How to do linear regression in pgfplots for data with nan?

Question

I am trying to use pgfplots to plot a linear regression to my data. If there is a nan in my data set, no fit is generated, with the warning:

NOTE: coordinate (1,nan) has been dropped because it is unbounded (in y).
(see also unbounded coords=jump).

and similar warnings for the other points in the set.

If I set an ignore chars and change the nan to that character, compiling fails with the error:

! Package pgfplots Error: input table '<inline_table>' has an unbalanced number
 of columns in row '2' (expected '2' cols; got '1'). Maybe the input table is c
orrupted? If you need unbalanced data, consider using 'nan' in empty cells (per
haps combined with 'unbounded coords=jump').

The following MWE reproduces this problem:

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\usepackage{pgfplotstable}
\pgfplotsset{compat=1.10}
\begin{document}
\begin{tikzpicture}
\begin{axis}
\addplot table[x=X,y={create col/linear regression={y=Y}}]{
X Y
1 nan
2 1
3 2
4 3
};
\end{axis}
\end{tikzpicture}
\end{document}

How can I have nan in my data set and still generate a fit, or how can I have an ignored character?

Answer 1

Currently, PGFPlots can't handle nan values when calculating the regression line. Adding this capability is quite simple though, only two lines need to be added to a (rather long) internal macro.

If you put the following code in your preamble after \usepackage{pgfplots}, your code will work:

