boot
Bootstrap Resampling
Generate R
bootstrap replicates of a statistic applied to data. Both
parametric and nonparametric resampling are possible. For the nonparametric
bootstrap, possible resampling methods are the ordinary bootstrap, the
balanced bootstrap, antithetic resampling, and permutation.
For nonparametric multi-sample problems stratified resampling is used:
this is specified by including a vector of strata in the call to boot.
Importance resampling weights may be specified.
- Keywords
- htest, nonparametric
Usage
boot(data, statistic, R, sim = "ordinary", stype = c("i", "f", "w"),
strata = rep(1,n), L = NULL, m = 0, weights = NULL,
ran.gen = function(d, p) d, mle = NULL, simple = FALSE, ...,
parallel = c("no", "multicore", "snow"),
ncpus = getOption("boot.ncpus", 1L), cl = NULL)
Arguments
- data
The data as a vector, matrix or data frame. If it is a matrix or data frame then each row is considered as one multivariate observation.
- statistic
A function which when applied to data returns a vector containing the statistic(s) of interest. When
sim = "parametric"
, the first argument tostatistic
must be the data. For each replicate a simulated dataset returned byran.gen
will be passed. In all other casesstatistic
must take at least two arguments. The first argument passed will always be the original data. The second will be a vector of indices, frequencies or weights which define the bootstrap sample. Further, if predictions are required, then a third argument is required which would be a vector of the random indices used to generate the bootstrap predictions. Any further arguments can be passed tostatistic
through the…
argument.- R
The number of bootstrap replicates. Usually this will be a single positive integer. For importance resampling, some resamples may use one set of weights and others use a different set of weights. In this case
R
would be a vector of integers where each component gives the number of resamples from each of the rows of weights.- sim
A character string indicating the type of simulation required. Possible values are
"ordinary"
(the default),"parametric"
,"balanced"
,"permutation"
, or"antithetic"
. Importance resampling is specified by including importance weights; the type of importance resampling must still be specified but may only be"ordinary"
or"balanced"
in this case.- stype
A character string indicating what the second argument of
statistic
represents. Possible values of stype are"i"
(indices - the default),"f"
(frequencies), or"w"
(weights). Not used forsim = "parametric"
.- strata
An integer vector or factor specifying the strata for multi-sample problems. This may be specified for any simulation, but is ignored when
sim = "parametric"
. Whenstrata
is supplied for a nonparametric bootstrap, the simulations are done within the specified strata.- L
Vector of influence values evaluated at the observations. This is used only when
sim
is"antithetic"
. If not supplied, they are calculated through a call toempinf
. This will use the infinitesimal jackknife provided thatstype
is"w"
, otherwise the usual jackknife is used.- m
The number of predictions which are to be made at each bootstrap replicate. This is most useful for (generalized) linear models. This can only be used when
sim
is"ordinary"
.m
will usually be a single integer but, if there are strata, it may be a vector with length equal to the number of strata, specifying how many of the errors for prediction should come from each strata. The actual predictions should be returned as the final part of the output ofstatistic
, which should also take an argument giving the vector of indices of the errors to be used for the predictions.- weights
Vector or matrix of importance weights. If a vector then it should have as many elements as there are observations in
data
. When simulation from more than one set of weights is required,weights
should be a matrix where each row of the matrix is one set of importance weights. Ifweights
is a matrix thenR
must be a vector of lengthnrow(weights)
. This parameter is ignored ifsim
is not"ordinary"
or"balanced"
.- ran.gen
This function is used only when
sim = "parametric"
when it describes how random values are to be generated. It should be a function of two arguments. The first argument should be the observed data and the second argument consists of any other information needed (e.g. parameter estimates). The second argument may be a list, allowing any number of items to be passed toran.gen
. The returned value should be a simulated data set of the same form as the observed data which will be passed tostatistic
to get a bootstrap replicate. It is important that the returned value be of the same shape and type as the original dataset. Ifran.gen
is not specified, the default is a function which returns the originaldata
in which case all simulation should be included as part ofstatistic
. Use ofsim = "parametric"
with a suitableran.gen
allows the user to implement any types of nonparametric resampling which are not supported directly.- mle
The second argument to be passed to
ran.gen
. Typically these will be maximum likelihood estimates of the parameters. For efficiencymle
is often a list containing all of the objects needed byran.gen
which can be calculated using the original data set only.- simple
logical, only allowed to be
TRUE
forsim = "ordinary", stype = "i", n = 0
(otherwise ignored with a warning). By default an
byR
index array is created: this can be large and ifsimple = TRUE
this is avoided by sampling separately for each replication, which is slower but uses less memory.- …
Other named arguments for
statistic
which are passed unchanged each time it is called. Any such arguments tostatistic
should follow the arguments whichstatistic
is required to have for the simulation. Beware of partial matching to arguments ofboot
listed above, and that arguments namedX
andFUN
cause conflicts in some versions of boot (but not this one).- parallel
The type of parallel operation to be used (if any). If missing, the default is taken from the option
"boot.parallel"
(and if that is not set,"no"
).- ncpus
integer: number of processes to be used in parallel operation: typically one would chose this to the number of available CPUs.
