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diff --git a/macros/latex/contrib/linearregression/linearregression.dtx b/macros/latex/contrib/linearregression/linearregression.dtx new file mode 100644 index 0000000000..2eeea126f8 --- /dev/null +++ b/macros/latex/contrib/linearregression/linearregression.dtx @@ -0,0 +1,1235 @@ +%\iffalse +% file: linearregression.dtx +% author: Battista Benciolini +% contact: benciolinibattista at gmail dot com +% date: see preamble +% +% process this file with pdflatex to obtain: +% +% - linearregression.pfd (full documentation, three pass needed) +% - mainlinearregression.tex (interactive main-program document) +% - linearregression.sty (package) +% - sampledata.txt (as the name says) +% +% The author would strongly appreciate to receive +% any comment, criticism and just usage report +% +%\fi +%\iffalse +%<*ins> +\begingroup +\input docstrip.tex +\keepsilent +\preamble +------------------------------------------------------------------------ +[2024-06-10] +This file is part of the (expanded) distribution of linearregression +The author of linearregression is Battista Benciolini +<benciolinibattista at gmail dot com > +------------------------------------------------------------------------ +The author would strongly appreciate to receive +any comment, criticism and just usage report +------------------------------------------------------------------------ +This program may be used, distributed and modified under +the conditions of the LaTeX Project Public License. +(see: http://www.latex-project.org/lppl.txt) +------------------------------------------------------------------------ +\endpreamble +\askforoverwritefalse +\generate{\file{linearregression.sty}{\from{linearregression.dtx}{package}}} +\generate{\file{mainlinearregression.tex}{\from{linearregression.dtx}{main}}} +\nopreamble\nopostamble +\generate{\file{sampledata.txt}{\from{linearregression.dtx}{data}}} +\endgroup +%</ins> +%\fi +%\iffalse +%<*driver> +\documentclass[a4paper,10pt]{ltxdoc} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[lite,nobysame,non-compressed-cites]{amsrefs} +\usepackage{amsmath,amssymb,amsfonts} +\usepackage{multicol} +\usepackage{linearregression} +\usepackage{graphics} +\DeclareRobustCommand*{\Ars}{\textsf{% +\lower -.48ex\hbox{\rotatebox{-20}{A}}\kern -.3em{rs}}% +\discretionary{-}{}{\kern -.05em}\TeX\discretionary{-}{}{% +\kern -.17em}\lower -.357ex\hbox{nica}}% excerpt from some GUIT sty file +\NewDocumentCommand\vect{m}{\underline{#1}} % vector +\NewDocumentCommand\barycenter{m}{\overline{#1}} % barycenter +\NewDocumentCommand\point{}{\vect{y}} % point +\NewDocumentCommand\coeff{}{\vect{x}} % direction +\NewDocumentCommand\dx{}{\vect\delta} % direction variation +\NewDocumentCommand\vv{}{\vect{v}} % barycentric coordinates +\NewDocumentCommand\trasp{}{^{\mathsf{T}}} % traspose +\NewDocumentCommand\Renne{}{\mathbb{R}^n} % vector space +\NewDocumentCommand\dor{}{f} % distance from origin +\NewDocumentCommand\ipoint{}{i} % index for points +\NewDocumentCommand\pointsum{}{\sum_{\ipoint=1}^m} % sum over points +\NewDocumentCommand\reff{m}{(\ref{#1})} % ref in ( ) +\NewDocumentCommand\matr{m}{{#1}} % matrix +\NewDocumentCommand\mC{}{\matr{C}} % matrix C +\NewDocumentCommand\mc{}{k} % elements of matrix C +\NewDocumentCommand\mL{}{\matr{\Lambda}} % matrix lambda +\NewDocumentCommand\mX{}{\matr{X}} % matrix X +\NewDocumentCommand\spm{}{\phantom{-}} % space for the sign +\NewDocumentCommand\ctext{}{caption} % caption (a variable !) +\NewDocumentCommand\matrixtwotwo{mmmm}{ % | 2 x 2 +\begin{pmatrix} #1 & #2 \\ #3 & #4 \end{pmatrix}} % | matrix +\DeclareMathOperator\tr{tr} % trace +\DeclareMathOperator\sgn{sgn} % signum +\title{Linear regression with \LaTeX} +\author{Battista Benciolini} +\NewDocumentCommand\titleauthorfootnote{}{\begingroup% Not an elegant solution +\let\thefootnote\relax % but it is ok at the moment +\footnote{Linear regression with LaTeX - available in CTAN}% +\footnote{Battista Benciolini - contact: benciolinibattista at gmail dot com}% +\endgroup\setcounter{footnote}{0}}% +\parindent=0pt +\begin{document} +\hypersetup{hidelinks} +\maketitle +\titleauthorfootnote +\tableofcontents +\vfill +\DocInput{linearregression.dtx} +\end{document} +%</driver> +%\fi +% +% \section{Introduction: first description of the problem\label{intro}} +% I start with a quote from \Ars\ (April 2021, number 31, page 73): +% \begin{quotation} +% The physicist Mario Rossi is investigating a phenomenon, +% presumably linear, and he performs measurements in his laboratory +% to verify his hypothesis; he measures the quantity $x$ which generates +% the phenomenon and he measures also one of the characteristics +% $y$ showed by the phenomenon under the effect of the stimulation $x$. +% \\ ... \par +% Subsequently Mario graphs the data of the table to judge if the points +% reasonably follow a linear trend or not; in this regard he computes the +% parameters of the regression line and he draws this line on the graph +% in order to judge the quality of the obtained results. +% \\ ... \par +% Being a \LaTeX\ user, he thinks to kill two birds with one stone: +% using \LaTeX\ to draw the graph with the experimental data consisting +% in the $x$, $y$ points and, at the same time, to compute the +% parameter $a$ e $b$ of the regression line $y = ax+b$, +% and finally to draw also this line on the same graph. +% \end{quotation} +% A summary description of the the problem is therefore the following. +% A set of data pairs is available and each pair is represented as a point +% in the plain. A straight line is searched that optimally approximates +% the points. The first step is therefore the choice of an optimality criterion. +% This choice is the topic of the next section. \par +% From the text we also know that the possible deviation of $y$ +% with respect to the model is quite larger than the uncertainty of $x$. +% \par +% After reading the description of the problem +% of Mario Rossi I tried to produce a solution. +% In this work I will use $y_1$ and $y_2$ +% instead of $x$ and $y$ for the two measured quantities that +% will become the first and second coordinate, or abscissa and ordinate, +% in the Cartesian plane. +% \par +% The problem can be treated as a mere problem of approximation or +% alternatively as an estimation problem in the frame of a +% probabilistic description of the uncertainty. The two treatments are +% conceptually different. The probabilistic treatment produces some more +% results, but the estimation of the parameters is the same. +% On the other hand the treatment as an approximation problem is in some sense +% more immediate and requires a less extended theoretical background. +% For this reason it will be preferred here. +% I consider the original problem and also a variation +% of it based on the assumption that the two variables are known with +% the same uncertainty. The two considered situations will prove +% to be quite different. +% +% \section{Geometric definition of there optimality criteria} +% \begin{figure} +% \setlength\unitlength{4cm} +% \begin{picture}(1, 0.7)(0.,0.) +% \multiput(0.,0.)