\makeatletter
% #1: keys
\def\pgfplotstable@linear@regression#1{%
    \begingroup
    \pgfqkeys{/pgfplots/table/create col/linear regression}{/pgf/fpu,#1}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/x}{\pgfplotstable@xsrc}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/y}{\pgfplotstable@ysrc}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/table}{\pgfplotstable@table}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/xmode}{\pgfplotstable@xmode}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/ymode}{\pgfplotstable@ymode}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/variance}{\pgfplotstable@variance@colname}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/variance list}{\pgfplotstable@variance@list}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/variance src}{\pgfplotstable@variance@table}%
    %
    \ifx\pgfplotstable@table\pgfutil@empty
        \pgfutil@ifundefined{pgfplotstablename}{}{% query the name of the actual table struct
            \let\pgfplotstable@table=\pgfplotstablename
        }%
    \fi
    \ifx\pgfplotstable@table\pgfutil@empty
        \pgfplots@error{Sorry, I couldn't determine a value for create col/linear regression/table. Which table should I load?}%
    \fi
    \ifx\pgfplotstable@xsrc\pgfutil@empty
        \pgfplotsifinaddplottablestruct{%
            \pgfutil@ifundefined{pgfplots@plot@tbl@x}{}{%
                \let\pgfplotstable@xsrc=\pgfplots@plot@tbl@x
                \ifx\pgfplotstable@ysrc\pgfutil@empty
                    \pgfplotstablegetcolsof\pgfplots@table
                    \ifnum\pgfplotsretval=2
                    \else
                        \pgfplotsthrow{invalid argument}{\pgfplotstable@ysrc}{Sorry, I don't which column should be used as `y' for the linear regression. Please provide 'linear regression={y=<colname>}'}\pgfeov%
                    \fi
                \fi
            }%
        }{}%
    \fi
    \ifx\pgfplotstable@xsrc\pgfutil@empty
        \def\pgfplotstable@xsrc{[index]0}%
    \fi
    \ifx\pgfplotstable@ysrc\pgfutil@empty
        \def\pgfplotstable@ysrc{[index]1}%
    \fi
    %
    \t@pgfplots@toka=\expandafter{\pgfplotstable@table}%
    \t@pgfplots@tokb=\expandafter{\pgfplotstable@xsrc}%
    \t@pgfplots@tokc=\expandafter{\pgfplotstable@ysrc}%
    \edef\pgfplots@loc@TMPa{{\the\t@pgfplots@tokb}\noexpand\of{\the\t@pgfplots@toka}}%
    \edef\pgfplots@loc@TMPb{{\the\t@pgfplots@tokc}\noexpand\of{\the\t@pgfplots@toka}}%
    \expandafter\pgfplotstablegetcolumn\pgfplots@loc@TMPa\to\pgfplotstable@X
    \expandafter\pgfplotstablegetcolumn\pgfplots@loc@TMPb\to\pgfplotstable@Y
    %
    \edef\pgfplotstable@xmode{\pgfplotstable@xmode}%
    \expandafter\pgfplotstable@linear@regression@prepare@mode\expandafter{\pgfplotstable@xmode}{x}%%
    \edef\pgfplotstable@ymode{\pgfplotstable@ymode}%
    \expandafter\pgfplotstable@linear@regression@prepare@mode\expandafter{\pgfplotstable@ymode}{y}%%
    %
    \ifx\pgfplotstable@variance@list\pgfutil@empty
        % check 'variance' key (loaded from table)
        \pgfplotslistnewempty\pgfplotstable@VARIANCE
        \ifx\pgfplotstable@variance@colname\pgfutil@empty
        \else
            \ifx\pgfplotstable@variance@table\pgfutil@empty
                \t@pgfplots@toka=\expandafter{\pgfplotstable@table}%
                \t@pgfplots@tokb=\expandafter{\pgfplotstable@variance@colname}%
                \edef\pgfplots@loc@TMPa{{\the\t@pgfplots@tokb}\noexpand\of{\the\t@pgfplots@toka}}%
                \expandafter\pgfplotstablegetcolumn\pgfplots@loc@TMPa\to\pgfplotstable@VARIANCE
            \else
                \t@pgfplots@toka=\expandafter{\pgfplotstable@variance@colname}%
                \t@pgfplots@tokb=\expandafter{\pgfplotstable@variance@table}%
                \edef\pgfplotstable@loc@TMPa{%
                    \noexpand\pgfplotstablegetcolumn{\the\t@pgfplots@toka}\noexpand\of{\the\t@pgfplots@tokb}\noexpand\to\noexpand\pgfplotstable@VARIANCE}%
                \pgfplotstable@loc@TMPa
            \fi
        \fi
    \else
        % load from list:
        \expandafter\pgfplotslistnew\expandafter\pgfplotstable@VARIANCE\expandafter{\pgfplotstable@variance@list}%
    \fi
    %
    \pgfplotslistnewempty\pgfplotstable@Xparsed
    %
