Bootstrapping of Time Series

Generate R bootstrap replicates of a statistic applied to a time series. The replicate time series can be generated using fixed or random block lengths or can be model based replicates.

tsboot(tseries, statistic, R, l = NULL, sim = "model",
       endcorr = TRUE, n.sim = NROW(tseries), orig.t = TRUE,
       ran.gen, ran.args = NULL, norm = TRUE, ...,
       parallel = c("no", "multicore", "snow"),
       ncpus = getOption("boot.ncpus", 1L), cl = NULL)

Arguments

tseries	A univariate or multivariate time series.
statistic	A function which when applied to `tseries` returns a vector containing the statistic(s) of interest. Each time `statistic` is called it is passed a time series of length `n.sim` which is of the same class as the original `tseries`. Any other arguments which `statistic` takes must remain constant for each bootstrap replicate and should be supplied through the ... argument to `tsboot`.
R	A positive integer giving the number of bootstrap replicates required.
sim	The type of simulation required to generate the replicate time series. The possible input values are `"model"` (model based resampling), `"fixed"` (block resampling with fixed block lengths of `l`), `"geom"` (block resampling with block lengths having a geometric distribution with mean `l`) or `"scramble"` (phase scrambling).
l	If `sim` is `"fixed"` then `l` is the fixed block length used in generating the replicate time series. If `sim` is `"geom"` then `l` is the mean of the geometric distribution used to generate the block lengths. `l` should be a positive integer less than the length of `tseries`. This argument is not required when `sim` is `"model"` but it is required for all other simulation types.
endcorr	A logical variable indicating whether end corrections are to be applied when `sim` is `"fixed"`. When `sim` is `"geom"`, `endcorr` is automatically set to `TRUE`; `endcorr` is not used when `sim` is `"model"` or `"scramble"`.
n.sim	The length of the simulated time series. Typically this will be equal to the length of the original time series but there are situations when it will be larger. One obvious situation is if prediction is required. Another situation in which `n.sim` is larger than the original length is if `tseries` is a residual time series from fitting some model to the original time series. In this case, `n.sim` would usually be the length of the original time series.
orig.t	A logical variable which indicates whether `statistic` should be applied to `tseries` itself as well as the bootstrap replicate series. If `statistic` is expecting a longer time series than `tseries` or if applying `statistic` to `tseries` will not yield any useful information then `orig.t` should be set to `FALSE`.
ran.gen	This is a function of three arguments. The first argument is a time series. If `sim` is `"model"` then it will always be `tseries` that is passed. For other simulation types it is the result of selecting `n.sim` observations from `tseries` by some scheme and converting the result back into a time series of the same form as `tseries` (although of length `n.sim`). The second argument to `ran.gen` is always the value `n.sim`, and the third argument is `ran.args`, which is used to supply any other objects needed by `ran.gen`. If `sim` is `"model"` then the generation of the replicate time series will be done in `ran.gen` (for example through use of `arima.sim`). For the other simulation types `ran.gen` is used for ‘post-blackening’. The default is that the function simply returns the time series passed to it.
ran.args	This will be supplied to `ran.gen` each time it is called. If `ran.gen` needs any extra arguments then they should be supplied as components of `ran.args`. Multiple arguments may be passed by making `ran.args` a list. If `ran.args` is `NULL` then it should not be used within `ran.gen` but note that `ran.gen` must still have its third argument.
norm	A logical argument indicating whether normal margins should be used for phase scrambling. If `norm` is `FALSE` then margins corresponding to the exact empirical margins are used.
...	Extra named arguments to `statistic` may be supplied here. Beware of partial matching to the arguments of `tsboot` listed above.
parallel, ncpus, cl	See the help for `boot`.

Value

An object of class "boot" with the following components.

If orig.t is TRUE then t0 is the result of statistic(tseries,...{}) otherwise it is NULL.

The results of applying statistic to the replicate time series.

The value of R as supplied to tsboot.

tseries

The original time series.

statistic

The function statistic as supplied.

sim

The simulation type used in generating the replicates.

endcorr

The value of endcorr used. The value is meaningful only when sim is "fixed"; it is ignored for model based simulation or phase scrambling and is always set to TRUE if sim is "geom".

n.sim

The value of n.sim used.

The value of l used for block based resampling. This will be NULL if block based resampling was not used.

ran.gen

The ran.gen function used for generating the series or for ‘post-blackening’.

ran.args

The extra arguments passed to ran.gen.

call

The original call to tsboot.

