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

\item{t0}{
If \code{orig.t} is \code{TRUE} then \code{t0} is the result of \code{statistic(tseries,\dots{})} 
otherwise it is \code{NULL}.
}
\item{t}{
The results of applying \code{statistic} to the replicate time series. 
}
\item{R}{
The value of \code{R} as supplied to \code{tsboot}.
}
\item{tseries}{
The original time series.
}
\item{statistic}{
The function \code{statistic} as supplied.
}
\item{sim}{
The simulation type used in generating the replicates.
}
\item{endcorr}{
The value of \code{endcorr} used.  The value is meaningful only when \code{sim} is
\code{"fixed"}; it is ignored for model based simulation or phase scrambling
and is always set to \code{TRUE} if \code{sim} is \code{"geom"}.
}
\item{n.sim}{
The value of \code{n.sim} used.
}
\item{l}{
The value of \code{l} used for block based resampling.  This will be \code{NULL} if 
block based resampling was not used.
}
\item{ran.gen}{
The \code{ran.gen} function used for generating the series or for "post-blackening".
}
\item{ran.args}{
The extra arguments passed to \code{ran.gen}.
}
\item{call}{
The original call to \code{tsboot}.
}}
\details{
If \code{sim} is \code{"fixed"} then each replicate time series is found by taking 
blocks of length \code{l}, from the original time series and putting them 
end-to-end until a new series of length \code{n.sim} is created.  When \code{sim} is 
\code{"geom"} a similar approach is taken except that now the block lengths are 
generated from a geometric distribution with mean \code{l}.   Post-blackening can 
be carried out on these replicate time series by including the function 
\code{ran.gen} in the call to \code{tsboot} and having \code{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 \code{boot} with the argument
\code{sim="parametric"}.
}
\references{
Davison, A.C. and Hinkley, D.V. (1997) 
\emph{Bootstrap Methods and Their Application}. Cambridge University Press.

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

Politis, D.N. and Romano, J.P. (1994) The stationary bootstrap. 
\emph{Journal of the American Statistical Association}, \bold{89}, 1303--1313.
}
\seealso{
  \code{\link{boot}}, \code{\link{arima.sim}}
}
\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$t[,1])
table(lynx.1b$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.
}
\keyword{nonparametric}
\keyword{ts}