Linear Prior with Autoregressive Errors

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

Usage

Lin_AR(
  n_coef = 2,
  s = 1,
  shape1 = 5,
  shape2 = 5,
  mean_slope = 0,
  sd_slope = 1,
  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 innovations in the AR process. Default is 1.
shape1, shape2: Parameters for beta-distribution prior for coefficients. Defaults are 5 and 5.
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.

Value

An object of class "bage_prior_linar".

Details

If Lin_AR() is used with an interaction, separate lines are constructed along the 'along' variable, within each combination of the 'by' variables.

The order of the autoregressive errors is controlled by the n_coef argument. The default is 2.

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.

Mathematical details

When Lin_AR() is used with a main effect,

$$\beta_1 = \alpha + \epsilon_1$$ $$\beta_j = \alpha + (j - 1) \eta + \epsilon_j, \quad j > 1$$ $$\alpha \sim \text{N}(0, 1)$$ $$\epsilon_j = \phi_1 \epsilon_{j-1} + \cdots + \phi_{\mathtt{n\_coef}} \epsilon_{j-\mathtt{n\_coef}} + \varepsilon_j$$ $$\varepsilon_j \sim \text{N}(0, \omega^2),$$

and when it is used with an interaction,

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

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

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

$$0.5 \phi_k - 0.5 \sim \text{Beta}(\mathtt{shape1}, \mathtt{shape2}).$$

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.

Examples

Lin_AR()
#>   Lin_AR() 
#>     n_coef: 2
#>          s: 1
#> mean_slope: 0
#>   sd_slope: 1
#>        min: -1
#>        max: 1
#>      along: NULL
#>        con: none
Lin_AR(n_coef = 3, s = 0.5, sd_slope = 2)
#>   Lin_AR(n_coef=3,s=0.5,sd_slope=2) 
#>     n_coef: 3
#>          s: 0.5
#> mean_slope: 0
#>   sd_slope: 2
#>        min: -1
#>        max: 1
#>      along: NULL
#>        con: none