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Autoregressive Prior

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

Use an autoregressive process to model a main effect, or use multiple autoregressive processes to model an interaction. Typically used with time effects or with interactions that involve time.

Usage

AR(
  n_coef = 2,
  s = 1,
  shape1 = 5,
  shape2 = 5,
  along = NULL,
  con = c("none", "by")
)

Arguments

n_coef

Number of lagged terms in the model, ie the order of the model. Default is 2.

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.

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, then separate AR processes are constructed along the 'along' variable, within each combination of the 'by' variables.

By default, the autoregressive processes have order 2. Alternative choices can be specified through the n_coef argument.

Argument s controls the size of innovations. Smaller values for s tend to give smoother estimates.

Mathematical details

When AR() is used with a main effect,

$$\beta_j = \phi_1 \beta_{j-1} + \cdots + \phi_{\mathtt{n\_coef}} \beta_{j-\mathtt{n\_coef}} + \epsilon_j$$ $$\epsilon_j \sim \text{N}(0, \omega^2),$$

and when it is used with an interaction,

$$\beta_{u,v} = \phi_1 \beta_{u,v-1} + \cdots + \phi_{\mathtt{n\_coef}} \beta_{u,v-\mathtt{n\_coef}} + \epsilon_{u,v}$$ $$\epsilon_{u,v} \sim \text{N}(0, \omega^2),$$

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, AR() derives a value for \(\omega\) that gives every element of \(\beta\) a marginal variance of \(\tau^2\). Parameter \(\tau\) has a half-normal prior

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

The correlation coefficients \(\phi_1, \cdots, \phi_{\mathtt{n\_coef}}\) each have prior

$$\phi_k \sim \text{Beta}(\mathtt{shape1}, \mathtt{shape2}).$$

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

  • AR() is based on the TMB function ARk

See also

  • AR1() Special case of AR(). Can be more numerically stable than higher-order models.

  • Lin_AR(), Lin_AR1() Straight line with AR errors

  • priors Overview of priors implemented in bage

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

Examples

AR(n_coef = 3)
#>   AR(n_coef=3) 
#>     n_coef: 3
#>        min: -1
#>        max: 1
#>          s: 1
#>      along: NULL
#>        con: none
AR(n_coef = 3, s = 2.4)
#>   AR(n_coef=3,s=2.4) 
#>     n_coef: 3
#>        min: -1
#>        max: 1
#>          s: 2.4
#>      along: NULL
#>        con: none
AR(along = "cohort")
#>   AR(along="cohort") 
#>     n_coef: 2
#>        min: -1
#>        max: 1
#>          s: 1
#>      along: cohort
#>        con: none