wmtsa (version 2.0-2)

wavFDPBand: Mid-octave 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 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

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="" 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.

Value

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

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

## 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)