(1.1,0){3}{\line(1,0){1}} +% \multiput(0.,0.)(1.1,0){3}{\line(0,1){1}} +% \multiput(1,1)(1.1,0){3}{\line(-1,0){1}} +% \multiput(1,1)(1.1,0){3}{\line(0,-1){1}} +% \thicklines +% \multiput(0.08,0.06)(1.1,0){3}{\line(4,3){0.88}} +% \multiput(0.26,0.09)(1.1,0){3}{\circle{0.03}} +% \multiput(0.45,0.65)(1.1,0){3}{\circle{0.03}} +% \multiput(0.92,0.44)(1.1,0){3}{\circle{0.03}} +% \put(0.26,0.09){\line(0,1){0.1050}} +% \put(0.45,0.65){\line(0,-1){0.3125}} +% \put(0.92,0.44){\line(0,1){0.2500}} +% \put(1.36,0.09){\line(-1,0){0.14}} +% \put(1.55,0.65){\line( 1,0){0.4167}} +% \put(2.02,0.44){\line(-1,0){0.3333}} +% \put(2.46,0.09){\line(-3,4){0.05}} +% \put(2.65,0.65){\line(3,-4){0.15}} +% \put(3.12,0.44){\line(-3,4){0.12}} +% \end{picture} +% \caption{The three kinds of segments +% used in the definition of the objective function} +% \label{fig:criteria} +% \end{figure} +% For each point given in the plane we can consider the corresponding point +% with the same abscissa and belonging to the line. +% Remember that the line is exactly what has to be determined. +% The distance between the given point and the just defined point on the line +% is a reasonable measure of the discrepancy between the empirical data and the +% corresponding theoretical model. +% The distances we are speaking about are the length of the segments shown +% in the leftmost scheme of figure (\ref{fig:criteria}). +% To obtain a global discrepancy measure that considers all the points +% at once we perform the sum of the squares of the lengths +% of the mentioned segments. It is now clear that the two coordinates +% of the points are treated quite differently and play a different role in +% the definition of the optimality criterion. This choice is reasonable when +% the measuring errors only (or mainly) affect the second coordinate. +% The optimal line is the line that minimize the just defined +% global discrepancy. The procedure for the determination of the optimal line +% is named linear regression. +% In this work it is named \textit{classical linear regression}. +% We can easily exchange the role of the two quantities, i.e.\ we can +% imagine that the first quantity is affected by errors. +% The problem is not conceptually different. The segments plotted in the +% central picture +% of figure (\ref{fig:criteria}) represent the discrepancy between +% the empirical data and the model. +% This other procedure is named \textit{classical linear regression +% with inverted role of the coordinates}.\par +% The situation is really different if the two coordinates have to be treated +% symmetrically. +% In this case the discrepancy between +% the empirical data and the model must be defined in a purely geometrical way. +% Just the line and the points enter in the definition without any special role +% for any predefined direction. With these requirements it is quite natural +% to use the distance of each point from the line. +% Remember that the distance of a point +% from a line is intended along the shortest path, i.e.\ measured in the +% direction orthogonal to the line itself. The rightmost scheme of +% figure (\ref{fig:criteria}) shows the segments that are considered. +% The global measure of discrepancy is again obtained as the sum +% of the squares of the length of the mentioned orthogonal segments. +% The procedure that obtain the optimal line that +% minimize the just defined global discrepancy is named +% \textit{symmetrical linear regression}.\par +% Some arguments of the present section will be repeated in section +% \ref{package} from the algebraic and computational point of view. +% +% \section{General information on the proposed solution, including limitations} +% The code that implements the solution is recorded in two files, that are +% a package (sty) file and a main interactive document. +% The file |linearregression.sty| provides several commands +% that can be used in any document. The file |mainlinearregression.tex| +% provides a simple interactive user interface. +% The package described in the sections \ref{manual} and +% \ref{package} (user manual and implementation) provides the +% functions that execute the various needed operations, i.e.\ +% data input, computations, printing the numerical results and +% generating a graphic representation of data and results. +% Some auxiliary functions complete the package. +% The design of the output (tables and plots) includes some arbitrary choices. +% The style of the graphic output is quite minimalist +% (e.g.:\ no colors, no variations of line styles).\par +% +% \section{Some comments about the programming aspect of the package +% and its documentation} +% Large part of the code is written using the |expl3| language. +% (Is it also named simply L3 ? Does expl still means experimental ?) +% I have tried to be compliant with the various recommendations and +% prescriptions for a correct use of the language, +% but I probably only partly succeeded.\par +% Different more elegant and more coherent solutions probably exist +% both for the general structure of the package and for some specific part +% of the code, but this is what I have been able to do. +% Some perhaps problematic aspects are mentioned here after\par +% Several used variables are global and they are accessed by various functions. +% This makes the various parts of the package +% quite connected to each other and creates strong dependencies. \par +% The layered programming style is only partially applied. +% The partition between document command and lower level functions is present, +% but part of the low level code is directly in the document commands. +% Variants are not used.