- cl
An optional parallel or snow cluster for use if
parallel = "snow"
. If not supplied, a cluster on the local machine is created for the duration of theboot
call.
Details
The statistic to be bootstrapped can be as simple or complicated as
desired as long as its arguments correspond to the dataset and (for
a nonparametric bootstrap) a vector of indices, frequencies or
weights. statistic
is treated as a black box by the
boot
function and is not checked to ensure that these
conditions are met.
The first order balanced bootstrap is described in Davison, Hinkley and Schechtman (1986). The antithetic bootstrap is described by Hall (1989) and is experimental, particularly when used with strata. The other non-parametric simulation types are the ordinary bootstrap (possibly with unequal probabilities), and permutation which returns random permutations of cases. All of these methods work independently within strata if that argument is supplied.
For the parametric bootstrap it is necessary for the user to specify
how the resampling is to be conducted. The best way of
accomplishing this is to specify the function ran.gen
which
will return a simulated data set from the observed data set and a
set of parameter estimates specified in mle
.
Value
The returned value is an object of class "boot"
, containing the
following components:
The observed value of statistic
applied to data
.
A matrix with sum(R)
rows each of which is a bootstrap replicate
of the result of calling statistic
.
The value of R
as passed to boot
.
The data
as passed to boot
.
The value of .Random.seed
when boot
started work.
The function statistic
as passed to boot
.
Simulation type used.
Statistic type as passed to boot
.
The original call to boot
.
The strata used. This is the vector passed to boot
, if it
was supplied or a vector of ones if there were no strata. It is not
returned if sim
is "parametric"
.
The importance sampling weights as passed to boot
or the empirical
distribution function weights if no importance sampling weights were
specified. It is omitted if sim
is not one of
"ordinary"
or "balanced"
.
If predictions are required (m > 0
) this is the matrix of
indices at which predictions were calculated as they were passed to
statistic. Omitted if m
is 0
or sim
is not
"ordinary"
.
The influence values used when sim
is "antithetic"
.
If no such values were specified and stype
is not "w"
then L
is returned as consecutive integers corresponding to
the assumption that data is ordered by influence values. This
component is omitted when sim
is not "antithetic"
.
The random generator function used if sim
is
"parametric"
. This component is omitted for any other value
of sim
.
The parameter estimates passed to boot
when sim
is
"parametric"
. It is omitted for all other values of
sim
.
Parallel operation
When parallel = "multicore"
is used (not available on Windows),
each worker process inherits the environment of the current session,
including the workspace and the loaded namespaces and attached
packages (but not the random number seed: see below).
More work is needed when parallel = "snow"
is used: the worker
processes are newly created R processes, and statistic
needs
to arrange to set up the environment it needs: often a good way to do
that is to make use of lexical scoping since when statistic
is
sent to the worker processes its enclosing environment is also sent.
(E.g. see the example for jack.after.boot
where
ancillary functions are nested inside the statistic
function.)
parallel = "snow"
is primarily intended to be used on
multi-core Windows machine where parallel = "multicore"
is not
available.
For most of the boot
methods the resampling is done in the
master process, but not if simple = TRUE
nor sim =
"parametric"
. In those cases (or where statistic
itself uses
random numbers), more care is needed if the results need to be
reproducible. Resampling is done in the worker processes by
censboot(sim = "wierd")
and by most of the schemes in
tsboot
(the exceptions being sim == "fixed"
and
sim == "geom"
with the default ran.gen
).
Where random-number generation is done in the worker processes, the
default behaviour is that each worker chooses a separate seed,
non-reproducibly. However, with parallel = "multicore"
or
parallel = "snow"
using the default cluster, a second approach
is used if RNGkind("L'Ecuyer-CMRG")
has been selected.
In that approach each worker gets a different subsequence of the RNG
stream based on the seed at the time the worker is spawned and so the
results will be reproducible if ncpus
is unchanged, and for
parallel = "multicore"
if parallel::mc.reset.stream()
is
called: see the examples for mclapply
.
Note that loading the parallel namespace may change the random seed, so for maximum reproducibility this should be done before calling this function.
References
There are many references explaining the bootstrap and its variations. Among them are :
Booth, J.G., Hall, P. and Wood, A.T.A. (1993) Balanced importance resampling for the bootstrap. Annals of Statistics, 21, 286--298.
Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.
Davison, A.C., Hinkley, D.V. and Schechtman, E. (1986) Efficient bootstrap simulation. Biometrika, 73, 555--566.
Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman & Hall.
Gleason, J.R. (1988) Algorithms for balanced bootstrap simulations. American Statistician, 42, 263--266.
