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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 to model a main effect, or use multiple random walks, each with their own seasonal effects, to model an interaction. Typically used with main effects or interactions that involve time.

Usage

RW_Seas(n_seas, s = 1, s_seas = 0, sd_seas = 1, along = NULL, zero_sum = FALSE)

Arguments

n_seas

Number of seasons

s

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

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.

zero_sum

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

Value

Object of class "bage_prior_rwseasvary" or "bage_prior_rwseasfix".

Details

If RW_Seas() is used with an interaction, separate series are used for the 'along' variable within each combination of the 'by' variables.

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

Argument n_seas controls the number of seasons. When using quarterly data, for instance, n_seas should be 4, and when using monthly data, n_seas should be 12.

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,

$$\beta_j = \alpha_j + \lambda_j$$ $$\alpha_j \sim \text{N}(\alpha_{j-1}, \tau^2)$$ $$\lambda_j \sim \text{N}(\lambda_{j-n}, \omega^2),$$

and when it is used with an interaction,

$$\beta_{u,v} = \alpha_{u,v} + \lambda_{u,v}$$ $$\alpha_{u,v} \sim \text{N}(\alpha_{u,v-1}, \tau^2),$$ $$\lambda_{u,v} \sim \text{N}(\lambda_{u,v-n}, \omega^2)$$

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;

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

  • \(n\) is n_seas.

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

Parameter \(\tau\) has a half-normal prior $$\tau \sim \text{N}^+(0, \text{s}^2),$$ where s is provided by the user.

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
#>     s_seas: 0
#>    sd_seas: 1
#>      along: NULL
#>   zero_sum: FALSE
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
#>     s_seas: 0.5
#>    sd_seas: 1
#>      along: NULL
#>   zero_sum: FALSE