RW2_Seas {bage} | R Documentation |
Second-Order Random Walk Prior with Seasonal Effect
Description
Use a second-oder random walk with seasonal effects as a model for a main effect, or use multiple second-order random walks, each with their own seasonal effects, as a model for an interaction. Typically used with temrs that involve time.
Usage
RW2_Seas(
n_seas,
s = 1,
sd = 1,
sd_slope = 1,
s_seas = 0,
sd_seas = 1,
along = NULL,
con = c("none", "by")
)
Arguments
n_seas |
Number of seasons |
s |
Scale for prior for innovations in
random walk. Default is |
sd |
Standard deviation
of initial value. Default is |
sd_slope |
Standard deviation
for initial slope of random walk. Default is |
s_seas |
Scale for innovations
in seasonal effects. Default is |
sd_seas |
Standard deviation for
initial values of seasonal effects.
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 |
Details
If RW2_Seas()
is used with an interaction,
a separate series is constructed within each
combination of the 'by' variables.
Argument s
controls the size of innovations in the random walk.
Smaller values for s
tend to produce smoother series.
Argument n_seas
controls the number of seasons.
When using quarterly data, for instance,
n_seas
should be 4
.
By default, the magnitude of seasonal effects
is fixed. However, setting s_seas
to a value
greater than zero produces seasonal effects
that evolve over time.
Value
Object of class
"bage_prior_rw2randomseasvary"
,
"bage_prior_rw2randomseasfix"
,
"bage_prior_rw2zeroseasvary"
, or
"bage_prior_rw2zeroseasfix"
.
Mathematical details
When RW2_Seas()
is used with a main effect,
\beta_j = \alpha_j + \lambda_j, \quad j = 1, \cdots, J
\alpha_1 \sim \text{N}(0, \mathtt{sd}^2)
\alpha_2 \sim \text{N}(0, \mathtt{sd\_slope}^2)
\alpha_j \sim \text{N}(2 \alpha_{j-1} - \alpha_{j-2}, \tau^2), \quad j = 3, \cdots, J
\lambda_j \sim \text{N}(0, \mathtt{sd\_seas}^2), \quad j = 1, \cdots, \mathtt{n\_seas} - 1
\lambda_j = -\sum_{s=1}^{\mathtt{n\_seas} - 1} \lambda_{j - s}, \quad j = \mathtt{n\_seas}, 2 \mathtt{n\_seas}, \cdots
\lambda_j \sim \text{N}(\lambda_{j-\mathtt{n\_seas}}, \omega^2), \quad \text{otherwise},
and when it is used with an interaction,
\beta_{u,v} = \alpha_{u,v} + \lambda_{u,v}, \quad v = 1, \cdots, V
\alpha_{u,1} \sim \text{N}(0, \mathtt{sd}^2)
\alpha_{u,2} \sim \text{N}(0, \mathtt{sd\_slope}^2)
\alpha_{u,v} \sim \text{N}(2 \alpha_{u,v-1} - \alpha_{u,v-2}, \tau^2), \quad v = 3, \cdots, V
\lambda_{u,v} \sim \text{N}(0, \mathtt{sd\_seas}^2), \quad v = 1, \cdots, \mathtt{n\_seas} - 1
\lambda_{u,v} = -\sum_{s=1}^{\mathtt{n\_seas} - 1} \lambda_{u,v - s}, \quad v = \mathtt{n\_seas}, 2 \mathtt{n\_seas}, \cdots
\lambda_{u,v} \sim \text{N}(\lambda_{u,v-\mathtt{n\_seas}}, \omega^2), \quad \text{otherwise},
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; and -
u
denotes position within the 'by' variable(s) of the interaction.
Parameter \omega
has a half-normal prior
\omega \sim \text{N}^+(0, \mathtt{s\_seas}^2)
.
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, \mathtt{s}^2)
.
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.
See Also
-
RW2()
Second-order random walk without seasonal effect -
RW_Seas()
Random walk with seasonal effect -
priors Overview of priors implemented in bage
-
set_prior()
Specify prior for intercept, main effect, or interaction -
Mathematical Details vignette
Examples
RW2_Seas(n_seas = 4) ## seasonal effects fixed
RW2_Seas(n_seas = 4, s_seas = 0.5) ## seasonal effects evolve
RW2_Seas(n_seas = 4, sd = 0) ## first term in random walk fixed at 0