
\section{Solving a Real Use Case: Scientific Data Analysis}

In this section, we assume that you did some scientific experiment. The
scientific experiment yielded three input data tables: one table for each
involved parameter $d=2$, $d=3$, $d=4$. The data tables contain ``degrees of
freedom'' and some accuracy measurement ``|l2_err|''. In addition, they might
contain some meta-data (in our case a column ``level''). For example, the data
table for $d=2$ might be stored in |data_d2.dat| and may contain
%
\begin{codeexample}[code only]
dof        l2_err     level
5          8.312e-02  2
17         2.547e-02  3
49         7.407e-03  4
129        2.102e-03  5
321        5.874e-04  6
769        1.623e-04  7
1793       4.442e-05  8
4097       1.207e-05  9
9217       3.261e-06  10
\end{codeexample}
%
The other two tables are similar, we provide them here to simplify the
reproduction of the examples. The table for $d=3$ is stored in |data_d3.dat|,
it is
%
\begin{codeexample}[code only]
dof        l2_err     level
7          8.472e-02  2
31         3.044e-02  3
111        1.022e-02  4
351        3.303e-03  5
1023       1.039e-03  6
2815       3.196e-04  7
7423       9.658e-05  8
18943      2.873e-05  9
47103      8.437e-06  10
\end{codeexample}
%
Finally, the last table is |data_d4.dat|
%
\begin{codeexample}[code only]
dof        l2_err     level
9          7.881e-02  2
49         3.243e-02  3
209        1.232e-02  4
769        4.454e-03  5
2561       1.551e-03  6
7937       5.236e-04  7
23297      1.723e-04  8
65537      5.545e-05  9
178177     1.751e-05  10
\end{codeexample}

What we want is to produce three plots, each dof versus |l2_err|, in a loglog
plot. We expect that the result is a line in a log--log plot, and we are
interested in its slope $\log e(N) = -a \log(N)$ because that characterizes our
experiment.


\subsection{Getting the Data into \TeX{}}
\label{sec:tut2:step1}

Our first step is to get one of our data tables into \PGFPlots{}. In addition,
we want axis descriptions for the $x$- and $y$-axes and a title on top of the
plot.

Our first version looks like
%
\begin{codeexample}[]
%\documentclass{article}
%\usepackage{pgfplots}
%\pgfplotsset{compat=1.5}

%\begin{document}

\begin{tikzpicture}
\begin{loglogaxis}[
    title=Convergence Plot,
    xlabel={Degrees of freedom},
    ylabel={$L_2$ Error},
]
    \addplot table {data_d2.dat};
\end{loglogaxis}
\end{tikzpicture}

%\end{document}
\end{codeexample}

Our example is similar to that of the lecture in Section~\ref{sec:tut1:step1}
in that it defines some basic axis descriptions by means of |title|, |xlabel|,
and |ylabel| and provides data using |\addplot table|. The only difference is
that we used |\begin{loglogaxis}| instead of |\begin{axis}| in order to
configure logarithmic scales on both axes. Note furthermore that we omitted any
options after |\addplot|. As explained in Section~\ref{sec:tut1:step1}, this
tells \PGFPlots{} to consult its |cycle list| to determine a suitable option
list.


\subsection{Adding the Remaining Data Files of Our Example.}
\label{sec:tut2:step2}

\PGFPlots{} accepts more than one |\addplot ... ;| command -- so we can just
add our remaining data files:
%
\begin{codeexample}[]
\begin{tikzpicture}
\begin{loglogaxis}[
    title=Convergence Plot,
    xlabel={Degrees of freedom},
    ylabel={$L_2$ Error},
]
    \addplot table {data_d2.dat};
    \addplot table {data_d3.dat};
    \addplot table {data_d4.dat};
\end{loglogaxis}
\end{tikzpicture}
\end{codeexample}

You might wonder how \PGFPlots{} chose the different line styles. And you might
wonder how to modify them. Well, if you simply write |\addplot| without options
in square brackets, \PGFPlots{} will automatically choose styles for that
specific plot. Here ``automatically'' means that it will consult its current
|cycle list|: a list of predefined styles such that every |\addplot| statement
receives one of these styles. This list is customizable to a high degree.

Instead of the |cycle list|, you can easily provide style options manually. If
you write

|\addplot[|\meta{options}|] ...|,

\PGFPlots{} will only use \meta{options} and will ignore its |cycle list|. If
you write a plus sign before the square brackets as in

|\addplot+[|\meta{options}|] ...|,

\PGFPlots{} will append \meta{options} to the automatically assigned cycle
list.


