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\title{Distance sampling analysis in unmarked}
\author{Richard Chandler\\USGS Patuxent Wildlife Research Center}


\begin{document}

\newcommand{\code}[1]{\texttt{\small{#1}}}
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\maketitle

\abstract{Distance sampling is a wildlife sampling technique used to
  estimate population size or density. Describing how density varies
  spatially is often of equal interest; however, conventional methods
  of analysis do not allow for explicit modeling of both density and
  detection probability. The function \code{distsamp} implements the
  multinomial-Poisson mixture model of Royle et. al
  \citep{royle_modeling_2004}, which was developed to overcome this
  limitation. This model requires that line- or point-transects are
  spatially replicated and that distance data are recorded in discrete
  intervals. The function \code{gdistsamp} extends this basic model,
  by introducing the parameter $\phi$, the probability of
  being available for detection \citep{chandlerEA_2011}. Furthermore,
  this function allows
  abundance to be modeled using the negative binomial distribution,
  which may be useful for dealing with over-dispersion. This document
  describes how to format data, fit models,
  and manipulate results in package \package{unmarked}. It does not
  cover the statistical theory and assumptions underlying distance
  sampling \citep{buckland_distsamp_2001}, which the user is expected
  to be familiar with. }

\section{Introduction}

Spatial variation in density is common to virtually all wildlife
populations, and describing this variation is a central objective of
much research. To accurately estimate and model density, it is often
necessary to account for individuals present but not
detected. Distance from observer is a ubiquitous source of variation
in detection probability, and thus distance sampling has become a
commonly used survey methodology. Numerous options exist for analyzing
distance sampling data, but here the focus is on the model of Royle et
al. \citep{royle_modeling_2004}, which assumes that multiple transects
have been surveyed and distance data are recorded in discrete
intervals. The details of the model formulation are as follows:

The latent transect-level abundance distribution is currently assumed
to be

\begin{equation}
  \label{eq:1}
  N_{i}\sim{Poisson}(\lambda)
  \quad  i=1,\dots,M\
\end{equation}

The detection process is modeled as

\begin{equation}
  \label{eq:2}
  y_{ij}\sim{Multinomial}(N_{i}, \pi_{ij})
  \quad  i=1,\dots,M\;j=1,\dots,J
\end{equation}

where $\pi_{ij}$ is the multinomial cell probability for transect $i$
in distance class $j$. These are computed by integrating a detection
function such as the half-normal (with shape parameter $\sigma$) over
each distance interval.

Parameters $\lambda$ and $\sigma$ can be vectors affected by
transect-specific covariates using the log link.

\section{Importing, formatting, and summarizing data}

The first step is to import the data into R. The simplest option is to
use the \code{read.csv} function to import a .csv file that has been
formatted so that each row represents a transect, and columns describe
either the number of individuals detected in each distance interval or
transect-specific covariates. Alternatively, if data were not recorded
in discrete distance intervals, a .csv file could be imported that
contains a row for each individual detected and columns for the
distances and transect names. This could then be converted to
transect-level data using the function \code{formatDistData}. For
example,

<<>>=
library(unmarked)
dists <- read.csv(system.file("csv", "distdata.csv", package="unmarked"))
head(dists, 10)
levels(dists$transect) <- c(levels(dists$transect), "g")
levels(dists$transect)
yDat <- formatDistData(dists, distCol="distance",
	transectNameCol="transect", dist.breaks=c(0, 5, 10, 15, 20))
yDat
@

Here we have created an object called yDat that contains counts for
each transect (row) in each distance interval (columns). Note the
method used to include transect "g", which was surveyd but where no
individuals were detected. It is important that all survyed transects
are included in the analysis.

Suppose there also exists transect-specific covariate data.

<<>>=
(covs <- data.frame(canopyHt = c(5, 8, 3, 2, 4, 7, 5),
	habitat = c('A','A','A','A','B','B','B'), row.names=letters[1:7]))
@

The function \code{unmarkedFrameDS} can now be used to organize these
data along with their metadata (study design (line- or
point-transect), distance class break points, transect lengths, and
units of measurement) into an object to be used as the \code{data}
argument in \code{distsamp}. By organizing the data this way, the user
does not need to repetitively specify these arguments during each call
to \code{distsamp}, thereby reducing the potential for errors and
facilitating data summary and manipulation.

<<>>=
umf <- unmarkedFrameDS(y=as.matrix(yDat), siteCovs=covs, survey="line",
	dist.breaks=c(0, 5, 10, 15, 20), tlength=rep(100, 7),
	unitsIn="m")
@

Note that there is no \code{obsCovs} argument, meaning that
distance-interval-level covariates cannot be included in the
analysis. The call to \code{unmarkedFrameDS} indicates that the data
were collected on seven line transects, each 100 meters long, and
detections were tabulated into distance intervals defined by the
\code{dist.breaks} cutpoints. It is important that both transect
lengths and distance break points are provided in the same units
specified by \code{unitsIn}.

