# How to plot this cascaded/waterfall python-type plot using TikZ

I would like to make a plot of this equation

P_n = \frac{n^a}{a!}\exp[-n]


for different n. How can I use this cascaded style from the TikZ/PGF examples, to plot P_n (y-axis) with a (x-axis), for different values of n (z-axis: the depth). The x-variable is integers, so I need a bar chart.

This post has received an answer here, but I wanted to resurrect the questions since the image style I would like is shown in this link. Instead of a continuous plot, I would have bar charts, but I would like an enveloping line for each waterfall plot that is tangential to all of the bar charts to clearly show the distributions.

Thanks for asking a new question! Note also that if you have more rows you'll need to extend the definition of \mycolor accordingly.

EDIT: Simplified the code and made it (as I think) much easier to customize by

• storing the function in myf. That is, if you have a different function, you'll only to adjust this bit.
• adding explanations in the code.
• replacing the single \pgfplotsinvokeforeach loop in a double loop over the parameters (a and n in your question).

Thats the code.

\documentclass[border=3.14mm]{standalone}
\usepackage{pgfplots}
% if you plot more than 6 rows, you'll need to extend the following
% definition accordingly. alternatively you could work with color maps here.
\newcommand{\mycolor}[1]{\ifcase#1
white%0
\or
blue%1
\or
red%2
\or
green!50!black%3
\or
yellow%4
\or
purple%5
\or
orange%6
\fi}
\pgfplotsset{compat=1.16}
\begin{document}
\tikzset{declare function={myf(\X,\Y)=((\Y)^\X/\X!)*exp(-\Y);}}
\def\amax{5} %<- maximal a
\def\SteepNess{48} % <- this value determines the shape of the Gaussians. if
% you have different function myf, you may need to adjust it
\begin{tikzpicture}
\begin{axis}[width=9cm,
set layers=standard,
domain=0:{\amax+1},
samples y=1,
view={40}{20},
xmin=-1,ymax=\amax+3,
%hide axis,
%xtick=\empty, ytick=\empty, ztick=\empty,
clip=false
]
\pgfmathtruncatemacro{\NumRows}{6}
\pgfplotsinvokeforeach{\NumRows,...,1}{ % your n will now be stored in #1
\draw [on layer=background, gray!20] (axis cs:0,#1,0) -- (axis cs:{\amax+1},#1,0);
}
\pgfmathtruncatemacro{\Xmax}{\NumRows*\amax}
\foreach \Y in {1,...,\NumRows}
{\xdef\myLst{(0,\Y,0)} % initialize the list for envelopes
\foreach \X in {1,...,\amax}
{
\pgfmathsetmacro{\mymax}{myf(\X,\Y)} %<- this is not absolutely necessary
\xdef\myLst{\myLst (\X,\Y,\mymax)} % build up the list
line width=0.7pt,on layer=main,draw=black,smooth,
domain=-0.4:0.4]  (x+\X,\Y,{myf(\X,\Y)*exp(-\SteepNess*x^2)});
}
\myplot
}
\xdef\myLst{\myLst (\amax+1,\Y,0)} % complete the list
% if you do not want to have the envelope, just comment the following out
\myplot
}
\end{axis}
\end{tikzpicture}
\end{document}


UPDATE: If you want to go to higher values of a and/or n, it might be worthwhile to rewrite your function, which is in the above of the form

(huge/large)*small


in the form

(not so huge/large)*((not so huge)*small) .


Here is an example. I do not claim that it is optimal, that's the first thing that came to my mind. Alternatives include to take the logarithms of subexpressions and taking the exponential at the end. (I compile with lualatex.)

\documentclass[border=3.14mm]{standalone}
\usepackage{pgfplots}
% if you plot more than 6 rows, you'll need to extend the following
% definition accordingly. alternatively you could work with color maps here.
\newcommand{\mycolor}[1]{\ifcase#1
white%0
\or
blue%1
\or
red%2
\or
green!50!black%3
\or
yellow%4
\or
purple%5
\or
orange%6
\or
cyan%7
\or
brown%8
\or
red!50!black%9
\or
blue!50!black%10
\fi}
\pgfplotsset{compat=1.16}
\begin{document}
\tikzset{declare function={myf(\X,\Y)=((\Y)^(\X/2)/\X!)*((\Y)^(\X/2)*exp(-\Y));}}
\def\amax{7} %<- maximal a
\pgfmathtruncatemacro{\NumRows}{7}
\def\SteepNess{48} % <- this value determines the shape of the Gaussians. if
% you have different function myf, you may need to adjust it
\begin{tikzpicture}
\begin{axis}[width=9cm,
set layers=standard,
domain=0:{\amax+1},
samples y=1,
view={40}{20},
xmin=-1,ymax=\amax+3,
%hide axis,
%xtick=\empty, ytick=\empty, ztick=\empty,
clip=false
]
\pgfplotsinvokeforeach{\NumRows,...,1}{ % your n will now be stored in #1
\draw [on layer=background, gray!20] (axis cs:0,#1,0) -- (axis cs:{\amax+1},#1,0);
}
\pgfmathtruncatemacro{\Xmax}{\NumRows*\amax}
\foreach \Y in {1,...,\NumRows}
{\xdef\myLst{(0,\Y,0)} % initialize the list for envelopes
\foreach \X in {1,...,\amax}
{
\pgfmathsetmacro{\mymax}{myf(\X,\Y)} %<- this is not absolutely necessary
\xdef\myLst{\myLst (\X,\Y,\mymax)} % build up the list
line width=0.7pt,on layer=main,draw=black,smooth,
domain=-0.4:0.4]  (x+\X,\Y,{myf(\X,\Y)*exp(-\SteepNess*x^2)});
}
\myplot
}
\xdef\myLst{\myLst (\amax+1,\Y,0)} % complete the list
% if you do not want to have the envelope, just comment the following out
\myplot
}
\end{axis}
\end{tikzpicture}
\end{document}


