Complex-Valued Wavelet Lifting Estimators of the Hurst Exponent for Irregularly Sampled Time Series

Description

Implementations of Hurst exponent estimators based on the relationship between wavelet lifting scales from complex-valued lifting schemes and wavelet energy.

Details

Package information:

This package exploits a complex-valued wavelet transform for irregularly spaced data to form wavelet-like scale-based energy measures for a time series. This is then used to estimate the Hurst exponent for real- and complex-valued time series. The main routines are

liftHurstC and liftHurstCC

Author(s)

Matt Nunes, Marina Knight

Maintainer: Matt Nunes <nunesrpackages@gmail.com>

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 <doi:10.1007/s11222-018-9820-8>.

For related literature on the lifting methodology adopted in the technique, see

Hamilton, J., Nunes, M. A., Knight, M. I. and Fryzlewicz, P. (2017) Complex-valued lifting and applications. Technom.,60 (1), 48–60.

For more information on long-memory processes, see e.g.

Beran, J. et al. (2013) Long-memory processes. Springer.
Lilly, J. M., Sykulski, A. M., Early, J. J. and Olhede, S. C. (2017) Fractional Brownian motion, the Mat\'ern process, and stochastic modeling of turbulent dispersion. Nonlin. Proc. Geophys., 24, 481–514.

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

Usage

liftHurstC(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

for more information.

for more information.

Details

Complex-valued wavelet lifting is performed on a 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. The coefficients are then 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. In effect, this is akin to performing two separate lifting transforms on the series, eliciting information from both parts of the complex-valued lifting transform. This procedure can be performed for multiple (random) lifting trajectories, each producing a slightly different estimate.

Value

The function returns a matrix of dimension length(cutoffs) x 5. The first column are the slopes of the regression fits for each cutoff, where the average is taken over the ntraj randomly generated lifting trajectories. Similarly, the second column represents the average Hurst exponent for the cutoffs over all lifting paths. The third column is the standard deviation of the ntraj Hurst estimates through performing non-decimated lifting. The fourth and fifth columns are the lower and upper values of the bootstrap confidence interval of the Hurst exponent estimate.

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 <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

library(fracdiff)

# simulate a long range dependent time series, and fake missingness
x<-fracdiff.sim(n = 200, d = 0.3)$series

missing<-sample(1:200,70,FALSE)
timeindex<-setdiff(1:200,missing)

Hestx<-liftHurstC(x[timeindex],grid=timeindex,ntraj=25)

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

for more information.

for more information.

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 <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)

Functions to perform summary calculations of wavelet scales and energies resulting from complex-valued lifting transforms.

Description

To estimate the slope of the relationship between wavelet scale and wavelet energy, choices have to be made as to how these quantities are computed. Examples of these choices are the functions listed here.

Usage

meanC(x) 
meanmoC(x) 

Arguments

Value

A numeric value corresponding to the average (modulus) squared detail coefficient or scaled average (modulus squared detail coefficient.

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 <doi:10.1007/s11222-018-9820-8>.

See Also

liftHurstC, liftHurstCC

Examples

x<-rnorm(50,30,2)
y<-rnorm(50,30,2)

z<-complex(real=x,imaginary=y)

# calculate the average squared value of complex-valued vector (i.e. energy)

meanC(z)