We can look at these data using a variety of methods.

<<umfhist,fig=TRUE,include=FALSE,width=4,height=4>>=
summary(umf)
hist(umf, xlab="distance (m)", main="", cex.lab=0.8, cex.axis=0.8)
@
\begin{figure}[!ht]
  \centering
  \includegraphics{distsamp-umfhist}
  \caption{Histogram of detection distances.}
  \label{fig:umfhist}
\end{figure}

\newpage
\section{Model fitting}

Now that we have put our data into an object of class
\code{unmarkedFrameDS}, we are ready to fit some models with
\code{distsamp}. The first argument is a \code{formula} which
specifies the detection covariates followed by the density (or
abundance) covariates. The only other required argument is the
\code{data}, but several other optional arguments exist. By default,
the half-normal detection function is used to model density in animals
/ ha. The detection function can be selected using the \code{keyfun}
argument. The response can be changed from ``density", to ``abund"
with the \code{output} argument. When modeling density, the output
units can be changed from h``a" to ``kmsq" using the \code{unitsOut}
argument. \code{distsamp} also includes the arguments \code{starts},
\code{method}, and \code{control}, which are common to all unmarked
fitting functions.

Below is a series of models that demonstrates \code{distsamp}'s
arguments and defaults.

<<>>=
hn_Null <- distsamp(~1~1, umf)
hn_Null <- distsamp(~1~1, umf, keyfun="halfnorm", output="density",
	unitsOut="ha")
haz_Null <- distsamp(~1~1, umf, keyfun="hazard")
hn_Hab.Ht <- distsamp(~canopyHt ~habitat, umf)
@

The first two models are the same, a null half-normal detection
function with density returned in animals / ha (on the log-scale). The
third model uses the hazard-rate detection function, and the fourth
model includes covariates affecting the Poisson mean ($\lambda$) and
the half-normal shape parameter ($\sigma$).


Once a model has been fit, typing its name will display parameter
estimate information and AIC. A summary method shows extra details
including the scale on which parameters were estimated and convergence
results.

<<>>=
haz_Null
@

\section{Manipulating results}

Back-transforming estimates to the original scale and obtaining
standard errors via the delta method is easily accomplished:

<<>>=
names(haz_Null)
backTransform(haz_Null, type="state")
backTransform(haz_Null, type="det")
backTransform(haz_Null, type="scale")
backTransform(linearComb(hn_Hab.Ht['det'], c(1, 5)))
@

The first back-transformation returns population density, since this
is the default state parameter modeled when \code{distsamp}'s
\code{output} argument is set to ``density". The second
back-transformation returns the hazard-rate shape parameter, and third
is the hazard-rate scale parameter. When covariates are present,
\code{backTransform} in conjunction with \code{linearComb} should be
used. Here, we requested the value of sigma when canopy height was 5
meters tall. Note that the intercept was included in the calculation
by setting the first value in the linear equation to 1.

Parameters that do not occur in the likelihood may also be of
interest. For example, the number of individuals occuring in the
sampled plots (local population size) is a fundamental parameter in
monitoring and conservation efforts. The following commands can be
used to derive this parameter from our model of density:

<<>>=
site.level.density <- predict(hn_Hab.Ht, type="state")$Predicted
plotArea.inHectares <- 100 * 40 / 10000
site.level.abundance <- site.level.density * plotArea.inHectares
(N.hat <- sum(site.level.abundance))
@

To describe the uncertainty of N.hat, or any other derived parameter,
we can use a parametric bootstrap approach. First we define a function
to estimate N.hat, and then we apply this function to numerous models
fit to data simulated from our original model.

<<>>=
getN.hat <- function(fit) {
    d <- predict(fit, type="state")$Predicted
    a <- d * (100 * 40 / 10000)
    N.hat <- c(N.hat = sum(a))
    return(N.hat)
    }
pb <- parboot(hn_Hab.Ht, statistic=getN.hat, nsim=25)
pb
@

Here, \code{t\_B} is an approximation of the sampling distribution for
\code{N.hat}, conditioned on our fitted model. Confidence intervals
can be calculated from the quantiles of \code{t\_B}. Note that in
practice nsim should be set to a much larger value and a
goodness-of-fit test should be performed before making inference from
a fitted model. Parameteric bootstrapping can be used for the latter
by supplying a fit statistic such as \code{SSE} instead of
\code{getN.hat}. See ?parboot and vignette('unmarked') for examples.

\section{Prediction and plotting}

A \code{predict} method exits for all \code{unmarkedFit} objects,
which is useful when multiple covariate combinations exist. This
method also facilitates plotting. Suppose we wanted model predictions
from the covariate model along the range of covariate values
studied. First we need to make new \code{data.frame}s holding the
desired covariate combinations. Note that column names must match
those of the original data, and factor variables must contain the same
levels.