Hall, P. (1989) Antithetic resampling for the bootstrap. Biometrika, 73, 713--724.
Hinkley, D.V. (1988) Bootstrap methods (with Discussion). Journal of the Royal Statistical Society, B, 50, 312--337, 355--370.
Hinkley, D.V. and Shi, S. (1989) Importance sampling and the nested bootstrap. Biometrika, 76, 435--446.
Johns M.V. (1988) Importance sampling for bootstrap confidence intervals. Journal of the American Statistical Association, 83, 709--714.
Noreen, E.W. (1989) Computer Intensive Methods for Testing Hypotheses. John Wiley & Sons.
See Also
boot.array
, boot.ci
,
censboot
, empinf
,
jack.after.boot
, tilt.boot
,
tsboot
.
Examples
# NOT RUN {
# Usual bootstrap of the ratio of means using the city data
ratio <- function(d, w) sum(d$x * w)/sum(d$u * w)
boot(city, ratio, R = 999, stype = "w")
# Stratified resampling for the difference of means. In this
# example we will look at the difference of means between the final
# two series in the gravity data.
diff.means <- function(d, f)
{ n <- nrow(d)
gp1 <- 1:table(as.numeric(d$series))[1]
m1 <- sum(d[gp1,1] * f[gp1])/sum(f[gp1])
m2 <- sum(d[-gp1,1] * f[-gp1])/sum(f[-gp1])
ss1 <- sum(d[gp1,1]^2 * f[gp1]) - (m1 * m1 * sum(f[gp1]))
ss2 <- sum(d[-gp1,1]^2 * f[-gp1]) - (m2 * m2 * sum(f[-gp1]))
c(m1 - m2, (ss1 + ss2)/(sum(f) - 2))
}
grav1 <- gravity[as.numeric(gravity[,2]) >= 7,]
boot(grav1, diff.means, R = 999, stype = "f", strata = grav1[,2])
# In this example we show the use of boot in a prediction from
# regression based on the nuclear data. This example is taken
# from Example 6.8 of Davison and Hinkley (1997). Notice also
# that two extra arguments to 'statistic' are passed through boot.
nuke <- nuclear[, c(1, 2, 5, 7, 8, 10, 11)]
nuke.lm <- glm(log(cost) ~ date+log(cap)+ne+ct+log(cum.n)+pt, data = nuke)
nuke.diag <- glm.diag(nuke.lm)
nuke.res <- nuke.diag$res * nuke.diag$sd
nuke.res <- nuke.res - mean(nuke.res)
# We set up a new data frame with the data, the standardized
# residuals and the fitted values for use in the bootstrap.
nuke.data <- data.frame(nuke, resid = nuke.res, fit = fitted(nuke.lm))
# Now we want a prediction of plant number 32 but at date 73.00
new.data <- data.frame(cost = 1, date = 73.00, cap = 886, ne = 0,
ct = 0, cum.n = 11, pt = 1)
new.fit <- predict(nuke.lm, new.data)
nuke.fun <- function(dat, inds, i.pred, fit.pred, x.pred)
{
lm.b <- glm(fit+resid[inds] ~ date+log(cap)+ne+ct+log(cum.n)+pt,
data = dat)
pred.b <- predict(lm.b, x.pred)
c(coef(lm.b), pred.b - (fit.pred + dat$resid[i.pred]))
}
nuke.boot <- boot(nuke.data, nuke.fun, R = 999, m = 1,
fit.pred = new.fit, x.pred = new.data)
# The bootstrap prediction squared error would then be found by
mean(nuke.boot$t[, 8]^2)
# Basic bootstrap prediction limits would be
new.fit - sort(nuke.boot$t[, 8])[c(975, 25)]
# Finally a parametric bootstrap. For this example we shall look
# at the air-conditioning data. In this example our aim is to test
# the hypothesis that the true value of the index is 1 (i.e. that
# the data come from an exponential distribution) against the
# alternative that the data come from a gamma distribution with
# index not equal to 1.
air.fun <- function(data) {
ybar <- mean(data$hours)
para <- c(log(ybar), mean(log(data$hours)))
ll <- function(k) {
if (k <= 0) 1e200 else lgamma(k)-k*(log(k)-1-para[1]+para[2])
}
khat <- nlm(ll, ybar^2/var(data$hours))$estimate
c(ybar, khat)
}
air.rg <- function(data, mle) {
# Function to generate random exponential variates.
# mle will contain the mean of the original data
out <- data
out$hours <- rexp(nrow(out), 1/mle)
out
}
air.boot <- boot(aircondit, air.fun, R = 999, sim = "parametric",
ran.gen = air.rg, mle = mean(aircondit$hours))
# The bootstrap p-value can then be approximated by
sum(abs(air.boot$t[,2]-1) > abs(air.boot$t0[2]-1))/(1+air.boot$R)
# }