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7.1 Model formulation

Based on the discussion in previous chapters, we can summarise the general ADAM ETS model. It is built upon the conventional model discussed in Section 4.5 but has several important differences, the most important of which is that it is formulated using lags of components rather than the transition of them over time (this was discussed in Chapter 5.1 for the pure additive model). General ADAM ETS model is formulated in the following way: \[\begin{equation} \begin{aligned} {y}_{t} = &w(\mathbf{v}_{t-\mathbf{l}}) + r(\mathbf{v}_{t-\mathbf{l}}) \epsilon_t \\ \mathbf{v}_{t} = &f(\mathbf{v}_{t-\mathbf{l}}) + g(\mathbf{v}_{t-\mathbf{l}}) \epsilon_t \end{aligned}, \tag{7.1} \end{equation}\] where \(\mathbf{v}_{t-\mathbf{l}}\) is the vector of lagged components and \(\mathbf{l}\) is the vector of lags, while all the other functions correspond to the ones used in (4.24). This model form is mainly useful for the formulation, rather than for the derivations, as discussed in Section 4.5. Not only it encompasses any pure models, it also allows formulating any of the mixed ones. For example, the ETS(M,A,M) will have the following values: \[\begin{equation*} \begin{aligned} w(\mathbf{v}_{t-\mathbf{l}}) = (l_{t-1}+b_{t-1}) s_{t-m}\text{, } & r(\mathbf{v}_{t-\mathbf{l}}) = w(\mathbf{v}_{t-\mathbf{l}}), \\ f(\mathbf{v}_{t-\mathbf{l}}) = \begin{pmatrix} l_{t-1} + b_{t-1} \\ b_{t-1} \\ s_{t-m} \end{pmatrix}\text{, } & g(\mathbf{v}_{t-\mathbf{l}}) = \begin{pmatrix} \alpha (l_{t-1} + b_{t-1}) \\ \beta (l_{t-1} + b_{t-1}) \\ \gamma s_{t-m} \end{pmatrix}, \\ \mathbf{v}_{t} = \begin{pmatrix} l_t \\ b_t \\ s_t \end{pmatrix}\text{, } & \mathbf{l} = \begin{pmatrix} 1 \\ 1 \\ m \end{pmatrix}, \\ \mathbf{v}_{t-\mathbf{l}} = \begin{pmatrix} l_{t-1} \\ b_{t-1} \\ s_{t-m} \end{pmatrix} \end{aligned}. \end{equation*}\] By inserting (??) in (7.1) we will get the classical ETS(M,A,M) model, mentioned in Section 3.5: \[\begin{equation} \begin{aligned} y_{t} = & (l_{t-1} + b_{t-1}) s_{t-m}(1 + \epsilon_t) \\ l_t = & (l_{t-1} + b_{t-1})(1 + \alpha \epsilon_t) \\ b_t = & b_{t-1} + (l_{t-1} + b_{t-1}) \beta \epsilon_t \\ s_t = & s_{t-m} (1 + \gamma \epsilon_t) \end{aligned}. \tag{7.2} \end{equation}\] The model (7.1) with different values for the functions is the basis of adam() function from smooth package.