\par +% One more remarks concern the documentation. +% I was uncertain about the opportunity of using the class |l3doc|. I decided to +% remain using |ltxdoc|. This is the reason why I do not use the environment +% |macro| and the command |\cs| in the documentation of some auxiliary +% functions named according with the |expl3| standard. +% (I have just an interim far from optimal solution +% for a reasonable formatting.) +% +% \section{A ready to use simple user interface\label{main}} +% The main file asks the user for the name of a +% file containing the data and generates a one (or two) page output. +%\iffalse +%<*main> +%\fi +% \begin{macrocode} +\documentclass[a4paper]{article} +\usepackage{lmodern} +\usepackage{linearregression} +\begin{document} +\pagestyle{empty} +\lraskfilename +\lrcomputation +\lrplot{12.0}{+}{+}{-}{-} +\lrprint +\end{document} +% \end{macrocode} +%\iffalse +%</main> +%\fi +% +% \section{A user manual for the package\label{manual}} +% The various analysis of a data set and the representation of the data +% and of the results is obtained with a sequence of several commands. +% The main operations are: +% (i) selection of the data file, (ii) data imput and computation, +% (iii) printing of a table, +% (iv) printing of a picture (that can be repeated with different parameters). +% It is generally convenient to put the table and the picture(s) +% in a proper floating environment. +% The commands for the four mentioned operations are described here after. +% The first needed operation is to set the name of the data file. +% This is done with the command \DescribeMacro{\lrfilename} +% \cs{lrfilename}\marg{file} that has a mandatory argument. +% The argument is the name of the data file. As an alternative the +% command \DescribeMacro{\lraskfilename} \cs{lraskfilename} can be used. +% It asks the user to type the name of the data file in the terminal. +% \par +% The macro \DescribeMacro{\lrcomputation} +% \cs{lrcomputation} reads the data +% and performs all the computations. +% The results of the computations remain available in internal +% variables and are then used by the macro that print them +% or generates a plot. +%\par +% The macro \DescribeMacro{\lrprint} +% \cs{lrprint} generates a table with all the estimated +% parameters and some information about the data. +% \par +% The macro \DescribeMacro{\lrplot} +% \cs{lrplot}\marg{imagewidth}\marg{key1}\marg{key2}\marg{key3}\marg{key4} +% really generates the plot. The first argument is the +% width of the plot, while the height is computed according +% to the distribution of the points. The other four arguments are referred +% to the data points, to the lines determined with classical regression, +% with classical regression with inverted role of the coordinates and +% with symmetric regression. +% The four items, i.e.\ the set of points and the three lines, are drawn +% or not according to the corresponding character found in |key|$i$. +% Each item is not plotted if the character is a |-|, it is plotted in any other +% case. Furthermore the lines are accompanied by a label made by the +% corresponding |key|, unless it is just a |+|. +% \par +% Few words are necessary about the format of the data file. +% Each record of the file hold the two values related to a point. +% The two values must be separated by any number (one is needed as a minimum) of +% space and comma characters. No character different from space +% can be accepted before the first value and after the second value. +% +% \section{An example\label{example}} +% The data reported here after will be available in |sampledata.txt| +% and will be used in the example presented in this section . +%\iffalse +%<*data> +%\fi +% \begin{multicols}{4} +% \begin{macrocode} +-0.546 0.107 + 1.093 -0.510 + 1.440 1.995 + 1.414 0.991 + 0.735 1.585 +-1.848 -0.235 +-0.203 -0.292 + 1.517 0.779 + 0.559 -1.341 +-0.462 -0.437 +-0.785 -0.661 +-0.558 0.397 + 0.181 -2.616 + 0.619 1.859 +-0.223 -1.915 + 0.629 -0.534 +-1.989 -2.300 +-0.241 1.098 +-0.931 -1.613 +-1.070 0.592 + 2.341 0.413 + 1.993 -0.111 +-2.357 -0.312 +-1.975 0.140 +% \end{macrocode} +% \end{multicols} +%\iffalse +%</data> +%\fi +% +% The analysis of the sample data and the generation of a numeric table +% is operated by a code similar to the following +% (see table \ref{tab:sampledata}). \\ +% |\lrfilename{sampledata.txt}| \\ |\lrcomputation| \\ +% |\begin{table}| \\ +% | \lrprint| \\ +% | \caption{Analysis of ... }| \\ |\label{tab:sampledata}\end{table}| +% \par +% The generation of some different graphical representation of the data and of +% the results is operated by a code similar to the following +% (see figures \ref{fig:sampledataB} ).\\ +% \RenewDocumentCommand\ctext{}{LEFT The three lines are obtained with the three +% optimality criteria. (AA) classical linear regression; (BB) classical linear +% regression with inverted role of the coordinates; (S) symmetric linear +% regression. RIGHT Data points and line estimated with +% symmetric linear regression.} +% |\begin{figure}|\\|\lrplot{10.}{-}{AA}{BB}{S}| \\ +% |\lrplot{10.}{+}{-}{-}{+}| +% \\ |\caption{|\ctext|}|\\ | \label{fig:sampledataB} \end{figure}| +% +% \lrfilename{sampledata.txt} \lrcomputation +% \begin{table} \lrprint \caption{Analysis of the sample data} +% \label{tab:sampledata} \end{table} +% \begin{figure} \lrplot{6.}{-}{AA}{BB}{S} \hfill \lrplot{6.}{+}{-}{-}{+} +% \caption{\ctext} \label{fig:sampledataB} \end{figure} +% +% \section{A package for linear regression +% and the theory behind it\label{package}} +%\iffalse +%<*package> +%\fi +% +% \subsection{Math preliminaries and notation \label{prelim}} +% The coordinates of a set of $m$ points on the plane are available. +% A straight line is searched that optimally approximates the points.\par +% The coordinates of a generic point are $y_1$ and $y_2$ +% and they are collected in the vector $\point$. +% Any given point is identified with the index $\ipoint$. +% (Explicit indices $(\dots)_1$ or $(\dots)_2$ always refer to the first +% or second coordinate of a point or to the first or second component +% of a vector in the plane. +% Symbolic index $(\dots)\ipoint$ always refers to the different points. Few +% formulas require both indices $(\dots)_{1\ipoint}$, $(\dots)_{2\ipoint}$.)