\subsection{Add a Legend and a Grid}
\label{sec:tut2:step3}

A legend is a text label explaining what the plots are. A legend can be
provided for one or more |\addplot| statements using the |legend entries| key:
%
\begin{codeexample}[]
\begin{tikzpicture}
\begin{loglogaxis}[
    title=Convergence Plot,
    xlabel={Degrees of freedom},
    ylabel={$L_2$ Error},
    grid=major,
    legend entries={$d=2$,$d=3$,$d=4$},
]
    \addplot table {data_d2.dat};
    \addplot table {data_d3.dat};
    \addplot table {data_d4.dat};
\end{loglogaxis}
\end{tikzpicture}
\end{codeexample}
%
Here, we assigned a comma-separated list of text labels, one for each of our
|\addplot| instructions. Note the use of math mode in the text labels. Note
that if any of your labels contains a comma, you have to surround the entry by
curly braces. For example, we could have used
|legend entries={{$d=2$},{$d=3$},{$d=4$}}| -- \PGFPlots{} uses these braces to
delimit arguments and strips them afterwards (this holds for any option, by the
way).

Our example also contains grid lines for which we used the |grid=major| key. It
activates major grid lines in all axes.

You might wonder how the text labels map to |\addplot| instructions. Well, they
are mapped by index. The first label is assigned to the first plot, the second
label to the second plot and so on. You can exclude plots from this counting if
you add the |forget plot| option to the plot (using |\addplot+[forget plot]|,
for example). Such plots are excluded from both cycle lists and legends.


\subsection{Add a Selected Fit-line}
\label{sec:tut2:step4}

Occasionally, one needs to compute linear regression lines through input
samples. Let us assume that we want to compute a fit line for the data in our
fourth data table (|data_d4.dat|). However, we assume that the interesting part
of the plot happens if the number of degrees of freedom reaches some asymptotic
limit (i.e.\@ is very large). Consequently, we want to assign a high
uncertainty to the first points when computing the fit line.

\PGFPlots{} offers to combine table input and mathematical expressions (note
that you can also type pure mathematic expressions, although this is beyond the
scope of this example). In our case, we employ this feature to create a
completely new column -- the linear regression line:

\begin{codeexample}[]
%\usepackage{pgfplotstable}
%...
\begin{tikzpicture}
\begin{loglogaxis}[
    title=Convergence Plot,
    xlabel={Degrees of freedom},
    ylabel={$L_2$ Error},
    grid=major,
    legend entries={$d=2$,$d=3$,$d=4$},
]
    \addplot table {data_d2.dat};
    \addplot table {data_d3.dat};
    \addplot table {data_d4.dat};
    \addplot table [
      x=dof,
      y={create col/linear regression={y=l2_err,
       variance list={1000,800,600,500,400,200,100}}}]
        {data_d4.dat};
\end{loglogaxis}
\end{tikzpicture}
\end{codeexample}

Note that we added a further package: |pgfplotstable|. It allows to postprocess
tables (among other things. It also has a powerful table typesetting toolbox
which rounds and formats numbers based on your input CSV file).

Here, we added a fourth plot to our axis. The first plot is also an
|\addplot table| statement as before -- and we see that it loads the data file
|data_d4.dat| just like the plot before. However, it has special keys which
control the coordinate input: |x=dof| means to load x coordinates from the
column named ``dof''. This is essentially the same as in all of our other plots
(because the ``dof'' column is the first column). It also uses
|y={create col/...}|. This lengthy statement defines a completely new column.
The |create col/linear regression| prefix is a key which can be used whenever
new table columns can be generated. As soon as the table is queried for the
first time, the statement is evaluated and then used for all subsequent rows.
The argument list for |create col/linear regression| contains the column name
for the function values |y=l2_err| which are to be used for the regression line
(the x arguments are deduced from |x=dof| as you guessed correctly). The
|variance list| option is optional. We use it to assign variances
(uncertainties) to the first input points. More precisely: the first
encountered data point receives a variance of 1000, the second 800, the third
600, and so on. The number of variances does not need to match up with the
number of points; \PGFPlots{} simply matches them with the first encountered
coordinates.

Note that since our |legend entries| key contains only three values, the
regression line has no legend entry. We could easily add one, if we wanted. We
can also use |\addplot+[forget plot] table[...]| to explicitly suppress the
generation of a legend as mentioned above.

Whenever \PGFPlots{} encounters mathematical expressions, it uses its built-in
floating point unit. Consequently, it has a very high data range -- and a
reasonable precision as well.


\subsection{Add an Annotation using \Tikz{}: a Slope Triangle}
\label{sec:tut2:step5}

Often, data requires interpretation -- and you may want to highlight particular
items in your plots. This ``highlight particular items'' requires to draw into
an axis, and it requires a high degree of flexibility. Users of \Tikz{} would
say that \Tikz{} is a natural choice -- and it is.

In our use case, we are interested in slopes. We may want to compare slopes of
different experiments. And we may want to show selected absolute values of
slopes.

Here, we use \Tikz{} to add custom annotations into a \PGFPlots{} axis. We
choose a particular type of a custom annotation: we want to mark two points on
a line plot. One way to do so would be to determine the exact coordinates and
to place a graphical element at this coordinate (which is possible using
|\draw ... (1e4,1e-5) ... ;|). Another (probably simpler) way is to use the
|pos| feature to identify a position ``25\% after the line started''.

