AR1 {bage} | R Documentation |
Autoregressive Prior of Order 1
Description
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.
Usage
AR1(
s = 1,
shape1 = 5,
shape2 = 5,
min = 0.8,
max = 0.98,
along = NULL,
con = c("none", "by")
)
Arguments
s |
Scale for the prior for the innovations.
Default is |
shape1 , shape2 |
Parameters for beta-distribution prior
for coefficients. Defaults are |
min , max |
Minimum and maximum values
for autocorrelation coefficient.
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,
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.
Value
An object of class "bage_prior_ar"
.
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, \mathtt{s}^2),
where s
is provided by the user.
Coefficient \phi
is constrained
to lie between min
and max
.
Its prior distribution is
\phi = (\mathtt{max} - \mathtt{min}) \phi' - \mathtt{min}
where
\phi' \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
when trying to obtain interpretable values
for main effects and interactions, it can be helpful to increase
identifiability through the use of constraints, specified through the
con
argument.
Current options for con
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
-
AR1()
is based on the TMB function AR1 The defaults for
min
andmax
are 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 -
Mathematical Details vignette
Examples
AR1()
AR1(min = 0, max = 1, s = 2.4)
AR1(along = "cohort")