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lucas_et_al_supplementarymaterial_2015-01-20.tex
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\clearpage
\section{Table of symbols}
%move to appendix
\begin{table}[h!]
\centering
\caption[List of symbols used to describe the gREM and simulations]{List of symbols used to describe the gREM and simulations. `-' means the quantity has no units.}
\begin{tabular}{@{}lll@{}}
\toprule
Symbol & Description & Units\\
\midrule
$\theta$ & Sensor width & \SI{}{\radian} \\
$\alpha$ & Animal signal width & \SI{}{\radian} \\
$x_i$ & Focal angle, $i \in \{1,2,3,4\} $ & \SI{}{\radian}\\
$r$ & Detection distance & \SI{}{\meter}\\
$\bar{p}$ & Average profile width & \SI{}{\meter}\\
$p$ & A specific profile width & \SI{}{\meter}\\
$v$ & Velocity & \SI{}{\meter\per\second}\\
$t$ & Time & \SI{}{\second}\\
$z$ & Number of detections & -\\
$D$ & Animal density & \SI{}{\per\meter\squared} \\
$T$ & Step length & \SI{}{\second}\\
$N$ & Number of steps per simulation & -\\
$d$ & Distance moved in a time step & \SI{}{\meter}\\
$S$ & Probability of remaining stationary & -\\
$A$ & Maximum turning angle & \SI{}{\radian}\\
\bottomrule
\end{tabular}
\label{t:paras}
\end{table}
\clearpage
\section{Supplementary methods}
\subsection{Introduction}
\lettr{T}hese supplementary methods derive all the models used.
For continuity, the gas model derivation is included here as well as in the main text.
The calculation of all integrals used in the gREM is included in the \emph{Python} script S3.
\input{REM-methods.tex}
\vspace{1cm}
\clearpage
\section{Supplementary information: Simulation model results of the gREM precision}
\setcounter{figure}{0}
\begin{figure}[h!]
\centering
\includegraphics[width=\textwidth]{imgs/ResultStandardDeviation.pdf}
\caption[gREM precision given a range of sensor and signal widths]{
Simulation model results of the gREM precision given a range of sensor and signal widths, shown by the standard deviation of the error between the estimated and true densities.
Standard deviations are shown from deep red to pink, representing high to low values between $0.483\times10^{-6}$ to $3.74\times10^{-6}$.
}
\label{f:StandardDeviation}
\end{figure}
\clearpage
\section{Supplementary information: Impact of parameter error}
\begin{figure}[h!]
\centering
{
\subfloat[\label{f:signal}]{
\includegraphics[width=0.4\textwidth]{imgs/AverageModelBias_callerror.pdf}
}
\subfloat[\label{f:sensor}]{
\includegraphics[width=0.4\textwidth]{imgs/AverageModelBias_cameraerror.pdf}
}
\subfloat[\label{f:radius}]{
\includegraphics[width=0.4\textwidth]{imgs/AverageModelBias_radiuserror.pdf}
}%%
\subfloat[\label{f:speed}]{
\includegraphics[width=0.4\textwidth]{imgs/AverageModelBias_speederror.pdf}
}
}%%
\caption[Model sensitivity to error in parameter estimates]{
Model sensitivity (for all gREM submodels) to error in estimates of A) signal width $\alpha$, B) sensor width $\theta$, C) detection distance $r$ and D) animal movement speed $v$.
Estimates are -10\% (red), -1\% (orange), 0\% (grey), +1\% (green) and +10\% (blue) of the true parameter value.
The black dashed line indicates zero error in density estimates.
The error bars 95\% confidence intervals across all simulations.
}
\label{f:sensitivity}
\end{figure}