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the double exponential model is so simple:

level: $s_t = \alpha x_t + (1-\alpha)(s_{t-1}+b_{t-1})$

trend: $b_t = \beta (s_t - s_{t-1}) + (1-\beta)b_{t-1}$

both intuitively weigh the new information vs the existing info.


2/3 of the triple exponential model (additive seasonality for simplicity) are also intuitive and simple:

seasonality: $c_t = \gamma(x_t - s_{t-1} - b_{t-1}) + (1-\gamma)c_{t-L}$

trend: $b_t = \beta (s_t - s_{t-1}) + (1-\beta)b_{t-1}$

but the level equation is so weird!

level: $s_t = \alpha (x_t-c_{t-L}) + (1-\alpha)(s_{t-1}+b_{t-1})$

why is the new information part taking into consideration only the seasonal component while the old information part takes into consideration only the trend component?

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it's not weird, it's just a weighted average between the seasonality adjusted observation: $x_t - c_{t-L}$ and the non-seasonal forecast for time t: $s_{t-1} + b_{t-1}$

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