\par +% With more then two points a criterion of best approximation +% is needed to select the optimal line that describes the data. \par +% Lower case symbols are used for scalars. Lower case underlined +% symbols are used for vectors in the plane. Upper case symbols +% are used for matrices. +% \par +% It is possible that certain data generate an ambiguity or a singularity +% in the computation. +% The following mathematical treatment of the problem +% do not mention these situations and the code does not deal with them. +% +% \subsection{Package declaration, required package and definition of variables} +% The various macro will be provided in a package file +% that is introduced as usual. Most of the macros require +% the \LaTeX3 syntax. +% \begin{macrocode} +\ProvidesPackage{linearregression}[2024-06-10] +\RequirePackage{pict2e} +\ExplSyntaxOn +% \end{macrocode} +% The variables used in the package are defined hereafter. +% \begin{macrocode} +\ior_new:N \g_BBLR_file_ior +\tl_new:N \g_BBLR_file_name_tl +\int_new:N \g_BBLR_number_of_points_int +\fp_new:N \g_BBLR_abscissa_fp +\fp_new:N \g_BBLR_ordinate_fp +\fp_new:N \g_BBLR_mean_abscissa_fp +\fp_new:N \g_BBLR_mean_ordinate_fp +\fp_new:N \g_BBLR_abscissa_SecOrdMoment_fp +\fp_new:N \g_BBLR_ordinate_SecOrdMoment_fp +\fp_new:N \g_BBLR_mixed_SecOrdMoment_fp +\fp_new:N \g_BBLR_slope_A_fp +\fp_new:N \g_BBLR_slope_B_fp +\fp_new:N \g_BBLR_slope_S_fp +\fp_new:N \g_BBLR_intercept_A_fp +\fp_new:N \g_BBLR_intercept_B_fp +\fp_new:N \g_BBLR_intercept_S_fp +\fp_new:N \g_BBLR_cos_fp +\fp_new:N \g_BBLR_sin_fp +\fp_new:N \g_BBLR_sig_sin_fp +\fp_new:N \g_BBLR_eig_diff_fp +\fp_new:N \g_BBLR_diag_diff_fp +\tl_new:N \g_BBLR_file_line_tl +\fp_new:N \g_BBLR_min_abscissa_fp +\fp_new:N \g_BBLR_min_ordinate_fp +\fp_new:N \g_BBLR_max_abscissa_fp +\fp_new:N \g_BBLR_max_ordinate_fp +\fp_new:N \g_BBLR_min_draw_abscissa_fp +\fp_new:N \g_BBLR_max_draw_abscissa_fp +\bool_new:N \g_BBLR_data_eof_bool +\int_new:N \g_BBLR_record_length_int +\int_new:N \g_BBLR_rec_count_int +\int_new:N \g_BBLR_first_separator_int +\int_new:N \g_BBLR_last_separator_int +\str_const:Nn \c_BBLR_space_str {~} +\str_const:Nn \c_BBLR_comma_str {,} +\str_const:Nn \c_BBLR_plus_str {+} +\str_const:Nn \c_BBLR_minus_str {-} +\bool_new:N \g_BBLR_plot_points_bool +\bool_new:N \g_BBLR_plot_lineA_bool +\bool_new:N \g_BBLR_plot_lineB_bool +\bool_new:N \g_BBLR_plot_lineS_bool +\fp_new:N \g_BBLR_base_fp +\fp_new:N \g_BBLR_height_fp +\fp_new:N \g_BBLR_Xbase_fp +\fp_new:N \g_BBLR_Xheight_fp +\fp_new:N \g_BBLR_Dabscissa_fp +\fp_new:N \g_BBLR_Dordinate_fp +\fp_new:N \g_BBLR_diameter_fp +\fp_gset:Nn \g_BBLR_diameter_fp{0.2} +\fp_new:N \g_BBLR_line_base_length_fp +\fp_new:N \g_BBLR_scale_factor_fp +\str_new:N \c_BBLR_point_code_str +\str_new:N \g_BBLR_labelA_str +\str_new:N \g_BBLR_labelB_str +\str_new:N \g_BBLR_labelS_str +% \end{macrocode} +% +% \subsection{Preparing data input} +% \begin{macro}{\lrfilename} +% The command \cs{lrfilename} records the file name passed as argument. +% \begin{macrocode} +\NewDocumentCommand{\lrfilename}{m}{ +\tl_gset:Nn \g_BBLR_file_name_tl {#1} +} +% \end{macrocode} +% \end{macro} +% \begin{macro}{\lraskfilename} +% The command \cs{lraskfilename} asks for the data file name from the terminal. +% \begin{macrocode} +\NewDocumentCommand{\lraskfilename}{}{ +\ior_get_term:nN {filename ? } \g_BBLR_file_name_tl +\tl_trim_spaces:N \g_BBLR_file_name_tl +} +% \end{macrocode} +% \end{macro} +% +% \subsection{Main command declaration, computation of +% first and second order moments} +% \begin{macro}{\lrcomputation} +% The command \cs{lrcomputation} reads the data file and +% performs all the relevant computations to solve the +% proposed problem. +% \begin{macrocode} +\NewDocumentCommand{\lrcomputation}{}{% +% \end{macrocode} +% +% In the sequel it will results that the first and second order moments +% of the data provide everything needed to solve the problem. +% The barycenter of the data is defined as +% \begin{equation} +% \barycenter{\point}=\frac{1}{m}\pointsum \point_\ipoint. +% \label{barycenter} \end{equation} +% It is convenient to scan the data to accumulate the sum +% that appears in \reff{barycenter}. +% The coordinates of each point are read from the file +% and they are immediately used. +% It is therefore not necessary to globally record the data. +% \begin{macrocode} +\bool_gset_false:N \g_BBLR_data_eof_bool +\int_zero:N \g_BBLR_number_of_points_int +\fp_zero:N \g_BBLR_mean_abscissa_fp +\fp_zero:N \g_BBLR_mean_ordinate_fp +\ior_open:Nn \g_BBLR_file_ior \g_BBLR_file_name_tl +\bool_until_do:Nn \g_BBLR_data_eof_bool { + \ior_str_get:NN \g_BBLR_file_ior \g_BBLR_file_line_tl + \if_eof:w \g_BBLR_file_ior + \bool_gset_true:N \g_BBLR_data_eof_bool + \else: + \int_incr:N \g_BBLR_number_of_points_int + \BBLR_decode_data: + \fp_gset:Nn \g_BBLR_mean_abscissa_fp + {\g_BBLR_mean_abscissa_fp + \g_BBLR_abscissa_fp} + \fp_gset:Nn \g_BBLR_mean_ordinate_fp + {\g_BBLR_mean_ordinate_fp + \g_BBLR_ordinate_fp} + \fi: +} +% \end{macrocode} +% Loop ended. Now close the file and divide by the number of points. +% \begin{macrocode} +\ior_close:N \g_BBLR_file_ior +\fp_gset:Nn \g_BBLR_mean_abscissa_fp +{\g_BBLR_mean_abscissa_fp / \g_BBLR_number_of_points_int} +\fp_gset:Nn \g_BBLR_mean_ordinate_fp +{\g_BBLR_mean_ordinate_fp / \g_BBLR_number_of_points_int} +% \end{macrocode} +% +% The barycentric coordinates are defined for each point +% \begin{equation} \vv_\ipoint= \point_\ipoint - \barycenter{\point} +% \label{residual} \end{equation} +% and the empirical dispersion matrix is defined as: +% \begin{equation} \mC=\frac{1}{m}\pointsum \vv_\ipoint\vv_\ipoint\trasp . +% \label{matrixC} \end{equation} +% Superscript as in $()\trasp$ means transpose. The elements of $\mC$ are the +% second order central moments and they are denoted as: +% \begin{equation} \mC=\matrixtwotwo{\mc_{11}}{\mc_{12}}{\mc_{12}}{\mc_{22}}. +% \label{matrixCc} \end{equation} +% A second scan of the data is performed to compute the +% sums that appears in \reff{matrixC} and to determine the +% the extremal values of the coordinates. Record scan can be regulated +% by a record counter, because the the number of points is now known. +% \begin{macrocode} +\fp_zero:N \g_BBLR_abscissa_SecOrdMoment_fp +\fp_zero:N \g_BBLR_ordinate_SecOrdMoment_fp +\fp_zero:N \g_BBLR_mixed_SecOrdMoment_fp +\fp_gset_eq:NN \g_BBLR_min_abscissa_fp \g_BBLR_mean_abscissa_fp +\fp_gset_eq:NN \g_BBLR_min_ordinate_fp \g_BBLR_mean_ordinate_fp +\fp_gset_eq:NN \g_BBLR_max_abscissa_fp \g_BBLR_mean_abscissa_fp +\fp_gset_eq:NN \g_BBLR_max_ordinate_fp \g_BBLR_mean_ordinate_fp +\ior_open:Nn \g_BBLR_file_ior \g_BBLR_file_name_tl +\int_zero:N \g_BBLR_rec_count_int +\int_do_until:nn +{\g_BBLR_rec_count_int = \g_BBLR_number_of_points_int} +{ + \ior_str_get:NN \g_BBLR_file_ior \g_BBLR_file_line_tl + \int_incr:N \g_BBLR_rec_count_int + \BBLR_decode_data: + \fp_gset:Nn \g_tmpa_fp + {\g_BBLR_abscissa_fp - \g_BBLR_mean_abscissa_fp} + \fp_gset:Nn \g_tmpb_fp + {\g_BBLR_ordinate_fp - \g_BBLR_mean_ordinate_fp} + \fp_gset:Nn \g_BBLR_abscissa_SecOrdMoment_fp + {\g_BBLR_abscissa_SecOrdMoment_fp + \g_tmpa_fp * \g_tmpa_fp} + \fp_gset:Nn \g_BBLR_mixed_SecOrdMoment_fp + {\g_BBLR_mixed_SecOrdMoment_fp + \g_tmpa_fp * \g_tmpb_fp} + \fp_gset:Nn \g_BBLR_ordinate_SecOrdMoment_fp + {\g_BBLR_ordinate_SecOrdMoment_fp + \g_tmpb_fp * \g_tmpb_fp} +\fp_gset:Nn \g_BBLR_min_abscissa_fp +{min(\g_BBLR_min_abscissa_fp, \g_BBLR_abscissa_fp)} +\fp_gset:Nn \g_BBLR_min_ordinate_fp +{min(\g_BBLR_min_ordinate_fp, \g_BBLR_ordinate_fp)} +\fp_gset:Nn \g_BBLR_max_abscissa_fp +{max(\g_BBLR_max_abscissa_fp, \g_BBLR_abscissa_fp)} +\fp_gset:Nn \g_BBLR_max_ordinate_fp +{max(\g_BBLR_max_ordinate_fp, \g_BBLR_ordinate_fp)} +} +\ior_close:N \g_BBLR_file_ior +\fp_gset:Nn \g_BBLR_abscissa_SecOrdMoment_fp +{\g_BBLR_abscissa_SecOrdMoment_fp / \g_BBLR_number_of_points_int} +\fp_gset:Nn \g_BBLR_mixed_SecOrdMoment_fp +{\g_BBLR_mixed_SecOrdMoment_fp / \g_BBLR_number_of_points_int} +\fp_gset:Nn \g_BBLR_ordinate_SecOrdMoment_fp +{\g_BBLR_ordinate_SecOrdMoment_fp / \g_BBLR_number_of_points_int} +\fp_gset:Nn \g_BBLR_Dabscissa_fp +{\g_BBLR_max_abscissa_fp - \g_BBLR_min_abscissa_fp } +\fp_gset:Nn \g_BBLR_Dordinate_fp +{\g_BBLR_max_ordinate_fp - \g_BBLR_min_ordinate_fp } +% \end{macrocode} +% A single pass algorithm exists, but it is numerically less stable. +% +% \subsection{Classical linear regression \label{classical}} +% A line in the plane is described by the equation +% \begin{equation} y_2=ay_1+b \label{eqab} \end{equation} +% that contains the parameters $a$ and $b$. +% For each point it is possible to define a distance or a discrepancy +% of the experimental data with respect to the model. +% In the given problem the second coordinate is much more affected by +% errors than the first coordinate. It is therefore reasonable +% to define the approximation error of each point as +% \begin{equation} e_\ipoint=y_{2\ipoint}-ay_{1\ipoint}-b +% \label{e}\end{equation} +% i.e.\ the difference between the empirical value $y_{2\ipoint}$ +% and its model counterpart $ay_{1\ipoint}+b$. +% The global discrepancy between the data and the model is measured by the +% least square objective function defined by: +% \begin{equation} \psi=\pointsum e_\ipoint^2 \label{psiab} \end{equation} +% and the parameters $a$ and $b$ will be determined +% just by the minimization of the function $\psi$ defined in \reff{psiab}. +% \par +% In the present treatment of the regression problem as a pure +% approximation problem the definition of $\psi$ in \reff{psiab} +% seams quite arbitrary. It is anyway a convenient choice. +% \par +% Expression \reff{e} can be rewritten in the different form +% \begin{equation} +% e_\ipoint=v_{2\ipoint}-av_{1\ipoint}+\barycenter{y}_2-a\barycenter{y}_1-b +% \label{e2}\end{equation} +% so that the function to be minimized can be expressed +% as the sum of two quadratic functions: +% \begin{equation} +% \psi= +% \pointsum (v_{2\ipoint}-av_{1\ipoint})^2+ +% m(\barycenter{y}_2-a\barycenter{y}_1-b)^2 +% \label{psiab2} \end{equation} +% and the minimum can be attained considering +% the two terms one at a time. +% The second term in the right-hand side of \reff{psiab2} +% vanishes if the choice of $b$ is: +% \begin{equation} b=\barycenter{y}_2-a\barycenter{y}_1. +% \label{estb} \end{equation} +% The first term in the right-hand side of \reff{psiab2} becomes: +% \begin{equation} \psi_{(a)}=m\left(\mc_{22}-2a\mc_{12}+a^2\mc_{11}\right). +% \label{parabola} \end{equation} +% Searching the minimum of $\psi$ w.r.t.\ $a$ is therefore the search +% of the abscissa of the vertex of a parabola +% with axis parallel to the second coordinated axis. +% The result is: +% \begin{equation} a=\mc_{12}/\mc_{11} +% \label{esta} \end{equation} +% Now the slope $a$ and the intercept $b$ can be actually computed. +% \begin{macrocode} +\fp_gset:Nn \g_BBLR_slope_A_fp +{\g_BBLR_mixed_SecOrdMoment_fp / \g_BBLR_abscissa_SecOrdMoment_fp } +\fp_gset:Nn \g_BBLR_intercept_A_fp +{\g_BBLR_mean_ordinate_fp - \g_BBLR_slope_A_fp * \g_BBLR_mean_abscissa_fp} +% \end{macrocode} +% \par +% The empirical data and the estimated values of $a$ and $b$ +% can be used to compute +% the value actually attained by the residuals $e_\ipoint$ and +% by the function $\psi$. Then the index +% \begin{equation} \hat\sigma_0^2=\psi/(m-2)\end{equation} +% can be used to evaluate the general quality of the data and of the model. +% This claim is clearly quite generic. A complete understanding +% of this evaluation would require to treat the linear regression +% problem in the framework of the probabilistic estimation theory. +% The used notation is derived from that theory.\par +% If the role of the two coordinates is exchanged the result +% for $a$ becomes (still with reference to \reff{eqab}) +% \begin{equation} a=\mc_{22}/\mc_{12}.\end{equation} +% A complete treatment of this different situation would include +% the redefinition of $e_\ipoint$ and of $\psi$. +% The slope and the intercept can be computed according with +% the different assumption. +% \begin{macrocode} +\fp_gset:Nn \g_BBLR_slope_B_fp +{\g_BBLR_ordinate_SecOrdMoment_fp / \g_BBLR_mixed_SecOrdMoment_fp} +\fp_gset:Nn \g_BBLR_intercept_B_fp +{\g_BBLR_mean_ordinate_fp - \g_BBLR_slope_B_fp * \g_BBLR_mean_abscissa_fp} +% \end{macrocode} +% +% \subsection{Symmetric linear regression \label{symmetric}} +% If both the coordinates of the experimental points are affected +% by the same uncertainty it is advisable to use a more symmetric +% optimality criterion and it is convenient to use a different model equation. +% \par +% The same line can be described by a different equation, i.e.\ +% \begin{equation} x_1y_1+x_2y_2=\dor \end{equation} +% or in vector form: +% \begin{equation} \coeff\trasp\point=\dor. \label{eqvx} \end{equation} +% The parameters in \reff{eqvx} +% are the scalar $\dor$ and the elements +% of the vector $\coeff$, i.e.\ $x_1$ and $x_2$. +% The line described by \reff{eqvx} is obviously +% invariant when the three parameters are simultaneously +% scaled by a constant. The normalization condition +% \begin{equation} \coeff\trasp\coeff=1, \label{norm} \end{equation} +% supplemented by $\dor\ge 0$, +% is quite convenient because the parameters will assume +% a significant geometrical meaning: +% $\coeff$ is the unit vector orthogonal to the line and $\dor$ +% is the distance of the line from the origin. +% The expression +% \begin{equation} d=\dor-\coeff\trasp\point \label{distance} \end{equation} +% is the distance of the generic point $\point$ from the line +% with a sign that is positive for points on the same side of the origin. +% \par +% The distance of each given point from the desired optimal line +% is denoted by $d_\ipoint$. +% It has a clear intrinsic geometrical meaning and it does not +% privileges one coordinate w.r.t.