liftHurstCC {CliftLRD}

Performs (non-decimated) complex-valued lifting based estimation of the Hurst exponent of a complex-valued time series

Description

The function exploits the linear relationship in complex-valued wavelet energy per scale to estimate the long range dependence parameter of an irregular (complex-valued) time series.

Usage

liftHurstCC(x, grid = 1:length(x), model = "FGN", ntraj = 50, cutoffs = 0, 
cut.fine = TRUE, efun = meanmoC, afun = idj, altype = 1, tail = TRUE, 
normalise = TRUE, level = 0.05, bc = TRUE, vc = TRUE, jsc = TRUE, 
BHonly=TRUE, ...) 

Arguments

x

A complex-valued time series, possibly irregularly spaced.

grid

The time samples corresponding to the time series x.

model

The underlying model the time series is assumed to follow. Possible values are "FBM", "FGN", and "ID".

ntraj

The number of lifting trajectories (bootstraps) used for the estimation of the Hurst exponent.

cutoffs

A vector indicating if the coarsest scales are to be removed when estimating the linear slope in the energy-scale relationship; for example if cutoffs = 0, all levels will be used in the estimation. For multiple cutoffs, all slopes will be calculated.

cut.fine

A boolean variable indicating if the finest (default) or coarsest scales are to be removed when estimating the linear slope in the energy-scale relationship; see cutoffs description for more information.

efun

A function indicating which measure of wavelet energy to use. The default is meanmoC, corresponding to the average (modulus) squared detail coefficient within a particular scale, scaled by (n-1). Another measure could be the traditional modulus squared mean, or the corresponding median calculation etc.

afun

A function indicating which measure of wavelet integral (scale) to use (corresponding to the x-axis of the energy-scale relationship). The default is idj, corresponding to the log2 of the dyadic wavelet scale, i.e an integer representing the scale.

altype

An integer indicating which type of artificial levels to be used, see
artificial.levels

for more information.

tail

A boolean variable indicating whether to amalgamate artificial levels with few coefficients, see artificial.levels

for more information.

normalise

A boolean variable indicating whether to normalise the detail coefficients by their individual (induced) standard deviations (computed using the diagonal of the complex-valued lifting matrix, see Hamilton et al. (2017) or fwtnppermC for more details).

level

The significance level for the bootstrap confidence interval of the Hurst exponent estimate.

bc

A boolean variable indicating whether bias-correction should take place or not, using the approximate chi^2 distribution of the wavelet energies.

vc

A boolean variable indicating whether a weighted linear regression should be used when estimating the Hurst exponent. If TRUE, the inverse of the variances of the approximate chi^2 distribution of the wavelet energies are used in the regression.

jsc

A boolean variable indicating whether the slope of the log-linear relationship between the artificial scales and the log of the integrals should be computed and used to reweight the estimate of the Hurst exponent.

BHonly

A boolean variable indicating whether only the Hurst estimate should be returned. If FALSE, then the (log) energies used in the calculation is also returned. Note that if bootstrapping is performed (tradonly=FALSE), then the energies for the last trajectory will be returned.

...

Any other optional arguments to the function fwtnpperm function from the nlt package, which performs wavelet lifting of the time series x according to a specified lifting trajectory.

Details

Complex-valued wavelet lifting is performed on a complex-valued time series to convert it into a set of complex-valued wavelet coefficients and corresponding lifting integrals, specific to when the data were "lifted" during the decomposition. By using the conjugate of the lifting matrix and also applying this to the series, we can obtain a second set of complex-valued wavelet coefficients. The coefficients are grouped into artificial levels, using the integrals to mimic the support of the wavelets in the classical wavelet setting, and therefore producing a notion of scale. The complex-valued coefficients in each artificial level are then used to compute values of the wavelet energy for a particular level. The (slope of the) linear relationship between the scales and their energies is then used in computing an estimate of the Hurst exponent for the series. Since we have two sets of complex coefficients, the estimation of the log-linear relationship can be performed twice and the esitmates averaged. This procedure can be performed for multiple (random) lifting trajectories, each producing a slightly different estimate.

Value

The function the estimate of the Hurst exponent for the series after averaging over all ntraj lifting paths involved in the nondecimated lifting.

If BHonly=FALSE, the routine also returns the energies and scales (on a log scale) which are used in the regression to estimate the decay properties of the spectrum (for the last lifting trajectory).

Author(s)

Matt Nunes, Marina Knight

References

Knight, M. I. and Nunes, M. A. (2018) Long memory estimation for complex-valued time series. Stat. Comput. (to appear). Online First Article: DOI 10.1007/s11222-018-9820-8.
Hamilton, J., Nunes, M. A., Knight, M. I. and Fryzlewicz, P. (2017) Complex-valued lifting and applications. Technom.,60 (1), 48–60.

For more details on the weighted linear regression and bias calculations, see e.g.

Veitch, D. and Abry, P. (1999) A Wavelet-Based Joint Estimator of the Parameters of Long-Range Dependence. IEEE Trans. Info. Theory 45 (3), 878–897.

See Also

artificial.levels, Hfrombeta, fwtnppermC

Examples

# generate a fake complex-valued series 
x<-complex(real=rnorm(150), imaginary=rnorm(150))

# perform lifting-based estimation of the Hurst exponent

Hestx<-liftHurstCC(x,ntraj=25)