Mid-octave spectral density function (SDF) estimation
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.
the fractional difference parameter. If the
scaling band estimates are desired (prompted by
setting n.sample > 0),
a vector containing the decomposition levels.
If n.sample <= 0, then
the levels may be given in any order and levels may be skipped.
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
the number of samples in the time series.
Although no time series is actually passed to
a logical flag. If
Estimates are made for the scaling filter band based upon an implicit assumption that the FD process is stationary (delta < 0.5).
a vector containing the mid-octave SDF estimates for an FD process.
D. B. Percival and A. T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000, 343–54.
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