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pic as plot mark

pgfplots add tag

JeT

I'd like to replace the mark of my plot with a mini brownian diffusion. 

I'll probably need to embed it in a `pic`.

I have the ingredients, I need a little help (some would say I am missing a pi(e)c(e)... ) on the recipe.

I don't look for the version below with one brownian  replicated 5 times and not consistent  with the measure on the main graph.

![MAinplot2.png](/image?hash=545e69920b248944cf8f79ba2abee0d2bd7bcf1be3ecfb4e724fba8e371ae384)

I imagine more something like (Poorly copied pasted) with one brownian per period, whose coordinates are consistent with the main plot.

![TAPic000.png](/image?hash=c45665aaad7f9cab50f178befe5b346433c671cd3c8585b7411954ef0a0ff929)

**Where do I need help ?**

I'd like each brownian 
 - to start at the mark 
 - have a width of one tick for the pic
 - diffusion being consistent with the main plot grid


Details in the MWE below.

```
\documentclass[tikz]{standalone}
\usepackage{pgfplots, pgfplotstable}
\usetikzlibrary{positioning}

\tikzset{
	declare function={
 	invgauss(\a,\b) = sqrt(-2*ln(\a))*cos(deg(2*pi*\b));
 }
} 

%-----------------------------------------------------------------
% Code from Jake @ TeX.SE to generate brownian motions
% If you have an idea to load to improve the compilation time
\makeatletter
\pgfplotsset{
 table/.cd,
 brownian motion/.style={
  create on use/brown/.style={
   create col/expr accum={
 (\coordindex>0)*(
  max(
   min(
 invgauss(rnd,rnd)*0.25*sqrt(1)+\pgfmathaccuma,
 \pgfplots@brownian@max
   ),
   \pgfplots@brownian@min
  )
 ) + (\coordindex<1)*\pgfplots@brownian@start
   }{\pgfplots@brownian@start}
  },
  y=brown, x expr={\coordindex},
  brownian motion/.cd,
  #1,
  /.cd
 },
 brownian motion/.cd,
   min/.store in=\pgfplots@brownian@min,
  min=-inf,
   max/.store in=\pgfplots@brownian@max,
   max=inf,		
   start/.store in=\pgfplots@brownian@start,
  start= 0 
}
\makeatother

%-----------------------------------------------------------------
% Create the pic that generates brownian motions
% I did my homework and read the pgfmanual.
% It enabled me to use parameters in `pic`
% The idea is to have flexibility on the brownian
% \pic{BM=color/Starting@y/nbs of paths/nbs of points in the path};
% "nbs of points in the path" represents the subdivision of 1 unit of time

\tikzset{pics/BM/.style args={#1/#2/#3/#4}{
code = {%
\pgfmathsetseed{3}
\pgfplotstablenew{#4}\wiener % Initialise an empty table with #4 steps for brownians
\node[](#1){
\begin{tikzpicture}
 \begin{axis}[
   axis line style={draw=none},
   tick style={draw=none},  
   xtick=\empty, ytick=\empty,
 % xlabel = subdivision of 1 unit of time,
 % ylabel = $S$,
 	]
 	\foreach \i in {1,...,#3}{% <- I can draw #3 paths
	% start is a paramater to make it start at the mark  on the main plot 	
 	\addplot[#1,smooth] table [brownian motion={start=#2}] {\wiener};
% 	\addplot[#1,smooth] table 
%	[brownian motion={start=#2,min=80,max=120}] {\wiener};
% no impact of min and max...	
 	}
 \end{axis}
\end{tikzpicture}
};
}}}


\begin{document}

% Main plot
\begin{tikzpicture}
	\begin{axis}[
 	xlabel = time,
 	ylabel = $S$
 ]
 
 %I'll apply it to a function later
 % \addplot+[domain=0:5] {f(x)}; 
 % But as a start, I set the list of points manually
 \addplot[color=blue,smooth,mark=*] coordinates {
 	(0,100)
 	(1,102)
 	(2,95)
 	(3,101)
 	(4,105)
 	(5,98)
 };

	\end{axis}
\end{tikzpicture}

% I'll display one consistent type
\begin{tikzpicture}
	\pic{BM=orange/100/10/200};	% First point 
\end{tikzpicture}
\begin{tikzpicture}
	\pic{BM=green/102/10/200};	% Second point
\end{tikzpicture}
\begin{tikzpicture}
	\pic{BM=blue/95/10/200}; % Third point
\end{tikzpicture}
\begin{tikzpicture}
	\pic{BM=orange/101/10/200};	% Fourth point
\end{tikzpicture}

	\end{document}
```

