Lin_AR1: Linear Prior with Autoregressive Errors of Order 1

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

Linear Prior with Autoregressive Errors of Order 1

Description

Use a line or lines with AR1 errors to model a main effect or interaction. Typically used with time.

Usage

Lin_AR1(
  s = 1,
  shape1 = 5,
  shape2 = 5,
  min = 0.8,
  max = 0.98,
  mean_slope = 0,
  sd_slope = 1,
  along = NULL,
  con = c("none", "by")
)

Arguments

s	Scale for the innovations in the AR process. 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.
mean_slope	Mean in prior for slope of line. Default is 0.
sd_slope	Standard deviation in the prior for the slope of the line. Larger values imply steeper slopes. Default is 1.
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 Lin_AR1() is used with an interaction, separate lines 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 the innovations. Smaller values tend to give smoother estimates.

Argument sd_slope controls the slopes of the lines. Larger values can give more steeply sloped lines.

Value

An object of class "bage_prior_linar".

Mathematical details

When Lin_AR1() is being used with a main effect,

\beta_1 = \alpha + \epsilon_1

\beta_j = \alpha + (j - 1) \eta + \epsilon_j, \quad j > 1

\alpha \sim \text{N}(0, 1)

\epsilon_j = \phi \epsilon_{j-1} + \varepsilon_j

\varepsilon \sim \text{N}(0, \omega^2),

and when it is used with an interaction,

\beta_{u,1} = \alpha_u + \epsilon_{u,1}

\beta_{u,v} = \eta (v - 1) + \epsilon_{u,v}, \quad v = 2, \cdots, V

\alpha_u \sim \text{N}(0, 1)

\epsilon_{u,v} = \phi \epsilon_{u,v-1} + \varepsilon_{u,v},

\varepsilon_{u,v} \sim \text{N}(0, \omega^2).

where

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

• j denotes position within the main effect;

• u denotes position within the 'along' variable of the interaction; and

• u denotes position within the 'by' variable(s) of the interaction.

The slopes have priors

\eta \sim \text{N}(\mathtt{mean\_slope}, \mathtt{sd\_slope}^2)

and

\eta_u \sim \text{N}(\mathtt{mean\_slope}, \mathtt{sd\_slope}^2).

Internally, Lin_AR1() derives a value for \omega that gives \epsilon_j or \epsilon_{u,v} a marginal variance of \tau^2. Parameter \tau has a half-normal prior

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

where a value for 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

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

See Also

• Lin_AR() Generalization of Lin_AR1()

• Lin() Line with independent normal errors

• AR1() AR1 process with no line

• priors Overview of priors implemented in bage

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

Examples

Lin_AR1()
Lin_AR1(min = 0, s = 0.5, sd_slope = 2)

bage documentation built on April 3, 2025, 8:53 p.m.