Block-dependent estimation of fractionally differenced (FD) model parameters


A discrete wavelet transform of the input series is used to calculate block-dependent estimates of the FD parameter, the variance of the FD parameter and the innovations variance. Both a maximum likelihood estimation (MLE) and weighted least squares estimation (WLSE) scheme are supported. If an MLE scheme is chosen, then the DWT is used for its ability to de-correlate long-memory processes. If a WLSE scheme is chosen, then the MODWT is used for its known statistical wavelet variance properties.


wavFDPBlock(x, wavelet="s8", levels=NULL, sdf=NULL,
    boundary=NULL, edof.mode=1,
    estimator="wlse", delta.range=c(-10.0,10.0),
    position=list(from=1,by=1,units=character()), units=character(),, documentation=character(), keep.series=FALSE)



a vector containing a uniformly-sampled real-valued time series.


a character string representing the different methods by which boundary wavelet coefficients and scaling coefficients are handled in calculating the FD model parameters. The options for this argument are dependent upon the estimator argument.

For the MLE case, the boundary options are:


Under a stationary FD process model, boundary wavelet and scaling coefficients are used in estimating the FD model parameters.


A stationary-nonstationary FD model assumes that the governing process may fall into the nonstationary regime and, accordingly, the boundary wavelet coefficients and scaling coefficients are excluded in estimating the FD model parameters.

For the WLSE case, the boundary options are:


Boundary wavelet coefficients are included in the estimate.


Boundary wavelet coefficients are excluded in the estimate.

The scaling coefficients are (always) excluded in weighted least squares estimates of FD model parameters. Default: "unbiased".


a two-element vector containing the search range for the FD parameter. Typically, the range [-10,10] is suitable for all physical systems. Default: c(-10 10).


a character string used to describe the input data. Default: character().


the mode by which the equivalent degrees of freedom are calculated. This argument is limited to 1,2, or 3 and is used only for the WLSE scheme. See wavEDOF for details. Default: 1.


a character string denoting the estimation method. Use "wlse" for a weighted least squares estimate and "mle" for a maximum likelihood estimate. Default: "wlse".


a logical value. If TRUE, the original series is preserved in the output object. Default: FALSE.


a vector containing the decomposition levels. The levels may be given in any order but must be positive. Default: 1:J where J is the maximum wavelet decomposition level at which there exists at least one interior wavelet coefficient.


a list containing the arguments from, by and to which describe the position(s) of the input data. All position arguments need not be specified as missing members will be filled in by their default values. Default: list(from=1, by=1, units=character()).


a vector containing a discretized approximation of the process spectral density function (SDF). The coefficients of this argument should correspond exactly with the normalized Fourier frequencies f=0, 1/P , 2/P, 3/P, ..., (M-1)/P, where P=2*(M-1) and M is the number of points in the SDF vector. For example, if the sdf vector contains five elements, the corresponding frequencies will be f=[0, 1/8, 1/4, 3/8, 1/2]. This argument is used only for the WLSE scheme when calculating EDOF mode 2 estimates. Default: NULL (EDOF mode 2 not used).

a character string representing the name of the input data. Default: character().


a string denoting the units of the time series. Default: character() (no units).


a character string denoting the filter type. See wavDaubechies for details. Default: "s8".


When estimator="mle" and boundary="stationary", the levels vector is forced to take on values [1,2,...,J] where J is the maximum number of levels in a full DWT. This is done because (in this case) the scaling coefficient and all wavelet coefficients are used to form the FD model parameter estimates.

In using the WLSE scheme it is recommended that only the unbiased estimator be used since the confidence intervals for the biased estimator have not been sufficiently studied.


an object of class wavFDP.


D. B. Percival and A. T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000, 340–92.

W. Constantine, D. B. Percival and P. G. Reinhall, Inertial Range Determination for Aerothermal Turbulence Using Fractionally Differenced Processes and Wavelets, Physical Review E, 2001, 64(036301), 12 pages.

See Also

wavEDOF, wavFDP, wavFDPTime, wavFDPBand, wavFDPSDF.


## perform a block-averaged MLE of FD parameters 
## for an FD(0.45, 1) realization over levels 1 
## through 6 using a stationary-nonstationary 
## FD model and Daubechies least asymmetric 
## 8-tap filters 
wavFDPBlock(fdp045, levels=1:6, wavelet="s8", est="mle", boundary="nonstationary")

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