262 lines
9.3 KiB
TeX
262 lines
9.3 KiB
TeX
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\section{Forecasting Accuracies during Peak Times}
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\label{peak_results}
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This appendix shows all result tables from the main text with the MASE
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averages calculated from time steps within peak times.
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Peaks are the times of the day where the typical customer has a lunch or
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dinner meal and defined to be either from 12 pm to 2 pm or from 6 pm to
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8 pm.
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While the exact decimals of the MASEs differ from the ones in the main
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text, the relative ranks of the forecasting methods are the same except in
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rare cases.
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\begin{center}
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\captionof{table}{Top-3 models by training weeks and average demand
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($1~\text{km}^2$ pixel size, 60-minute time steps)}
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\label{t:results:a}
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\begin{tabular}{|c|c|*{12}{c|}}
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\hline
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\multirow{3}{*}{\rotatebox{90}{\thead{Training}}}
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& \multirow{3}{*}{\rotatebox{90}{\thead{Rank}}}
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& \multicolumn{3}{c|}{\thead{No Demand}}
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& \multicolumn{3}{c|}{\thead{Low Demand}}
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& \multicolumn{3}{c|}{\thead{Medium Demand}}
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& \multicolumn{3}{c|}{\thead{High Demand}} \\
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~ & ~
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& \multicolumn{3}{c|}{(0 - 2.5)}
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& \multicolumn{3}{c|}{(2.5 - 10)}
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& \multicolumn{3}{c|}{(10 - 25)}
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& \multicolumn{3}{c|}{(25 - $\infty$)} \\
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\cline{3-14}
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~ & ~
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& Method & MASE & $n$
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& Method & MASE & $n$
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& Method & MASE & $n$
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& Method & MASE & $n$ \\
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\hline \hline
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\multirow{3}{*}{3} & 1
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& \textbf{\textit{trivial}}
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& 0.794 & \multirow{3}{*}{\rotatebox{90}{4586}}
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& \textbf{\textit{hsma}}
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& 0.817 & \multirow{3}{*}{\rotatebox{90}{2975}}
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& \textbf{\textit{hsma}}
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& 0.838 & \multirow{3}{*}{\rotatebox{90}{2743}}
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& \textbf{\textit{rtarima}}
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& 0.871 & \multirow{3}{*}{\rotatebox{90}{2018}} \\
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~ & 2
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& \textit{hsma} & 0.808 & ~
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& \textit{hses} & 0.847 & ~
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& \textit{hses} & 0.851 & ~
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& \textit{rtses} & 0.872 & ~ \\
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~ & 3
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& \textit{pnaive} & 0.938 & ~
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& \textit{hets} & 0.848 & ~
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& \textit{hets} & 0.853 & ~
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& \textit{rtets} & 0.874 & ~ \\
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\hline
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\multirow{3}{*}{4} & 1
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& \textbf{\textit{trivial}}
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& 0.791 & \multirow{3}{*}{\rotatebox{90}{4532}}
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& \textbf{\textit{hsma}}
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& 0.833 & \multirow{3}{*}{\rotatebox{90}{3033}}
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& \textbf{\textit{hsma}}
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& 0.839 & \multirow{3}{*}{\rotatebox{90}{2687}}
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& \textbf{\textit{vrfr}}
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& 0.848 & \multirow{3}{*}{\rotatebox{90}{2016}} \\
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~ & 2
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& \textit{hsma} & 0.794 & ~
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& \textit{hses} & 0.838 & ~
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& \textit{hses} & 0.847 & ~
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& \textbf{\textit{rtarima}} & 0.851 & ~ \\
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~ & 3
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& \textit{pnaive} & 0.907 & ~
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& \textit{hets} & 0.841 & ~
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& \textit{hets} & 0.851 & ~
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& \textit{rtses} & 0.857 & ~ \\
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\hline
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\multirow{3}{*}{5} & 1
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& \textbf{\textit{trivial}}
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& 0.782 & \multirow{3}{*}{\rotatebox{90}{4527}}
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& \textbf{\textit{hsma}}
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& 0.844 & \multirow{3}{*}{\rotatebox{90}{3055}}
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& \textbf{\textit{hsma}}
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& 0.841 & \multirow{3}{*}{\rotatebox{90}{2662}}
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& \textbf{\textit{vrfr}}
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& 0.849 & \multirow{3}{*}{\rotatebox{90}{2019}} \\
