This book is in Open Review. I want your feedback to make the book better for you and other readers. To add your annotation, select some text and then click the on the pop-up menu. To see the annotations of others, click the button in the upper right hand corner of the page

## 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 $$K$$: $$$y_{t} = \frac{v_{K,t} - \theta_K \epsilon_{t}}{\eta_K}. \tag{12.17}$$$ 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: $$$\mathrm{E}(v_{i,t}|t) = \frac{\eta_i}{\eta_K} \mathrm{E}(v_{K,t}|t) \text{ for } t=\{-i+1, -i+2, \dots, 0\}. \tag{12.18}$$$

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.