regression.holiday() R function from [bsts]

Regression Based Holiday Models

Add a regression-based holiday model to the state specification. 1.1

Details

The model assumes that

y_t = \beta_{d(t)} + \epsilon_t %y[t] = beta[d(t)] + observation_error

The regression state model assumes vector of regression coefficients $beta$ contains elements $beta[d] ~ N(0, sigma)$ .

The HierarchicalRegressionHolidayModel assumes $beta$ is composed of holiday-specific sub-vectors $beta[h, ] ~ N(b0, V)$ , where each $beta[h,]$ contains coefficients describing the days in the influence window of holiday h. The hierarchical version of the model treats $b0$ and $V$ as parameters to be learned, with prior distributions

b_0 \sim N(\bar b, \Omega)b0 ~ N(b.bar, Omega)

and

V \sim IW(\nu, S)V ~ IW(nu, S).

where $IW$ represents the inverse Wishart distribution.


AddRegressionHoliday(
    state.specification = NULL,
    y,
    holiday.list,
    time0 = NULL,
    prior = NULL,
    sdy = sd(as.numeric(y), na.rm = TRUE))

AddHierarchicalRegressionHoliday(
    state.specification = NULL,
    y,
    holiday.list,
    coefficient.mean.prior = NULL,
    coefficient.variance.prior = NULL,
    time0 = NULL,
    sdy = sd(as.numeric(y), na.rm = TRUE))

Arguments

state.specification: A list of state components that you wish to add to. If omitted, an empty list will be assumed.
holiday.list: A list of objects of type Holiday. The width of the influence window should be the same number of days for all the holidays in this list. If the data contains many instances of holidays with different window widths, then multiple instances HierarchicalRegressionHolidayModel can be used as long as all holidays in the same state component model have the same sized window width.
y: The time series to be modeled, as a numeric vector convertible to xts. This state model assumes y

contains daily data.
prior: An object of class NormalPrior

describing the expected variation among daily holiday effects.
coefficient.mean.prior: An object of type MvnPrior giving the hyperprior for the average effect of a holiday in each day of the influence window.
coefficient.variance.prior: An object of type InverseWishartPrior describing the prior belief about the variation in holiday effects from one holiday to the next.
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].
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

Returns a list with the elements necessary to specify a local linear trend state model.

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

Examples


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)

## In a real example you'd want more than 100 MCMC iterations.
niter <- 100

## Fit the model
ss <- AddLocalLevel(list(), y)
ss <- AddRegressionHoliday(ss, y, holiday.list = holiday.list)
model <- bsts(y, state.specification = ss, niter = niter)

## Plot all model state components.
plot(model, "comp")

## Plot the specific holiday state component.
plot(ss[[2]], model)

## Try again with some shrinkage.  With only 3 holidays there won't be much
## shrinkage.  
ss2 <- AddLocalLevel(list(), y)

## Plot the specific holiday state component.
ss2 <- AddHierarchicalRegressionHoliday(ss2, y, holiday.list = holiday.list)
model2 <- bsts(y, state.specification = ss2, niter = niter)

plot(model2, "comp")
plot(ss2[[2]], model2)

regression.holiday function