# FDWhittle: Estimate the Hurst coefficient by Whittle's method In fractal: Fractal Time Series Modeling and Analysis

## Description

Using an estimate of the spectral density function for an input time series, Whittle's method fits the parameters of a specified SDF model to the data by optimizing an appropriate functional. In this case, the SDF for a fractionally differenced (FD) process model is used and an estimate of (delta), the FD parameter, is returned.

## Usage

 ```1 2``` ```FDWhittle(x, method="continuous", dc=FALSE, freq.max=0.5, delta.min=-1,delta.max=2.5, sdf.method="direct", ...) ```

## Arguments

 `x` a vector containing a uniformly-sampled real-valued time series. `...` optional SDF estimation arguments passed directly to the `SDF` function. See help documentation for the `SDF` function for more information. `dc` a logical value. If `FALSE`, the DC component of the SDF (corresponding to the sample mean of the series) is not used in optimizing the Whittle functional. Default: `FALSE`. `delta.max` the maximum value for the FD parameter to use in the constrained optimization problem. Default: `2.5`. `delta.min` the minimum value for the FD parameter to use in the constrained optimization problem. Default: `-1`. `freq.max` the largerst normalized frequency of the SDFs use in the analysis. Default: `0.25`. `method` a character string indicating the method to be used in estimating the Hurst coefficient (H). Choices are: `"continuous"`Whittle's method using a continuous model approach to form the optimization functional. This functional is subsequently implemented via a discrete form of the SDF for an FD process. `"discrete"`Whittle's method using (directly) a discrete form of the SDF for an FD process. Default: `"continuous"`. `sdf.method` a character string denoting the method to use in estimating the SDF. Choices are `"direct"`, `"lag window"`, `"wosa"` (Welch's Overlapped Segment Averaging), `"multitaper"`. See help documentation for the `SDF` function for more information. Default: `"direct"`.

## Value

estimate of the FD parameter of the time series.

## References

M. S. Taqqu and V. Teverovsky, On Estimating the Intensity of Long- Range Dependence in Finite and Infinite Variance Time Series (1998), in A practical Guide to Heavy Tails: Statistical Techniques and Applications, pp. 177–217, Birkhauser, Boston.

`hurstSpec`, `FDSimulate`.

## Examples

 ```1 2 3 4``` ```set.seed(100) walk <- cumsum(rnorm(1024)) FDWhittle(walk, method="discrete", sdf.method="multitaper") FDWhittle(walk, method="continuous", sdf.method="multitaper") ```

### Example output

```Loading required package: splus2R