## 12.2 Estimation of multiple seasonal model

Estimating a multiple seasonal ETS model is challenging because it implies a large optimisation task. The number of parameters related to seasonal components is equal in general to $$\sum_{j=1}^n m_j + n$$: $$\sum_{j=1}^n m_j$$ initial values and $$n$$ smoothing parameters. For example, in case of hourly data, a triple seasonal model for hours of day, hours of week and hours of year will have: $$m_1 = 24$$, $$m_2 = 24 \times 7 = 168$$ and $$m_3= 7 \times 24 \times 365 = 61320$$, resulting overall in $$24 + 168 + 61320 + 3 = 61498$$ parameters related to seasonal components to estimate. This is not a trivial task and would take hours to converge to optimum unless the pre-initials (Section 11.4) are already close to optimum. So, if you want to construct multiple seasonal ADAM ETS model, it makes sense to use a different initialisation (see discussion in Section 11.4), reducing the number of estimated parameters. A possible solution in this case is backcasting (Section 11.4.1). The number of parameters in our example would reduce from 61498 to 3, substantially speeding up the model estimation process.

Another consideration is a fitting model to the data. In the conventional ETS, the size of the transition matrix is equal to the number of initial parameters, which makes it too slow to be practical on high-frequency data (multiplying a $$61498 \times 61498$$ matrix by a vector is a challenging task even for modern computers). But due to the lagged structure of the ADAM (discussed in Section 5), construction of multiple seasonal models does not take as much time for ADAM ETS because we end up multiplying a matrix of $$3 \times 3$$ by a vector with three rows (skipping level and trend, which would add two more elements). So, in ADAM, the main computational burden comes from recursive relation in the state space model’s transition equation because this operation needs to be repeated at least $$T$$ times, whatever the sample size $$T$$ is. As a result, you would want to get to the optimum with as few iterations as possible, not needing to refit the model with different parameters to the same data many times. This gives another motivation for reducing the number of parameters to estimate (and thus for using backcasting).

Another potential simplification would be to use deterministic seasonality for some seasonal frequencies. The possible solution, in this case, is to use explanatory variables (Section 10) for the higher frequency states (see discussion in Section 12.3) or use multiple seasonal ETS, setting some of smoothing parameters equal to zero.

Finally, given that we deal with large samples, some states of ETS might become more reactive than needed, having higher than required smoothing parameters. One of the possible ways to overcome this limitation is by using multistep loss functions(Section 11.3). For example, Kourentzes and Trapero (2018) showed that using such loss functions as TMSE (from Subsection 11.3.2) in the estimation of ETS models on high-frequency data leads to improvements in accuracy due to the shrinkage of parameters towards zero, mitigating the potential overfitting issue. The only problem with this approach is that it is more computationally expensive and thus would take more time (at least $$h$$ times more, where $$h$$ is the length of the forecast horizon) than the conventional likelihood estimation.