\ the other. +% The function to be minimized by the optimal line is +% \begin{equation} +% \phi=\frac{1}{m}\pointsum d_\ipoint^2. \label{phi1} \end{equation} +% The parameters of \reff{eqvx} are determined by the minimization +% of the function $\phi$ that can be expressed as: +% \begin{equation} +% \phi=\frac{1}{m}\pointsum (\coeff\trasp\point_{\ipoint}-\dor)^{2} +% \label{phi2} \end{equation} +% and then, after some algebraic manipulations: +% \begin{equation} +% \phi=\coeff\trasp\mC\coeff+(\dor-\coeff\trasp\barycenter{\point})^2 +% \label{phi3}. \end{equation} +% The function $\phi$ is composed (as it was the function $\psi$) by the sum +% of two parts. The second term in the right-hand side of \reff{phi3} +% vanishes if the choice of $\dor$ is: +% \begin{equation} \dor=\coeff\trasp\barycenter{\point}. +% \label{estd} \end{equation} +% Then it is necessary to minimize the function +% \begin{equation} \phi_{(\coeff)} = \coeff\trasp\mC\coeff +% \label{quadraticfun} \end{equation} +% with the constrain $\coeff\trasp\coeff=1$. +% It can be proved that the function $\phi_{(\coeff)}$ +% is stationary if $\coeff$ is an eigenvector of \mC. \par +% The function $\phi_{(\coeff)}$ and the constrain must be combined +% using a Lagrange multiplier: +% \begin{equation} +% \Phi= \coeff\trasp\mC\coeff+\lambda(1-\coeff\trasp\coeff). +% \label{Phi} \end{equation} +% Then the stationarity points of $\Phi$ must be determined. +% Equating to zero the derivatives of $\Phi$ gives +% \begin{equation} +% \mC\coeff=\lambda\coeff +% \label{auto} \end{equation} +% i.e.\ $\coeff$ is an eigenvector of $\mC$. \par +% The same result is obtained with the following argument. +% The function $\phi_{(\coeff)}$ is stationary if its first variation +% is zero. The variation of $\coeff$ is named $\dx$ . +% It must respect the constrain, that becomes $\dx\trasp\coeff=0$. +% The first variation of $\phi_{(\coeff)}$ is $2\dx\trasp\mC\coeff$, +% and it is zero if and only if the following implication is valid: +% $\dx\trasp\coeff=0 \implies \dx\trasp\mC\coeff=0$, +% and the implication is valid if and only if the vector +% $\mC\coeff$ has the same direction of $\coeff$, i.e.\ if +% $\coeff$ is an eigenvector of $\mC$. +% \par +% The result on the optimal line +% can be described geometrically in the following way: +% (i) the optimal line includes the barycenter of the data; +% (ii) the optimal line is orthogonal to the eigenvector of +% $\mC$ corresponding to the minimum eigenvalue.\par +% The obtained result is also valid in $\Renne$. +% A set of points in $\Renne$ must be approximated by an $(n-1)$-dimensional +% affine subspace. (Other more general situations can be considered.) +% \par +% The trace of the matrix $\mC$, denoted as $\tr(\mC)$, is a measure of the +% global dispersion of the set of points. +% The minimum eigenvalue $\lambda_{\textrm{min}}$ of $\mC$ is a measure +% of the dispersion of the set of points with +% respect to the optimal affine subspace. Therefore the index +% \begin{equation} \frac{n\lambda_{\textrm{min}}}{\tr(\mC)} +% \end{equation} +% can be used as an indicator of the relative residual +% dispersion of the data around the optimal line. +% The defined index is dimensionless and it is +% always between $0$ and $1$. +% \par +% For the actual computation of $\coeff$ it is convenient to consider +% the spectral factorization of the matrix $\mC$, i.e.\ +% $\mC=\mX\mL\mX\trasp$ where $\mL$ is a diagonal matrix +% whose diagonal elements are the eigenvalues of $\mC$ +% and $\mX$ is an orthonormal matrix whose columns are +% the eigenvectors of $\mC$. The spectral factorization exists +% for any symmetric matrix, but it is specially simple for +% a $2\times 2$ matrix. +% \begin{equation} +% \matrixtwotwo{\mc_{11}}{\mc_{12}}{\mc_{12}}{\mc_{22}}= +% \matrixtwotwo{c}{-s}{s}{\spm c} +% \matrixtwotwo{\lambda_1}{0}{0}{\lambda_2} +% \matrixtwotwo{\spm c}{s}{-s}{c} +% \label{spectral}\end{equation} +% The eigenvalues can be easily obtained because +% their sum is the trace of $\mC$ +% \begin{equation} +% \lambda_1 + \lambda_2 = \mc_{11}+\mc_{22} +% \label{Sum}\end{equation} +% and their product +% is the determinant of the same matrix. +% Therefore after some manipulations it results: +% \begin{equation} +% \lambda_1 - \lambda_2 = \sqrt{(\mc_{11}-\mc_{22})^2+4\mc_{12}^2} +% \label{Difference}\end{equation} +% and the two eigenvalues are then immediately obtained. \par +% It is convenient to compute the difference of the two diagonal elements +% of the dispersion matrix and the difference of its eigenvalues. +% \begin{macrocode} +\fp_gset:Nn \g_BBLR_diag_diff_fp +{\g_BBLR_abscissa_SecOrdMoment_fp - \g_BBLR_ordinate_SecOrdMoment_fp} +\fp_gset:Nn \g_BBLR_eig_diff_fp +{sqrt(\g_BBLR_diag_diff_fp * \g_BBLR_diag_diff_fp + + 4 * \g_BBLR_mixed_SecOrdMoment_fp * \g_BBLR_mixed_SecOrdMoment_fp)} +% \end{macrocode} +% The computation of $c$ and $s$ is obtained from \reff{spectral} +% taking into account that $c^2+s^2=1$. +% From \reff{spectral} it results: +% \begin{equation} \mc_{11}-\mc_{22}=(\lambda_1-\lambda_2)(c^2-s^2) +% \label{Cos2A}\end{equation} +% and also +% \begin{equation} \mc_{12}=(\lambda_1-\lambda_2)cs +% \label{Sin2A}\end{equation} +% that is only used to determine the sign of $cs$. +% The expression for the parameters $c$ and $s$ are: +% \begin{equation} +% c=\sqrt{\frac{1}{2}+\frac{\mc_{11}-\mc_{22}}{2(\lambda_1-\lambda_2)}} +% \label{cos}\end{equation} +% \begin{equation} +% s=\sgn(\mc_{12}) +% \sqrt{\frac{1}{2}-\frac{\mc_{11}-\mc_{22}}{2(\lambda_1-\lambda_2)}} +% \label{sin}\end{equation} +% The parameters $s$ and $c$ are the sine and cosine +% of the angle between the axis of $y_1$ and the eigenvector +% corresponding to the maximum eigenvalue. \par +% They are computed using the already defined elements. +% \begin{macrocode} +\fp_gset:Nn \g_BBLR_cos_fp% +{sqrt((1 + \g_BBLR_diag_diff_fp / \g_BBLR_eig_diff_fp) / 2)} +\fp_gset:Nn \g_BBLR_sig_sin_fp {\fp_sign:n {\g_BBLR_mixed_SecOrdMoment_fp}} +\fp_gset:Nn \g_BBLR_sin_fp +{\g_BBLR_sig_sin_fp*sqrt((1-\g_BBLR_diag_diff_fp / \g_BBLR_eig_diff_fp) / 2)} +% \end{macrocode} +% The vector $\coeff$ is : +% \begin{equation} +% \coeff=\sgn(-s\barycenter{y}_1+c\barycenter{y}_2) +% \begin{pmatrix} -s \\ c\end{pmatrix}. +% \label{xhat}\end{equation} +% +%\par +% The parameter $a$ of model \reff{eqab} can be obtained as: +% \begin{equation} +% a=s/c +% \end{equation} +% Now the slope and the intercept of the optimal line corresponding to the +% symmetric criterion can be computed. +% +% \begin{macrocode} +\fp_gset:Nn \g_BBLR_slope_S_fp +{\g_BBLR_sin_fp / \g_BBLR_cos_fp } +\fp_gset:Nn \g_BBLR_intercept_S_fp +{\g_BBLR_mean_ordinate_fp - \g_BBLR_slope_S_fp * \g_BBLR_mean_abscissa_fp} +} +% \end{macrocode} +% +% The theoretical treatment of the proposed problem and the +% implementation of its numerical solution end