View source: R/bage_prior-constructors.R
RW2 | R Documentation |
Use a second-order random walk as a model for a main effect, or use multiple second-order random walks as a model for an interaction. A second-order random walk is a random walk with drift where the drift term varies. It is typically used with terms that involve age or time, where there are sustained trends upward or downward.
RW2(s = 1, sd = 1, sd_slope = 1, along = NULL, con = c("none", "by"))
s |
Scale for the prior for the innovations.
Default is |
sd |
Standard deviation
of initial value. Default is |
sd_slope |
Standard deviation
of initial slope. Default is |
along |
Name of the variable to be used as the 'along' variable. Only used with interactions. |
con |
Constraints on parameters.
Current choices are |
If RW2()
is used with an interaction,
a separate random walk is constructed
within each combination of the
'by' variables.
Argument s
controls the size of innovations.
Smaller values for s
tend to give smoother series.
Argument sd
controls variance in
initial values. Setting sd
to 0
fixes
initial values at 0
.
Argument sd_slope
controls variance in the
initial slope.
An object of class "bage_prior_rw2random"
or "bage_prior_rw2zero"
.
When RW2()
is used with a main effect,
\beta_1 \sim \text{N}(0, \mathtt{sd}^2)
\beta_2 \sim \text{N}(\beta_1, \mathtt{sd\_slope}^2)
\beta_j \sim \text{N}(2 \beta_{j-1} - \beta_{j-2}, \tau^2), \quad j = 2, \cdots, J
and when it is used with an interaction,
\beta_{u,1} \sim \text{N}(0, \mathtt{sd}^2)
\beta_{u,2} \sim \text{N}(\beta_{u,1}, \mathtt{sd\_slope}^2)
\beta_{u,v} \sim \text{N}(2\beta_{u,v-1} - \beta_{u,v-2}, \tau^2), \quad v = 3, \cdots, V
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.
Parameter \tau
has a half-normal prior
\tau \sim \text{N}^+(0, \mathtt{s}^2)
.
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.
RW()
Random walk
RW2_Seas()
Second order random walk with seasonal effect
AR()
Autoregressive with order k
AR1()
Autoregressive with order 1
Sp()
Smoothing via splines
SVD()
Smoothing over age via singular value decomposition
priors Overview of priors implemented in bage
set_prior()
Specify prior for intercept,
main effect, or interaction
RW2()
RW2(s = 0.5)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.