The wavelet and scaling filters used for wavelet decompositions are nominally
associated with approximate bandpass filters. Specifically, at decomposition
level j, the wavelet transform coefficients correspond approximately to the
normalized frequency range of [ 1/2^(j+1), 1 /2^j ]. The square of the wavelet
coefficients are used to form the so-called wavelet variance (or wavelet
spectrum) which is seen as a regularization of the SDF. Under an assumed FD
process, this function estimates the mid-octave SDF values. The estimates are
calculated assuming that the wavelet transform filters form perfect
(rectangular) passbands. Decomposition levels 1 and 2 are calculated using a
second order Taylor series expansion about the mid-octave frequencies while, for
levels greater than 2, a small angle approximation ( sin( pi * f ) ~ pi * f ) is
used to develop a closed form solution which is a function of FD model
parameters as well as the mid-octave frequencies.
OPTIONAL ARGUMENTS:
levels
A vector containing the decomposition levels. If n.sample <= 0, then the levels
may be given in any order and levels may be skipped. If, however, n.sample > 0,
then levels must contain the values 1, 2, 3, ..., J where J is the maximum
wavelet transform decomposition level. Default: 1:5.
delta
The fractional difference parameter. If the scaling band estimates are desired
(prompted by setting n.sample > 0), then delta must be less than 0.5 since the
formulae for calculating the scaling band estimates implicitly assume
stationarity. Default: 0.4.
n.sample
The number of samples in the time series. Although no time series is actually
passed to the wavFDPBand function, the n.sample argument is used in estimating
the mid-octave SDF value over the band of frequencies which are nominally
associated with the scaling filter in a wavelet transform. If n.sample > 0, this
function will append the estimate of the average SDF value over the scaling band
to the wavelet octave estimates. If n.sample <= 0, only the wavelet octave
estimates are returned. Default: 1024.
scaling
A logical flag. If TRUE, the mid-octave value of the FDP SDF octave
corresponding to the scaling coefficients is also returned. Default: TRUE.
method
A character string denoting the method to be used for estimating the average
spectral density values at the center frequency (on a log scale) of each DWT
octave. The choices are ``integration": Numerical integration of the standard
FDP spectral density function. ``bandpass": A small angle approximation to the
standard FDP spectral desnity functions for decomposition levels j >= 3 in
combination with a Taylor series approximation for levels j = 1,2 Default:
``bandpass".
VALUE:
result
A vector containing the mid-octave SDF estimates for an FD process.
DETAILS:
Estimates are made for the scaling filter band based upon an implicit assumption
that the FD process is stationary (delta < 0.5).
REFERENCES:
(1) D. B. Percival and A. T. Walden, ``Wavelet Methods for Time Series Analysis'',
Cambridge University Press, 2000, 343-54.
SEE ALSO:
,
,
,
.
EXAMPLES:
## calculate the mid-octave SDF values for
## an FD process over various wavelet bands
wavFDPBand( levels = c(1, 3, 5:7), delta = 0.45,
+ scaling = F )