Based on the result of Section~\ref{sec:tut2:step4}, we find
%
\begin{codeexample}[]
%\usepackage{pgfplotstable}
%...
\begin{tikzpicture}
\begin{loglogaxis}[
    title=Convergence Plot,
    xlabel={Degrees of freedom},
    ylabel={$L_2$ Error},
    grid=major,
    legend entries={$d=2$,$d=3$,$d=4$},
]
    \addplot table {data_d2.dat};
    \addplot table {data_d3.dat};
    \addplot table {data_d4.dat};
    \addplot table [
      x=dof,
      y={create col/linear regression={y=l2_err,
       variance list={1000,800,600,500,400,200,100}}}]
        {data_d4.dat}
            % save two points on the regression line
            % for drawing the slope triangle
            coordinate [pos=0.25] (A)
            coordinate [pos=0.4]  (B)
   ;
    % save the slope parameter:
    \xdef\slope{\pgfplotstableregressiona}

    % draw the opposite and adjacent sides
    % of the triangle
    \draw (A) -| (B)
         node [pos=0.75,anchor=west]
         {\pgfmathprintnumber{\slope}};
\end{loglogaxis}
\end{tikzpicture}
\end{codeexample}

The example is already quite involved since we added complexity in every step.
Before we dive into the details, let us take a look at a simpler example:
%
\begin{codeexample}[]
\begin{tikzpicture}
\begin{loglogaxis}
    \addplot table {data_d2.dat}
       coordinate [pos=0.25] (A)
       coordinate [pos=0.4]  (B)
    ;

    \draw [-stealth] (A) -| (B);

    \node [pin=0:Special.] at (769,1.623e-04) {};
\end{loglogaxis}
\end{tikzpicture}
\end{codeexample}
%
Here, we see two annotation concepts offered by \PGFPlots{}: the first is to
insert drawing commands right after an |\addplot| command (but before the
closing semicolon). The second is to add standard \Tikz{} commands, but use
designated \PGFPlots{} coordinates. Both are \Tikz{} concepts. The first is
what we want here: we want to identify two coordinates which are ``somewhere''
on the line. In our case, we define two named coordinates: coordinate $A$ at
25\% of the line and coordinate $B$ at 40\% of the line. Then, we use
\verb#\draw (A) -| (B)# to draw a triangle between these two points. The second
is only useful if we know some absolute coordinates in advance.

Coming back to our initial approach with the regression line, we see that it
uses the first concept: it introduces named coordinates after |\addplot|, but
before the closing semicolon. The statement |\xdef\slope| introduces a new
macro. It contains the (expanded due to the ``eXpanded DEFinition'') value of
|\pgfplotstableregressiona| which is the slope of the regression line. In
addition to the slope triangle, we also add a node in which we typeset that
value using |\pgfmathprintnumber|.

Note that the example above is actual a ``happy case'': it can happen easily
that labels which are added inside of the axis environment are clipped away:
%
\begin{codeexample}[]
\begin{tikzpicture}
\begin{loglogaxis}[
    tiny,
]
    \addplot table {data_d2.dat}
        node [pos=1,pin=0:Special.] {}
    ;
\end{loglogaxis}
\end{tikzpicture}
\end{codeexample}
%
The example above combines the |pos| label placement with the node's label.
Note that the small style |tiny| installs a \PGFPlots{} preset which is better
suited for very small plots -- it is one of the many supported scaling
parameters. The problem here is apparent: the text of our extra node is clipped
away. Depending on your data, you have a couple of solutions here:
%
\begin{itemize}
    \item use |clip=false| to disable clipping of plot paths at all,
    \item use |clip mode=individual| to enable clipping only for plot paths,
    \item draw the node outside of the axis environment but inside of the
        picture environment.
\end{itemize}
%
The first attempt works quite well for most figures:
%
\begin{codeexample}[]
\begin{tikzpicture}
\begin{loglogaxis}[
    tiny,
    clip=false,
]
    \addplot table {data_d2.dat}
        node [pos=1,pin=0:Special.] {}
    ;
\end{loglogaxis}
\end{tikzpicture}
\end{codeexample}
%
Note that this approach in which the nodes are placed before the closing
semicolon implies that nodes inherit the axis line style and color.


\subsection{Summary}

We learned how to define a (logarithmic) axis, and how to assign basic axis
descriptions. We also saw once more how to use one or more |\addplot table|
commands to load table data into \PGFPlots{}. We took a brief look into
regression and \Tikz{} drawing annotations.

We also encountered the |tiny| style which is one of the ways to customize the
size of an axis. Others are |width|, |height|, the other predefined size styles
like |normalsize|, |small|, or |footnotesize|, and the two different scaling
modes |/pgfplots/scale| and |/tikz/scale| (the first scales only the axis, the
second also the labels).

Next steps might be how to visualize functions using line plots, how to align
adjacent graphics properly (even if the axis descriptions vary), how to employ
scatter plots of \PGFPlots{}, or how to draw functions of two variables.

%The steps are explained in all detail in the reference
%Section~\ref{sec:pgfplots:reference}.
