\subsection{Fitting Lines -- Regression} \label{sec:linefitting} { \pgfkeys{ /pgfmanual/gray key prefixes={/pgfplots/table}, } This section documents the attempts of \PGFPlots\ to fit lines to input coordinates. \PGFPlots\ currently supports |create col/linear regression| applied to columns of input tables. The feature relies on \PGFPlotstable, it is actually implemented as a table postprocessing method. \begin{stylekey}{/pgfplots/table/create col/linear regression=\marg{key-value-config}}% \pgfkeys{ /pgfmanual/gray key prefixes={/pgfplots/table/create col/linear regression/}, /pdflinks/search key prefixes in/.add={/pgfplots/table/create col/linear regression/,}{}, } A style for use in |\addplot table| which computes a linear (least squares) regression $y(x) = a \cdot x + b$ using the sample data $(x_i,y_i)$ which has to be specified inside of \meta{key-value-config} (see below). It creates a new column on-the-fly which contains the values $y(x_i) = a \cdot x_i + b$. The values $a$ and $b$ will be stored (globally) into \declareandlabel{\pgfplotstableregressiona} and \declareandlabel{\pgfplotstableregressionb}. \begin{codeexample}[] \begin{tikzpicture} \begin{axis}[legend pos=outer north east] \addplot table[row sep=\\] {% plot X versus Y. This is original data. X Y\\ 1 1 \\ 2 4\\ 3 9\\ 4 16\\ 5 25\\ 6 36\\ }; \addplot table[row sep=\\, y={create col/linear regression={y=Y}}] % compute a linear regression from the input table { X Y\\ 1 1 \\ 2 4\\ 3 9\\ 4 16\\ 5 25\\ 6 36\\ }; %\xdef\slope{\pgfplotstableregressiona} %<-- might be handy occasionally \addlegendentry{$y(x)$} \addlegendentry{% $\pgfmathprintnumber{\pgfplotstableregressiona} \cdot x \pgfmathprintnumber[print sign]{\pgfplotstableregressionb}$} \end{axis} \end{tikzpicture} \end{codeexample} The example above has two plots: one showing the data and one containing the |linear regression| line. We use |y={create col/linear regression={}}| here, which means to create a new column\footnote{The \texttt{y=\{create col/} feature is available for any other \PGFPlotstable\ postprocessing style, see the \texttt{create on use} documentation in the \PGFPlotstable\ manual.} containing the regression values automatically. As arguments, we need to provide the $y$ column name explicitly\footnote{In fact, \PGFPlots\ sees that there are only two columns and uses the second as default. But you need to provide it if there are at least 3 columns.}. The $x$ value is determined from context: |linear regression| is evaluated inside of |\addplot table|, so it uses the same $x$ as |\addplot table| (i.e.\ if you write |\addplot table[x=|\marg{col name}|]|, the regression will also use \marg{col name} as its |x| input). Furthermore, it shows the line parameters $a$ and $b$ in the legend. The following \meta{key-value-config} keys are accepted as comma--separated list: \begin{key}{% /pgfplots/table/create col/linear regression/table=\marg{\textbackslash macro {\normalfont or} file name} (initially empty)} Provides the table from where to load the |x| and |y| columns. It defaults to the currently processed one, i.e.\ to the value of |\pgfplotstablename|. \end{key} \begin{keylist}{% /pgfplots/table/create col/linear regression/x=\marg{column} (initially empty), /pgfplots/table/create col/linear regression/y=\marg{column} (initially empty)} Provides the source of $x_i$ and $y_i$ data, respectively. The argument \meta{column} is usually a column name of the input table, yet it can also contains |[index]|\meta{integer} to designate column indizes (starting with $0$), |create on use| specifications or |alias|es (see the \PGFPlotstable\ manual for details on |create on use| and |alias|). The initial configuration (an empty value) checks the context where the |linear regression| is evaluated. If it is evaluated inside of |\pgfplotstabletypeset|, it uses the first and second table columns. If it is evaluated inside of |\addplot table|, it uses the same $x$ input as the |\addplot table| statement. The |y| key needs to be provided explicitly (unless the table has only two columns). \end{keylist} \begin{keylist}{% /pgfplots/table/create col/linear regression/xmode=\mchoice{auto,linear,log} (initially auto), /pgfplots/table/create col/linear regression/ymode=\mchoice{auto,linear,log} (initially auto)} Enables or disables processing of logarithmic coordinates. Logarithmic processing means to apply $\ln$ before computing the regression line and $\exp$ afterwards. The choice |auto| checks if the column is evaluated inside of a \PGFPlots\ axis. If so, it uses the axis scaling of the embedding axis. Otherwise, it uses |linear|. In case of logarithmic coordinates, the |log basis x| and |log basis y| keys determine the basis. \begin{codeexample}[] \begin{tikzpicture} \begin{loglogaxis} \addplot table[x=dof,y=error2] {pgfplotstable.example1.dat}; \addlegendentry{$y(x)$} \addplot table[ x=dof, y={create col/linear regression={y=error2}}] {pgfplotstable.example1.dat}; % might be handy occasionally: %\xdef\slope{\pgfplotstableregressiona} \addlegendentry{slope $\pgfmathprintnumber{\pgfplotstableregressiona}$} \end{loglogaxis} \end{tikzpicture} \end{codeexample} The (commented) line containing |\slope| is explained above; it allows to remember different regression slopes in our example. \end{keylist} \begin{keylist}{% /pgfplots/table/create col/linear regression/variance list=\marg{list} (initially empty),% /pgfplots/table/create col/linear regression/variance=\marg{column name} (initially empty)% } Both keys allow to provide uncertainties (variances) to single data points. A high (relative) variance indicates an unreliable data point, a value of $1$ is standard. The |variance list| key allows to provide variances directly as comma--separated list, for example |variance list={1000,1000,500,200,1,1}|. The |variance| key allows to load values from a table \meta{column name}. Such a column name is (initially, see below) loaded from the same table where data points have been found. The \meta{column name} may also be a |create on use| name. \begin{codeexample}[] \begin{tikzpicture} \begin{loglogaxis} \addplot table[x=dof,y=error2] {pgfplotstable.example1.dat}; \addlegendentry{$y(x)$} \addplot table[ x=dof, y={create col/linear regression={ y=error2, variance list={1000,800,600,500,400}} } ] {pgfplotstable.example1.dat}; \addlegendentry{slope $\pgfmathprintnumber{\pgfplotstableregressiona}$} \end{loglogaxis} \end{tikzpicture} \end{codeexample} If both, |variance list| and |variance| are given, the first one will be preferred. Note that it is not necessary to provide variances for every data point. \end{keylist} \begin{key}{/pgfplots/table/create col/linear regression/variance src=\marg{\textbackslash table {\normalfont or} file name} (initially empty)} Allows to load the |variance| from another table. The initial setting is empty. It is acceptable if the |variance| column in the external table has fewer entries than expected, in this case, only the first ones will be used. \end{key} \end{stylekey} \paragraph{Limitations:} Currently, \PGFPlots\ supports only linear regression, and it only supports regression together with |\addplot table|. Furthermore, long input tables might need quite some time. }