Filling specified area by random dots in TikZ
To fill the area between two curves with random points, you can plot the average of the two functions and jitter the markers using random noise with a range that's equal to the distance between the curves.
To demonstrate, here's what the plot for the random dots looks like without the noise component:
In this case, the distance between the two curves is constant, so the noise can be generated using a uniform distribution centered halfway between the curves with a range equal to the offset between the curves:
To get a denser pattern, simply increase the number of samples:
The same approach works for more complicated functions:
However, things get more complicated when you have two curves that aren't a constant distant apart: In that case, the density of the random dots will not be constant throughout the domain:
To achieve a constant density, you'll need to adjust the horizontal distribution of the random points. So instead of the points being a noisy function of x, we'll use a parametric equation ( x(t), 0.5*a(x(t)) + 0.5*b(x(t)) + U(a(x(t)), b(x(t))).
The horizontal distance between two adjacent points has to be inversely proportional to the distance between the two bounding curves (since you need more points to fill a wider band).
To find the horizontal positions for the points that fulfill that requirements, you can use the antiderivative of the inverse of the difference of the two bounding curves.
In this case, for example, you would need the antiderivative of f(x) = 1/(a(x)-b(x)) = 1 / (2*(x/4)^2+1). I think I should be able to do this by hand, but I cheated and used Wolfram Alpha. The required function is F(x) = 2 sqrt(2) (atan(x/(2 sqrt(2)))). In order to stretch this function over the plot domain, you can normalize it using the factor x_max/F(x_max). The function for the horizontal positions of the points is thus
F*(t) = 2 sqrt(2) (atan(t/(2*sqrt(2)))) * 5 / 2.99
Using this to plot the random dots without the noise component, we get:
Adding the noise:
Code for the two parallel lines:
\documentclass[tikz, border=5mm]{standalone}
\usepackage{pgfplots}
\usepackage{amsmath}
\begin{document}
\begin{tikzpicture}[
declare function={a(\x)=0.75*\x-2;},
declare function={b(\x)=0.75*\x-1;}
]
\begin{axis}[
domain=0:5,
axis lines=middle,
axis equal image,
xtick=\empty, ytick=\empty,
enlargelimits=true,
clip mode=individual, clip=false
]
\addplot [red, only marks, mark=*, samples=300, mark size=0.75]
{0.5*(a(x)+b(x)) + 0.5*rand*(a(x)-b(x))};
\addplot [thick] {a(x)};
\addplot [thick] {b(x)};
\end{axis}
\end{tikzpicture}
\end{document}
Code for the parabolas:
\documentclass[tikz, border=5mm]{standalone}
\usepackage{pgfplots}
\usepackage{amsmath}
\begin{document}
\begin{tikzpicture}[
declare function={a(\x)=-(0.5*\x-1.5)^2+1;},
declare function={b(\x)=-(0.5*\x-1.5)^2;},
]
\begin{axis}[
domain=1:5,
axis lines=middle,
axis equal image,
xtick=\empty, ytick=\empty,
enlargelimits=true,
clip mode=individual, clip=false
]
\addplot [red, only marks, mark=*, samples=300, mark size=0.75]
{0.5*(a(x)+b(x)) + 0.5*rand*(a(x)-b(x))};
\addplot [thick] {a(x)};
\addplot [thick] {b(x)};
\end{axis}
\end{tikzpicture}
\end{document}
Code for the funnel:
\documentclass[tikz, border=5mm]{standalone}
\usepackage{pgfplots}
\usepackage{amsmath}
\begin{document}
\begin{tikzpicture}[
declare function={a(\x)=(\x/4)^2;},
declare function={b(\x)=-(\x/4)^2-1;},
declare function={f(\x) = 2*sqrt(2)*rad(atan(\x/(2*sqrt(2))))*5/2.99;}
]
\begin{axis}[
domain=0:5, xmin=0,
axis lines=middle,
axis equal image,
xtick=\empty, ytick=\empty,
enlargelimits=true,
clip mode=individual, clip=false
]
\addplot [red, only marks, mark=*, samples=500, mark size=0.75]
(f(x), {0.5*(a(x)+b(x)) + rand * ( a(f(x)) - b(f(x))) / 2});
\addplot [thick] {a(x)};
\addplot [thick] {b(x)};
\end{axis}
\end{tikzpicture}
\end{document}
Original answer:
You can use a plot to draw random dots that lie inside the band:
\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\draw [-latex, thick] (0, 0) -- (0, 6) node [above] {\Large{$e$}};
\draw [-latex, thick] (0, 3) -- (6, 3) node [right] {\Large{$\widehat{Y}$}};
\draw [thick] (0, 0.5) -- (5, 4);
\draw [thick] (0, 1.75) -- (5, 5.25);
\draw plot [only marks, mark=*, mark size=0.5, domain=0:5, samples=700] (\x,{rnd*1.25+3.5/5*\x+0.5});
\end{tikzpicture}
\end{document}
One way to get random points that are distributed in a visually appealing manner is to use Poisson disc sampling. This technique is used in cartography to generate patterns for representing sand, for example.
The method isn't particularly complicated, but too computationally demanding to be implemented directly in LaTeX (I'd be happy to be proven wrong on this one). What you can do is to generate the points using an external tool, and then filter and plot them using PGFPlots.
Here's an example of how to do this using the Python code from http://code-spot.co.za/2010/04/07/poisson-disk-sampling-example-code/.
