tsboot: Bootstrapping of Time Series

Description

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.

Usage

tsboot(tseries, statistic, R, l=NULL, sim="model", endcorr=TRUE, 
       n.sim=length(tseries), orig.t=TRUE, ran.gen=NULL, 
       ran.args=NULL, norm=TRUE, ...)

Arguments

tseries

A univariate or multivariate time series. The time series can be formed by the functions rts or cts (S-Plus version 3.2 and later) or the earlier ts function. Irregular time series, with class "its", m

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

Value

An object of class "boot" with the following components.
t0If orig.t is TRUE then t0 is the result of statistic(tseries,...)
otherwise it is NULL.

item

R
sim
l
endcorr
n.sim
orig.t
ran.gen
ran.args
norm
...
t
R
tseries
statistic
sim
endcorr
n.sim
l
ran.gen
ran.args
call

code

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

Run this code

library(ts)
data(lynx)
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$t[,1])
table(lynx.1b$t[,1])
table(lynx.2$t[,1])
table(lynx.2b$t[,1])
table(lynx.3$t[,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.

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