121 lines
4.4 KiB
TeX
121 lines
4.4 KiB
TeX
\section{Enhancing Forecasting Models with External Data}
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\label{enhanced_feats}
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In this appendix, we show how the feature matrix in Sub-section
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\ref{ml_models} can be extended with features other than historical order
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data.
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Then, we provide an overview of what external data we tried out as predictors
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in our empirical study.
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\subsection{Enhanced Feature Matrices}
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Feature matrices can naturally be extended by appending new feature columns
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$x_{t,f}$ or $x_f$ on the right where the former represent predictors
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changing throughout a day and the latter being static either within a
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pixel or across a city.
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$f$ refers to an external predictor variable, such as one of the examples
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listed below.
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In the SVR case, the columns should be standardized before fitting as external
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predictors are most likely on a different scale than the historic order
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data.
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Thus, for a matrix with seasonally-adjusted order data $a_t$ in it, an
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enhanced matrix looks as follows:
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$$
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\vec{y}
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=
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\begin{pmatrix}
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a_T \\
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a_{T-1} \\
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\dots \\
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a_{H+1}
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\end{pmatrix}
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~~~~~
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\mat{X}
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=
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\begin{bmatrix}
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a_{T-1} & a_{T-2} & \dots & a_{T-H} & ~~~
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& x_{T,A} & \dots & x_{B} & \dots \\
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a_{T-2} & a_{T-3} & \dots & a_{T-(H+1)} & ~~~
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& x_{T-1,A} & \dots & x_{B} & \dots \\
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\dots & \dots & \dots & \dots & ~~~
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& \dots & \dots & \dots & \dots \\
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a_H & a_{H-1} & \dots & a_1 & ~~~
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& x_{H+1,A} & \dots & x_{B} & \dots
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\end{bmatrix}
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$$
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\
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Similarly, we can also enhance the tabular matrices from
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\ref{tabular_ml_models}.
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The same comments as for their pure equivalents in Sub-section \ref{ml_models}
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apply, in particular, that ML models trained with an enhanced matrix can
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process real-time data without being retrained.
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\subsection{External Data in the Empirical Study}
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\label{external_data}
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In the empirical study, we tested four groups of external features that we
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briefly describe here.
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\vskip 0.1in
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\textbf{Calendar Features}:
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\begin{itemize}
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\item Time of day (as synthesized integers: e.g., 1,050 for 10:30 am,
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or 1,600 for 4 pm)
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\item Day of week (as one-hot encoded booleans)
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\item Work day or not (as booleans)
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\end{itemize}
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\vskip 0.1in
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\textbf{Features derived from the historical Order Data}:
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\begin{itemize}
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\item Number of pre-orders for a time step (as integers)
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\item 7-day SMA of the percentages of discounted orders (as percentages):
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The platform is known for running marketing campaigns aimed at
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first-time customers at irregular intervals. Consequently, the
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order data show a wave-like pattern of coupons redeemed when looking
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at the relative share of discounted orders per day.
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\end{itemize}
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\vskip 0.1in
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\textbf{Neighborhood Features}:
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\begin{itemize}
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\item Ambient population (as integers) as obtained from the ORNL LandScan
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database
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\item Number of active platform restaurants (as integers)
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\item Number of overall restaurants, food outlets, retailers, and other
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businesses (as integers) as obtained from the Google Maps and Yelp
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web services
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\end{itemize}
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\vskip 0.1in
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\textbf{Real-time Weather} (raw data obtained from IBM's
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Wunderground database):
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\begin{itemize}
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\item Absolute temperature, wind speed, and humidity
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(as decimals and percentages)
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\item Relative temperature with respect to 3-day and 7-day historical
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means (as decimals)
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\item Day vs. night defined by sunset (as booleans)
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\item Summarized description (as indicators $-1$, $0$, and $+1$)
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\item Lags of the absolute temperature and the summaries covering the
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previous three hours
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\end{itemize}
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\vskip 0.1in
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Unfortunately, we must report that none of the mentioned external data
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improved the accuracy of the forecasts.
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Some led to models overfitting the data, which could not be regulated.
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Manual tests revealed that real-time weather data are the most promising
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external source.
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Nevertheless, the data provided by IBM's Wunderground database originate from
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weather stations close to airports, which implies that we only have the
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same aggregate weather data for the entire city.
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If weather data is available on a more granular basis in the future, we see
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some potential for exploitation.
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