[Tex/LaTex] Plot to illustrate secant lines

Question

I am trying for teaching purpose to explain how basically linear regression might work and how it could be extended. Since I am not the TeXpert I fear that my idea might be tricky and more difficult than it probably can be. I have seen many nice plots with tikz and pgfplots.

I searched around and found finally the following image.

This image looks quite appropriate, cause I can explain how the regression might look like. But the problem is that I cannot manipulate, e.g. draw grid, change the naming of x_0 resp \epsilon, change the red line to a dotted one, add other points than P and Q, etc. If the function itself is x^2 or some mixture, I don't mind for explanation. My main objective is to illustrate the basic ideas. And if I would have a sample, then I could extend it little bit further to mean, variance, etc.

Is anybody willing to give me a helping hand? I don't mind if somebody could point me to existing samples. Would be fine as well.

Answer 1

With TikZ it is really easy. I used Plain TeX, so you will need to \input tikz.tex instead of \usepackage{tikz} and instead of \begin{document}...\end{document} you issue \bye at the end of your document.

Typeset the following code with pdftex

\input tikz.tex
\nopagenumbers% for cropping
\usetikzlibrary{arrows,intersections}
\tikzpicture[
        thick,
        >=stealth',
        dot/.style = {
            draw,
            fill=white,
            circle,
            inner sep=0pt,
            minimum size=4pt
        }
    ]
    \coordinate (O) at (0,0);
    \draw[->] (-0.3,0) -- (8,0) coordinate[label={below:$x$}] (xmax);
    \draw[->] (0,-0.3) -- (0,5) coordinate[label={right:$f(x)$}] (ymax);
    \path[name path=x] (0.3,0.5) -- (6.7,4.7);
    \path[name path=y] plot[smooth] coordinates {(-0.3,2) (2,1.5) (4,2.8) (6,5)};
    \scope[name intersections={of=x and y,name=i}]
        \fill[gray!20] (i-1) -- (i-2 |- i-1) -- (i-2) -- cycle;
        \draw (0.3,0.5) -- (6.7,4.7) node[pos=0.8,below right] {Sekante};
        \draw[red] plot[smooth] coordinates {(-0.3,2) (2,1.5) (4,2.8) (6,5)};
        \draw (i-1) node[dot,label={above:$P$}] (i-1) {} -- node[left] {$f(x_0)$} (i-1 |- O) node[dot,label={below:$x_0$}] {};
        \path (i-2) node[dot,label={above:$Q$}] (i-2) {} -- (i-2 |- i-1) node[dot] (i-12) {};
        \draw (i-12) -- (i-12 |- O) node[dot,label={below:$x_0 + \varepsilon$}] {};
        \draw[blue,<->] (i-2) -- node[right] {$f(x_0 + \varepsilon) - f(x_0)$} (i-12);
        \draw[blue,<->] (i-1) -- node[below] {$\varepsilon$} (i-12);
        \path (i-1 |- O) -- node[below] {$\varepsilon$} (i-2 |- O);
        \draw[gray] (i-2) -- (i-2 -| xmax);
        \draw[gray,<->] ([xshift=-0.5cm]i-2 -| xmax) -- node[fill=white] {$f(x_0 + \varepsilon)$}  ([xshift=-0.5cm]xmax);
    \endscope
\endtikzpicture
\bye

This will produce the following output (cropped)

As intersections are computed by the intersections library this solution is adaptive. In this code, not actually the point Q is moved, but one of the points on the red line, which causes Q to move (if you look closely you can see, that the red line gets an ugly bump while Q moves right).

My workflow for creating animations is the following:

• Modify the source file, such that for each variation a seperate page in the output is created (in most cases using the PGF \foreach loop, like here)
• Crop the resulting PDF using Heiko Oberdiek's pdfcrop.
• Import the cropped PDF in GIMP.
• In GIMP: Reverse the layer order and export as .gif with the option As Animation checked and a delay of 200 milliseconds (otherwise it is too fast for me).

The following contains the code used to create the animation. I marked the extra and modified lines needed in contrast to the above code.

\input tikz.tex
\nopagenumbers% for cropping
\usetikzlibrary{arrows,intersections}
\foreach \Q in {4,4.1,4.2,...,5,4.9,4.8,...,4.1} {%<-- added
\tikzpicture[
        thick,
        >=stealth',
        dot/.style = {
            draw,
            fill=white,
            circle,
            inner sep=0pt,
            minimum size=4pt
        }
    ]
    \coordinate (O) at (0,0);
    \draw[->] (-0.3,0) -- (8,0) coordinate[label={below:$x$}] (xmax);
    \draw[->] (0,-0.3) -- (0,5) coordinate[label={right:$f(x)$}] (ymax);
    \path[name path=x] (0.3,0.5) -- (6.7,4.7);
    \path[name path=y] plot[smooth] coordinates {(-0.3,2) (2,1.5) (\Q,2.8) (6,5)};%<-- modified
    \scope[name intersections={of=x and y,name=i}]
        \fill[gray!20] (i-1) -- (i-2 |- i-1) -- (i-2) -- cycle;
        \draw (0.3,0.5) -- (6.7,4.7) node[pos=0.8,below right] {Sekante};
        \draw[red] plot[smooth] coordinates {(-0.3,2) (2,1.5) (\Q,2.8) (6,5)};%<-- modified
        \draw (i-1) node[dot,label={above:$P$}] (i-1) {} -- node[left] {$f(x_0)$} (i-1 |- O) node[dot,label={below:$x_0$}] {};
        \path (i-2) node[dot,label={above:$Q$}] (i-2) {} -- (i-2 |- i-1) node[dot] (i-12) {};
        \draw (i-12) -- (i-12 |- O) node[dot,label={below:$x_0 + \varepsilon$}] {};
        \draw[blue,<->] (i-2) -- node[right] {$f(x_0 + \varepsilon) - f(x_0)$} (i-12);
        \draw[blue,<->] (i-1) -- node[below] {$\varepsilon$} (i-12);
        \path (i-1 |- O) -- node[below] {$\varepsilon$} (i-2 |- O);
        \draw[gray] (i-2) -- (i-2 -| xmax);
        \draw[gray,<->] ([xshift=-0.5cm]i-2 -| xmax) -- node[fill=white] {$f(x_0 + \varepsilon)$}  ([xshift=-0.5cm]xmax);
    \endscope
\endtikzpicture
\eject%<-- added
}%<-- added
\bye