hwt {locits}

Compute a Haar wavelet transform for data of arbitrary n length

Description

Function computes Haar wavelet and scaling function coefficients for data set of any length. Algorithm computes every possible coefficient that it can for both decimated and nondecimated versions of the transform.

Usage

hwt(x, type = c("wavelet", "station"), reindex = FALSE)

Arguments

x

A vector of length n, where n is a positive integer. This is the data that you wish to compute the Haar wavelet transform for.

type

The type of transform, either the decimated or nondecimated algorithm.

reindex

If TRUE then the routine attempts to match scales with the usual dyadic transform, wd. If FALSE then the coefficients that are returned are "as is"

Details

Essentially, this algorithm attempts to compute every possible Haar wavelet coefficient. For example, if the length of the input series was 6 then this means that three coefficients at the finest scale can be computed using the first, second and third pair of input data points using the weights c(1, -1)/sqrt(2). However, from the three coefficients that result from this, there is only one pair, so only one "next coarser" coefficient can be computed.

The reindex option is subtle. Essentially, it tries to ensure that the returned coefficients end up at the same scales as if a data set of the next highest dyadic length was analyzed by the wd function. E.g. if the length of the series was 10 then with reindex=FALSE (default) only three levels are returned for each of the wavelet and scaling coefficients. If reindex=TRUE then the number of levels returned would be as if wd analysed a data set of length 16 (the smallest dyadic number larger than 10). The wd levels would be zero to three and this is what would be returned in this function if reindex=TRUE. However, note, in this case, the coarsest level coefficient happens to be NULL (or not computable). One can view the algorithm as computing a partial transform of 10 of the 16 elements and substituting NA for anything it can't compute.

Value

An object of class hwtANYN which is a list with the following components.

c

The scaling function coefficients. This is a list of length nlevels which contains the scaling function coefficients. The coarsest scale coefficients are to be found in the lowest-indexed slots of the list (e.g. c[[1]]) and increasing slot index corresponds to finer scales. So, c[[length(c)]] corresponds to the finest coefficients. Note, an entry in the slot can also be NULL. This indicates that no coefficients could be calculated at this scale, usually the coarsest.

d

A for c but for wavelet coefficients.

nlevels

The number of scale levels in the Haar wavelet decomposition. if reindex=TRUE then this number will be the log to base 2 of the smallest power of two larger than the length of the input vector x.

type

Whether a decimated wavelet transform has been computed ("wavelet") or a nondecimated transform ("station"). Note, the name of the argument "station" has been chosen to coincide with the type in the regular wavelet transform computed by wd.

reindex

Either TRUE or FALSE. If TRUE then the scale levels correspond directly to those computed by wd, the regular wavelet transform. If FALSE then the levels returned in c and d are just indexed from the first non-null level.

Author(s)

G. P. 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

hwtos, plot.hwtANYN, print.hwtANYN, summary.hwtANYN

Examples

#
# Generate test data set of length 5 (note, NOT a power of two)
#
v2 <- rnorm(5)
#
# Compute its Haar transform
#
v2hwt <- hwt(v2)
#
# How many levels does it have?
#
nlevelsWT(v2hwt)
#
# What are the coarsest scale wavelet coefficients
#
v2hwt$d[[1]]
#
# What are the finest scale scaling function coefficients
#
v2hwt$c[[nlevels(v2hwt$c)-1]]