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~ & 2
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& \textit{hsma} & 0.802 & ~
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& \textit{hses} & 0.851 & ~
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& \textit{hets} & 0.844 & ~
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& \textbf{\textit{rtarima}} & 0.851 & ~ \\
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~ & 3
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& \textit{pnaive} & 0.888 & ~
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& \textit{hets} & 0.863 & ~
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& \textit{hses} & 0.845 & ~
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& \textit{vsvr} & 0.853 & ~ \\
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\hline
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\multirow{3}{*}{6} & 1
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& \textbf{\textit{trivial}}
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& 0.743 & \multirow{3}{*}{\rotatebox{90}{4470}}
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& \textbf{\textit{hsma}}
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& 0.843 & \multirow{3}{*}{\rotatebox{90}{3086}}
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& \textbf{\textit{hsma}}
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& 0.841 & \multirow{3}{*}{\rotatebox{90}{2625}}
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& \textbf{\textit{vrfr}}
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& 0.844 & \multirow{3}{*}{\rotatebox{90}{2025}} \\
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~ & 2
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& \textit{hsma} & 0.765 & ~
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& \textit{hses} & 0.853 & ~
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& \textit{hses} & 0.844 & ~
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& \textbf{\textit{hets}} & 0.847 & ~ \\
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~ & 3
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& \textit{pnaive} & 0.836 & ~
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& \textit{hets} & 0.861 & ~
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& \textit{hets} & 0.844 & ~
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& \textit{vsvr} & 0.849 & ~ \\
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\hline
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\multirow{3}{*}{7} & 1
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& \textbf{\textit{trivial}}
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& 0.728 & \multirow{3}{*}{\rotatebox{90}{4454}}
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& \textbf{\textit{hsma}}
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& 0.855 & \multirow{3}{*}{\rotatebox{90}{3132}}
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& \textbf{\textit{hets}}
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& 0.843 & \multirow{3}{*}{\rotatebox{90}{2597}}
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& \textbf{\textit{hets}}
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& 0.839 & \multirow{3}{*}{\rotatebox{90}{2007}} \\
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~ & 2
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& \textit{hsma} & 0.744 & ~
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& \textit{hses} & 0.862 & ~
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& \textit{hsma} & 0.845 & ~
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& \textbf{\textit{vrfr}} & 0.842 & ~ \\
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~ & 3
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& \textit{pnaive} & 0.812 & ~
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& \textit{hets} & 0.868 & ~
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& \textbf{\textit{vsvr}} & 0.849 & ~
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& \textit{vsvr} & 0.846 & ~ \\
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\hline
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\multirow{3}{*}{8} & 1
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& \textbf{\textit{trivial}}
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& 0.736 & \multirow{3}{*}{\rotatebox{90}{4402}}
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& \textbf{\textit{hsma}}
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& 0.865 & \multirow{3}{*}{\rotatebox{90}{3159}}
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& \textbf{\textit{hets}}
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& 0.843 & \multirow{3}{*}{\rotatebox{90}{2575}}
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& \textbf{\textit{hets}}
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& 0.837 & \multirow{3}{*}{\rotatebox{90}{2002}} \\
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~ & 2
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& \textit{hsma} & 0.759 & ~
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& \textit{hets} & 0.874 & ~
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& \textbf{\textit{vsvr}} & 0.848 & ~
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& \textbf{\textit{vrfr}} & 0.841 & ~ \\
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~ & 3
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& \textit{pnaive} & 0.820 & ~
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& \textit{hses} & 0.879 & ~
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& \textit{hsma} & 0.850 & ~
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& \textit{vsvr} & 0.847 & ~ \\
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\hline
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\end{tabular}
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\end{center}
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\begin{center}
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\captionof{table}{Ranking of benchmark and horizontal models
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($1~\text{km}^2$ pixel size, 60-minute time steps):
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the table shows the ranks for cases with $2.5 < ADD < 25$
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(and $25 < ADD < \infty$ in parentheses if they differ)}
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\label{t:hori:a}
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\begin{tabular}{|c|ccc|cccccccc|}
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\hline
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\multirow{2}{*}{\rotatebox{90}{\thead{\scriptsize{Training}}}}
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& \multicolumn{3}{c|}{\thead{Benchmarks}}
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& \multicolumn{8}{c|}{\thead{Horizontal (whole-day-ahead)}} \\
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\cline{2-12}
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~ & \textit{naive} & \textit{fnaive} & \textit{paive}
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& \textit{harima} & \textit{hcroston} & \textit{hets} & \textit{hholt}
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& \textit{hhwinters} & \textit{hses} & \textit{hsma} & \textit{htheta} \\
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\hline \hline
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3 & 11 & 7 (2) & 8 (5) & 5 (7) & 4 & 3
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& 9 (10) & 10 (9) & 2 (6) & 1 & 6 (8) \\
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4 & 11 & 7 (2) & 8 (3) & 5 (6) & 4 (5) & 3 (1)
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& 9 (10) & 10 (9) & 2 (8) & 1 (4) & 6 (7) \\
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5 & 11 & 7 (2) & 8 (4) & 5 (3) & 4 (9) & 3 (1)
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& 9 (10) & 10 (5) & 2 (8) & 1 (6) & 6 (7) \\
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6 & 11 & 8 (5) & 9 (6) & 5 (4) & 4 (7) & 2 (1)
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& 10 & 7 (2) & 3 (8) & 1 (9) & 6 (3) \\
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7 & 11 & 8 (5) & 10 (6) & 5 (4) & 4 (7) & 2 (1)
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& 9 (10) & 7 (2) & 3 (8) & 1 (9) & 6 (3) \\
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8 & 11 & 9 (5) & 10 (6) & 5 (4) & 4 (7) & 2 (1)
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& 8 (10) & 7 (2) & 3 (8) & 1 (9) & 6 (3) \\
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\hline
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\end{tabular}
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\end{center}
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\
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\begin{center}
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\captionof{table}{Ranking of classical models on vertical time series
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($1~\text{km}^2$ pixel size, 60-minute time steps):
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the table shows the ranks for cases with $2.5 < ADD < 25$
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(and $25 < ADD < \infty$ in parentheses if they differ)}
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\label{t:vert:a}
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\begin{tabular}{|c|cc|ccccc|ccccc|}
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\hline
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\multirow{2}{*}{\rotatebox{90}{\thead{\scriptsize{Training}}}}
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& \multicolumn{2}{c|}{\thead{Benchmarks}}
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& \multicolumn{5}{c|}{\thead{Vertical (whole-day-ahead)}}
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& \multicolumn{5}{c|}{\thead{Vertical (real-time)}} \\
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\cline{2-13}
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~ & \textit{hets} & \textit{hsma} & \textit{varima} & \textit{vets}
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& \textit{vholt} & \textit{vses} & \textit{vtheta} & \textit{rtarima}
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& \textit{rtets} & \textit{rtholt} & \textit{rtses} & \textit{rttheta} \\
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\hline \hline
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3 & 2 (10) & 1 (7) & 6 (4) & 8 (6) & 10 (9)
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& 7 (5) & 11 (12) & 4 (1) & 5 (3) & 9 (8) & 3 (2) & 12 (11) \\
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4 & 2 (7) & 1 (10) & 6 (4) & 8 (6) & 10 (9)
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& 7 (5) & 12 (11) & 3 (1) & 5 (3) & 9 (8) & 4 (2) & 11 (12) \\
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5 & 2 (3) & 1 (10) & 7 (5) & 8 (7) & 10 (9)
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& 6 & 11 & 4 (1) & 5 (4) & 9 (8) & 3 (2) & 12 \\
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6 & 2 (1) & 1 (10) & 6 (5) & 8 (7) & 10 (9)
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& 7 (6) & 11 (12) & 3 (2) & 5 (4) & 9 (8) & 4 (3) & 12 (11) \\
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7 & 2 (1) & 1 (10) & 8 (5) & 7 & 10 (9)
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& 6 & 11 (12) & 5 (2) & 4 & 9 (8) & 3 & 12 (11) \\
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8 & 2 (1) & 1 (9) & 8 (5) & 7 & 10 (8)
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& 6 & 12 (10) & 5 (2) & 4 & 9 (6) & 3 & 11 \\
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\hline
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\end{tabular}
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\end{center}
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\
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\pagebreak
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\begin{center}
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\captionof{table}{Ranking of ML models on vertical time series
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($1~\text{km}^2$ pixel size, 60-minute time steps):
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the table shows the ranks for cases with $2.5 < ADD < 25$
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(and $25 < ADD < \infty$ in parentheses if they differ)}
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\label{t:ml:a}
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\begin{tabular}{|c|cccc|cc|}
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\hline
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\multirow{2}{*}{\rotatebox{90}{\thead{\scriptsize{Training}}}}
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& \multicolumn{4}{c|}{\thead{Benchmarks}}
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& \multicolumn{2}{c|}{\thead{ML}} \\
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\cline{2-7}
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~ & \textit{fnaive} & \textit{hets} & \textit{hsma}
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& \textit{rtarima} & \textit{vrfr} & \textit{vsvr} \\
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\hline \hline
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3 & 6 & 2 (5) & 1 (3) & 3 (1) & 5 (2) & 4 \\
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4 & 6 (5) & 2 (3) & 1 (6) & 3 (2) & 5 (1) & 4 \\
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5 & 6 (5) & 2 (4) & 1 (6) & 4 (2) & 5 (1) & 3 \\
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6 & 6 (5) & 2 & 1 (6) & 4 & 5 (1) & 3 \\
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7 & 6 (5) & 2 (1) & 1 (6) & 4 & 5 (2) & 3 \\
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8 & 6 (5) & 2 (1) & 1 (6) & 4 & 5 (2) & 3 \\
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\hline
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\end{tabular}
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\end{center}
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\
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