here. +% \end{macro} +% +% \subsection{Print of table of results} +% \begin{macro}{\lrprint} +% The command \cs{lrprint} prints some info on the data +% and the results of the computations in tabular form. +% \begin{macrocode} +\NewDocumentCommand{\lrprint}{}{ +\begin{center} +\begin{tabular}{| l | r |} \hline + Data~File: & \g_BBLR_file_name_tl \\ \hline + Number~of~points: & \int_use:N\g_BBLR_number_of_points_int \\ \hline + Mean~values~of~the~coordinates: &% + $\fp_use:N \g_BBLR_mean_abscissa_fp$ \\ & + $\fp_use:N \g_BBLR_mean_ordinate_fp$ \\ \hline + Minimum~values~of~the~coordinates: &% + $\fp_use:N \g_BBLR_min_abscissa_fp$ \\ & + $\fp_use:N \g_BBLR_min_ordinate_fp$ \\ \hline + Maximum~values~of~the~coordinates: &% + $\fp_use:N \g_BBLR_max_abscissa_fp$ \\ & + $\fp_use:N \g_BBLR_max_ordinate_fp$ \\ \hline + {Second~order~moments}\phantom{xxxxxxxxx}{abscissa} &% + $\fp_use:N \g_BBLR_abscissa_SecOrdMoment_fp$ \\ + \multicolumn{1}{|r|}{mixed} & % +$\fp_use:N \g_BBLR_mixed_SecOrdMoment_fp$ ~ \\ +\multicolumn{1}{|r|}{ordinate} & % +$\fp_use:N \g_BBLR_ordinate_SecOrdMoment_fp$ \\ \hline + Slope~and~intercept~of~optimal~line & $\fp_use:N \g_BBLR_slope_A_fp$ \\ + (estimated~with~errors~in~ordinate)&$\fp_use:N \g_BBLR_intercept_A_fp$\\ \hline + Slope~and~intercept~of~optimal~line & $\fp_use:N \g_BBLR_slope_B_fp$ \\ + (estimated~with~errors~in~abscissa)&$\fp_use:N \g_BBLR_intercept_B_fp$\\ \hline + Components~of~unit~vector~along~the~line & $\fp_use:N \g_BBLR_cos_fp$ \\ + & $\fp_use:N \g_BBLR_sin_fp$ \\ + Slope~and~intercept~of~optimal~line &$\fp_use:N \g_BBLR_slope_S_fp$ \\ +(estimated~with~symmetric~regression) & + $\fp_use:N \g_BBLR_intercept_S_fp$\\ \hline +\end{tabular} +\end{center} +} +% \end{macrocode} +% \end{macro} +% +% \subsection{Plot of points and lines} +% \begin{macro}{\lrplot} +% The command \cs{lrplot} produce a framed plot of the data +% and of the regression line(s). The size of the plot and its actual +% content are determined by the arguments. +% \begin{macrocode} +\NewDocumentCommand{\lrplot}{mmmmm}{% +% \end{macrocode} +% The plotting area is divided into a main plotting area for +% the representation of points and line(s) and a small surrounding free space. +% The height is computed taking into account the distribution of the points. +% \begin{macrocode} +\fp_gset:Nn \g_BBLR_base_fp {#1} +\fp_gset:Nn \g_BBLR_Xbase_fp {\g_BBLR_base_fp - 0.6} +\fp_gset:Nn \g_BBLR_scale_factor_fp{\g_BBLR_Xbase_fp / \g_BBLR_Dabscissa_fp} +\fp_gset:Nn \g_BBLR_Xheight_fp {\g_BBLR_Dordinate_fp * \g_BBLR_scale_factor_fp} +\fp_gset:Nn \g_BBLR_height_fp {\g_BBLR_Xheight_fp + 0.6} +% \end{macrocode} +% The information about the items to be plotted is in the remaining arguments. +% \begin{macrocode} +\str_gset:Nn \g_BBLR_point_code_str {#2} +\str_gset:Nn \g_BBLR_labelA_str {#3} +\str_gset:Nn \g_BBLR_labelB_str {#4} +\str_gset:Nn \g_BBLR_labelS_str {#5} +\bool_gset:Nn \g_BBLR_plot_points_bool +{!(\str_if_eq_p:NN \g_BBLR_point_code_str \c_BBLR_minus_str)} +\bool_gset:Nn \g_BBLR_plot_lineA_bool +{!(\str_if_eq_p:NN \g_BBLR_labelA_str \c_BBLR_minus_str)} +\bool_gset:Nn \g_BBLR_plot_lineB_bool +{!(\str_if_eq_p:NN \g_BBLR_labelB_str \c_BBLR_minus_str)} +\bool_gset:Nn \g_BBLR_plot_lineS_bool +{!(\str_if_eq_p:NN \g_BBLR_labelS_str \c_BBLR_minus_str)} +% \end{macrocode} +% The unit of length is $1$ centimeter. The plotting area is framed. +% \begin{macrocode} +\setlength{\unitlength}{1.0cm} +\fp_gset:Nn \g_tmpa_fp {\g_BBLR_Xbase_fp +0.2} +\fp_gset:Nn \g_tmpb_fp {\g_BBLR_Xheight_fp +0.1} +\begin{picture}(\fp_use:N\g_BBLR_base_fp,\fp_use:N\g_BBLR_height_fp)(-0.3,-0.3) +\put(-0.1,-0.1){\line(1,0){\fp_use:N\g_tmpa_fp}} +\put(-0.1,\fp_use:N\g_tmpb_fp){\line(1,0){\fp_use:N\g_tmpa_fp}} +\fp_gset:Nn \g_tmpa_fp {\g_tmpa_fp -0.1} +\fp_gset:Nn \g_tmpb_fp {\g_tmpb_fp +0.1} +\put(-0.1,-0.1){\line(0,1){\fp_use:N\g_tmpb_fp}} +\put(\fp_use:N\g_tmpa_fp,-0.1){\line(0,1){\fp_use:N\g_tmpb_fp}} +% \end{macrocode} +% The plot of points and line(s) is obtained using auxiliary functions. +% \begin{macrocode} +\thicklines +\bool_if:nT {\g_BBLR_plot_points_bool}{\BBLR_plot_points:} +\bool_if:nT {\g_BBLR_plot_lineA_bool}{ +\BBLR_draw_line:NNN \g_BBLR_slope_A_fp\g_BBLR_intercept_A_fp\g_BBLR_labelA_str} +\bool_if:nT {\g_BBLR_plot_lineB_bool}{ +\BBLR_draw_line:NNN \g_BBLR_slope_B_fp\g_BBLR_intercept_B_fp\g_BBLR_labelB_str} +\bool_if:nT {\g_BBLR_plot_lineS_bool}{ +\BBLR_draw_line:NNN \g_BBLR_slope_S_fp\g_BBLR_intercept_S_fp\g_BBLR_labelS_str} +\end{picture} +}% +% \end{macrocode} +% \end{macro} +% +% \subsection{Functions for internal use} +% The functions listed here after are for internal +% use and are just minimally documented. \par +% The function |\BBLR_decode_data:| +% \marginpar{\raggedleft\texttt{ +% \textbackslash{}BBLR\textunderscore{}decode\textunderscore{}data:}} +% extract two numeric values from the string read from the file. +% Some tricky actions are necessary because +% a so called csv file sometime do not contains the separating commas. +% \begin{macrocode} +\cs_new_protected:Nn \BBLR_decode_data: { +\tl_trim_spaces:N \g_BBLR_file_line_tl +\int_gzero:N \g_tmpa_int +\int_gzero:N \g_BBLR_first_separator_int +\int_gzero:N \g_BBLR_last_separator_int +\int_gset:Nn \g_BBLR_record_length_int { +\str_count:N \g_BBLR_file_line_tl} +\str_map_variable:NNn \g_BBLR_file_line_tl \g_tmpa_str { +\int_gincr:N \g_tmpa_int +\bool_lazy_or:nnTF +{\str_if_eq_p:NN \g_tmpa_str \c_BBLR_comma_str} +{\str_if_eq_p:NN \g_tmpa_str \c_BBLR_space_str} +{\int_gset_eq:NN \g_BBLR_last_separator_int \g_tmpa_int +\int_if_zero:nTF {\g_BBLR_first_separator_int} +{\int_gset_eq:NN \g_BBLR_first_separator_int \g_tmpa_int +}{\prg_do_nothing:} +}{\prg_do_nothing:} +} +\int_gincr:N \g_BBLR_last_separator_int +\int_gdecr:N \g_BBLR_first_separator_int +\fp_gset:Nn \g_BBLR_abscissa_fp{ +\str_range:Nnn \g_BBLR_file_line_tl{1}{\g_BBLR_first_separator_int}} +\fp_gset:Nn \g_BBLR_ordinate_fp{ +\str_range:Nnn \g_BBLR_file_line_tl +{\g_BBLR_last_separator_int}{\g_BBLR_record_length_int}} +} +% \end{macrocode} +% The function |\BBLR_plot_points:| \marginpar{\raggedleft\texttt{ +% \textbackslash{}BBLR\textunderscore{}plot\textunderscore{}points:}} +% scans the data file to read the coordinates and +% it draws a circle for each point. +% +% \begin{macrocode} +\cs_new_protected:Nn \BBLR_plot_points: { +\ior_open:Nn \g_BBLR_file_ior \g_BBLR_file_name_tl +\int_zero:N \g_BBLR_rec_count_int +\int_do_until:nn +{\g_BBLR_rec_count_int = \g_BBLR_number_of_points_int} +{ + \ior_str_get:NN \g_BBLR_file_ior \g_BBLR_file_line_tl + \int_incr:N \g_BBLR_rec_count_int + \BBLR_decode_data: + \fp_gset:Nn \g_tmpa_fp{(\g_BBLR_abscissa_fp-\g_BBLR_min_abscissa_fp)* + \g_BBLR_scale_factor_fp} + \fp_gset:Nn \g_tmpb_fp{(\g_BBLR_ordinate_fp-\g_BBLR_min_ordinate_fp)* + \g_BBLR_scale_factor_fp} + \put(\fp_use:N\g_tmpa_fp, \fp_use:N\g_tmpb_fp){ + {\circle*{\fp_use:N\g_BBLR_diameter_fp}}} +} +\ior_close:N \g_BBLR_file_ior +} +% \end{macrocode} +% The function |\BBLR_draw_line:NNN| \marginpar{\raggedleft\texttt{ +% \textbackslash{}BBLR\textunderscore{}draw\textunderscore{}line:NNN}} +% draws the line. The first two parameters given as arguments +% are the slope and the intercept. The third parameter is a label. +% \par The next code finds the intersection of the line with the plotting area. +% \begin{macrocode} +\cs_new_protected:Nn \BBLR_draw_line:NNN { +\fp_gset:Nn \fp_tmpa_fp {#1 * \g_BBLR_min_abscissa_fp + #2 } +\fp_compare:nTF{\fp_tmpa_fp > \g_BBLR_max_ordinate_fp}{ +\fp_gset:Nn \g_BBLR_min_draw_abscissa_fp {(\g_BBLR_max_ordinate_fp -#2) / #1} +}{ +\fp_compare:nTF{\fp_tmpa_fp < \g_BBLR_min_ordinate_fp}{ +\fp_gset:Nn \g_BBLR_min_draw_abscissa_fp {(\g_BBLR_min_ordinate_fp - #2) / #1} +}{ +\fp_gset:Nn \g_BBLR_min_draw_abscissa_fp { \g_BBLR_min_abscissa_fp } +}} +\fp_gset:Nn \fp_tmpa_fp {#1 * \g_BBLR_max_abscissa_fp + #2 } +\fp_compare:nTF{\fp_tmpa_fp > \g_BBLR_max_ordinate_fp}{ +\fp_gset:Nn \g_BBLR_max_draw_abscissa_fp {(\g_BBLR_max_ordinate_fp -#2) / #1} +}{ +\fp_compare:nTF{\fp_tmpa_fp < \g_BBLR_min_ordinate_fp}{ +\fp_gset:Nn \g_BBLR_max_draw_abscissa_fp { (\g_BBLR_min_ordinate_fp - #2) / #1} +}{ +\fp_gset:Nn \g_BBLR_max_draw_abscissa_fp { \g_BBLR_max_abscissa_fp } +}} +% \end{macrocode} +% Some parameters (i.e.\ starting point and base-length) +% are computed and the line is drawn. +% \begin{macrocode} +\fp_gset:Nn \fp_tmpa_fp {(\g_BBLR_min_draw_abscissa_fp - +\g_BBLR_min_abscissa_fp)* \g_BBLR_scale_factor_fp} +\fp_gset:Nn \fp_tmpb_fp {(#1 * \g_BBLR_min_draw_abscissa_fp + #2 - +\g_BBLR_min_ordinate_fp)* \g_BBLR_scale_factor_fp} +\fp_gset:Nn \fp_BBLR_line_base_length_fp{(\g_BBLR_max_draw_abscissa_fp - +\g_BBLR_min_draw_abscissa_fp) * \g_BBLR_scale_factor_fp} +\put(\fp_use:N\fp_tmpa_fp, \fp_use:N\fp_tmpb_fp){ +\line(1.,\fp_use:N #1){\fp_use:N\fp_BBLR_line_base_length_fp}} +% \end{macrocode} +%The third parameter is used as a label, if it is not a |+|. +% \begin{macrocode} +\bool_if:nF {\str_if_eq_p:NN #3 \c_BBLR_plus_str}{ +\fp_gset:Nn \fp_tmpa_fp +{0.08 * \g_BBLR_min_draw_abscissa_fp + 0.92 * \g_BBLR_max_draw_abscissa_fp} +\fp_gset:Nn \fp_tmpb_fp {#1 * \fp_tmpa_fp + #2 } +\fp_gset:Nn \fp_tmpa_fp +{(\fp_tmpa_fp-\g_BBLR_min_abscissa_fp)*\g_BBLR_scale_factor_fp ++ 0.3 * #1 /sqrt(1.+#1*#1)} +\fp_gset:Nn \fp_tmpb_fp +{(\fp_tmpb_fp-\g_BBLR_min_ordinate_fp)* \g_BBLR_scale_factor_fp +- 0.3 /sqrt(1.+#1*#1)} +\put(\fp_use:N\fp_tmpa_fp, \fp_use:N\fp_tmpb_fp){#3} +} +} +% \end{macrocode} +% +% \begin{macrocode} +\ExplSyntaxOff +% \end{macrocode} +% +% +%\iffalse +%</package> +%\fi +% +% \section{Acknowledgments} +% The colleagues Paolo Zatelli, Alfonso Vitti and Giulia Graldi +% read some preliminary version +% of this text and suggested several improvements. \par +% +% \section{About the references} +% \subsection*{Mathematics} +% The books by Lang \cite{Lang} and by Strang \cite{Strang} give +% all the background on linear algebra.\par +% The texts by Sansò \cites{Sanso1, Sanso2} (in italian) treat the +% teory of probability and its application to metrology. +% See: |http://www.geolab.polimi.it/text-books/|.\par +% The paper by Karl Pearson \cite{Pearson} is the oldest text that +% I have found on the symmetric regression, or total regression. +% \subsection*{Programming} +% The two documents \cites{L3A, L3B} are the fountamental and official guide +% for \LaTeX3 programming. The books by Donald Knuth \cites{Knuth} +% and Leslie Lamport \cites{Lamport} are still essential references. +% The papers by Enrico Gregorio \cites{egreg1, egreg2, egreg3, egreg4, egreg5} +% explain some general and some special aspect of \LaTeX3 programming. +% +% \section{References} +% \begin{biblist}[\normalsize] +% \bib{egreg1}{article}{ +% author={Gregorio, Enrico}, +% journal={ArsTeXnica}, +% number={14},pages={41\ndash 47}, date={2012}, +% title={\LaTeX3: un nuovo gioco per i maghi e per diventarlo}, +% } +% \bib{egreg2}{article}{ +% author={Gregorio, Enrico}, +% journal={ArsTeXnica}, +% number={22},pages={69\ndash 77}, date={2016}, +% title={Liste, cicli, \LaTeX3}, +% } +% \bib{egreg3}{article}{ +% author={Gregorio, Enrico}, +% journal={ArsTeXnica}, +% number={24},pages={37\ndash 44}, date={2017}, +% title={Condizionali in \LaTeX}, +% } +% \bib{egreg4}{article}{ +% author={Gregorio, Enrico}, +% journal={ArsTeXnica}, +% number={30},pages={36\ndash 45}, date={2020}, +% title={Funzioni e |expl3|}, +% } +% \bib{egreg5}{article}{ +% author={Gregorio, Enrico}, +% journal={TUGboat}, +% volume={41},number={3},pages={299\ndash 307}, date={2020}, +% title={Functions and |expl3|}, +% } +% \bib{Knuth}{book}{ +% author={Knuth, Donald}, +% title={The TeXbook}, +% date={1986}, +% publisher={American Mathematical Society and Addison-Wesley}, +% } +% \bib{Lang}{book}{ +% author={Lang, Serge}, +% title={Linear Algebra}, +% date={1987}, +% publisher={Springer-Verlag}, +% place={Berlin Heidelberg}, +% } +% \bib{Lamport}{book}{ +% author={Lamport, Leslie}, +% title={LaTeX - A document preparation system (2nd ed.\ )}, +% date={1994}, +% publisher={Addison-Wesley}, +% note={something interesting in the fist edition, too}, +% } +% \bib{L3A}{article}{ +% title={The |expl3| package and LaTeX3 programming}, +% author={The LaTeX project team}, date={2024}, +% note={file: |expl3.pdf| available in CTAN in l3kernel}, +% } +% \bib{L3B}{article}{ +% title={The \LaTeX3 interface}, +% author={The LaTeX project team}, date={2024}, +% note={file: |interface3.pdf| available in CTAN in l3kernel} +% } +% \bib{Pearson}{article}{ +% title={On lines and planes of closest fit to systems of points in space}, +% author={Pearson, Karl}, date={1901}, +% journal={Philosophical Magazine}, +% volume={2},number={11},pages={559\ndash 572}, +% } +% \bib{Sanso1}{book}{ +% author={Sansò, Fernando}, +% title={Elementi di teoria della probabilità}, +% date={1996}, +% publisher={Città-Studi}, +% place={Milano}, +% } +% \bib{Sanso2}{book}{ +% author={Sansò, Fernando}, +% title={La teoria della stima}, +% date={1996}, +% publisher={Città-Studi}, +% place={Milano}, +% } +% \bib{Strang}{book}{ +% author={Strang, Gilbert}, +% title={Introduction to linear algebra}, +% date={2009}, +% publisher={Wellesley-Cambridge press,}, +% } +% \end{biblist} +% +% \par\vfill\centerline{\small ***}\vfill +% \end{document} +% +%\iffalse +% END OF FILE linearregression.dtx +%\fi |