Damped Random Walk Prior

Use a damped random walk as a model for a main effect, or use multiple damped random walks as a model for an interaction. Typically used with terms that involve time, particularly in forecasts. Damping often improves forecast accuracy.

Usage

DRW(
  s = 1,
  sd = 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.

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.

Value

An object of class "bage_prior_drwrandom" or "bage_prior_drwzero".

Details

If DRW() 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 produce smoother series.

Argument sd controls variance in initial values. Setting sd to 0 fixes initial values at 0.

Mathematical details

When DRW() is used with a main effect,

$$\beta_1 \sim \text{N}(0, \mathtt{sd}^2)$$ $$\beta_j \sim \text{N}(\phi \beta_{j-1}, \tau^2), \quad j > 1$$

and when it is used with an interaction,

$$\beta_{u,1} \sim \text{N}(0, \mathtt{sd}^2)$$ $$\beta_{u,v} \sim \text{N}(\phi \beta_{u,v-1}, \tau^2), \quad v > 1$$

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.

DRW() has the same basic structure as AR1(). However, in DRW(), \(\tau\) controls the variance of the innovations, but in AR1() \(\tau\) controls the marginal variance of each \(\beta_j\) or \(\beta_{u,v}\).

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

• DRW2() Damped second-order random walk

• RW() Random walk, without damping

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

• RW_Seas() Random walk with seasonal effect

• AR() Autoregressive with order k

• AR1() Autoregressive with order 1

• Sp() Smoothing via splines

• SVD() Smoothing over age using singular value decomposition

• priors Overview of priors implemented in bage

• set_prior() Specify prior for intercept, main effect, or interaction

• Mathematical Details vignette

Examples

DRW()
#>   DRW() 
#>          s: 1
#>         sd: 1
#>     shape1: 5
#>     shape2: 5
#>        min: 0.8
#>        max: 0.98
#>      along: NULL
#>        con: none
DRW(min = 0, max = 1)
#>   DRW(min=0,max=1) 
#>          s: 1
#>         sd: 1
#>     shape1: 5
#>     shape2: 5
#>        min: 0
#>        max: 1
#>      along: NULL
#>        con: none
DRW(sd = 0)
#>   DRW(sd=0) 
#>          s: 1
#>         sd: 0
#>     shape1: 5
#>     shape2: 5
#>        min: 0.8
#>        max: 0.98
#>      along: NULL
#>        con: none