, width= 5in, angle = -90]{alfaro.timing.plot.1.ps}
  \caption{Results of the first timing experiment (alfaro)}
  \label{fig:alfaro.timing.plot.1}
\end{figure}

\subsection{Second experiment}
\label{sec:second-experiment}

The second experiment shows a clear U shape pattern in the ``M + R/W''
and a slightly decreasing pattern in the ``R/W'' as the number subsets
increases. The results are shown in
Figure~\ref{fig:alfaro.timing.plot.2}.

\begin{figure}[!h]
  \centering
  \includegraphics[height = 6.5in, width= 5in, angle = -90]{alfaro.timing.plot.2.ps}
  \caption{Results of the second timing experiment (alfaro)}
  \label{fig:alfaro.timing.plot.2}
\end{figure}

\subsubsection{Third experiment}
\label{sec:third-experiment}

The hypothesis we want to test with this experiment is if a fix size
of number of columns per subset of $2^{11}$ was optimal for any value
of size of the data. I was sure that I was the only one using the
cluster and I generated the data in order. It was Friday November 10,
2012; from 3pm to 9:20pm.

I started by running and timing the glmfit for different values of
data size, ranging from $2^{20}$ to $2^{30}$, increasing in multiples
of 2. In all the following timings it assumed that the data has
$2^{4}$ columns.

The next step was to compare these timings with the number of columns
per subset being $2^{10}$ and $2^{12}$. Since for a number of columns
per subset of $2^{11}$ the last two sizes of the data, $2^{29}$ and
$2^{30}$, took so much time (13.26 and 25.64 minutes, respectively), I
did not include this data sizes for timings of $2^{10}$ and
$2^{12}$. The results are given in
Table~\ref{tab:alfaro.experiment.3}.

As for the hypothesis it only holds for data set of size $2^{24}$,
$2^{25}$ and $2^{26}$. Moreover, it seems like increasing the total
size of the data increases the value of the size of the subset needed
to obtain the minimum.

Finally, I noticed that the time to produce the data set very big
after running the glmfit for the size of the subset equal to $2^{11}$,
so for $2^{10}$ and $2^{12}$ I also measure the time it took to
produce the data. Surprisingly, it takes a longer time to produce the
data set with $2^{10}$ columns per set than the one with $2^{11}$, for
the case of $2^{28}$ columns. To confirm that this was not just a
something by chance, I ran the creation of the data again for the case
of $2^{10}$ columns per subset, obtaining the same outcome. The
results are given in
Table~\ref{tab:alfaro.timing.creation.experiment.3}.

\begin{table}[!h]
  \centering
  \begin{tabular}[!h]{cccc}
    \toprule
    Data set size &Subset size = $2^{10}$  &Subset size = $2^{11}$
    &Subset size = $2^{12}$\\
    \midrule
    $2^{20}$& $18.362$ &  $18.513$    &$20.429$ \\	
    $2^{21}$& $23.730$ &  $24.463$    &$22.474$ \\	
    $2^{22}$& $25.494$ &  $26.162$    &$27.471$ \\	
    $2^{23}$& $29.375$ &  $28.461$    &$28.450$ \\	
    $2^{24}$& $39.404$ &  $39.002$    &$39.487$ \\	
    $2^{25}$& $60.452$ &  $58.152$    &$59.035$ \\	
    $2^{26}$& $109.609$&  $103.685$   &$108.452$\\ 
    $2^{27}$& $201.882$&  $204.450$   &$192.075$\\ 
    $2^{28}$& $434.615$&  $399.709$   &$396.032$\\ 
    $2^{29}$&          &  $795.645$   &         \\
    $2^{30}$&          &  $1538.438$  &         \\
    \bottomrule
  \end{tabular}
  \caption{Time it takes to run a glmfit for the third experiment, in
    seconds (alfaro). Data set size is the total number of columns. Subset size  is the
    number of columns per subset.}
  \label{tab:alfaro.experiment.3}
\end{table}

\begin{table}[!h]
  \centering
  \begin{tabular}[!h]{cccc}
    \toprule
    Data set size & Subset size = $2^{10}$ & Subset size =
    $2^{10}$&  Subset size = $2^{12}$\\
    \midrule
    $2^{20}$ & $15.384$  &  $15.518$  &  $13.836$ \\ 
    $2^{21}$ & $21.447$  &  $21.364$  &  $14.455$ \\ 
    $2^{22}$ & $31.399$  &  $31.359$  &  $17.374$ \\ 
    $2^{23}$ & $53.463$  &  $53.421$  &  $24.422$ \\ 
    $2^{24}$ & $93.907$  &  $98.523$  &  $36.561$ \\ 
    $2^{25}$ & $187.461$ &  $178.767$ &  $70.642$ \\ 
    $2^{26}$ & $352.554$ &  $349.744$ &  $124.192$\\
    $2^{27}$ & $685.747$ &  $686.239$ &  $235.133$\\
    $2^{28}$ & $1354.750$&  $1360.969$&  $464.937$\\
    \bottomrule
  \end{tabular}
  \caption{Time it takes to produce the data set for experiment 3, in
    seconds (alfaro). Data set size is the total number of
    columns. Subset size is the number of columns per subset.}
  \label{tab:alfaro.timing.creation.experiment.3}
\end{table}


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