\section{Enhancing Forecasting Models with External Data}
\label{enhanced_feats}

In this appendix, we show how the feature matrix in Sub-section
    \ref{ml_models} can be extended with features other than historical order
    data.
Then, we provide an overview of what external data we tried out as predictors
    in our empirical study.

\subsection{Enhanced Feature Matrices}

Feature matrices can naturally be extended by appending new feature columns
    $x_{t,f}$ or $x_f$ on the right where the former represent predictors
    changing throughout a day and the latter being static either within a
    pixel or across a city.
$f$ refers to an external predictor variable, such as one of the examples
    listed below.
In the SVR case, the columns should be standardized before fitting as external
    predictors are most likely on a different scale than the historic order
    data.
Thus, for a matrix with seasonally-adjusted order data $a_t$ in it, an
    enhanced matrix looks as follows:

$$
\vec{y}
=
\begin{pmatrix}
    a_T \\
    a_{T-1} \\
    \dots \\
    a_{H+1}
\end{pmatrix}
~~~~~
\mat{X}
=
\begin{bmatrix}
    a_{T-1}         & a_{T-2} & \dots & a_{T-H}     & ~~~
        & x_{T,A}   & \dots   & x_{B} & \dots \\
    a_{T-2}         & a_{T-3} & \dots & a_{T-(H+1)} & ~~~
        & x_{T-1,A} & \dots   & x_{B} & \dots \\
    \dots           & \dots   & \dots & \dots       & ~~~
        & \dots     & \dots   & \dots & \dots \\
    a_H             & a_{H-1} & \dots & a_1         & ~~~
        & x_{H+1,A} & \dots   & x_{B} & \dots
\end{bmatrix}
$$
\

Similarly, we can also enhance the tabular matrices from
    \ref{tabular_ml_models}.
The same comments as for their pure equivalents in Sub-section \ref{ml_models}
    apply, in particular, that ML models trained with an enhanced matrix can
    process real-time data without being retrained.
    
\subsection{External Data in the Empirical Study}
\label{external_data}

In the empirical study, we tested four groups of external features that we
    briefly describe here.

\vskip 0.1in

\textbf{Calendar Features}:
\begin{itemize}
    \item Time of day (as synthesized integers: e.g., 1,050 for 10:30 am,
                       or 1,600 for 4 pm)
    \item Day of week (as one-hot encoded booleans)
    \item Work day or not (as booleans)
\end{itemize}

\vskip 0.1in

\textbf{Features derived from the historical Order Data}:
\begin{itemize}
    \item Number of pre-orders for a time step (as integers)
    \item 7-day SMA of the percentages of discounted orders (as percentages):
          The platform is known for running marketing campaigns aimed at
          first-time customers at irregular intervals. Consequently, the
          order data show a wave-like pattern of coupons redeemed when looking
          at the relative share of discounted orders per day.
\end{itemize}

\vskip 0.1in

\textbf{Neighborhood Features}:
\begin{itemize}
    \item Ambient population (as integers) as obtained from the ORNL LandScan
          database
    \item Number of active platform restaurants (as integers)
    \item Number of overall restaurants, food outlets, retailers, and other
          businesses (as integers) as obtained from the Google Maps and Yelp
          web services
\end{itemize}

\vskip 0.1in

\textbf{Real-time Weather} (raw data obtained from IBM's
                            Wunderground database):
\begin{itemize}
    \item Absolute temperature, wind speed, and humidity
          (as decimals and percentages)
    \item Relative temperature with respect to 3-day and 7-day historical
          means (as decimals)
    \item Day vs. night defined by sunset (as booleans)
    \item Summarized description (as indicators $-1$, $0$, and $+1$)
    \item Lags of the absolute temperature and the summaries covering the
          previous three hours
\end{itemize}

\vskip 0.1in

Unfortunately, we must report that none of the mentioned external data
    improved the accuracy of the forecasts.
Some led to models overfitting the data, which could not be regulated.
Manual tests revealed that real-time weather data are the most promising
    external source.
Nevertheless, the data provided by IBM's Wunderground database originate from
    weather stations close to airports, which implies that we only have the
    same aggregate weather data for the entire city.
If weather data is available on a more granular basis in the future, we see
    some potential for exploitation.