# wavFDPTime: Instantaneous estimation of fractionally differenced model... In wmtsa: Wavelet Methods for Time Series Analysis

## Description

The MODWT is used to calculate instantaneous estimates of the FD parameter, the variance of the FD parameter and the innovations variance. The user can select between maximum likelihood and least squares estimators. Localized estimates may also be formed by using multiple chi-squared degrees of freedom in estimating the FD model parameters.

## Usage

 ```1 2 3 4 5``` ```wavFDPTime(x, wavelet="s8", levels=NULL, biased=FALSE, estimator="mle", dof.order=0, delta.range=c(-10.0,10.0), position=list(from=1,by=1,units=character()), units=character(), title.data=character(), documentation=character(), keep.series=FALSE) ```

## Arguments

 `x` a vector containing a uniformly-sampled real-valued time series. `biased` a logical flag used to choose between denoting biased or unbiased estimates. Biased estimates are those which use all available levels in calculating the FD model parameters. Unbiased estimates are calculated with only those wavelet coefficients not subject to circular filter operations, i.e. only the interior wavelet coefficients are used in calculating unbiased estimates. Default: `TRUE`. `delta.range` 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)`. `documentation` a character string used to describe the input `x`. Default: `character()`. `dof.order` the degree of freedom (DOF) order. The number of chi-square DOFs used in estimating the FD parameters is equal to 2 * dof.order + 1 where necessarily dof.order > 0. As the order increases, the estimates will become smoother but less localized in time. Default: `0`. `estimator` a character string denoting the estimation method. Use "lse" for least squares estimates and "mle" for maximum likelihood estimates. Default: `"lse"`. `keep.series` a logical value. If `TRUE`, the original series is preserved in the output object. Default: `FALSE`. `levels` 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. `position` a `list` containing the arguments `from, by` and `to` which describe the position(s) of the input `x`. 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())`. `title.data` a character string representing the name of the input `x`. Default: `character()`. `units` a string denoting the units of the time series. Default: `character()` (no units). `wavelet` a character string denoting the filter type. See `wavDaubechies` for details. Default: `"s8"`.

## Value

an object of class `wavFDP`.

## References

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.

`wavFDP`, `wavFDPBlock`, `wavFDPBand`, `wavFDPSDF`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```## perform a unbiased instantaneous LSE of FD ## parameters for an FD(0.45, 1) realization ## over levels 1 through 6 using Daubechies ## least asymmetric 8-tap filters. Use a zeroth ## order DOF (equivalent to 1 chi-square DOF) z <- wavFDPTime(fdp045, levels=1:6, wavelet="s8", est="lse", biased=FALSE) ## display the results print(z) ## plot the results plot(z) ## plot the results with the confidence intervals ## centered about the mean (known) value of the ## the FD parameter plot(z, mean.delta=0.45) ```