Wavelet lifting estimators of the Hurst exponent for regularly and irregularly sampled time series

Description

Implementations of Hurst exponent estimators based on the relationship between wavelet lifting scales and wavelet energy

Details

This package exploits a 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. The main routine is

liftHurst

Author(s)

Marina Knight, Guy Nason, Matt Nunes

Maintainer: Matt Nunes <nunesrpackages@gmail.com>

References

Knight, M. I, Nason, G. P. and Nunes, M. A. (2017) A wavelet lifting approach to long-memory estimation. Stat. Comput. 27 (6), 1453–1471. DOI 10.1007/s11222-016-9698-2.

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

Nunes, M. A., Knight, M. I and Nason, G. P. (2006) Adaptive lifting for nonparametric regression. Stat. Comput. 16 (2), 143–159.

Knight, M. I. and Nason, G. P. (2009) A 'nondecimated' wavelet transform. Stat. Comput. 19 (1), 1–16.

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

Beran, J. et al. (2013) Long-memory processes. Springer.

Compute Hurst exponent from wavelet scale - energy regression slope

Description

Uses the slope of the relationship between wavelet scale and wavelet energy to compute an estimate of the Hurst exponent

Usage

Hfrombeta(beta, model = c("FBM","FGN","ID"))

Arguments

Details

There is a theoretical linear relationship growth in the (log) wavelet energy for increasing wavelet scale. This corresponds to the decay in the autocorrelation of a (long range dependent) time series being analysed, and therefore the Hurst exponent, H. The specific relation to H is dependent to the assumed model; in particular for a Fractional Brownian motion, the relationship between H and the slope is H = abs(beta - 1)/2, whereas for Fractional Gaussian noise or dth order Fractional differenced series, the relationship is H = (beta+1)/2.

Value

Author(s)

Matt Nunes

References

Knight, M. I, Nason, G. P. and Nunes, M. A. (2017) A wavelet lifting approach to long-memory estimation. Stat. Comput. 27 (6), 1453–1471. DOI 10.1007/s11222-016-9698-2.

Beran, J. et al. (2013) Long-Memory Processes. Springer.

See Also

liftHurst

Examples

Hfrombeta(0.8,model="FGN")

artificial.levels

Description

This function splits the coefficients into levels according to either (i) increasing quantiles of the removed interval lengths or (ii) dyadic splitting relative to a fixed lowest scale

Usage

artificial.levels(y, rem, time, tail = TRUE, type = 1)

Arguments

Details

The function computes the so-called artificial levels of a set of removed integrals and corresponding detail coefficients, to mimic the dyadic level splitting in a classical wavelet framework. Details on the "usual" quantile-based splitting can be found in artlev. If type==2 or type==3, the artificial levels are defined by intervals of the form [a0 2^j,a0 2^(j-1) ) as described in Jansen et al. (2009), with a0 = 0.5 for type==2 and set to the minimum sampling interval for type==3. The amalgamation of coarser artificial levels prevents variable energies at coarser scales affecting the predicted relationship between the wavelet scales and their corresponding energies.

Value

Author(s)

Matt Nunes, Marina Knight

References

Jansen, M, Nason, G. P. and Silverman, B. W. (2009) Multiscale methods for data on graphs and irregular multidimensional situations. J. Roy. Stat. Soc. B 71, Part 1, 97–125.

See Also

liftHurst

Examples

#create test signal data
#
library(adlift)
x<-runif(100)
y<-make.signal2("blocks",x=x)
#
#perform forward transform...
#
out<-fwtnp(x,y,LocalPred=AdaptNeigh,neighbours=2)
#
al<-artificial.levels(out$lengthsremove,out$removelist, x, type = 1)
#
#
# the indices of removelist split into levels:
al
#

bootstrap confidence interval calculation

Description

This function uses the Hurst exponent estimates from different lifting trajectories to form a bootstrap confidence interval

Usage

bootci(x, level = 0.05)

Arguments

Value

A vector of length 2, indicating the lower and upper confidence interval values.

Author(s)

Matt Nunes

References

Knight, M. I, Nason, G. P. and Nunes, M. A. (2017) A wavelet lifting approach to long-memory estimation. Stat. Comput. 27 (6), 1453–1471. DOI 10.1007/s11222-016-9698-2.

See Also

liftHurst

Examples

x<-rnorm(100,0.7,0.24)	# vector representing Hurst estimates from NLT

bootci(x)

Functions to perform summary calculations of wavelet scales and energies.

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

idj(x, j) 
meanj(x, j) 
medj(x, j) 
mean2(x) 
mad2(x) 
meanmo(x) 

Arguments

Value

A numeric value corresponding to the average squared detail coefficient, squared mean absolute deviation, median scale etc.

Author(s)

Matt Nunes

References

Knight, M. I, Nason, G. P. and Nunes, M. A. (2017) A wavelet lifting approach to long-memory estimation. Stat. Comput. 27 (6), 1453–1471. DOI 10.1007/s11222-016-9698-2.

See Also

liftHurst

Examples

x<-rnorm(50,30,2)

# calculate the average squared value of x (i.e. energy)

mean2(x)

Performs (non-decimated) lifting based estimation of the Hurst exponent

Description

The function exploits the linear relationship in wavelet energy per scale to estimate the long range dependence parameter of a irregular time series.

Usage

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

Arguments

for more information.

for more information.

Details

Wavelet lifting is performed on a time series to convert it into a set of 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 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. This procedure can be performed for multiple (random) lifting trajectories, each producing a slightly different estimate.

Value

If tradonly=TRUE, the function returns a matrix of dimension length(cutoffs) x 2. The first column are the slopes of the regression fits for each cutoff, whereas the second column are the corresponding estimates for the Hurst exponent.

If tradonly=FALSE, 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), as well as the weights used in the regression (if vc = TRUE). if jsc = TRUE, the slope of the integral log-linear relationship is also returned.

Author(s)

Marina Knight, Matt Nunes

References

Knight, M. I, Nason, G. P. and Nunes, M. A. (2017) A wavelet lifting approach to long-memory estimation. Stat. Comput. 27 (6), 1453–1471. DOI 10.1007/s11222-016-9698-2.

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, fwtnpperm

Examples

library(fracdiff)

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

# perform lifting-based estimation of the Hurst exponent

Hestx<-liftHurst(x, tradonly=TRUE)

# An example with missingness
x1<-fracdiff.sim(n = 500, d = 0.3)$series
gap1<-30:40
gap2<-77:90
gap3<-146:166
timeindex<-setdiff(1:500,c(gap1,gap2,gap3))

Hestx1<-liftHurst(x1[timeindex],grid=timeindex, tradonly=TRUE)