Use a random walk with seasonal effects to model a main effect, or use multiple random walks, each with their own seasonal effects, to model an interaction. Typically used with main effects or interactions that involve time.
Details
If RW_Seas() is used with an interaction,
separate series are used for
the "along" variable within
each combination of the
"by" variables.
Argument s controls the size of innovations in the random walk.
Smaller values for s tend to give smoother series.
Argument n_seas controls the number of seasons.
When using quarterly data, for instance,
n_seas should be 4, and when using
monthly data, n_seas should be 12.
By default, the magnitude of seasonal effects
can change over time. However, setting s_seas
to 0 produces seasonal effects that are fixed,
eg where "January" effect is the same every year,
the "Feburary" effect is the same every year, and so on.
Mathematical details
When RW_Seas() is used with a main effect,
$$\beta_j = \alpha_j + \lambda_j$$ $$\alpha_j \sim \text{N}(\alpha_{j-1}, \tau^2)$$ $$\lambda_j \sim \text{N}(\lambda_{j-n}, \omega^2),$$
and when it is used with an interaction,
$$\beta_{u,v} = \alpha_{u,v} + \lambda_{u,v}$$ $$\alpha_{u,v} \sim \text{N}(\alpha_{u,v-1}, \tau^2),$$ $$\lambda_{u,v} \sim \text{N}(\lambda_{u,v-n}, \omega^2)$$
where
\(\pmb{\beta}\) is the main effect or interaction;
\(\alpha_j\) or \(\alpha_{u,v}\) is an element of the random walk;
\(\lambda_j\) or \(\lambda_{u,v}\) is an element of the seasonal effect;
\(j\) denotes position within the main effect;
\(v\) denotes position within the "along" variable of the interaction;
\(u\) denotes position within the "by" variable(s) of the interaction; and
\(n\) is
n_seas.
Parameter \(\omega\) has a half-normal prior
$$\omega \sim \text{N}^+(0, \text{s\_seas}^2),$$
where s_seas is provided by the user. If
s_seas is set to 0, then \(\omega\) is 0,
and the seasonal effects are fixed over time.
Parameter \(\tau\) has a half-normal prior
$$\tau \sim \text{N}^+(0, \text{s}^2),$$
where s is provided by the user.
See also
RW()Random walk without seasonal effectRW2_Seas()Second-order random walk, with seasonal effectpriors Overview of priors implemented in bage
set_prior()Specify prior for intercept, main effect, or interaction