latex_figure.tex

\begin{figure}[htb!]
  \begin{centering}
  \begin{tikzpicture}
   \begin{polaraxis}[xmin=0,xmax=180,ymax=1,
    ytick align=outside,
    yticklabel style={
        anchor=north,
        yshift=-2*\pgfkeysvalueof{/pgfplots/major tick length}
    }]
	\addplot+[mark=f,domain=0:180,samples=600] 
		{cos(x-90)*(sin(2*x)*cos(2*x) + cos(0.3*x + 1) - sin(30*x)*0.05 + cos(100*x)*0.02)/1.75 }; 
    \addplot+[mark=g,domain=0:180,samples=600] 
		{cos(x-90)*(sin(2*x)*cos(2*x) + cos(0.3*x + 1))/1.75 };
    \addplot+[dashed, polar comb,domain=0:180, samples=6,black,opacity=0.75,mark=x]
	    {cos(x-90)*(sin(2*x)*cos(2*x) + cos(0.3*x + 1) - sin(30*x)*0.05 + cos(100*x)*0.02)/1.75  };
	\addplot+[dashed, polar comb,domain=10:45, samples=3,black,opacity=0.75,mark=x]
	    {cos(x-90)*(sin(2*x)*cos(2*x) + cos(0.3*x + 1) - sin(30*x)*0.05 + cos(100*x)*0.02)/1.75  };
	\addplot+[dashed, polar comb,domain=100:130, samples=4,black,opacity=0.75,mark=x]
	    {cos(x-90)*(sin(2*x)*cos(2*x) + cos(0.3*x + 1) - sin(30*x)*0.05 + cos(100*x)*0.02 )/1.75 };
	\addplot+[dashed, polar comb,domain=105:125, samples=3,black,opacity=0.75,mark=x]
	    {cos(x-90)*(sin(2*x)*cos(2*x) + cos(0.3*x + 1) - sin(30*x)*0.05 + cos(100*x)*0.02 )/1.75 };
	\legend{$f$, $g$}
	\end{polaraxis}
  \end{tikzpicture}\par
  \end{centering}
  \caption{Approximating a complicated function $f$ by a simpler function $g$ and sampling $f$ based on the probability distribution of $g$. Note how we are more likely to sample the $f$ in directions where it has a large value.\protect\footnotemark}
  \label{fig:importance-sampling}
\end{figure}