A discrete wavelet transform is used to estimate the FD parameter, the variance
of the FD parameter and the innovations variance for a given time series. Both a
maximum likelihood estimation (MLE) and weighted least squares estimation (WLSE)
scheme are available. If an MLE scheme is chosen, then the DWT is used for its
ability to de-correlate long-memory processes. If a WLSE scheme is chosen, then
the MODWT is used for its known statistical wavelet variance properties.
REQUIRED ARGUMENTS:
x
A vector containing a uniformly-sampled real-valued time series.
OPTIONAL ARGUMENTS:
levels
A vector containing the decomposition levels. The levels may be given in any
order but must be positive. Default: 1:J where J is the maximum wavelet
decomposition level at which there exists at least one interior wavelet
coefficient.
filter
A character string denoting the filter type. See wavDaubechies for details.
Default: ``s8".
estimator
A character string denoting the estimation method. Use ``wlse" for a weighted
least squares estimate and ``mle" for a maximum likelihood estimate. Default:
``wlse".
boundary
A character string representing the different methods by which boundary wavelet
coefficients and scaling coefficients are handled in calculating the FD model
parameters. The options for this argument are dependent upon the estimator
argument. For the MLE case, the boundary options are:
``stationary": Under a
stationary FD process model, boundary wavelet and scaling coefficients are used
in estimating the FD model parameters.
``nonstationary": A
stationary-nonstationary FD model assumes that the governing process may fall
into the nonstationary regime and, accordingly, the boundary wavelet
coefficients and scaling coefficients are excluded in estimating the FD model
parameters.
For the WLSE case, the boundary options are:
``biased":
Boundary wavelet coefficients are included in the estimate.
``unbiased":
Boundary wavelet coefficients are excluded in the estimate.
The scaling
coefficients are (always) excluded in weighted least squares estimates of FD
model parameters. Default: ``unbiased".
edof
The mode by which the equivalent degrees of freedom are calculated. This
argument is limited to 1,2, or 3 and is used only for the WLSE scheme. See
wavEDOF for details. Default: 1.
sdf
A vector containing a discretized approximation of the process spectral density
function (SDF). The coefficients of this argument should correspond exactly with
the normalized Fourier frequencies f = [0, 1/P , 2/P, 3/P, ..., (M-1)/P] where P
= 2*(M-1) and M is the number of points in the SDF vector. For example, if the
sdf vector contains five elements, the corresponding frequencies will be f = [0,
1/8, 1/4, 3/8, 1/2]. This argument is used only for the WLSE scheme when
calculating EDOF mode 2 estimates. Default: empty vector.
delta.range
A two-element vector containing the search range for the FD parameter.
Typically, the range [-10,10] is suitable for all physical systems. Default:
c(-10 10).
VALUE:
result
An object of class WaveletFDP.
DETAILS:
When estimator = ``mle" and boundary = ``stationary", the levels vector is
forced to take on values 1,2,...,J where J is the maximum number of levels in
a full DWT. This is done because (in this case) the scaling coefficient and all
wavelet coefficients are used to form the FD model parameter estimates.
In using the WLSE scheme it is recommended that only the unbiased estimator be
used since the confidence intervals for the biased estimator have not been
sufficiently studied.
REFERENCES:
(1) D. B. Percival and A. T. Walden, ``Wavelet Methods for Time Series Analysis'',
Cambridge University Press, 2000, 340-92.
(2) W. Constantine, D. B. Percival and P. G. Reinhall, ``Inertial Range
Determination for Aerothermal Turbulence Using Fractionally Differenced
Processes and Wavelets'', Physical Review E, 2001, 64(036301), 12 pages.
SEE ALSO:
,
,
,
,
.
EXAMPLES:
## perform a block-averaged MLE of FD
## parameters for an FD( 0.45, 1 ) realization
## over levels 1 through 6 using a
## stationary-nonstationary FD model and
## Daubechies least asymmetric 8-tap filters
wavFDPBlock( fdp045, levels = 1:6, wavelet = "s8",
est = "mle", boundary = "nonstationary" )