Details

If sim is "fixed" then each replicate time series is found by taking blocks of length l, from the original time series and putting them end-to-end until a new series of length n.sim is created. When sim is "geom" a similar approach is taken except that now the block lengths are generated from a geometric distribution with mean l. Post-blackening can be carried out on these replicate time series by including the function ran.gen in the call to tsboot and having tseries as a time series of residuals.

Model based resampling is very similar to the parametric bootstrap and all simulation must be in one of the user specified functions. This avoids the complicated problem of choosing the block length but relies on an accurate model choice being made.

Phase scrambling is described in Section 8.2.4 of Davison and Hinkley (1997). The types of statistic for which this method produces reasonable results is very limited and the other methods seem to do better in most situations. Other types of resampling in the frequency domain can be accomplished using the function boot with the argument sim = "parametric".

References

Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

Kunsch, H.R. (1989) The jackknife and the bootstrap for general stationary observations. Annals of Statistics, 17, 1217--1241.

Politis, D.N. and Romano, J.P. (1994) The stationary bootstrap. Journal of the American Statistical Association, 89, 1303--1313.

Examples

lynx.fun <- function(tsb) {
     ar.fit <- ar(tsb, order.max = 25)
     c(ar.fit$order, mean(tsb), tsb)
}

# the stationary bootstrap with mean block length 20
lynx.1 <- tsboot(log(lynx), lynx.fun, R = 99, l = 20, sim = "geom")

# the fixed block bootstrap with length 20
lynx.2 <- tsboot(log(lynx), lynx.fun, R = 99, l = 20, sim = "fixed")

# Now for model based resampling we need the original model
# Note that for all of the bootstraps which use the residuals as their
# data, we set orig.t to FALSE since the function applied to the residual
# time series will be meaningless.
lynx.ar <- ar(log(lynx))
lynx.model <- list(order = c(lynx.ar$order, 0, 0), ar = lynx.ar$ar)
lynx.res <- lynx.ar$resid[!is.na(lynx.ar$resid)]
lynx.res <- lynx.res - mean(lynx.res)

lynx.sim <- function(res,n.sim, ran.args) {
     # random generation of replicate series using arima.sim 
     rg1 <- function(n, res) sample(res, n, replace = TRUE)
     ts.orig <- ran.args$ts
     ts.mod <- ran.args$model
     mean(ts.orig)+ts(arima.sim(model = ts.mod, n = n.sim,
                      rand.gen = rg1, res = as.vector(res)))
}

lynx.3 <- tsboot(lynx.res, lynx.fun, R = 99, sim = "model", n.sim = 114,
                 orig.t = FALSE, ran.gen = lynx.sim,
                 ran.args = list(ts = log(lynx), model = lynx.model))

#  For "post-blackening" we need to define another function
lynx.black <- function(res, n.sim, ran.args) {
     ts.orig <- ran.args$ts
     ts.mod <- ran.args$model
     mean(ts.orig) + ts(arima.sim(model = ts.mod,n = n.sim,innov = res))
}

# Now we can run apply the two types of block resampling again but this
# time applying post-blackening.
lynx.1b <- tsboot(lynx.res, lynx.fun, R = 99, l = 20, sim = "fixed",
                  n.sim = 114, orig.t = FALSE, ran.gen = lynx.black,
                  ran.args = list(ts = log(lynx), model = lynx.model))

lynx.2b <- tsboot(lynx.res, lynx.fun, R = 99, l = 20, sim = "geom",
                  n.sim = 114, orig.t = FALSE, ran.gen = lynx.black,
                  ran.args = list(ts = log(lynx), model = lynx.model))

# To compare the observed order of the bootstrap replicates we
# proceed as follows.
table(lynx.1$t[, 1])
#> 
#>  2  3  4  5  7  8  9 10 11 13 
#> 21 18 39  3  6  2  1  2  5  2 
table(lynx.1b$t[, 1])
#> 
#>  2  3  4  5  6  7  8  9 10 11 12 13 14 15 
#>  2  2 27  1  2 10  1  2  4 37  3  4  3  1 
table(lynx.2$t[, 1])
#> 
#>  2  3  4  5  6  7  9 11 
#> 15 18 58  2  1  1  1  3 
table(lynx.2b$t[, 1])
#> 
#>  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 
#>  2  1 17  3  1 11  5  4  1 43  7  1  1  1  1 
table(lynx.3$t[, 1])
#> 
#>  2  4  5  6  7  8  9 10 11 12 14 15 16 
#>  5 11  6  4  6  3  2  3 50  5  1  2  1 
# Notice that the post-blackened and model-based bootstraps preserve
# the true order of the model (11) in many more cases than the others.