# Random Walk Holiday State Model Adds a random walk holiday state model to the state specification. This model says [REMOVE_ME]$$ %y_t = \alpha_{d(t), t} + \epsilon_t%y[t] = alpha[d(t), t] + observation_error, [REMOVE_ME_2]$$ where there is one element in $alpha[, t]$ for each day in the holiday influence window. The transition equation is [REMOVE_ME]$$ %\alpha_{d(t+1), t+1} = \alpha_{d(t+1), t} + \epsilon_{t+1}%alpha[d(t+1), t+1] = alpha[d(t+1), t] + state_error [REMOVE_ME_2]$$ if t+1 occurs on day d(t+1) of the influence window, and [REMOVE_ME]$$ \alpha_{d(t+1), t+1} = \alpha_{d(t+1), t} %%alpha[d(t+1), t+1] = alpha[d(t+1), t]% [REMOVE_ME_2]$$ otherwise. 1.1 ## Description Adds a random walk holiday state model to the state specification. This model says $$ %y_t = \alpha_{d(t), t} + \epsilon_t%y[t] = alpha[d(t), t] + observation_error, $$ where there is one element in $alpha[, t]$ for each day in the holiday influence window. The transition equation is $$ %\alpha_{d(t+1), t+1} = \alpha_{d(t+1), t} + \epsilon_{t+1}%alpha[d(t+1), t+1] = alpha[d(t+1), t] + state_error $$ if t+1 occurs on day d(t+1) of the influence window, and $$ \alpha_{d(t+1), t+1} = \alpha_{d(t+1), t} %%alpha[d(t+1), t+1] = alpha[d(t+1), t]% $$ otherwise. ```r AddRandomWalkHoliday(state.specification = NULL, y, holiday, time0 = NULL, sigma.prior = NULL, initial.state.prior = NULL, sdy = sd(as.numeric(y), na.rm = TRUE)) ``` ## Arguments - `state.specification`: A list of state components that you wish augment. If omitted, an empty list will be assumed. - `y`: The time series to be modeled, as a numeric vector convertible to `xts`. This state model assumes `y` contains daily data. - `holiday`: An object of class `Holiday` describing the influence window of the holiday being modeled. - `time0`: An object convertible to `Date` containing the date of the initial observation in the training data. If omitted and `y` is a `zoo` or `xts` object, then `time0` will be obtained from the index of `y[1]`. - `sigma.prior`: An object created by `SdPrior` describing the prior distribution for the standard deviation of the random walk increments. - `initial.state.prior`: An object created using `NormalPrior`, describing the prior distribution of the the initial state vector (at time 1). - `sdy`: The standard deviation of the series to be modeled. This will be ignored if `y` is provided, or if all the required prior distributions are supplied directly. ## Returns A list describing the specification of the random walk holiday state model, formatted as expected by the underlying C++ code. ## References Harvey (1990), "Forecasting, structural time series, and the Kalman filter", Cambridge University Press. Durbin and Koopman (2001), "Time series analysis by state space methods", Oxford University Press. ## Author(s) Steven L. Scott steve.the.bayesian@gmail.com ## See Also `bsts`. `RegressionHolidayStateModel` `HierarchicalRegressionHolidayStateModel` ## Examples ```r trend <- cumsum(rnorm(730, 0, .1)) dates <- seq.Date(from = as.Date("2014-01-01"), length = length(trend), by = "day") y <- zoo(trend + rnorm(length(trend), 0, .2), dates) AddHolidayEffect <- function(y, dates, effect) { ## Adds a holiday effect to simulated data. ## Args: ## y: A zoo time series, with Dates for indices. ## dates: The dates of the holidays. ## effect: A vector of holiday effects of odd length. The central effect is ## the main holiday, with a symmetric influence window on either side. ## Returns: ## y, with the holiday effects added. time <- dates - (length(effect) - 1) / 2 for (i in 1:length(effect)) { y[time] <- y[time] + effect[i] time <- time + 1 } return(y) } ## Define some holidays. memorial.day <- NamedHoliday("MemorialDay") memorial.day.effect <- c(.3, 3, .5) memorial.day.dates <- as.Date(c("2014-05-26", "2015-05-25")) y <- AddHolidayEffect(y, memorial.day.dates, memorial.day.effect) presidents.day <- NamedHoliday("PresidentsDay") presidents.day.effect <- c(.5, 2, .25) presidents.day.dates <- as.Date(c("2014-02-17", "2015-02-16")) y <- AddHolidayEffect(y, presidents.day.dates, presidents.day.effect) labor.day <- NamedHoliday("LaborDay") labor.day.effect <- c(1, 2, 1) labor.day.dates <- as.Date(c("2014-09-01", "2015-09-07")) y <- AddHolidayEffect(y, labor.day.dates, labor.day.effect) ## The holidays can be in any order. holiday.list <- list(memorial.day, labor.day, presidents.day) number.of.holidays <- length(holiday.list) ## In a real example you'd want more than 100 MCMC iterations. niter <- 100 ss <- AddLocalLevel(list(), y) ss <- AddRandomWalkHoliday(ss, y, memorial.day) ss <- AddRandomWalkHoliday(ss, y, labor.day) ss <- AddRandomWalkHoliday(ss, y, presidents.day) model <- bsts(y, state.specification = ss, niter = niter, seed = 8675309) ## Plot model components. plot(model, "comp") ## Plot the effect of the specific state component. plot(ss[[2]], model) ```

add.random.walk.holiday function