Mid-octave spectral density function (SDF) estimation

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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 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.

Usage

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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 delta must be less than 0.5 since the formulae for calculating the scaling band estimates implicitly assume stationarity. Default: 0.4.

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.

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 density functions for decomposition levels j >= 3 in combination with a Taylor series approximation for levels j=1,2.

Default: "bandpass".

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.

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 mid-octave 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

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## 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=FALSE)

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