BootWPTOS {BootWPTOS}

Compute test of stationarity for time series via bootstrap wavelet packet method.

Description

Computes b-spectrum (wavelet packet periodogram) for predefined set of wavelet packets on time series. Then applies bootstrap method to resample new versions of the (assumed for the test) stationary series and retests the series. If the value of the test statistic is out of line (bigger) than the resampled test statistics then the series might well not be stationary.

Usage

BootWPTOS(x, levs, indices, filter.number = 1, family = "DaubExPhase",
	Bsims = 200, lapplyfn = lapply, ret.all = FALSE)

Arguments

x

The time series you wish to test. Length is a power of two

levs

The levels of the wavelet packets that you want to involve in the test.

indices

The indices of the wavelet packets that you want to involve in the test.

filter.number

The filter number of the wavelet that underlies the wavelet packet.

family

The family of the wavelet that underlies the wavelet packet used for the test.

Bsims

The number of bootstrap simulations.

lapplyfn

By default this argument is the lapply function which operates sequentially. However, if you have the parallel package you could supply the mclapply function which processes list elements in parallel on a multicore machines.

ret.all

If FALSE then the results of the test are returned in the standard R htest object, and can be printed and manipulated by those standard tools. If TRUE then a list is returned with information about the test.

Details

Function computes a test statistic for test of stationarity on a time series. Then successive bootstrap realizations are drawn using the surrogate function. If the original time series WAS stationary then surrogate causes stationary draws with the same spectral characteristic as the data to be produced (with Gaussian marginals). Under the null hypothesis the time series is assumed stationary and so the distribution of all of the test statistics should be the same and the p-value of the test statistic be uniformly distributed. If the series is nonstationary then the value of the statistic is likely to be bigger on the first computed test statistic on the dats and much bigger than all the others. We can work out a bootstrap p-value by counting how many resampled test statistics are bigger than the one computed on the data.

Value

Normally a list, cast as a htest class object with the following components:

statistic

The test statistic computed on the data

p.value

The bootstrap p.value of the test.

method

The name of the method of this test statistic.

data.name

The name of the data set tested.

Bootvals

The remaining bootstrap generated test statistics.

Author(s)

G.P. Nason

References

Cardinali, A. and Nason, G.P. (2016) Practical Powerful Wavelet Packet Tests for Second-Order Stationarity. Applied and Computational Harmonic Analysis, 2016 doi:10.1016/j.acha.2016.06.006

See Also

WPts, WPTOSpickout

Examples

#
# Generate a stationary time series (e.g. iid standard normals)
#
x <- rnorm(512)
#
# What would be the finest scale?
#
J <- IsPowerOfTwo(length(x))
J
#[1] 9
#
# So, in WaveThresh there are 9 scales indexed 0 to 8.
#
# Let us test x for stationarity
#
# The finest scale wavelets (or packets) are at scale 8
# The next finest scale is 7.
#
# Wavelets themselves are always indexed 1, father wavelets 0.
# We don't tend to use father wavelets for stationary testing.
#
# There are 2^j packets at scale J-j (so 2 at the finest [father and
# mother], 4 at the next finest [father=0, mother=1, packets 2 and 3].
#
# Let's just look at the finest scale wavelet (8,1) and the next finest
# scale wavelet (7,1) and two other wavelet packets  (7,2) and (7,3)
#
x.test <- BootWPTOS(x=x, levs=c(8,7,7,7), indices=c(1,1,2,3), Bsims=30)
#
# Note: Bsims=30 is almost certainly too small (but it is small here because
# on installation R run these examples and I don't want it to take too long.
# 100+ is almost certainly necessary, and probably 500+ useful and 1000+
# to be "sure". If you can load the multicore library then you can
# replace lapplyfn=lapply with lapplyfn=mclapply to get a parallel processing
#
# What are the results of our test?
#
x.test
#
#	WPBootTOS test of stationarity
#
#data:  x
#= 1.8096, p-value = 0.7
#
# So, the p-value is > 0.05 so this test indicates that there is
# no evidence for non-stationarity. Running it for 1000 bootstrap simulations
# gave a p-vale of 0.736.
#
# The next example is nonstationary. However, after the series has been
# generated you should plot it. The second half has a different variance
# to the first half but it is very difficult, usually, to identify the
# different variances on a plot.
#
x2 <- c(rnorm(256), rnorm(256,sd=1.4))
#
# Let's do a test, but involve ALL non-father-wavelet packets from scales
# 8, 7, 6 and 5.
#
x2.test <- BootWPTOS(x=x2, levs=c(8,7,7,7,rep(6,7), rep(5,15)),
	indices=c(1,1,2,3, 1:7, 1:15), Bsims=30)
x2.test
#
#	WPBootTOS test of stationarity
#
#data:  x2
#= 5.4362, p-value < 2.2e-16
#
# So, strong evidence for nonstationarity because p.value < 0.05 (much less
# than!). Again here we've only use 30 bootstrap simulations and this is
# probably too small. Using Bsims=1000 and mclapply (for speed) gave a p-value
# of 0.002, so still assessed to be nonstationary, but we have more confidence
# in the answer.