[Tex/LaTex] How to plot this cascaded/waterfall python-type plot using TikZ

tikz-pgftikz-styles

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.

Best Answer

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
\edef\myplot{\noexpand\addplot3 [fill=\mycolor{\Y},fill opacity=0.4,
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
\edef\myplot{\noexpand\addplot3[smooth,fill=\mycolor{\Y},fill opacity=0.1] coordinates{\myLst};}
% 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
\edef\myplot{\noexpand\addplot3 [fill=\mycolor{\Y},fill opacity=0.4,
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
\edef\myplot{\noexpand\addplot3[smooth,fill=\mycolor{\Y},fill opacity=0.1] coordinates{\myLst};}
% 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
\edef\myplot{\noexpand\addplot3 [fill=\mycolor{\Y},fill opacity=0.4,
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
\edef\myplot{\noexpand\addplot3[smooth,tension=0.2,fill=\mycolor{\Y},fill opacity=0.1] coordinates{\myLst};}
% if you do not want to have the envelope, just comment the following out
\myplot
}
\end{axis}
\end{tikzpicture}
\end{document}

Related Solutions

[Tex/LaTex] Draw Pareto chart with pgfplots

You can add a new plot with a different plot type by using \addplot [<plot type>] ...;.

If you want to add a line plot to a column plot, you would use \addplot [sharp plot] ...;.

If you want to keep the current color cycle uninterrupted, add a + in front of the options: \addplot +[sharp plot] ...;

To calculate the cumulated sums automatically, you can make use of the PGFplotstable package that ships with PGFplots. It allows you to save data into tables which can then be fed to PGFplots using \addplot table {<\tablemacro>}. You create the table using

\pgfplotstableread{
Answer Count
no 7
yes 3
undecided 1
}\results

where the first data line is the header that assigns the column names, and the \results is the macro the table is saved to. To create a new column that contains the cumulated results, you issue

\pgfplotstableset{
    create on use/Cumulated/.style={
        create col/expr={
            \pgfmathaccuma + \thisrow{Count}
        }
    }
}

which tells PGFplotstable that whenever you access the column Cumulated (which does not exist yet), it will automatically create a new column consisting of the sum of \pgfmathaccuma (initially empty) and the current value of Count.

In your plot, you would then just say

\addplot [sharp plot] table [y=Cumulated] {\results};

Here's the complete code:

\documentclass[11pt]{amsart}
\usepackage{pgfplots}
\usepackage{pgfplotstable}

\begin{document}

\pgfplotstableread{
Answer Count
no 7
yes 3
undecided 1
}\results

\pgfplotstableset{
    create on use/Cumulated/.style={
        create col/expr={
            \pgfmathaccuma + \thisrow{Count}
        }
    }
}

\begin{tikzpicture}
  \begin{axis}[
    ybar, ymin=0,
    ylabel={\#participants},
    symbolic x coords={no,yes,undecided},
    xtick=data
    ]
    \addplot [fill=gray!30, nodes near coords] table {\results};
    \addplot [sharp plot,mark=*] table [y=Cumulated] {\results};
  \end{axis}
\end{tikzpicture}

\end{document}

[Tex/LaTex] Add axis label to bar chart using tikz

You just need xlabel by analogy with ylabel. The labels on the x-axis are just tick labels, like those on the y-axis. The fact that they happen to be words rather than numbers doesn't prevent you from also labelling the axis as a whole, just as you can label the y-axis ;). At least, it seems to work for me:

labelled axes

\documentclass[tikz,border=10pt]{standalone}
\usepackage{pgfplots}
\usepackage{pgfplotstable}
\pgfplotsset{compat=newest}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
    ybar=12pt,
    ymin=0,ymax=150,
    xtick=data,
    enlarge x limits={abs=2cm},
    symbolic x coords={blond, brown},
    bar width = 20pt,
    ylabel= number,
    xlabel= hair colour,
        ytick align=outside,
        ytick pos=left,
        major x tick style = transparent,
        legend style={at={(0.04,0.96)},anchor=north west, font=\footnotesize, legend cell align=left,},
        ]
    \addplot[ybar,fill=blue, area legend] coordinates {
        (blond,20)
        (brown,100)};
    \addplot[ybar,fill=purple, area legend] coordinates {
        (blond,110)
        (brown,105)};
 \legend{Boys, Girls}
\end{axis}
\end{tikzpicture}

\end{document}

Best Answer

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[Tex/LaTex] Draw Pareto chart with pgfplots

[Tex/LaTex] Add axis label to bar chart using tikz

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