69 lines
3.3 KiB
TeX
69 lines
3.3 KiB
TeX
\subsubsection{Autoregressive Integrated Moving Averages.}
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\label{arima}
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\cite{box1962}, \cite{box1968}, and more papers by the same authors in the
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1960s introduce a type of model where observations correlate with their
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neighbors and refer to them as autoregressive integrated moving average
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(ARIMA) models for stationary time series.
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For a thorough overview, we refer to \cite{box2015} and \cite{brockwell2016}.
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A time series $y_t$ is stationary if its moments are independent of the
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point in time where it is observed.
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A typical example is a white noise $\epsilon_t$ series.
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Therefore, a trend or seasonality implies non-stationarity.
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\cite{kwiatkowski1992} provide a test to check the null hypothesis of
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stationary data.
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To obtain a stationary time series, one chooses from several techniques:
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First, to stabilize a changing variance (i.e., heteroscedasticity), one
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applies a Box-Cox transformation (e.g., $log$) as first suggested by
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\cite{box1964}.
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Second, to factor out a trend (or seasonal) pattern, one computes differences
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of consecutive (or of lag $k$) observations or even differences thereof.
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Third, it is also common to pre-process $y_t$ with one of the decomposition
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methods mentioned in Sub-section \ref{stl} below with an ARIMA model
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then trained on an adjusted $y_t$.
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In the autoregressive part, observations are modeled as linear combinations of
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its predecessors.
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Formally, an $AR(p)$ model is defined with a drift term $c$, coefficients
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$\phi_i$ to be estimated (where $i$ is an index with $0 < i \leq p$), and
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white noise $\epsilon_t$ like so:
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$
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AR(p): \ \
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y_t = c + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \dots + \phi_p y_{t-p}
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+ \epsilon_t
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$.
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The moving average part considers observations to be regressing towards a
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linear combination of past forecasting errors.
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Formally, a $MA(q)$ model is defined with a drift term $c$, coefficients
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$\theta_j$ to be estimated, and white noise terms $\epsilon_t$ (where $j$
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is an index with $0 < j \leq q$) as follows:
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$
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MA(q): \ \
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y_t = c + \epsilon_t + \theta_1 \epsilon_{t-1} + \theta_2 \epsilon_{t-2}
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+ \dots + \theta_q \epsilon_{t-q}
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$.
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Finally, an $ARIMA(p,d,q)$ model unifies both parts and adds differencing
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where $d$ is the degree of differences and the $'$ indicates differenced
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values:
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$
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ARIMA(p,d,q): \ \
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y'_t = c + \phi_1 y'_{t-1} + \dots + \phi_p y'_{t-p} + \theta_1 \epsilon_{t-1}
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+ \dots + \theta_q \epsilon_{t-q} + \epsilon_{t}
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$.
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$ARIMA(p,d,q)$ models are commonly fitted with maximum likelihood estimation.
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To find an optimal combination of the parameters $p$, $d$, and $q$, the
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literature suggests calculating an information theoretical criterion
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(e.g., Akaike's Information Criterion) that evaluates the fit on
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historical data.
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\cite{hyndman2008a} provide a step-wise heuristic to choose $p$, $d$, and $q$,
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that also decides if a Box-Cox transformation is to be applied, and if so,
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which one.
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To obtain a one-step-ahead forecast, the above equation is reordered such
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that $t$ is substituted with $T+1$.
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For forecasts further into the future, the actual observations are
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subsequently replaced by their forecasts.
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Seasonal ARIMA variants exist; however, the high frequency $k$ in the kind of
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demand a UDP faces typically renders them impractical as too many
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coefficients must be estimated.
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