Use an autoregressive process of order 1 to model a main effect, or use multiple AR1 processes to model an interaction. Typically used with time effects or with interactions that involve time.
Details
If AR() is used with an interaction,
separate AR processes are constructed along
the "along" variable, within each combination of the
"by" variables.
Arguments min and max can be used to specify
the permissible range for autocorrelation.
Argument s controls the size of innovations. Smaller values
for s tend to give smoother estimates.
Mathematical details
When AR1() is used with a main effect,
$$\beta_j = \phi \beta_{j-1} + \epsilon_j$$ $$\epsilon_j \sim \text{N}(0, \omega^2),$$
and when it is used with an interaction,
$$\beta_{u,v} = \phi \beta_{u,v-1} + \epsilon_{u,v}$$ $$\epsilon_{u,v} \sim \text{N}(0, \omega^2),$$
where
\(\pmb{\beta}\) is the main effect or interaction;
\(j\) denotes position within the main effect;
\(v\) denotes position within the "along" variable of the interaction; and
\(u\) denotes position within the "by" variable(s) of the interaction.
Internally, AR1() derives a value for \(\omega\) that
gives every element of \(\beta\) a marginal
variance of \(\tau^2\). Parameter \(\tau\)
has a half-normal prior
$$\tau \sim \text{N}^+(0, \text{s}^2),$$
where s is provided by the user.
Coefficient \(\phi\) is constrained
to lie between min and max.
Its prior distribution is
$$\phi = (\text{max} - \text{min}) \phi' - \text{min}$$
where
$$\phi' \sim \text{Beta}(2, 2).$$
References
AR1()is based on the TMB function AR1The defaults for
minandmaxare based on the defaults forforecast::ets().
See also
AR()Generalization ofAR1()priors Overview of priors implemented in bage
set_prior()Specify prior for intercept, main effect, or interaction