Top Answer

user 3.14159

This is an answer that literally uses `pic`s as plot marks. The `pic` has various options which I hope are somewhat selfexplanatory. Please note that generally you will be better off if you dial a rather recent compatibility mode of `pgfplots`. I chose `\pgfplotsset{compat=1.17}` and there is no guarantee that the following works with ancient versions. To make a `pic` a plot mark, one can use something of the sort
```
\pgfdeclareplotmark{Brownian}{\pic[every mark]{Brownian};}
```
The `every mark` key is used to allow the user to pass styles to the `pic` using the `mark options` key, so this works just like any other plot mark. The `axis direction cs:` prefix is crucial to have  reasonable pics inside an `axis` environment. This also means that you cannot use them outside an `axis` unless you take extra steps, see the PS below for one out of several options to accomplish this. Finally, we use `scatter classes` to define different plot marks with different `mark options`, they can then simply be added in square brackets after the coordinates, as the following example illustrates.
```
\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.17}
\tikzset{pics/Brownian/.style={code={
	\tikzset{Brownian/.cd,#1}%
    \def\pv##1{\pgfkeysvalueof{/tikz/Brownian/##1}}%
	\foreach \XXX in {1,...,\pv{N}}	
	{
	\draw[pic actions] (axis direction cs:0,0) 
	foreach \YYY in {1,...,\pv{M}} 
	{-- ++ (axis direction cs:{\pv{dx}},{\pv{dy}*(rnd-0.5)})}
	;}
	}},Brownian/.cd,N/.initial=25,M/.initial=25,dx/.initial=0.04,dy/.initial=0.25}
\begin{document}

\pgfdeclareplotmark{Brownian}{\pic[every mark]{Brownian};}

% Main plot
\begin{tikzpicture}
	\begin{axis}[xmax=6,
	scatter/classes={
        a={mark=Brownian,mark options={red,Brownian/dy=0.4,Brownian/N=40}},
        b={mark=Brownian,mark options={blue}},
        c={mark=Brownian,mark options={orange,Brownian/dy=0.12}}
},
 	xlabel = time,
 	ylabel = $S$
 ]
 \addplot[color=blue,smooth, scatter,
        scatter src=explicit symbolic,] coordinates {
 	(0,100) [c]
 	(1,102) [b]
 	(2,95) [a]
 	(3,101) [a]
 	(4,105) [b]
 	(5,98) [c]
 };
\end{axis}
\end{tikzpicture}
\end{document}
```
![Screen Shot 2021-02-21 at 10.52.46 AM.png](/image?hash=7aeff7d1ef95bef48985ec906bf564c4cab8eb838254ac75ff932837d1e566e7)

P.S. If one wants to create a `pic` that works inside *and* outside of an `axis` environment, one can use the `\pgfplotsifinaxis` macro. There are other options available, but since this is not central to the current question I won't spell them out since I did not test any of them.
```
\tikzset{pics/Brownian/.style={code={
	\tikzset{Brownian/.cd,#1}%
    \def\pv##1{\pgfkeysvalueof{/tikz/Brownian/##1}}%
	\foreach \XXX in {1,...,\pv{N}}	
	{\pgfplotsifinaxis{%
	\draw[pic actions] (axis direction cs:0,0) 
	foreach \YYY in {1,...,\pv{M}} 
	{-- ++ (axis direction cs:{\pv{dx}},{\pv{dy}*(rnd-0.5)})};}{%
	\draw[pic actions] (0,0) 
	foreach \YYY in {1,...,\pv{M}} 
	{-- ++ ({\pv{dx}},{\pv{dy}*(rnd-0.5)})};}{%	
	}}
	}},Brownian/.cd,N/.initial=25,M/.initial=25,dx/.initial=0.04,dy/.initial=0.25}
```

I do not remember what I did back then but I would not longer create the tables with `\xdef`. The `/.list` key handler allows one to avoid all this nonsense. (I should have known but should also mention that various self-declared experts insisted on the foreach loops introducing groups being an absolute necessity...)

Next exercise for me. Adapt your `Brownian/.style` to [your (other) iconic answer](https://tex.stackexchange.com/questions/468706/brownian-motion-normal-distribution-3d). As you'll see, the `\mytabone,\mytwo...` prevent the use of loops. If I don't manage on my own by the end of the week I'll ask for your help. To be continued :)

I am asking about the random walk because if this is all that's required there is an arguably simpler and faster way to do the Brownians. ``` \documentclass[tikz]{standalone} \begin{document} \begin{tikzpicture}[pics/Brownian/.style={code={ \tikzset{Brownian/.cd,#1}% \def\pv##1{\pgfkeysvalueof{/tikz/Brownian/##1}}% \foreach \XXX in {1,...,\pv{N}} { \draw[pic actions] (0,0) foreach \YYY in {1,...,40} {-- ++ ({\pv{dx}},{\pv{dy}*(rnd-0.5)})} ;} }},Brownian/.cd,N/.initial=50,M/.initial=50,dx/.initial=0.04,dy/.initial=0.1] \path (0,2) pic{Brownian} (0,0) pic[blue]{Brownian={N=60,dx=0.06}}; \end{tikzpicture} \end{document} ```

