hwtos2 {locits}

Test of second-order stationarity using wavelets.

Description

The main function to perform a test of second-order stationarity as outlined in Nason (2012). Essentially, this routine computes an evolutionary wavelet spectral estimate and then computes the Haar wavelet coefficients of each scale of the spectral estimate. Any large Haar coefficients are indicative of nonstationarity. A multiple hypothesis test assesses whether any of the Haar coefficients are large enough to reject the null hypothesis of stationarity.

Usage

hwtos2(x, alpha = 0.05, filter.number = 1, family = "DaubExPhase",
    lowlev = 3, WTscale = NULL, maxSD = NULL, verbose = FALSE,
    silent = FALSE, UseCForVarip2 = TRUE, OPLENGTH = 1e+05)

Arguments

x

The time series you want to test for second order stationarity. This should be a stochastic series. The function will report an error if x is a constant function. The function might not work properly if it contains a significant trend or patches of non-stochastic observations.

alpha

The overall (nominal) size of the test.

filter.number

The index number of the wavelet used to compute the evolutionary spectral estimate with.

family

The family of wavelet used to compute the evolutionary spectral estimate.

lowlev

Do not compute Haar wavelet coefficients on evolutionary wavelet spectra at level lower than lowlev.

WTscale

The theory of the test shows that the Haar wavelet coefficients of the raw wavelet periodogram are asymptotically normal as long as the scale of the Haar wavelet is ‘coarse’ enough. Roughly, speaking WTscale is internally coded to be the log of the square root of T, the length of the series (J/2), but you can set another value.

maxSD

As part of its execution, this function computes an evolutionary wavelet spectral estimate from the time series. Since the test is based on the assumption of stationarity, the EWS is averaged over time. There will be J = \log_2 T scale levels and, if maxSD = NULL then all of the J levels get used for later functions, such as computing the variance of Haar wavelet coefficients. This argument permits you to restrict the number of coarse scales going into further calculations (e.g. removes the coarser scales from further examination). Mostly, the default will be fine and maximises the use of the available information.

verbose

If TRUE then informative error messages are printed. If FALSE they are not.

silent

If TRUE then no informative messages are printed. If FALSE then a limited amout of informative is printed.

UseCForVarip2

If TRUE then fast C code is use to compute wavelet coefficients' variance. If FALSE then R code is used wholly throughout, but the execution will be much slower.

OPLENGTH

The PsiJ and PsiJmat routines both used preallocated storage. This argument can be provided to increase the amount of storage. Note, you should not need to change this unless the routine as whole stops and tells you to rerun it with increased storage.

Details

This function looks at the Haar wavelet coefficients of an evolutionary wavelet spectrum. This is a modification of the principle of von Sachs and Neumann (2000) which worked with the Haar wavelet coefficients of a local Fourier spectrum.

See also, the stationarity test which implements the Priestley-Subba Rao (1969) test. This function is contained in the fractal package.

Value

An object of class tos, a list containing the following components:

nreject

The number of FDR rejections

rejpval

The p-value associated with FDR rejections

spvals

A vector of p-values from all of the tests, sorted in ascending order.

sTS

A vector of sorted test statistics from all of the tests, sorted into the same order as spvals

AllTS

A list containing all of the test statistics. The first entry contains test statistics corresponding to the coarsest scale, the last entry corresponds to the finest scale. Each component in the list is either empty (because the scale was omitted because it was less than lowlev) or contains a wd class object. The wd class object contains the test statistics for each Haar wavelet coefficient (not the coefficients). Hence, the value of the test statistic for any scale/location or level of the wavelet periodogram can easily be extracted.

AllPVal

As AllTS except the values stored are the p-values, not the test statistics.

alpha

The nominal size of the overall hypothesis test.

x

The original time series that was analyzed

xSD

A vector containing J levels, which is the EWS estimate averaged across time.

Author(s)

Guy Nason.

References

Nason, G.P. (2013) A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series. J. R. Statist. Soc. B, 75, 879-904. doi:10.1111/rssb.12015

Priestley, M.B. and Subba Rao (1969) A test for non-stationarity of time series. J. R. Statist. Soc. B, 31, 140-149.

von Sachs, R. and Neumann, M.H. (2000) A wavelet-based test for stationarity. J. Time Ser. Anal., 21, 597-613.

See Also

varip2, stationarity

Examples

#
# First, test a set of iid Gaussians: should be stationary!
#
hwtos2(rnorm(256))
#  8    7    6    5    4    3  
#Class 'tos' : Stationarity Object :
#       ~~~~  : List with 9 components with names
#              nreject rejpval spvals sTS AllTS AllPVal alpha x xSD 
#
#
#summary(.):
#----------
#There are  186  hypothesis tests altogether
#There were  0  FDR rejects
#No p-values were smaller than the FDR val of:   
#Using Bonferroni rejection p-value is  0.0002688172 
#And there would be  0  rejections.
#
#  NOTE: the summary indicates that nothing was rejected: hence stationary!
#
# Second, example. Concatenated Gaussians with different variances
#
hwtos2(c(rnorm(256), rnorm(256,sd=2)))
#  9    8    7    6    5    4    3  
#Class 'tos' : Stationarity Object :
#       ~~~~  : List with 9 components with names
#              nreject rejpval spvals sTS AllTS AllPVal alpha x xSD 
#
#
#summary(.):
#----------
#There are  441  hypothesis tests altogether
#There were  5  FDR rejects
#The rejection p-value was  3.311237e-06 
#Using Bonferroni rejection p-value is  0.0001133787 
#And there would be  5  rejections.
#Listing FDR rejects... (thanks Y&Y!)
#P: 5 HWTlev:  0  indices on next line...[1] 1
#P: 6 HWTlev:  0  indices on next line...[1] 1
#P: 7 HWTlev:  0  indices on next line...[1] 1
#P: 8 HWTlev:  0  indices on next line...[1] 1
#P: 9 HWTlev:  0  indices on next line...[1] 1
#
# NOTE: This time 5 Haar wavelet coefficients got rejected: hence series
# is not stationary.