AR {bage} | R Documentation |
Autoregressive Prior
Description
Use an autoregressive process to model a main effect, or use multiple autoregressive processes to model an interaction. Typically used with time effects or with interactions that involve time.
Usage
AR(
n_coef = 2,
s = 1,
shape1 = 5,
shape2 = 5,
along = NULL,
con = c("none", "by")
)
Arguments
n_coef |
Number of lagged terms in the
model, ie the order of the model. Default is |
s |
Scale for the prior for the innovations.
Default is |
shape1 , shape2 |
Parameters for beta-distribution prior
for coefficients. Defaults are |
along |
Name of the variable to be used as the 'along' variable. Only used with interactions. |
con |
Constraints on parameters.
Current choices are |
Details
If AR()
is used with an interaction, then
separate AR processes are constructed along
the 'along' variable, within each combination of the
'by' variables.
By default, the autoregressive processes
have order 2. Alternative choices can be
specified through the n_coef
argument.
Argument s
controls the size of innovations.
Smaller values for s
tend to give smoother estimates.
Value
An object of class "bage_prior_ar"
.
Mathematical details
When AR()
is used with a main effect,
\beta_j = \phi_1 \beta_{j-1} + \cdots + \phi_{\mathtt{n\_coef}} \beta_{j-\mathtt{n\_coef}} + \epsilon_j
\epsilon_j \sim \text{N}(0, \omega^2),
and when it is used with an interaction,
\beta_{u,v} = \phi_1 \beta_{u,v-1} + \cdots + \phi_{\mathtt{n\_coef}} \beta_{u,v-\mathtt{n\_coef}} + \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, AR()
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, \mathtt{s}^2).
The correlation coefficients \phi_1, \cdots, \phi_{\mathtt{n\_coef}}
each have prior
\phi_k \sim \text{Beta}(\mathtt{shape1}, \mathtt{shape2}).
Constraints
With some combinations of terms and priors, the values of the intercept, main effects, and interactions are are only weakly identified. For instance, it may be possible to increase the value of the intercept and reduce the value of the remaining terms in the model with no effect on predicted rates and only a tiny effect on prior probabilities. This weak identifiability is typically harmless. However, in some applications, such as forecasting, or when trying to obtain interpretable values for main effects and interactions, it can be helpful to increase identifiability through the use of constraints.
Current options for constraints are:
-
"none"
No constraints. The default. -
"by"
Only used in interaction terms that include 'along' and 'by' dimensions. Within each value of the 'along' dimension, terms across each 'by' dimension are constrained to sum to 0.
References
-
AR()
is based on the TMB function ARk
See Also
-
AR1()
Special case ofAR()
. Can be more numerically stable than higher-order models. -
priors Overview of priors implemented in bage
-
set_prior()
Specify prior for intercept, main effect, or interaction
Examples
AR(n_coef = 3)
AR(n_coef = 3, s = 2.4)
AR(along = "cohort")