Dear Marmot, I played with your **great** answer. Elegant and simple to adapt. Merci :) Yes it's just several sets of random walks, each set starting at a new point on the graph. If I may explain the context (pls don't bet your 401k based on this). - The brownian is what the model thinks how an asset will move in the next period of time. - The main plot is actually what happens in reality. ![MAinplot4.png](/image?hash=123d0d73d7566a94a4e3c9bf5a774a350ad39b0a10dac5e9adff72362802f7b3) This graph shows the moments when model and reality disconnect (the following orange mark is not on any path of the previous BM), the model needs adapting for the next period of time (i.e. the next BM will be wider or narrower). In practice, when an asset goes up, volatility (variance, spread of the diffusion) goes down (and vice versa). Graphically, it means that the brownian at next point will be - all the wider as it's far below the previous point. - all the more narrow as it's far above the previous point. What I miss, is to include the variance to reflect that dynamic (asset goes up, volatility goes down, following brownian adapts). `invgauss(rnd,rnd)*0.25*sqrt(1)+\pgfmathaccuma` is hardcoded to 25% of variance on 1 unit of time. I need to work on that :)

Here is something to play with but it does not fulfill all requirements. To improve it would be good if you could explain what the Brownian should be. Just several random walks from a common starting point? ``` \documentclass[tikz]{standalone} \usepackage{xsavebox} \usepackage{pgfplots, pgfplotstable} \tikzset{ declare function={ invgauss(\a,\b) = sqrt(-2*ln(\a))*cos(deg(2*pi*\b)); } } %----------------------------------------------------------------- % Code from Jake @ TeX.SE to generate brownian motions % If you have an idea to load to improve the compilation time \makeatletter \pgfplotsset{ table/.cd, brownian motion/.style={ create on use/brown/.style={ create col/expr accum={ (\coordindex>0)*( max( min( invgauss(rnd,rnd)*0.25*sqrt(1)+\pgfmathaccuma, \pgfplots@brownian@max ), \pgfplots@brownian@min ) ) + (\coordindex<1)*\pgfplots@brownian@start }{\pgfplots@brownian@start} }, y=brown, x expr={\coordindex}, brownian motion/.cd, #1, /.cd }, brownian motion/.cd, min/.store in=\pgfplots@brownian@min, min=-inf, max/.store in=\pgfplots@brownian@max, max=inf, start/.store in=\pgfplots@brownian@start, start= 0 } \makeatother %----------------------------------------------------------------- % Create the pic that generates brownian motions % I did my homework and read the pgfmanual. % It enabled me to use parameters in `pic` % The idea is to have flexibility on the brownian % \pic{BM=color/Starting@y/nbs of paths/nbs of points in the path}; % "nbs of points in the path" represents the subdivision of 1 unit of time \tikzset{pics/BM/.style args={#1/#2/#3/#4}{ code = {% \pgfmathsetseed{3} \pgfplotstablenew{#4}\wiener % Initialise an empty table with #4 steps for brownians \begin{axis}[hide axis,enlargelimits=false,#1] \foreach \i in {1,...,#3}{% <- I can draw #3 paths % start is a paramater to make it start at the mark on the main plot \addplot[#1,smooth] table [brownian motion={start=#2}] {\wiener}; % \addplot[#1,smooth] table % [brownian motion={start=#2,min=80,max=120}] {\wiener}; % no impact of min and max... } \end{axis} }}} \begin{xlrbox}{BrownianRed} \scalebox{0.12}{\begin{tikzpicture} \pic{BM={draw=red}/100/10/200}; % First point \end{tikzpicture}} \end{xlrbox} \begin{xlrbox}{BrownianBlue} \scalebox{0.12}{\begin{tikzpicture} \pic{BM={draw=blue}/100/10/200}; % First point \end{tikzpicture}} \end{xlrbox} \begin{xlrbox}{BrownianOrange} \scalebox{0.12}{\begin{tikzpicture} \pic{BM={draw=orange}/100/10/200}; % First point \end{tikzpicture}} \end{xlrbox} \begin{document} \pgfdeclareplotmark{BrownianRed}{\pgftext[left]{\xusebox{BrownianRed}}} \pgfdeclareplotmark{BrownianBlue}{\pgftext[left]{\xusebox{BrownianBlue}}} \pgfdeclareplotmark{BrownianOrange}{\pgftext[left]{\xusebox{BrownianOrange}}} % Main plot \begin{tikzpicture} \begin{axis}[xmax=6, scatter/classes={ a={mark=BrownianBlue}, b={mark=BrownianRed}, c={mark=BrownianOrange} }, xlabel = time, ylabel = $S$ ] \addplot[color=blue,smooth, scatter, scatter src=explicit symbolic,] coordinates { (0,100) [c] (1,102) [b] (2,95) [a] (3,101) [a] (4,105) [b] (5,98) [c] }; \end{axis} \end{tikzpicture} \end{document} ```

Yes, that would work if you always want to have the same random plot. You could also use a `\savebox` (I think). If you want each time a separate one, you need to create them. Please clarify if you want each time the same one. BTW, you need to add something of the sort ``` \tikzset{declare function={invgauss(\a,\b) = sqrt(-2*ln(\a))*cos(deg(2*pi*\b));}} ``` to your code so that it runs through.

1 Answer