    \pgfmathfloatcreate{0}{0.0}{0}%
    \let\pgfplotstable@S=\pgfmathresult
    \let\pgfplotstable@Sxx=\pgfmathresult
    \let\pgfplotstable@Sx=\pgfmathresult
    \let\pgfplotstable@Sy=\pgfmathresult
    \let\pgfplotstable@Sxy=\pgfmathresult
    \pgfutil@loop
    \pgfplotslistcheckempty\pgfplotstable@X
    \ifpgfplotslistempty
        \pgfplots@loop@CONTINUEfalse
    \else
        \pgfplots@loop@CONTINUEtrue
    \fi
    \ifpgfplots@loop@CONTINUE
        \pgfplotslistpopfront\pgfplotstable@X\to\pgfplotstable@x
        \pgfplotslistpopfront\pgfplotstable@Y\to\pgfplotstable@y
        %
        \pgfplotstableparsex{\pgfplotstable@x}%
        \let\pgfplotstable@x=\pgfmathresult
        \expandafter\pgfplotslistpushback\pgfmathresult\to\pgfplotstable@Xparsed
        \pgfplotstableparsey{\pgfplotstable@y}%
        \let\pgfplotstable@y=\pgfmathresult
        \pgfmathfloatifflags{\pgfplotstable@y}{3}{}{% <---- This is new. The "3" stands for "nan"
        %
        \pgfplotslistcheckempty\pgfplotstable@VARIANCE
        \ifpgfplotslistempty
            \pgfmathfloatcreate{1}{1.0}{0}%
            \let\pgfplotstable@invsqr=\pgfmathresult
        \else
            \pgfplotslistpopfront\pgfplotstable@VARIANCE\to\pgfplotstable@variance
            \pgfmathfloatparsenumber{\pgfplotstable@variance}%
            \let\pgfplotstable@variance=\pgfmathresult
            \pgfmathfloatmultiply@{\pgfplotstable@variance}{\pgfplotstable@variance}%
            \let\pgfplotstable@sqr=\pgfmathresult
            \pgfmathfloatreciprocal@{\pgfplotstable@sqr}%
            \let\pgfplotstable@invsqr=\pgfmathresult
        \fi
        %
        \pgfmathfloatadd@{\pgfplotstable@S}{\pgfplotstable@invsqr}%
        \let\pgfplotstable@S=\pgfmathresult
        %
        \pgfmathfloatmultiply@{\pgfplotstable@x}{\pgfplotstable@invsqr}%
        \let\pgfplots@table@accum=\pgfmathresult
        \pgfmathfloatadd@{\pgfplotstable@Sx}{\pgfplots@table@accum}%
        \let\pgfplotstable@Sx=\pgfmathresult
        %
        \pgfmathfloatmultiply@{\pgfplotstable@x}{\pgfplots@table@accum}%
        \let\pgfplots@table@accum=\pgfmathresult
        \pgfmathfloatadd@{\pgfplotstable@Sxx}{\pgfplots@table@accum}%
        \let\pgfplotstable@Sxx=\pgfmathresult
        %
        \pgfmathfloatmultiply@{\pgfplotstable@y}{\pgfplotstable@invsqr}%
        \let\pgfplots@table@accum=\pgfmathresult
        \pgfmathfloatadd@{\pgfplotstable@Sy}{\pgfplots@table@accum}%
        \let\pgfplotstable@Sy=\pgfmathresult
        %
        \pgfmathfloatmultiply@{\pgfplotstable@x}{\pgfplots@table@accum}%
        \let\pgfplots@table@accum=\pgfmathresult
        \pgfmathfloatadd@{\pgfplotstable@Sxy}{\pgfplots@table@accum}%
        \let\pgfplotstable@Sxy=\pgfmathresult
        }% <---- This is new.
    \pgfutil@repeat
    %
    \pgfmathparse{\pgfplotstable@S * \pgfplotstable@Sxx - \pgfplotstable@Sx *\pgfplotstable@Sx}%
    \let\pgfplotstable@delta=\pgfmathresult
    %
    \pgfmathparse{(\pgfplotstable@S * \pgfplotstable@Sxy - \pgfplotstable@Sx * \pgfplotstable@Sy) / \pgfplotstable@delta}%
    \let\pgfplotstable@a=\pgfmathresult
    %
    \pgfmathparse{(\pgfplotstable@Sxx * \pgfplotstable@Sy - \pgfplotstable@Sx * \pgfplotstable@Sxy) / \pgfplotstable@delta}%
    \let\pgfplotstable@b=\pgfmathresult
    %
    \pgfplotslistnewempty\pgfplotstable@RESULT
    \pgfplotslistforeachungrouped\pgfplotstable@Xparsed\as\pgfplotstable@x{%
        \pgfmathfloatmultiply@{\pgfplotstable@x}{\pgfplotstable@a}%
        \let\pgfplotstable@tmp=\pgfmathresult
        \pgfmathfloatadd@{\pgfplotstable@tmp}{\pgfplotstable@b}%
        \ifx\pgfplotstableparseylogbase\pgfutil@empty
        \else
            \pgfplotstableparseyinv@{\pgfmathresult}%
        \fi
        \pgfmathfloattosci{\pgfmathresult}%
        \expandafter\pgfplotslistpushback\pgfmathresult\to\pgfplotstable@RESULT
    }%
    \pgfmathfloattosci\pgfplotstable@a
    \let\pgfplotstable@a=\pgfmathresult
    %
    \pgfmathfloattosci\pgfplotstable@b
    \let\pgfplotstable@b=\pgfmathresult
    %
    \global\let\pgfplotstableregressiona\pgfplotstable@a%
    \global\let\pgfplotstableregressionb\pgfplotstable@b%
    \let\pgfplotsretval=\pgfplotstable@RESULT
    \pgfmath@smuggleone\pgfplotsretval
    \endgroup
}%
\makeatother

