Midoctave spectral density function (SDF) estimation
Description
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 socalled wavelet variance (or wavelet spectrum) which is seen as a regularization of the SDF. Under an assumed FD process, this function estimates the midoctave 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 midoctave 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 midoctave frequencies.
Usage
1 2  wavFDPBand(delta=1/4, method="bandpass", scaling=TRUE,
levels=1:5, n.sample=n.sample < 2^(max(levels)+1))

Arguments
delta 
the fractional difference parameter. If the
scaling band estimates are desired (prompted by
setting n.sample > 0),
then 
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, 
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
Default: 
n.sample 
the number of samples in the time series.
Although no time series is actually passed to
the 
scaling 
a logical flag. If

Details
Estimates are made for the scaling filter band based upon an implicit assumption that the FD process is stationary (delta < 0.5).
Value
a vector containing the midoctave SDF estimates for an FD process.
References
D. B. Percival and A. T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000, 343–54.
See Also
wavFDPBlock
, wavFDPTime
, wavVar
, wavFDPSDF
.
Examples
1 2 3  ## calculate the midoctave SDF values for an FD
## process over various wavelet bands
wavFDPBand(levels=c(1, 3, 5:7), delta=0.45, scaling=FALSE)
