58 lines
2.4 KiB
TeX
58 lines
2.4 KiB
TeX
\section{Tabular and Real-time Forecasts without Retraining}
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\label{tabular_ml_models}
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Regarding the structure of the feature matrix for the ML models in Sub-section
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\ref{ml_models}, we provide an alternative approach that works without
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the STL method.
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Instead of decomposing a time series and arranging the resulting
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seasonally-adjusted time series $a_t$ into a matrix $\mat{X}$, one can
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create a matrix with two types of feature columns mapped to the raw
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observations in $\vec{y}$:
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While the first group of columns takes all observations of the same time of
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day over a horizon of, for example, one week ($n_h=7$), the second group
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takes all observations covering a pre-defined time horizon, for example
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$3$ hours ($n_r=3$ for 60-minute time steps), preceding the time step to
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be fitted.
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Thus, we exploit the two-dimensional structure of time tables as well, and
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conceptually model historical and recent demand.
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The alternative feature matrix appears as follows where the first three
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columns are the historical and the last three the recent demand features:
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$$
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\vec{y}
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=
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\begin{pmatrix}
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y_T \\
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y_{T-1} \\
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\dots \\
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y_{1+n_hH}
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\end{pmatrix}
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~~~~~
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\mat{X}
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=
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\begin{bmatrix}
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y_{T-H} & y_{T-2H} & \dots & y_{T-n_hH}
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& y_{T-1} & y_{T-2} & \dots & y_{T-n_r} \\
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y_{T-1-H} & y_{T-1-2H} & \dots & y_{T-1-n_hH}
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& y_{T-2} & y_{T-3} & \dots & y_{T-n_r-1} \\
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\dots & \dots & \dots & \dots
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& \dots & \dots & \dots & \dots \\
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y_{1+(n_h-1)H} & y_{1+(n_h-2)H} & \dots & y_1
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& y^*_{1+n_hH-1} & y^*_{1+n_hH-2} & \dots & y^*_{1+n_hH-n_r}
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\end{bmatrix}
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$$
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\
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Being a detail, we note that the recent demand features lying on the end of
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the previous day are set to $0$, which is shown with the $^*$ notation
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above.
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This alignment of the undecomposed order data $y_t$ ensures that the ML
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models learn the two seasonal patterns independently.
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The parameters $n_h$ and $n_r$ must be adapted to the data, but we found the
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above values to work well.
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As such matrices resemble time tables, we refer to them as tabular.
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However, we found the ML models with vertical time series to outperform the
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tabular ML models, which is why we disregarded them in the study.
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This tabular form could be beneficial for UDPs with a demand that exhibits
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a weaker seasonality such as a meal delivery platform.
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