Full code:

\documentclass[tikz, border=5mm]{standalone}
\usepackage{pgfplots}
\usepackage{pgfplotstable}
\pgfplotsset{compat=1.10}

\makeatletter
% #1: keys
\def\pgfplotstable@linear@regression#1{%
    \begingroup
    \pgfqkeys{/pgfplots/table/create col/linear regression}{/pgf/fpu,#1}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/x}{\pgfplotstable@xsrc}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/y}{\pgfplotstable@ysrc}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/table}{\pgfplotstable@table}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/xmode}{\pgfplotstable@xmode}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/ymode}{\pgfplotstable@ymode}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/variance}{\pgfplotstable@variance@colname}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/variance list}{\pgfplotstable@variance@list}%
    \pgfkeysgetvalue{/pgfplots/table/create col/linear regression/variance src}{\pgfplotstable@variance@table}%
    %
    \ifx\pgfplotstable@table\pgfutil@empty
        \pgfutil@ifundefined{pgfplotstablename}{}{% query the name of the actual table struct
            \let\pgfplotstable@table=\pgfplotstablename
        }%
    \fi
    \ifx\pgfplotstable@table\pgfutil@empty
        \pgfplots@error{Sorry, I couldn't determine a value for create col/linear regression/table. Which table should I load?}%
    \fi
    \ifx\pgfplotstable@xsrc\pgfutil@empty
        \pgfplotsifinaddplottablestruct{%
            \pgfutil@ifundefined{pgfplots@plot@tbl@x}{}{%
                \let\pgfplotstable@xsrc=\pgfplots@plot@tbl@x
                \ifx\pgfplotstable@ysrc\pgfutil@empty
                    \pgfplotstablegetcolsof\pgfplots@table
                    \ifnum\pgfplotsretval=2
                    \else
                        \pgfplotsthrow{invalid argument}{\pgfplotstable@ysrc}{Sorry, I don't which column should be used as `y' for the linear regression. Please provide 'linear regression={y=<colname>}'}\pgfeov%
                    \fi
                \fi
            }%
        }{}%
    \fi
    \ifx\pgfplotstable@xsrc\pgfutil@empty
        \def\pgfplotstable@xsrc{[index]0}%
    \fi
    \ifx\pgfplotstable@ysrc\pgfutil@empty
        \def\pgfplotstable@ysrc{[index]1}%
    \fi
    %
    \t@pgfplots@toka=\expandafter{\pgfplotstable@table}%
    \t@pgfplots@tokb=\expandafter{\pgfplotstable@xsrc}%
    \t@pgfplots@tokc=\expandafter{\pgfplotstable@ysrc}%
    \edef\pgfplots@loc@TMPa{{\the\t@pgfplots@tokb}\noexpand\of{\the\t@pgfplots@toka}}%
    \edef\pgfplots@loc@TMPb{{\the\t@pgfplots@tokc}\noexpand\of{\the\t@pgfplots@toka}}%
    \expandafter\pgfplotstablegetcolumn\pgfplots@loc@TMPa\to\pgfplotstable@X
    \expandafter\pgfplotstablegetcolumn\pgfplots@loc@TMPb\to\pgfplotstable@Y
    %
    \edef\pgfplotstable@xmode{\pgfplotstable@xmode}%
    \expandafter\pgfplotstable@linear@regression@prepare@mode\expandafter{\pgfplotstable@xmode}{x}%%
    \edef\pgfplotstable@ymode{\pgfplotstable@ymode}%
    \expandafter\pgfplotstable@linear@regression@prepare@mode\expandafter{\pgfplotstable@ymode}{y}%%
    %
    \ifx\pgfplotstable@variance@list\pgfutil@empty
