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Random Walk Prior with Seasonal Effect

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

Use a random walk with seasonal effects as a model for a main effect, or use multiple random walks, each with their own seasonal effects, as a model for an interaction. Typically used with terms that involve time.

Usage

RW_Seas(
  n_seas,
  s = 1,
  sd = 1,
  s_seas = 0,
  sd_seas = 1,
  along = NULL,
  con = c("none", "by")
)

Arguments

n_seas

Number of seasons

s

Scale for prior for innovations in random walk. Default is 1.

sd

Standard deviation of initial value. Default is 1. Can be 0.

s_seas

Scale for innovations in seasonal effects. Default is 0.

sd_seas

Standard deviation for initial values of seasonal effects. 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

Object of class "bage_prior_rwrandomseasvary", "bage_prior_rwrandomseasfix", "bage_prior_rwzeroseasvary", or "bage_prior_rwzeroseasfix".

Details

If RW_Seas() is used with an interaction, a separate series is constructed within each combination of the 'by' variables.

Argument s controls the size of innovations in the random walk. Smaller values for s tend to produce smoother series.

Argument sd controls variance in initial values of the random walk. sd can be 0.

Argument n_seas controls the number of seasons. When using quarterly data, for instance, n_seas should be 4.

By default, the magnitude of seasonal effects is fixed. However, setting s_seas to a value greater than zero produces seasonal effects that evolve over time.

Mathematical details

When RW_Seas() is used with a main effect,

$$\beta_j = \alpha_j + \lambda_j, \quad j = 1, \cdots, J$$ $$\alpha_1 \sim \text{N}(0, \mathtt{sd}^2)$$ $$\alpha_j \sim \text{N}(\alpha_{j-1}, \tau^2), \quad j = 2, \cdots, J$$ $$\lambda_j \sim \text{N}(0, \mathtt{sd\_seas}^2), \quad j = 1, \cdots, \mathtt{n\_seas} - 1$$ $$\lambda_j = -\sum_{s=1}^{\mathtt{n\_seas} - 1} \lambda_{j - s}, \quad j = \mathtt{n\_seas}, 2 \mathtt{n\_seas}, \cdots$$ $$\lambda_j \sim \text{N}(\lambda_{j-\mathtt{n\_seas}}, \omega^2), \quad \text{otherwise},$$

and when it is used with an interaction,

$$\beta_{u,v} = \alpha_{u,v} + \lambda_{u,v}, \quad v = 1, \cdots, V$$ $$\alpha_{u,1} \sim \text{N}(0, \mathtt{sd}^2)$$ $$\alpha_{u,v} \sim \text{N}(\alpha_{u,v-1}, \tau^2), \quad v = 2, \cdots, V$$ $$\lambda_{u,v} \sim \text{N}(0, \mathtt{sd\_seas}^2), \quad v = 1, \cdots, \mathtt{n\_seas} - 1$$ $$\lambda_{u,v} = -\sum_{s=1}^{\mathtt{n\_seas} - 1} \lambda_{u,v - s}, \quad v = \mathtt{n\_seas}, 2 \mathtt{n\_seas}, \cdots$$ $$\lambda_{u,v} \sim \text{N}(\lambda_{u,v-\mathtt{n\_seas}}, \omega^2), \quad \text{otherwise},$$

where

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

  • \(\alpha_j\) or \(\alpha_{u,v}\) is an element of the random walk;

  • \(\lambda_j\) or \(\lambda_{u,v}\) is an element of the seasonal effect;

  • \(j\) 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.

Parameter \(\omega\) has a half-normal prior $$\omega \sim \text{N}^+(0, \mathtt{s\_seas}^2).$$ If s_seas is set to 0, then \(\omega\) is 0, and seasonal effects are time-invariant.

Parameter \(\tau\) has a half-normal prior $$\tau \sim \text{N}^+(0, \mathtt{s}^2).$$

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.

See also

  • RW() Random walk without seasonal effect

  • RW2_Seas() Second-order random walk with seasonal effect

  • priors Overview of priors implemented in bage

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

Examples

RW_Seas(n_seas = 4)               ## seasonal effects fixed
#>   RW_Seas(n_seas=4) 
#>     n_seas: 4
#>          s: 1
#>         sd: 1
#>     s_seas: 0
#>    sd_seas: 1
#>      along: NULL
#>        con: none
RW_Seas(n_seas = 4, s_seas = 0.5) ## seasonal effects evolve
#>   RW_Seas(n_seas=4,s_seas=0.5) 
#>          n: NULL
#>          s: 1
#>         sd: 1
#>     s_seas: 0.5
#>    sd_seas: 1
#>      along: NULL
#>        con: none
RW_Seas(n_seas = 4, sd = 0)       ## first term in random walk fixed at 0
#>   RW_Seas(n_seas=4,sd=0) 
#>     n_seas: 4
#>          s: 1
#>         sd: 0
#>     s_seas: 0
#>    sd_seas: 1
#>      along: NULL
#>        con: none