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Autoregressive Prior of Order 1

Source: R/bage_prior-constructors.R
AR1.Rd

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 1.

shape1, shape2

Parameters for beta-distribution prior for coefficients. Defaults are 5 and 5.

min, max

Minimum and maximum values for autocorrelation 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_ar".

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.

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 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.

References

  • AR1() is based on the TMB function AR1

  • The defaults for min and max are based on the defaults for forecast::ets().

See also

Examples

AR1()
#>   AR1() 
#>        min: 0.8
#>        max: 0.98
#>          s: 1
#>      along: NULL
#>        con: none
AR1(min = 0, max = 1, s = 2.4)
#>   AR1(s=2.4,min=0,max=1) 
#>        min: 0
#>        max: 1
#>          s: 2.4
#>      along: NULL
#>        con: none
AR1(along = "cohort")
#>   AR1(along="cohort") 
#>        min: 0.8
#>        max: 0.98
#>          s: 1
#>      along: cohort
#>        con: none