# In triple exponential smoothing, what is the proper formula for recalculating gamma (seasonality)?

A pretty targeted but precise question --

In triple exponential smoothing (which there are many combinations of additive, multiplicative).

What is the proper formula for calculating the new seasonality for each observation?

I've seen two different formulas online:

$s_{t}=\gamma\left(\frac{y_{t}}{\ell_{t-1}+b_{t-1}}\right)+\left(1-\gamma\right)\left(s_{t-m}\right)$

[above is assuming additive trend, it would multiply the previous trend otherwise)

or

$s_{t}=\gamma\left(\frac{y_{t}}{\ell_{t-1}}\right)+\left(1-\gamma\right)\left(s_{t-m}\right)$

Aka in the first function, the previous trend is used in the calculation of the new seasonal factor, whereas in the second, it is omitted.

Which is correct? Thanks.

The first one should be used if there is trend. The second one assumes there is no trend. An alternative that can be used whether or not there is trend is $$s_t = \gamma\left(\frac{y_t}{\ell_t}\right) + (1-\gamma) (s_{t-m}).$$ where $\ell_t$ is the level component, $s_t$ is the seasonal component and $m$ is the period of seasonality.