How can two bivariate normal distributions be plotted in one plot?

Question

I would like to visualize the following two bivariate normal distributions:

I found Jakes awesome answer and adjusted it so that the plots get rotated the correct way.

Current plot

Question

How can I make the plot better viewable? (E.g. only displaying the function that is currently on top but keeping the correct color?)

MWE

% Thanks to Jake for the template
% https://tex.stackexchange.com/a/31715/5645
\documentclass{standalone}

\usepackage{pgfplots}

\begin{document}

\pgfplotsset{
colormap={whitered}{color(0cm)=(white); color(1cm)=(orange!75!red)},
colormap={whiteblue}{color(0cm)=(white); color(1cm)=(blue)},
}

\begin{tikzpicture}[
    declare function={mu11=60;},
    declare function={mu12=20;},
    declare function={sigma11=5;},
    declare function={sigma12=5;},
    declare function={mu21=70;},
    declare function={mu22=40;},
    declare function={sigma21=5;},
    declare function={sigma22=5;},
    declare function={rho=0.8;},
    declare function={normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));},
    declare function={bivar(\ma,\sa,\mb,\sb,\rho)=
        1/(2*pi*\sa*\sb*\rho) * exp(-((x-\ma)^2/\sa^2 + (y-\mb)^2/\sb^2 - (2*\rho*(x-\ma)*(y-\mb))/(\sa*\sb)))/(2*(1-\rho*\rho));}]
\begin{axis}[
    width=15cm,
    view={-15}{70},
    enlargelimits=false,
    grid=major,
    domain=40:90,
    y domain=0:60,
    samples=26,
    xlabel=$x_1$,
    ylabel=$x_2$,
    zlabel={$P$},
    colorbar,
    colorbar style={
        at={(1,0)},
        anchor=south west,
        height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
        title={$P(x_1,x_2)$}
    }
]
\addplot3 [surf, opacity=0.8,fill opacity=0.9,colormap={whitered}{color(0cm)=(white); color(1cm)=(orange!75!red)}, samples=50] {bivar(mu11,sigma11,mu12,sigma12,rho)};
\addplot3 [surf, opacity=0.5,fill opacity=0.3,colormap={whiteblue}{color(0cm)=(white); color(1cm)=(blue)}, samples=50] {bivar(mu21,sigma21,mu22,sigma22,rho)};

\draw [black!50] (axis cs:-1,0,0) -- (axis cs:4,0,0);
\draw [black!50] (axis cs:0,-1,0) -- (axis cs:0,4,0);

\node at (axis cs:-1,1,0.18) [pin=165:$P(x_1)$] {};
\node at (axis cs:1.5,4,0.32) [pin=-15:$P(x_2)$] {};
\end{axis}
\end{tikzpicture}
\end{document}

Answer 1

Since PGFPlots can't combine the plots from different \addplot commands, you'll have to use a single \addplot command.

To only show the larger of the two distributions at each point, you can use

\addplot3 [...] {
max(
    bivar(mu11,sigma11,mu12,sigma12,rho),
    bivar(mu21,sigma21,mu22,sigma22,rho)
)};

To use different colours for the two distributions, you can create a single colormap that runs from blue to white to orange, and map one of the distributions to the negative domain and the other to the positive domain using

\addplot3 [..., 
colormap={bluewhitered}{color(0cm)=(blue); color(0.5cm)=(white); color(1cm)=(orange!75!red)},
point meta={
(
    bivar(mu11,sigma11,mu12,sigma12,rho)>
    bivar(mu21,sigma21,mu22,sigma22,rho)?
    bivar(mu11,sigma11,mu12,sigma12,rho):
    -bivar(mu21,sigma21,mu22,sigma22,rho)
)   
}
] { ... };

\documentclass{standalone}

\usepackage{pgfplots}

\begin{document}

\pgfplotsset{
colormap={whitered}{color(0cm)=(white); color(1cm)=(orange!75!red)},
colormap={whiteblue}{color(0cm)=(white); color(1cm)=(blue)},
}

\begin{tikzpicture}[
    declare function={mu11=60;},
    declare function={mu12=20;},
    declare function={sigma11=5;},
    declare function={sigma12=5;},
    declare function={mu21=70;},
    declare function={mu22=40;},
    declare function={sigma21=5;},
    declare function={sigma22=5;},
    declare function={rho=0.8;},
    declare function={normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));},
    declare function={bivar(\ma,\sa,\mb,\sb,\rho)=
        1/(2*pi*\sa*\sb*\rho) * exp(-((x-\ma)^2/\sa^2 + (y-\mb)^2/\sb^2 - (2*\rho*(x-\ma)*(y-\mb))/(\sa*\sb)))/(2*(1-\rho*\rho));}]
\begin{axis}[
    width=15cm,
    view={-15}{70},
    enlargelimits=false,
    grid=major,
    domain=40:90,
    y domain=0:60,
    samples=50,
    xlabel=$x_1$,
    ylabel=$x_2$,
    zlabel={$P$},
    colorbar,
    colorbar style={
        at={(1.1,0)},
        anchor=south west,
        height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
        title={$P(x_1,x_2)$}
    }
]
\addplot3 [
    surf,
    colormap={bluewhitered}{color(0cm)=(blue); color(0.5cm)=(white); color(1cm)=(orange!75!red)},
    point meta={
    (
        bivar(mu11,sigma11,mu12,sigma12,rho)>
        bivar(mu21,sigma21,mu22,sigma22,rho)?
        bivar(mu11,sigma11,mu12,sigma12,rho):
        -bivar(mu21,sigma21,mu22,sigma22,rho)
    )   
    }
    ] {
    max(
        bivar(mu11,sigma11,mu12,sigma12,rho),
        bivar(mu21,sigma21,mu22,sigma22,rho)
    )};

\draw [black!50] (axis cs:-1,0,0) -- (axis cs:4,0,0);
\draw [black!50] (axis cs:0,-1,0) -- (axis cs:0,4,0);

\node at (axis cs:-1,1,0.18) [pin=165:$P(x_1)$] {};
\node at (axis cs:1.5,4,0.32) [pin=-15:$P(x_2)$] {};
\end{axis}
\end{tikzpicture}
\end{document}