<<>>=
head(habConstant <- data.frame(canopyHt = seq(2, 8, length=20),
	habitat=factor("A", levels=c("A", "B"))))
(htConstant <- data.frame(canopyHt = 5,
	habitat=factor(c("A", "B"))))
@

Now \code{predict} can be used to estimate density and $\sigma$ for
each row of our new \code{data.frame}s.

<<>>=
(Elambda <- predict(hn_Hab.Ht, type="state", newdata=htConstant,
	appendData=TRUE))
head(Esigma <- predict(hn_Hab.Ht, type="det", newdata=habConstant,
	appendData=TRUE))
@

Once predictions have been made, plotting is
straight-forward. Figure~\ref{fig:predplot}a, shows density as a
function of habitat type, and Figure~\ref{fig:predplot}b shows that
$\sigma$ is not related to canopy height.

<<predplot,fig=TRUE,include=FALSE,width=6,height=3>>=
par(mfrow=c(1, 2))
with(Elambda, {
	x <- barplot(Predicted, names=habitat, xlab="Habitat",
		ylab="Density (animals / ha)", ylim=c(0, 25), cex.names=0.7,
        cex.lab=0.7, cex.axis=0.7)
	arrows(x, Predicted, x, Predicted+SE, code=3, angle=90, length=0.05)
	box()
	})
with(Esigma, {
	plot(canopyHt, Predicted, type="l", xlab="Canopy height",
		ylab=expression(paste("Half-normal shape parameter (", sigma, ")",
        sep="")), ylim=c(0, 20), cex=0.7, cex.lab=0.7,
    cex.axis=0.7)
	lines(canopyHt, Predicted+SE, lty=2)
	lines(canopyHt, Predicted-SE, lty=2)
	})
@
\begin{figure}[!ht]
  \centering
  \includegraphics{distsamp-predplot}
  \caption{Predicted covariate relatonships.}
  \label{fig:predplot}
\end{figure}


Plots of the detection function parameters can be less informative
than plots of the detection functions themselves. To do the latter, we
can plug predicted values of $\sigma$ at given covariate values into
the \code{gxhn} function. For instance, Figure~\ref{fig:detplot}a
shows to the half-normal function at a canopy height of 2m. This was
plotted by setting $\sigma$ to 10.8, the predicted value shown
above. The available detection functions are described on the
\code{detFuns} help page. Probability density functions such as
\code{dxhn} can be plotted with the distance histogram using the
\code{hist} method for \code{unmarkedFitDS} objects
(Figure\ref{fig:detplot}b). This only works for models without
detection covariates; however, probability density functions at
specific covariate values can be added in a fashion similar to that
above (Figure\ref{fig:detplot}b).


<<detplot,fig=TRUE,include=FALSE,width=6,height=3>>=
par(mfrow=c(1, 2))
plot(function(x) gxhn(x, sigma=10.8), 0, 20, xlab="Distance (m)",
	ylab="Detection prob. at 2m canopy ht.", cex.lab=0.7,
    cex.axis=0.7, las=1)
hist(hn_Null, xlab="Distance (m)", ylab="Probability density", main="",
    ylim=c(0, 0.1), cex.lab=0.7, cex.axis=0.7, las=1)
plot(function(x) dxhn(x, sigma=10.8), 0, 20, add=TRUE, col="blue")
plot(function(x) dxhn(x, sigma=9.9), 0, 20, add=TRUE, col="green")
legend('topright', c("Canopy ht. = 2m", "Null", "Canopy ht. = 8m"),
	col=c("blue", "black", "green"), lty=1, cex=0.4)
@
\begin{figure}[!ht]
  \centering
  \includegraphics{distsamp-detplot}
  \caption{Detection and probability density functions.}
  \label{fig:detplot}
\end{figure}

\newpage

\section{Model extensions}

A common criticism of distance sampling is that all individuals must
be available for detection. Similarly, the probability of detecting an
individual a distance of 0 must be 1. These assumptions often cannot
be met. For instance, when counting cues such as bird songs or whale
blows, the probability that an individual will produce a cue (and thus
be available for detection) is rarely 1 during the sampling
interval. Recently developed methods \citep{chandlerEA_2011}
allow for the estimation of the
probability of being available for detection $\phi$. To do so,
replicate distance sampling observations must be collected at each
transect. These replicates could be collected using repeated visits or
multiple observers working independently. Implementation of this model
in \package{unmarked} is accomplished using the \code{gdistsamp}
function. The function also provides the option to model abundance
using the negative binomial distribution. Formatting data and
specifying models is similar to methods described above and is more
fully outlined in the help pages.

\section{Conclusion}

This document has emphasized methods tailored to distance sampling
analysis; however, the more general methods available in package
\package{unmarked} can also be applied to models fitted using
\code{distsamp} and \code{gdistsamp}.
For example, model-selection and model-averaging can be
accomplished using the \code{fitList} function and the \code{modSel}
and \code{predict} methods.


\bibliography{unmarked}

\end{document}
