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5.1 Model formulation

The pure additive case is interesting, because this is the group of models that has closed forms for both conditional mean and variance. It is formulated in the following way: \begin{aligned} {y}_{t} = &\mathbf{w}' \mathbf{v}_{t-\boldsymbol{l}} + \epsilon_t \\ \mathbf{v}_{t} = &\mathbf{F} \mathbf{v}_{t-\boldsymbol{l}} + \mathbf{g} \epsilon_t \end{aligned}, \tag{5.2} where $$\mathbf{w}$$ is the measurement vector, $$\mathbf{F}$$ is the transition matrix and $$\mathbf{g}$$ is the persistence vector. An example of a pure additive model is ETS(A,A,A), for which we have the following values: \begin{aligned} \mathbf{w} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, & \mathbf{F} = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \\ \mathbf{g} = \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix}, & \mathbf{v}_{t} = \begin{pmatrix} l_t \\ b_t \\ s_t \end{pmatrix}, & \boldsymbol{l} = \begin{pmatrix} 1 \\ 1 \\ m \end{pmatrix} \end{aligned}. \tag{5.3} By inserting these values in the equation (5.2), we will obtain the model discussed in the ETS Taxonomy section: \begin{aligned} y_{t} = & l_{t-1} + b_{t-1} + s_{t-m} + \epsilon_t \\ l_t = & l_{t-1} + b_{t-1} + \alpha \epsilon_t \\ b_t = & b_{t-1} + \beta \epsilon_t \\ s_t = & s_{t-m} + \gamma \epsilon_t \end{aligned}. \tag{5.4} Just to compare, the conventional ETS(A,A,A), formulated according to (4.21) would have the following transition matrix: $$$\mathbf{F} = \begin{pmatrix} 1 & 1 & \mathbf{0}'_{m-1} & 0 \\ 0 & 1 & \mathbf{0}'_{m-1} & 0 \\ 0 & 0 & \mathbf{0}'_{m-1} & 1 \\ \mathbf{0}_{m-1} & \mathbf{0}_{m-1} & \mathbf{I}_{m-1} & \mathbf{0}_{m-1} \end{pmatrix}, \tag{5.5}$$$ where $$\mathbf{I}_{m-1}$$ is the identity matrix of the size $$(m-1) \times (m-1)$$ and $$\mathbf{0}_{m-1}$$ is the vector of zeroes of size $$m-1$$. The model (5.2) is more parsimonious and simplifies some of the calculations, making it realistic, for example, to apply models to data with large frequency $$m$$ (e.g. 24, 48, 52, 365).