Generate and save the points covering the required domain and range:
import poisson_disk
import numpy as np
points = np.array(poisson_disk.sample_poisson_uniform(5, 5, 0.1, 20))
np.savetxt('poissonpoints.txt', points)
Then filter and plot the points in PGFPlots. For this, you can use the expression
y filter/.code={
\pgfmathparse{\pgfmathresult / and(\pgfmathresult<a(x), \pgfmathresult>b(x))}
}
which makes use of the fact that the conditional in the divisor evaluates to zero when a point is outside the bounding curves, resulting in a division by zero which causes PGFPlots to discard the point.
\documentclass[tikz, border=5mm]{standalone}
\usepackage{pgfplots}
\usepackage{amsmath}
\begin{document}
\begin{tikzpicture}[
declare function={a(\x)=(\x/4)^2;},
declare function={b(\x)=-(\x/4)^2-1;},
]
\begin{axis}[
domain=0:5, xmin=0,
axis lines=middle,
axis equal image,
xtick=\empty, ytick=\empty,
enlargelimits=true,
clip mode=individual, clip=false
]
\addplot [thick] {a(x)};
\addplot [thick] {b(x)};
\addplot [only marks, draw=none, mark size=0.5,
y filter/.code={
\pgfmathparse{\pgfmathresult/and(\pgfmathresult<a(x), \pgfmathresult>b(x))}
}
] table [y expr=\thisrowno{1}-3] {poissonpoints.txt};
\end{axis}
\end{tikzpicture}
\end{document}
A second solution, based on How to draw a rectangle filled with random size circular pattern? it works by clipping the area of the plot.
Dots' aspect and number can be customized through some keys.
\documentclass[tikz]{standalone}
\usepackage{tikz}
\newif\ifpointsamewidth
\pgfkeys{/tikz/.cd,
height rect/.store in=\height,
height rect=10,
width rect/.store in=\width,
width rect=10,
num points/.store in=\numpoints,
num points=100,
point width/.store in=\maxpointwidth,
point width=1.5pt,
use points equal width/.is if=pointsamewidth,
use points equal width=false,
point style type/.is choice,
point style type/fill/.style={fill=\pointcolor},
point style type/radial shade/.style={inner color=\pointinnercolor,outer color=\pointoutercolor},
point style type/vertical shade/.style={top color=\pointtopcolor,bottom color=\pointbottomcolor},
point style type/horizontal shade/.style={left color=\pointleftcolor,right color=\pointrightcolor},
point style/.store in=\pointstyle,
point style={/tikz/point style type/fill},
point fill color/.store in=\pointcolor,
point fill color=blue!30,
point inner color/.store in=\pointinnercolor,
point inner color=white,
point outer color/.store in=\pointoutercolor,
point outer color=blue!30,
point top color/.store in=\pointtopcolor,
point top color=white,
point bottom color/.store in=\pointbottomcolor,
point bottom color=blue!30,
point left color/.store in=\pointleftcolor,
point left color=blue!30,
point right color/.store in=\pointrightcolor,
point right color=white,
}
\pgfkeys{/tikz/random point diagram/.code={
\path (0,0) rectangle (\width,\height);
\foreach \point in {1,...,\numpoints}{
\pgfmathparse{random()}
\pgfmathsetmacro\xpos{\width*\pgfmathresult}
\pgfmathparse{random()}
\pgfmathsetmacro\ypos{\height*\pgfmathresult}
\ifpointsamewidth%
\node[circle,inner sep=\maxpointwidth, \pointstyle] (point-\point) at (\xpos,\ypos) {};
\else%
\pgfmathparse{random()}
\pgfmathsetmacro\pointwidth{\maxpointwidth*\pgfmathresult}
\node[circle,inner sep=\pointwidth pt, \pointstyle] (point-\point) at (\xpos,\ypos) {};
\fi
}
}
}
% that's just an alias for \node
\makeatletter
\def\drawdiagram{\tikz@path@overlay{node}}
\makeatother
\begin{document}
\begin{tikzpicture}
\draw [-latex, thick] (0, 0) -- (0, 6) node [above] {\Large{$e$}};
\draw [-latex, thick] (0, 3) -- (6, 3) node [right] {\Large{$\widehat{Y}$}};
\draw [thick] (0, 0.5) -- (5, 4);
\draw [thick] (0, 1.75) -- (5, 5.25);
\clip (5, 4) -- (0, 0.5) -- (0, 1.75) -- (5, 5.25) --cycle;
\drawdiagram[
num points=600,
point fill color=black,
random point diagram] at (0, 1.75){};
\end{tikzpicture}
\end{document}
The result:
The previous solution has dots with random width. A new key has been introduced to get rid of this behaviour: use points equal width. The second picture adopts this feature:
\begin{tikzpicture}
\draw [-latex, thick] (0, 0) -- (0, 6) node [above] {\Large{$e$}};
\draw [-latex, thick] (0, 3) -- (6, 3) node [right] {\Large{$\widehat{Y}$}};
\draw [thick] (0.5, 2.25) arc (160:20:2.5cm);
\draw [thick] (0.5, 1.25) arc (160:20:2.5cm);
\clip (0.5, 2.25) arc (160:20:2.5cm)--++(0,-1) arc(20:160:2.5cm);
\drawdiagram[
num points=600,
use points equal width,
point width=0.75pt,
point fill color=black,
random point diagram] at (0.5, 1.25){};
\end{tikzpicture}
Same concept as before: draw your graph then clip it. Starting the random point diagram in the lower angle ensures to have the random dots within the plot (if the plot exceeds a 10 x 10 grid, change height and width rect.
The result:
