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tex/3_mod/6_decomp.tex

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+\subsection{Time Series Decomposition}
+\label{decomp}
+
+Concerning the time table in Figure \ref{f:timetable}, a seasonal demand
+    pattern is inherent to both horizontal and vertical time series.
+First, the weekday influences if people eat out or order in with our partner
+    receiving more orders on Thursday through Saturday than the other four
+    days.
+This pattern is part of both types of time series.
+Second, on any given day, demand peaks occur around lunch and dinner times.
+This only regards vertical series.
+Statistical analyses show that horizontally sliced time series indeed exhibit
+    a periodicity of $k=7$, and vertically sliced series only yield a seasonal
+    component with a regular pattern if the periodicity is set to the product
+    of the number of weekdays and the daily time steps indicating a distinct
+    intra-day pattern per weekday.
+
+Figure \ref{f:stl} shows three exemplary STL decompositions for a
+    $1~\text{km}^2$ pixel and a vertical time series with 60-minute time steps
+    (on the x-axis) covering four weeks:
+With the noisy raw data $y_t$ on the left, the seasonal and trend components,
+    $s_t$ and $t_t$, are depicted in light and dark gray for increasing $ns$
+    parameters.
+The plots include (seasonal) na\"{i}ve forecasts for the subsequent test day
+    as dotted lines.
+The remainder components $r_t$ are not shown for conciseness.
+The periodicity is set to $k = 7 * 12 = 84$ as our industry partner has $12$
+    opening hours per day.
+
+\begin{center}
+\captionof{figure}{STL decompositions for a medium-demand pixel with hourly
+                   time steps and periodicity $k=84$}
+\label{f:stl}
+\includegraphics[width=.95\linewidth]{static/stl_gray.png}
+\end{center}
+
+As described in Sub-section \ref{stl}, with $k$ being implied by the
+    application, at the very least, the length of the seasonal smoothing
+    window, represented by the $ns$ parameter, must be calibrated by the
+    forecaster:
+It controls how many past observations go into each smoothened $s_t$.
+Many practitioners, however, skip this step and set $ns$ to a big number, for
+    example, $999$, then referred to as "periodic."
+For the other parameters, it is common to use the default values as
+    specified in \cite{cleveland1990}.
+The goal is to find a decomposition with a regular pattern in $s_t$.
+In Figure \ref{f:stl}, this is not true for $ns=7$ where, for
+    example, the four largest bars corresponding to the same time of day a
+    week apart cannot be connected by an approximately straight line.
+On the contrary, a regular pattern in the most extreme way exists for
+    $ns=999$, where the same four largest bars are of the same height.
+This observation holds for each time step of the day.
+For $ns=11$, $s_t$ exhibits a regular pattern whose bars adapt over time:
+The pattern is regular as bars corresponding to the same time of day can be
+    connected by approximately straight lines, and it is adaptive as these
+    lines are not horizontal.
+The trade-off between small and large values for $ns$ can thus be interpreted
+    as allowing the average demand during peak times to change over time:
+If demand is intermittent at non-peak times, it is reasonable to expect the
+    bars to change over time as only the relative differences between peak and
+    non-peak times impact the bars' heights with the seasonal component being
+    centered around $0$.
+To confirm the goodness of a decomposition statistically, one way is to verify
+    that $r_t$ can be modeled as a typical error process like white noise
+    $\epsilon_t$.
+
+However, we suggest an alternative way of calibrating the STL method in an
+    automated fashion based on our unified CV approach.
+As hinted at in Figure \ref{f:stl}, we interpret an STL decomposition as a
+    forecasting method on its own by just adding the (seasonal) na\"{i}ve
+    forecasts for $s_t$ and $t_t$ and predicting $0$ for $r_t$.
+Then, the $ns$ parameter is tuned just like a parameter for an ML model.
+To the best of our knowledge, this has not yet been proposed before.
+Conceptually, forecasting with the STL method can be viewed as a na\"{i}ve
+    method with built-in smoothing, and it outperformed all other
+    benchmark methods in all cases.