61 lines
3.6 KiB
TeX
61 lines
3.6 KiB
TeX
\subsubsection{Seasonal and Trend Decomposition using Loess}
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\label{stl}
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A time series $y_t$ may exhibit different types of patterns; to fully capture
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each of them, the series must be decomposed.
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Then, each component is forecast with a distinct model.
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Most commonly, the components are the trend $t_t$, seasonality $s_t$, and
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remainder $r_t$.
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They are themselves time series, where only $s_t$ exhibits a periodicity $k$.
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A decomposition may be additive (i.e., $y_t = s_t + t_t + r_t$) or
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multiplicative (i.e., $y_t = s_t * t_t * r_t$); the former assumes that
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the effect of the seasonal component is independent of the overall level
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of $y_t$ and vice versa.
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The seasonal component is centered around $0$ in both cases such that its
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removal does not affect the level of $y_t$.
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Often, it is sufficient to only seasonally adjust the time series, and model
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the trend and remainder together, for example, as $a_t = y_t - s_t$ in the
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additive case.
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Early approaches employed moving averages (cf., Sub-section \ref{ets}) to
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calculate a trend component, and, after removing that from $y_t$, averaged
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all observations of the same seasonal lag to obtain the seasonal
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component.
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The downsides of this are the subjectivity in choosing the window lengths for
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the moving average and the seasonal averaging, the incapability of the
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seasonal component to vary its amplitude over time, and the missing
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handling of outliers.
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The X11 method developed at the U.S. Census Bureau and described in detail by
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\cite{dagum2016} overcomes these disadvantages.
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However, due to its background in economics, it is designed primarily for
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quarterly or monthly data, and the change in amplitude over time cannot be
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controlled.
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Variants of this method are the SEATS decomposition by the Bank of Spain and
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the newer X13-SEATS-ARIMA method by the U.S. Census Bureau.
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Their main advantages stem from the fact that the models calibrate themselves
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according to statistical criteria without manual work for a statistician
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and that the fitting process is robust to outliers.
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\cite{cleveland1990} introduce a seasonal and trend decomposition using a
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repeated locally weighted regression - the so-called Loess procedure - to
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smoothen the trend and seasonal components, which can be viewed as a
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generalization of the methods above and is denoted by the acronym STL.
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In contrast to the X11, X13, and SEATS methods, the STL supports seasonalities
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of any lag $k$ that must, however, be determined with additional
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statistical tests or set with out-of-band knowledge by the forecaster
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(e.g., hourly demand data implies $k = 24 * 7 = 168$ assuming customer
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behavior differs on each day of the week).
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Moreover, the seasonal component's rate of change, represented by the $ns$
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parameter and explained in detail with Figure \ref{f:stl} in Section
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\ref{decomp}, must be set by the forecaster as well, while the trend's
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smoothness may be controlled via setting a non-default window size.
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Outliers are handled by assignment to the remainder such that they do not
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affect the trend and seasonal components.
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In particular, the manual input needed to calibrate the STL explains why only
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the X11, X13, and SEATS methods are widely used by practitioners.
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However, the widespread adoption of concepts like cross-validation (cf.,
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Sub-section \ref{cv}) in recent years enables the usage of an automated
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grid search to optimize the parameters.
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The STL's usage within a grid search is facilitated even further by its being
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computationally cheaper than the other methods discussed.
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