# Equation with summation - regression plotting - pgfplots

I am trying to 3D plot the below equation in tikz. I have basic premise setup like this, with sample set as datatable.

\documentclass{article}
\usepackage{pgfplots}

\pgfplotsset{compat=1.15}
\begin{document}

% sample points
\pgfplotstableread{
X Y Z m
2.2 14 0 0
2.7 23 0 0
3 13 0 0
3.55 22 0 0
4 15 0 0
4.5 20 0 0
4.75 28 0 0
5.5 23 0 0
}\datatablet

% Trying to recreate gnuplot here
% https://en.wikipedia.org/wiki/Partial_derivative
\begin{tikzpicture}
\begin{axis}[
grid=both,
ztick={0,4,...,10},
zmin = 0, zmax = 10,
point meta min=0,
point meta max=10,
colormap/cool,
view={30}{30}  %tune here to change viewing angle
]

\addplot3[surf,shader=faceted, domain=-2:2] {(x - y)^2};

\end{axis}
\end{tikzpicture}

\end{document}


What is confusing me is, the summation part on RHS. How do I compute that? For each x and y value, I need to take table entry one after one, compute, sum up and then proceed for plotting. I could not think of a simple for loop scenario to do this. Kindly help.

Below is the expected image (this was drawn using python): Here is a simple document on how I constructed it in python if it could be hlepful (especially for that looping and summation part)

• Did you try to implement the sum via tikzmath, as done here? – marmot Oct 25 '18 at 11:29
• not yet, that was only single series $x_i$, now I have $y_i$, so yet to try that. – Parthiban Rajendran Oct 25 '18 at 12:46
• Implementing the y list is straightforward. What prevents me from writing an answer is that there is a clash between fpu and math, see e.g. here. That is, I am able to implement the function and to compute points but cannot plot the function. If I switch off fpu, then the dimensions become too large. – marmot Oct 25 '18 at 17:56

## 1 Answer

To my own surprise, it is possible to implement such sums, and it is not even particularly difficult. I focus on the sum. The way to go is to just patch the sum together in a loop and let pgfplots parse it. Amazingly, this works.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\pgfplotstableread{
X Y Z m
2.2 14 0 0
2.7 23 0 0
3 13 0 0
3.55 22 0 0
4 15 0 0
4.5 20 0 0
4.75 28 0 0
5.5 23 0 0
}\datatablet
% from https://tex.stackexchange.com/a/445369/121799
\newcommand*{\ReadOutElement}{%
\pgfplotstablegetelem{#2}{#3}\of{#1}%
\let#4\pgfplotsretval
}
% based on https://tex.stackexchange.com/a/307032/121799
% and https://tex.stackexchange.com/a/451326/121799
\newcommand{\GetColumn}{
\pgfplotstablegetrowsof{\datatablet}
\pgfmathtruncatemacro{\rownumber}{\pgfplotsretval-1}
\xdef#2{0} % add a zero to have some 0th entry
\foreach \XX in {0,...,\rownumber}
{
\ReadOutElement{\datatablet}{\XX}{#1}{\tmp}
\xdef#2{#2,\tmp}
}
}
% read out x and y values
\GetColumn{X}{\xvalues}
\xdef\xvalues{{\xvalues}}
\GetColumn{Y}{\yvalues}
\xdef\yvalues{{\yvalues}}
% define the single terms in the sum: \x and \y represent x_i and y_i
% while \a and \b stand for \beta_0 and \beta_1, respectively
\tikzset{
declare function={myn(\x,\y,\a,\b)=(\y-(\a+\b*\x))*(\y-(\a+\b*\x));
}}

\pgfplotstablegetrowsof{\datatablet}
\pgfmathtruncatemacro{\rownumber}{\pgfplotsretval}
\pgfmathsetmacro{\tmpx}{\xvalues}
\pgfmathsetmacro{\tmpy}{\yvalues}
\xdef\tmpN{myn(\tmpx,\tmpy,x,y)}
\foreach \X in {2,...,\rownumber}
{
\pgfmathsetmacro{\tmpx}{\xvalues[\X]}
\pgfmathsetmacro{\tmpy}{\yvalues[\X]}
\xdef\tmpN{\tmpN+myn(\tmpx,\tmpy,x,y)}
}
\typeout{N(x,y)=\tmpN}
\begin{tikzpicture}
\begin{axis}[
grid=both,
colormap/cool,
view={30}{30}  %tune here to change viewing angle
]
\addplot3[surf,shader=faceted, domain=-40:40,domain y=-40:40] {-1*(\tmpN)};
\end{axis}
\end{tikzpicture}
\end{document} Up to some x dir=reverse and/or y dir=reverse, which one might add, this looks awfully similar to your python plot. The performance is also a bit better than what one gets with tikzmath, which I couldn't use here anyway because of a clash with fpu.

... and this is a slightly shorter version doing the same thing.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\pgfplotstableread{
X Y Z m
2.2 14 0 0
2.7 23 0 0
3 13 0 0
3.55 22 0 0
4 15 0 0
4.5 20 0 0
4.75 28 0 0
5.5 23 0 0
}\datatablet
% from https://tex.stackexchange.com/a/445369/121799
\newcommand*{\ReadOutElement}{%
\pgfplotstablegetelem{#2}{#3}\of{#1}%
\let#4\pgfplotsretval
}
% define the single terms in the sum: \x and \y represent x_i and y_i
% while \a and \b stand for \beta_0 and \beta_1, respectively
\tikzset{
declare function={myn(\x,\y,\a,\b)=(\y-(\a+\b*\x))*(\y-(\a+\b*\x));
}}

\pgfplotstablegetrowsof{\datatablet}
\pgfmathtruncatemacro{\rownumber}{\pgfplotsretval-1}
\ReadOutElement{\datatablet}{0}{X}{\tmp}
\xdef\tmpx{\tmp}
\ReadOutElement{\datatablet}{0}{Y}{\tmp}
\xdef\tmpy{\tmp}
\xdef\tmpN{myn(\tmpx,\tmpy,x,y)}
\foreach \X in {1,...,\rownumber}
{
\ReadOutElement{\datatablet}{\X}{X}{\tmp}
\xdef\tmpx{\tmp}
\ReadOutElement{\datatablet}{\X}{Y}{\tmp}
\xdef\tmpy{\tmp}\xdef\tmpN{\tmpN+myn(\tmpx,\tmpy,x,y)}
}
\typeout{N(x,y)=\tmpN}
\begin{tikzpicture}
\begin{axis}[
grid=both,
colormap/cool,
view={30}{30}  %tune here to change viewing angle
]
\addplot3[surf,shader=faceted, domain=-40:40,domain y=-40:40] {-1*(\tmpN)};
\end{axis}
\end{tikzpicture}
\end{document}

• oh thank you very much for this attempt marmot. I was thinking of exploring tikzmath, but would have not slightest clue if it failed due to my limited knowledge and would have spend many more time pondering about it. your timely support highly helps me make progress and focus more on my content than on bringing it in a graph. – Parthiban Rajendran Oct 26 '18 at 5:45