Autoregressive Prior

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, along = NULL, zero_sum = FALSE)

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.

along

Name of the variable to be used as the 'along' variable. Only used with interactions.

zero_sum

If TRUE, values must sum to 0 within each combination of the 'by' variables. Default is FALSE.

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.

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,

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

and when it is used with an interaction,

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

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;

• \(u\) denotes position within the 'by' variable(s) of the interaction; and

• \(n\) is n_coef.

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

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

where s is provided by the user.

The autocorrelation coefficients \(\phi_1, \cdots, \phi_n\) are restricted to values between -1 and 1 that jointly lead to a stationary model. The quantity \(r = \sqrt{\phi_1^2 + \cdots + \phi_n^2}\) has the boundary-avoiding prior

$$r \sim \text{Beta}(2, 2).$$

References

• AR() is based on the TMB function ARk

See also

• AR1() Special case of AR()

• 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
#>   zero_sum: FALSE
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
#>   zero_sum: FALSE
AR(along = "cohort")
#>   AR(along="cohort") 
#>     n_coef: 2
#>        min: -1
#>        max: 1
#>          s: 1
#>      along: cohort
#>   zero_sum: FALSE