        % check 'variance' key (loaded from table)
        \pgfplotslistnewempty\pgfplotstable@VARIANCE
        \ifx\pgfplotstable@variance@colname\pgfutil@empty
        \else
            \ifx\pgfplotstable@variance@table\pgfutil@empty
                \t@pgfplots@toka=\expandafter{\pgfplotstable@table}%
                \t@pgfplots@tokb=\expandafter{\pgfplotstable@variance@colname}%
                \edef\pgfplots@loc@TMPa{{\the\t@pgfplots@tokb}\noexpand\of{\the\t@pgfplots@toka}}%
                \expandafter\pgfplotstablegetcolumn\pgfplots@loc@TMPa\to\pgfplotstable@VARIANCE
            \else
                \t@pgfplots@toka=\expandafter{\pgfplotstable@variance@colname}%
                \t@pgfplots@tokb=\expandafter{\pgfplotstable@variance@table}%
                \edef\pgfplotstable@loc@TMPa{%
                    \noexpand\pgfplotstablegetcolumn{\the\t@pgfplots@toka}\noexpand\of{\the\t@pgfplots@tokb}\noexpand\to\noexpand\pgfplotstable@VARIANCE}%
                \pgfplotstable@loc@TMPa
            \fi
        \fi
    \else
        % load from list:
        \expandafter\pgfplotslistnew\expandafter\pgfplotstable@VARIANCE\expandafter{\pgfplotstable@variance@list}%
    \fi
    %
    \pgfplotslistnewempty\pgfplotstable@Xparsed
    %
    \pgfmathfloatcreate{0}{0.0}{0}%
    \let\pgfplotstable@S=\pgfmathresult
    \let\pgfplotstable@Sxx=\pgfmathresult
    \let\pgfplotstable@Sx=\pgfmathresult
    \let\pgfplotstable@Sy=\pgfmathresult
    \let\pgfplotstable@Sxy=\pgfmathresult
    \pgfutil@loop
    \pgfplotslistcheckempty\pgfplotstable@X
    \ifpgfplotslistempty
        \pgfplots@loop@CONTINUEfalse
    \else
        \pgfplots@loop@CONTINUEtrue
    \fi
    \ifpgfplots@loop@CONTINUE
        \pgfplotslistpopfront\pgfplotstable@X\to\pgfplotstable@x
        \pgfplotslistpopfront\pgfplotstable@Y\to\pgfplotstable@y
        %
        \pgfplotstableparsex{\pgfplotstable@x}%
        \let\pgfplotstable@x=\pgfmathresult
        \expandafter\pgfplotslistpushback\pgfmathresult\to\pgfplotstable@Xparsed
        \pgfplotstableparsey{\pgfplotstable@y}%
        \let\pgfplotstable@y=\pgfmathresult
        \pgfmathfloatifflags{\pgfplotstable@y}{3}{}{% <---- This is new. The "3" stands for "nan"
        %
        \pgfplotslistcheckempty\pgfplotstable@VARIANCE
        \ifpgfplotslistempty
            \pgfmathfloatcreate{1}{1.0}{0}%
            \let\pgfplotstable@invsqr=\pgfmathresult
        \else
            \pgfplotslistpopfront\pgfplotstable@VARIANCE\to\pgfplotstable@variance
            \pgfmathfloatparsenumber{\pgfplotstable@variance}%
            \let\pgfplotstable@variance=\pgfmathresult
            \pgfmathfloatmultiply@{\pgfplotstable@variance}{\pgfplotstable@variance}%
            \let\pgfplotstable@sqr=\pgfmathresult
            \pgfmathfloatreciprocal@{\pgfplotstable@sqr}%
            \let\pgfplotstable@invsqr=\pgfmathresult
        \fi
        %
        \pgfmathfloatadd@{\pgfplotstable@S}{\pgfplotstable@invsqr}%
        \let\pgfplotstable@S=\pgfmathresult
        %
        \pgfmathfloatmultiply@{\pgfplotstable@x}{\pgfplotstable@invsqr}%
        \let\pgfplots@table@accum=\pgfmathresult
        \pgfmathfloatadd@{\pgfplotstable@Sx}{\pgfplots@table@accum}%
