# why is the level equation in the holt winters triple exponential model different from the other two?

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?

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: $$s_{t-1} + b_{t-1}$$