cnlt.univ {CNLTtsa}

Performs 'nondecimated' complex-valued wavelet lifting for univariate time series analysis

Description

The forward complex-valued lifting transform for decomposing a signal of interest is dependent on the trajectory (lifting order) used in the forward lifting transform. This procedure uses trajectory bootstrapping to provide (complex-valued) time-scale information at all times and scales

Usage

cnlt.univ(x, f, P = 100, nkeep = 2, verbose = TRUE, ...)

Arguments

x

A vector of grid values. Can be of any length, not necessarily equally spaced.

f

A vector of function values corresponding to x. Must be of the same length as x.

P

Number of trajectories to be used in the nondecimated lifting transform.

nkeep

Number of scaling points we want at the end of the transform. The usual choice is nkeep=2.

verbose

Indicates whether useful messages should be printed to the console during the procedure.

...

Any other arguments to be passed to fwtnppermC, see the function documentation for more details.

Details

Essentially, this function applies the forward complex wavelet lifting transform fwtnppermC P times, each with a different random lifting trajectory. This results in P sets of complex-valued detail coefficients, along with their associated scales. This information is stored in order to compute the periodogram for (x,f). The “degree of asymmetry" in the prediction is also recorded. This is the ratio between the maximum distance to the removed point to one of its neighbours and the minimum distance from the removed point to one of its neighbours, see Chapter 5.3 in Sanderson (2010) for more details.

Value

An object of class cnlt (subclasses: univ, SG). A list with components:

x

The sampling grid corresponding to f used for the decomposition.

det1

A list, entry i corresponding to detail coefficients associated to point x_i.

lre

A list, entry i corresponding to the scales (integrals) when lifting point x_i.

lreA

A list, entry i corresponding to the degree of asymmetry of the neighbourhood used in the prediction step of point x_i, see description above.

Warning

Using a large number of trajectories for long datasets could take a long time!

Author(s)

Jean Hamilton, Matt Nunes

References

Hamilton, J., Nunes, M. A., Knight, M. I. and Fryzlewicz, P. (2018) Complex-valued wavelet lifting and applications. Technometrics, 60 (1), 48-60, DOI 10.1080/00401706.2017.1281846.

Sanderson, J. (2010) Wavelet methods for time series with bivariate observations and irregular sampling grids. PhD Thesis, University of Bristol, UK.

For the real-valued equivalent procedure, see also
Knight, M. I., Nunes, M. A. and Nason, G. P. (2012) Spectral Estimation for Locally Stationary Time Series with Missing Observations. Stat. Comput. 22 (4), 877–895.

See Also

fwtnppermC, link{get.percoeffsC.biv}

Examples

x<-sort(runif(100))

y <-sin(2*pi*(1/25)*x) + sin(2*pi*(1/50)*x)

## Not run: 
xy.dec<-cnlt.univ(x,y,P=300)

xy.dec$det[[1]][1:10]

## End(Not run)