        \let\pgfplotstable@Sx=\pgfmathresult
        %
        \pgfmathfloatmultiply@{\pgfplotstable@x}{\pgfplots@table@accum}%
        \let\pgfplots@table@accum=\pgfmathresult
        \pgfmathfloatadd@{\pgfplotstable@Sxx}{\pgfplots@table@accum}%
        \let\pgfplotstable@Sxx=\pgfmathresult
        %
        \pgfmathfloatmultiply@{\pgfplotstable@y}{\pgfplotstable@invsqr}%
        \let\pgfplots@table@accum=\pgfmathresult
        \pgfmathfloatadd@{\pgfplotstable@Sy}{\pgfplots@table@accum}%
        \let\pgfplotstable@Sy=\pgfmathresult
        %
        \pgfmathfloatmultiply@{\pgfplotstable@x}{\pgfplots@table@accum}%
        \let\pgfplots@table@accum=\pgfmathresult
        \pgfmathfloatadd@{\pgfplotstable@Sxy}{\pgfplots@table@accum}%
        \let\pgfplotstable@Sxy=\pgfmathresult
        }% <---- This is new.
    \pgfutil@repeat
    %
    \pgfmathparse{\pgfplotstable@S * \pgfplotstable@Sxx - \pgfplotstable@Sx *\pgfplotstable@Sx}%
    \let\pgfplotstable@delta=\pgfmathresult
    %
    \pgfmathparse{(\pgfplotstable@S * \pgfplotstable@Sxy - \pgfplotstable@Sx * \pgfplotstable@Sy) / \pgfplotstable@delta}%
    \let\pgfplotstable@a=\pgfmathresult
    %
    \pgfmathparse{(\pgfplotstable@Sxx * \pgfplotstable@Sy - \pgfplotstable@Sx * \pgfplotstable@Sxy) / \pgfplotstable@delta}%
    \let\pgfplotstable@b=\pgfmathresult
    %
    \pgfplotslistnewempty\pgfplotstable@RESULT
    \pgfplotslistforeachungrouped\pgfplotstable@Xparsed\as\pgfplotstable@x{%
        \pgfmathfloatmultiply@{\pgfplotstable@x}{\pgfplotstable@a}%
        \let\pgfplotstable@tmp=\pgfmathresult
        \pgfmathfloatadd@{\pgfplotstable@tmp}{\pgfplotstable@b}%
        \ifx\pgfplotstableparseylogbase\pgfutil@empty
        \else
            \pgfplotstableparseyinv@{\pgfmathresult}%
        \fi
        \pgfmathfloattosci{\pgfmathresult}%
        \expandafter\pgfplotslistpushback\pgfmathresult\to\pgfplotstable@RESULT
    }%
    \pgfmathfloattosci\pgfplotstable@a
    \let\pgfplotstable@a=\pgfmathresult
    %
    \pgfmathfloattosci\pgfplotstable@b
    \let\pgfplotstable@b=\pgfmathresult
    %
    \global\let\pgfplotstableregressiona\pgfplotstable@a%
    \global\let\pgfplotstableregressionb\pgfplotstable@b%
    \let\pgfplotsretval=\pgfplotstable@RESULT
    \pgfmath@smuggleone\pgfplotsretval
    \endgroup
}%
\makeatother

\begin{document}
\begin{tikzpicture}
\begin{axis}
\addplot [only marks] table {
X Y
1 nan
2 1
3 2.5
4 3
};
\addplot [no markers] table[x=X,y={create col/linear regression={y=Y}}]{
X Y
1 nan
2 1
3 2.5
4 3
};
\end{axis}
\end{tikzpicture}
\end{document}

Answer 2

Run with xelatex. nan is defined as a PostScript function which deletes the x value:

\RequirePackage{filecontents}
\documentclass[pstricks]{standalone}
\usepackage{pst-plot}
\begin{filecontents*}{LSM.data}
X Y
1 nan
1 1
3 1.8
4 3.3
\end{filecontents*}

\begin{document}

\begin{pspicture}(-1,-2)(5,4)
\psaxes{->}(4.5,3.5)
\readdata[ignoreLines=1]{\data}{LSM.data}
\pstVerb{/nan { pop } def }
\listplot[plotstyle=LSM,PstDebug=1,linecolor=blue]{\data}% PstDebug to print the equation
\listplot[plotstyle=dots,linecolor=blue]{\data}
\end{pspicture}

\end{document}