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12.4 Initialisation of ADAM ARIMA

Each state \(v_{i,t}\) needs to be initialised with \(i\) values (e.g. 1 for the first state, 2 for the second etc). This leads in general to more initial values for states than the SSARIMA from I. Svetunkov and Boylan (2020b): \(\frac{K(K+1)}{2}\) instead of \(K\). However, this formulation has a more compact transition matrix, leading to computational improvements in terms of applying the model to the data with large \(K\) (e.g. multiple seasonalities). Besides, we can reduce the number of initial seeds to estimate either by using a different initialisation procedure (e.g. backcasting) or estimating directly \(y_t\) and \(\epsilon_t\) for \(t=\{-K+1, -K+2, \dots, 0\}\) to obtain the initials for each state via the formula (10.5). In order to reduce the number of estimated parameters to \(K\), we can take the conditional expectations for the states, in which case we will have: \[\begin{equation*} \mathrm{E}(v_{i,t} | t) = \eta_i y_{t} \text{ for } t=\{-K+1, -K+2, \dots, 0\}, \end{equation*}\] and then use these expectations for the initialisation of ARIMA. A the same time, we can express the actual value in terms of the state and error from (10.2) for the last state \(K\): \[\begin{equation} y_{t} = \frac{v_{K,t} - \theta_K \epsilon_{t}}{\eta_K}. \tag{12.17} \end{equation}\] We select the last state \(K\) because it has the highest number of initials to estimate among all states. We can then insert the value (12.17) in each formula for each state for \(i=\{1, 2, \dots, K-1\}\) and take their expectations: \[\begin{equation} \mathrm{E}(v_{i,t}|t) = \frac{\eta_i}{\eta_K} \mathrm{E}(v_{K,t}|t) \text{ for } t=\{-i+1, -i+2, \dots, 0\}. \tag{12.18} \end{equation}\]

So the process then comes to estimating the initials states of \(v_{K,t}\) for \(t=\{-K+1, -K+2, \dots, 0\}\) and then propagating them to the other states. However, this strategy will not work for models, when ARI polynomials are shorter than MA ones, i.e. $ {j=0}^n (P_j + D_j) m_j < {j=0}^n Q_j m_j $. But using the same principle of initialisation via the conditional expectation, we can set the initial MA states to zero and estimate only ARI states.

References

Svetunkov, Ivan, and John E. 2020b. “State-space ARIMA for supply-chain forecasting.” International Journal of Production Research 58 (3): 818–27. doi:10.1080/00207543.2019.1600764.