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## 12.1 Model formulation

James W. Taylor (2003a) proposed an exponential smoothing model with double seasonality and applied it to energy data. Since then, the topic was developed by Gould et al. (2008), Taylor (2008), Taylor (2010), De Livera (2010) and De Livera et al. (2011). In this chapter we will discuss some of the proposed models, how they relate to the ADAM framework and can be implemented. Roughly, the idea of a model with multiple seasonalities is in introducing additional seasonal components. For the general framework this means that the state vector (for example, in a model with trend and seasonality) becomes: \[\begin{equation} \mathbf{v}_t' = \begin{pmatrix} l_t & b_t & s_{1,t} & s_{2,t} & \dots & s_{n,t} \end{pmatrix}, \tag{12.1} \end{equation}\] where \(n\) is the number of seasonal components (e.g. hour of day, hour of week and hour of year components). The lag matrix in this case becomes: \[\begin{equation} \mathbf{l}'=\begin{pmatrix}1 & 1 & m_1 & m_2 & \dots & m_n \end{pmatrix}, \tag{12.2} \end{equation}\] where \(m_i\) is the \(i\)-th seasonal periodicity. While, in theory there can be combinations between additive and multiplicative seasonal components, we argue that such a mixture does not make sense, and the components should align with each other. This means that in case of ETS(M,N,M), all seasonal components should be multiplicative, while in ETS(A,A,A) they should be additive. This results fundamentally in two types of models:

Additive seasonality: \[\begin{equation} \begin{aligned} & {y}_{t} = \check{y}_t + s_{1,t-m_1} + \dots + s_{n,t-m_n} \epsilon_t \\ & \vdots \\ & s_{1,t} = s_{1,t-m_1} + \gamma_1 \epsilon_t \\ & \vdots \\ & s_{n,t} = s_{n,t-m_n} + \gamma_n \epsilon_t \end{aligned}, \tag{12.3} \end{equation}\] where \(\check{y}_t\) is the point value based on all non-seasonal components (e.g. \(\check{y}_t=l_{t-1}\) in case of no trend model) and \(\gamma_i\) is the \(i\)-th seasonal smoothing parameter.

Multiplicative seasonality: \[\begin{equation} \begin{aligned} & {y}_{t} = \check{y}_t \times s_{1,t-m_1} \times \dots \times s_{n,t-m_n} \times(1+\epsilon_t) \\ & \vdots \\ & s_{1,t} = s_{1,t-m_1} (1 + \gamma_1 \epsilon_t) \\ & \vdots \\ & s_{n,t} = s_{n,t-m_n} (1+ \gamma_n \epsilon_t) \end{aligned}. \tag{12.4} \end{equation}\]

Depending on a specific model, the number of seasonal components can be 1, 2, 3 or more (although more than 3 might not make much sense from modelling point of view). De Livera (2010) introduced components based on fourier terms, updated over time via smoothing parameters. This feature is not yet fully supported in `adam()`

, but it is possible to substitute some of seasonal components (especially those that have fractional periodicity) with fourier terms via explanatory variables and update them over time. The explanatory variables idea was discussed in the Chapter 10.