78 lines
4 KiB
TeX
78 lines
4 KiB
TeX
\subsubsection{Na\"{i}ve Methods, Moving Averages, and Exponential Smoothing.}
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\label{ets}
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Simple forecasting methods are often employed as a benchmark for more
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sophisticated ones.
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The so-called na\"{i}ve and seasonal na\"{i}ve methods forecast the next time
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step in a time series, $y_{T+1}$, with the last observation, $y_T$,
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and, if a seasonal pattern is present, with the observation $k$ steps
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before, $y_{T+1-k}$.
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As variants, both methods can be generalized to include drift terms in the
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presence of a trend or changing seasonal amplitude.
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If a time series exhibits no trend, a simple moving average (SMA) is a
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generalization of the na\"{i}ve method that is more robust to outliers.
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It is defined as follows: $\hat{y}_{T+1} = \frac{1}{h} \sum_{i=T-h}^{T} y_i$
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where $h$ is the horizon over which the average is calculated.
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If a time series exhibits a seasonal pattern, setting $h$ to a multiple of the
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periodicity $k$ suffices that the forecast is unbiased.
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Starting in the 1950s, another popular family of forecasting methods,
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so-called exponential smoothing methods, was introduced by
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\cite{brown1959}, \cite{holt1957}, and \cite{winters1960}.
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The idea is that forecasts $\hat{y}_{T+1}$ are a weighted average of past
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observations where the weights decay over time; in the case of the simple
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exponential smoothing (SES) method we obtain:
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$
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\hat{y}_{T+1} = \alpha y_T + \alpha (1 - \alpha) y_{T-1}
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+ \alpha (1 - \alpha)^2 y_{T-2}
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+ \dots + \alpha (1 - \alpha)^{T-1} y_{1}
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$
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where $\alpha$ (with $0 \le \alpha \le 1$) is a smoothing parameter.
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Exponential smoothing methods are often expressed in an alternative component
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form that consists of a forecast equation and one or more smoothing
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equations for unobservable components.
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Below, we present a generalization of SES, the so-called Holt-Winters'
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seasonal method, in an additive formulation.
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$\ell_t$, $b_t$, and $s_t$ represent the unobservable level, trend, and
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seasonal components inherent in $y_t$, and $\beta$ and $\gamma$ complement
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$\alpha$ as smoothing parameters:
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\begin{align*}
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\hat{y}_{t+1} & = \ell_t + b_t + s_{t+1-k} \\
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\ell_t & = \alpha(y_t - s_{t-k}) + (1 - \alpha)(\ell_{t-1} + b_{t-1}) \\
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b_t & = \beta (\ell_{t} - \ell_{t-1}) + (1 - \beta) b_{t-1} \\
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s_t & = \gamma (y_t - \ell_{t-1} - b_{t-1}) + (1-\gamma)s_{t-k}
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\end{align*}
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With $b_t$, $s_t$, $\beta$, and $\gamma$ removed, this formulation reduces to
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SES.
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Distinct variations exist: Besides the three components, \cite{gardner1985}
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add dampening for the trend, \cite{pegels1969} provides multiplicative
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formulations, and \cite{taylor2003} adds dampening to the latter.
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The accuracy measure commonly employed is the sum of squared errors between
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the observations and their forecasts.
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Originally introduced by \cite{assimakopoulos2000}, \cite{hyndman2003} show
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how the Theta method can be regarded as an equivalent to SES with a drift
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term.
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We mention this method here only because \cite{bell2018} emphasize that it
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performs well at Uber.
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However, in our empirical study, we find that this is not true in general.
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\cite{hyndman2002} introduce statistical processes, so-called innovations
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state-space models, to generalize the methods in this sub-section.
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They call this family of models ETS as they capture error, trend, and seasonal
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terms.
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Linear and additive ETS models have a structure like so:
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\begin{align*}
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y_t & = \vec{w} \cdot \vec{x}_{t-1} + \epsilon_t \\
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\vec{x_t} & = \mat{F} \vec{x}_{t-1} + \vec{g} \epsilon_t
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\end{align*}
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$y_t$ denote the observations as before while $\vec{x}_t$ is a state vector of
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unobserved components.
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$\epsilon_t$ is a white noise series and the matrix $\mat{F}$ and the vectors
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$\vec{g}$ and $\vec{w}$ contain a model's coefficients.
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Just as the models in the next sub-section, ETS models are commonly fitted
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with maximum likelihood and evaluated using information theoretical
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criteria against historical data.
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We refer to \cite{hyndman2008b} for a thorough summary.
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