`addSeason2formula.Rd`

This function helps to construct a `formula`

object that
can be used in a call to `hhh4`

to model
seasonal variation via a sum of sine and cosine terms.

`addSeason2formula(f = ~1, S = 1, period = 52, timevar = "t")`

- f
formula that the seasonal terms should be added to, defaults to an intercept

`~1`

.- S
number of sine and cosine terms. If

`S`

is a vector, unit-specific seasonal terms are created.- period
period of the season, defaults to 52 for weekly data.

- timevar
the time variable in the model. Defaults to

`"t"`

.

The function adds the seasonal terms
$$
\sum_{s=1}^\code{S} \gamma_s \sin(\frac{2\pi s}{\code{period}} t)
+\delta_s \cos(\frac{2\pi s}{\code{period}} t),
$$
where \(\gamma_s\) and \(\delta_s\) are the unknown
parameters and \(t\), \(t = 1, 2, \ldots\) denotes the time
variable `timevar`

, to an existing formula `f`

.

Note that the seasonal terms can also be expressed as
$$\gamma_{s} \sin(\frac{2\pi s}{\code{period}} t) + \delta_{s} \cos(\frac{2\pi s}{\code{period}} t) =
A_s \sin(\frac{2\pi s}{\code{period}} t + \epsilon_s)$$
with amplitude \(A_s=\sqrt{\gamma_s^2 +\delta_s^2}\)
and phase shift \(\tan(\epsilon_s) = \delta_s / \gamma_s\).
The amplitude and phase shift can be obtained from a fitted
`hhh4`

model via `coef(..., amplitudeShift = TRUE)`

,
see `coef.hhh4`

.

Returns a `formula`

with the seasonal terms added and
its environment set to `.GlobalEnv`

.
Note that to use the resulting formula in `hhh4`

,
a time variable named as specified by the argument `timevar`

must
be available.

M. Paul, with contributions by S. Meyer

```
# add 2 sine/cosine terms to a model with intercept and linear trend
addSeason2formula(f = ~ 1 + t, S = 2)
# the same for monthly data
addSeason2formula(f = ~ 1 + t, S = 2, period = 12)
# different number of seasons for a bivariate time series
addSeason2formula(f = ~ 1, S = c(3, 1), period = 52)
```