DRW2: Damped Second-Order Random Walk Prior

In bage: Bayesian Estimation and Forecasting of Age-Specific Rates

Damped Second-Order Random Walk Prior

Description

Use a damped 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 damped second-order random walk is a random walk with drift where the drift term varies, with a tendency to converge on zero. It is typically used with terms that involve time, where there are sustained trends upward or downward. Damping often improves forecast accuracy.

Usage

DRW2(
  s = 1,
  sd = 1,
  sd_slope = 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 1.
sd	Standard deviation of initial value. Default is 1. Can be 0.
sd_slope	Standard deviation of initial slope. Default is 1.
shape1, shape2	Parameters for beta-distribution prior for damping coefficient. Defaults are 5 and 5.
min, max	Minimum and maximum values for damping coefficient. Defaults are 0.8 and 0.98.
along	Name of the variable to be used as the 'along' variable. Only used with interactions.
con	Constraints on parameters. Current choices are "none" and "by". Default is "none". See below for details.

Details

If DRW2() is used with an interaction, a separate damped random walk is constructed within each combination of the 'by' variables.

Arguments min and max can be used to control the amount of damping that occurs.

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.

Value

An object of class "bage_prior_drw2random" or "bage_prior_drw2zero".

Mathematical details

When DRW2() 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}(\beta_{j-1} + \phi (\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}(\beta_{u,v-1} + \phi (\beta_{u,v-1} - \beta_{u,v-2}), \tau^2), \quad v = 3, \cdots, V

where

• \pmb{\beta} is the main effect or interaction;

• \phi is the damping coefficient;

• 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.

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}).

Standard deviation \tau has a half-normal prior

\tau \sim \text{N}^+(0, \mathtt{s}^2),

where s is provided by the user.

Constraints

The specification of constraints is likely to change in future versions of bage.

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

• DRW() Damped first-order random walk

• RW2() Second-order random walk, without damping

• 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

• Mathematical Details vignette

Examples

DRW2()
DRW2(s = 0.5)
DRW2(min = 0, max = 1)

bage documentation built on Feb. 22, 2026, 5:07 p.m.