The model, implemented in this function, is discussed in Svetunkov & Boylan
(2019).
The basic ARIMA(p,d,q) used in the function has the following form:
\((1 - B)^d (1 - a_1 B - a_2 B^2 - ... - a_p B^p) y_[t] = (1 + b_1 B +
b_2 B^2 + ... + b_q B^q) \epsilon_[t] + c\)
where \(y_[t]\) is the actual values, \(\epsilon_[t]\) is the error term,
\(a_i, b_j\) are the parameters for AR and MA respectively and \(c\) is
the constant. In case of non-zero differences \(c\) acts as drift.
This model is then transformed into ARIMA in the Single Source of Error
State space form (proposed in Snyder, 1985):
\(y_{t} = w' v_{t-l} + \epsilon_{t}\)
\(v_{t} = F v_{t-l} + g_t \epsilon_{t}\)
where \(v_{t}\) is the state vector (defined based on
orders) and \(l\) is the vector of lags, \(w_t\) is the
measurement vector (with explanatory variables if provided), \(F\)
is the transition matrix, \(g_t\) is the persistence vector
(which includes explanatory variables if they were used).
Due to the flexibility of the model, multiple seasonalities can be used. For
example, something crazy like this can be constructed:
SARIMA(1,1,1)(0,1,1)[24](2,0,1)[24*7](0,0,1)[24*30], but the estimation may
take a lot of time... If you plan estimating a model with more than one
seasonality, it is recommended to use msarima instead.
The model selection for SSARIMA is done by the auto.ssarima function.
For some more information about the model and its implementation, see the
vignette: vignette("ssarima","smooth")
The function constructs bunch of ARIMAs in Single Source of Error
state space form (see ssarima documentation) and selects the
best one based on information criterion. The mechanism is described in
Svetunkov & Boylan (2019).
Due to the flexibility of the model, multiple seasonalities can be used. For
example, something crazy like this can be constructed:
SARIMA(1,1,1)(0,1,1)[24](2,0,1)[24*7](0,0,1)[24*30], but the estimation may
take a lot of time... It is recommended to use auto.msarima in
cases with more than one seasonality and high frequencies.
For some more information about the model and its implementation, see the
vignette: vignette("ssarima","smooth")
The model, implemented in this function, is discussed in Svetunkov & Boylan
(2019).
The basic ARIMA(p,d,q) used in the function has the following form:
\((1 - B)^d (1 - a_1 B - a_2 B^2 - ... - a_p B^p) y_[t] = (1 + b_1 B +
b_2 B^2 + ... + b_q B^q) \epsilon_[t] + c\)
where \(y_[t]\) is the actual values, \(\epsilon_[t]\) is the error term,
\(a_i, b_j\) are the parameters for AR and MA respectively and \(c\) is
the constant. In case of non-zero differences \(c\) acts as drift.
This model is then transformed into ARIMA in the Single Source of Error
State space form (proposed in Snyder, 1985):
\(y_{t} = o_{t} (w' v_{t-l} + x_t a_{t-1} + \epsilon_{t})\)
\(v_{t} = F v_{t-l} + g \epsilon_{t}\)
\(a_{t} = F_{X} a_{t-1} + g_{X} \epsilon_{t} / x_{t}\)
Where \(o_{t}\) is the Bernoulli distributed random variable (in case of
normal data equal to 1), \(v_{t}\) is the state vector (defined based on
orders) and \(l\) is the vector of lags, \(x_t\) is the
vector of exogenous parameters. \(w\) is the measurement vector,
\(F\) is the transition matrix, \(g\) is the persistence
vector, \(a_t\) is the vector of parameters for exogenous variables,
\(F_{X}\) is the transitionX matrix and \(g_{X}\) is the
persistenceX matrix.
Due to the flexibility of the model, multiple seasonalities can be used. For
example, something crazy like this can be constructed:
SARIMA(1,1,1)(0,1,1)[24](2,0,1)[24*7](0,0,1)[24*30], but the estimation may
take some finite time... If you plan estimating a model with more than one
seasonality, it is recommended to consider doing it using msarima.
The model selection for SSARIMA is done by the auto.ssarima function.
For some more information about the model and its implementation, see the
vignette: vignette("ssarima","smooth")