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\subsection{Gridification, Time Tables, and Time Series Generation}
\label{grid}
The platform's tabular order data are sliced with respect to both location and
time and then aggregated into time series where an observation tells
the number of orders in an area for a time step/interval.
Figure \ref{f:grid} shows how the orders' delivery locations are each
matched to a square-shaped cell, referred to as a pixel, on a grid
covering the entire service area within a city.
This gridification step is also applied to the pickup locations separately.
The lower-left corner is chosen at random.
\cite{winkenbach2015} apply the same gridification idea and slice an urban
area to model a location-routing problem, and \cite{singleton2017} portray
it as a standard method in the field of urban analytics.
With increasing pixel sizes, the time series exhibit more order aggregation
with a possibly stronger demand pattern.
On the other hand, the larger the pixels, the less valuable become the
generated forecasts as, for example, a courier sent to a pixel
preemptively then faces a longer average distance to a restaurant in the
pixel.
\begin{center}
\captionof{figure}{Gridification for delivery locations in Paris with a pixel
size of $1~\text{km}^2$}
\label{f:grid}
\includegraphics[width=.8\linewidth]{static/gridification_for_paris_gray.png}
\end{center}
After gridification, the ad-hoc orders within a pixel are aggregated by their
placement timestamps into sub-daily time steps of pre-defined lengths
to obtain a time table as exemplified in Figure \ref{f:timetable} with
one-hour intervals.
\begin{center}
\captionof{figure}{Aggregation into a time table with hourly time steps}
\label{f:timetable}
\begin{tabular}{|c||*{9}{c|}}
\hline
\backslashbox{Time}{Day} & \makebox[2em]{\ldots}
& \makebox[3em]{Mon} & \makebox[3em]{Tue}
& \makebox[3em]{Wed} & \makebox[3em]{Thu}
& \makebox[3em]{Fri} & \makebox[3em]{Sat}
& \makebox[3em]{Sun} & \makebox[2em]{\ldots} \\
\hline
\hline
11:00 & \ldots & $y_{11,Mon}$ & $y_{11,Tue}$ & $y_{11,Wed}$ & $y_{11,Thu}$
& $y_{11,Fri}$ & $y_{11,Sat}$ & $y_{11,Sun}$ & \ldots \\
\hline
12:00 & \ldots & $y_{12,Mon}$ & $y_{12,Tue}$ & $y_{12,Wed}$ & $y_{12,Thu}$
& $y_{12,Fri}$ & $y_{12,Sat}$ & $y_{12,Sun}$ & \ldots \\
\hline
\ldots & \ldots & \ldots & \ldots & \ldots
& \ldots & \ldots & \ldots & \ldots & \ldots \\
\hline
20:00 & \ldots & $y_{20,Mon}$ & $y_{20,Tue}$ & $y_{20,Wed}$ & $y_{20,Thu}$
& $y_{20,Fri}$ & $y_{20,Sat}$ & $y_{20,Sun}$ & \ldots \\
\hline
21:00 & \ldots & $y_{21,Mon}$ & $y_{21,Tue}$ & $y_{21,Wed}$ & $y_{21,Thu}$
& $y_{21,Fri}$ & $y_{21,Sat}$ & $y_{21,Sun}$ & \ldots \\
\hline
\ldots & \ldots & \ldots & \ldots & \ldots
& \ldots & \ldots & \ldots & \ldots & \ldots \\
\hline
\end{tabular}
\end{center}
\
Consequently, each $y_{t,d}$ in Figure \ref{f:timetable} is the number of
all orders within the pixel for the time of day $t$ and day of week
$d$ ($y_t$ and $y_{t,d}$ are the same but differ in that the latter
acknowledges a 2D view).
The same trade-off as with gridification applies:
The shorter the interval, the weaker is the demand pattern to be expected in
the time series due to less aggregation while longer intervals lead to
less usable forecasts.
We refer to time steps by their start time, and their number per day, $H$,
is constant.
Given a time table as in Figure \ref{f:timetable} there are two ways to
generate a time series by slicing:
\begin{enumerate}
\item \textbf{Horizontal View}:
Take only the order counts for a given time of the day
\item \textbf{Vertical View}:
Take all order counts and remove the double-seasonal pattern induced
by the weekday and time of the day with decomposition
\end{enumerate}
Distinct time series are retrieved by iterating through the time tables either
horizontally or vertically in increments of a single time step.
Another property of a generated time series is its length, which, following
the next sub-section, can be interpreted as the sum of the production
training set and the test day.
In summary, a distinct time series is generated from the tabular order data
based on a configuration of parameters for the dimensions pixel size,
number of daily time steps $H$, shape (horizontal vs. vertical), length,
and the time step to be predicted.