Before we can address the identification problem in the context of
  the identification workflow, we must define how images are grouped
  into occurrences.
In this section we propose a clustering algorithm to accomplish
  this task.

\paragraph{Occurrence definition}
The Darwin Core defines an \occurrence{} as a collection of evidence
  that shows an organism exists within specific location and span of
  time~\cite{wieczorek_darwin_2012}.
For our purposes this amounts to a cluster of images localized in space
  and time.
We propose that the \occurrence{} grouping algorithm should perform
  agglomerative clustering on the GPS coordinates and time specified in
  the image metadata.

\paragraph{Space-time image distance}
Towards this goal we define a space-time feature $\g_i$ for each image
  $i$, and a pairwise distance, $\Delta(\g_i, \g_j)$, between these
  features.
This feature will a two dimensional feature tuple, %
$\g_i = \paren{\time_i, \gps_i}$, where the first component is the
  POSIX timestamp $\time_i$, and the second component is a GPS coordinate %
$\gps_i = \brak{\lat_i, \lon_i}^{T}$, where the angles of latitude and
  longitude are measured in radians.
To compute this distance between two images $\g_i$ and $\g_j$ we first
  compute the distance in each component of the feature tuple.
The difference in time is the absolute value of the timedelta,  %
$\Delta_t(\g_i, \g_j) = \abs{\time_i - \time_j}$, which is in seconds.

% DISTANCE BETWEEN TWO IMAGES (space and final)
Next, the distance in space is computed by approximating the Earth as a
  sphere.
In general, the distance between two points on a sphere with radius $r$
  is a function of inverse haversines, and is expressed as:
\begin{equation}\label{eqn:geodistance}
    d(\gps_i, \gps_j, r) =
        2 r \asin{\sqrt{
            \haversine{\lat_i - \lat_j} +
            \haversine{\lon_i - \lon_j} +
            \cos\paren{\lat_i} \cos\paren{\lat_j}}}
\end{equation}
In the previous equation, $\haversine{\theta} = \haversineFULL{\theta}$
  is the half vertical sine function.
Thus, we arrive at the spatial distance between two images by
  estimating the radius of the earth to be $r=6367$ kilometers.
\begin{equation}
    \Delta_s(\g_i, \g_j) = d(\gps_i, \gps_j, 6367).
\end{equation}
This results in distance in seconds and a distance in kilometers, which
  are in incompatible units.
To combine these distances we convert kilometers to seconds by
  heuristically estimating the walking speed, $S$, of an animal (for
  zebras we use $S=2\sciE{-3}$ kilometers per second).
This allows us to cancel kilometers from the expression and express GPS
  distance as a unit of time:
$\frac{\Delta_s(\g_i, \g_j)}{S}$.
This distance can be interpreted as the total amount of time it would
  take an animal to move between two points.
The total distance between two images is the sum of these components.
\begin{equation}\label{eqn:imgdist}
    \Delta(\g_i, \g_j) =
        \Delta_t(\g_i, \g_j) + \frac{\Delta_s(\gps_i, \gps_j)}{S}
\end{equation}
Notice that if there is no difference in GPS location, then this
  measure becomes to a distance in time.

\paragraph{Clustering procedure}
Having defined pairwise a distance between two images, we proceed to
  describe the agglomerative clustering algorithm.
There are two inputs to the agglomerative clustering algorithm:
(1) The matrix of pairwise distance between images, and
(2) the minimum distance threshold between two images.
The matrix of distances is computed using~\cref{eqn:imgdist}, and we
  set the distance threshold to $600$ seconds.
Any pair of images that is within this threshold connected via a
  linkage matrix.
Connected components in this matrix form the final clusters that we use
  as \occurrences{}.

\paragraph{Discussion of occurrences}
These computed \occurrences{} are valuable measurements for multiple
  components of the IBEIS software.
At its core an \occurrence{} describes \wquest{when} a group of animals
  was seen and \wquest{where} that group was seen.
However, to answer the questions like \wquest{how many} animals there
  were, \wquest{who} an animal is, \wquest{who else} is an animal with,
  and \wquest{where else} have these animals been seen, the annotations
  in the \occurrence{} must be grouped into individual \encounters{} and
  then matched against the \masterdatabase{}.
The next section describes the first of these procedures:
the \intraoccurrence{} identification algorithm that produces
  \encounters{}.