Just for fun: if you wish to have bars instead of Gaussians (Gauss was a mathematician and in a bar one gets good drinks ;-), and also do not want the envelop touch the axis, do

\documentclass[border=3.14mm]{standalone}
\usepackage{pgfplots}
% if you plot more than 6 rows, you'll need to extend the following
% definition accordingly. alternatively you could work with color maps here.
\newcommand{\mycolor}[1]{\ifcase#1
white%0
\or
blue%1
\or
red%2
\or
green!50!black%3
\or
yellow%4
\or
purple%5
\or
orange%6
\or
cyan%7
\or
brown%8
\or
red!50!black%9
\or
blue!50!black%10
\fi}
\pgfplotsset{compat=1.16}
\begin{document}
\tikzset{declare function={myf(\X,\Y)=((\Y)^(\X/2)/\X!)*((\Y)^(\X/2)*exp(-\Y));}}
\def\amax{7} %<- maximal a
\pgfmathtruncatemacro{\NumRows}{7}
\pgfmathsetmacro{\myBarWidth}{0.2}
\def\SteepNess{48} % <- this value determines the shape of the Gaussians. if
% you have different function myf, you may need to adjust it
\begin{tikzpicture}
\begin{axis}[width=9cm,
set layers=standard,
domain=0:{\amax+1},
samples y=1,
view={40}{20},
xmin=-1,ymax=\amax+3,
%hide axis,
%xtick=\empty, ytick=\empty, ztick=\empty,
clip=false
]
\pgfplotsinvokeforeach{\NumRows,...,1}{ % your n will now be stored in #1
\draw [on layer=background, gray!20] (axis cs:0,#1,0) -- (axis cs:{\amax+1},#1,0);
}
\pgfmathtruncatemacro{\Xmax}{\NumRows*\amax}
\foreach \Y in {1,...,\NumRows}
{\xdef\myLst{(1,\Y,0)} % initialize the list for envelopes
\foreach \X in {1,...,\amax}
{
\pgfmathsetmacro{\mymax}{myf(\X,\Y)} %<- this is not absolutely necessary
\xdef\myLst{\myLst (\X,\Y,\mymax)} % build up the list
line width=0.7pt,on layer=main,draw=black] coordinates {(-\myBarWidth+\X,\Y,0)
(-\myBarWidth+\X,\Y,{myf(\X,\Y)*exp(-\SteepNess*0^2)})
(\myBarWidth+\X,\Y,{myf(\X,\Y)*exp(-\SteepNess*0^2)})
(\myBarWidth+\X,\Y,0)};}
\myplot
}
\xdef\myLst{\myLst (\amax,\Y,0)} % complete the list
% if you do not want to have the envelope, just comment the following out
\myplot
}
\end{axis}
\end{tikzpicture}
\end{document}


• You're welcome. This looks great! Is it possible to add a waveform that envelopes each of the coloured distributions?
– Sid
Aug 7, 2018 at 18:08
• Fantastic! These waterfall plots are quite common in python's matplotlib package. Hopefully, many others can make use of this for a more native plot to LaTeX!
– Sid
Aug 7, 2018 at 18:50
• image link is broken for me
– qwr
Aug 7, 2018 at 21:36
• @marmot, if you look closely, some of the envelopes have some kink along the waveform. Also if you change a for, say, 10, then the waveforms don't follow the height of the bars.
– Sid
Aug 7, 2018 at 22:19
• @Sid You can adjust the tension of these plots. Or one would have to resort to more elaborate means, at the moment it is really just a smooth interpolation of the maxima with tension=1. And I cannot plot your function to \amax=10, how do you compile it? I get an ! Arithmetic overflow. error.
– user121799
Aug 7, 2018 at 22:29