# RFhurst: Hurst coefficient In RandomFields: Simulation and Analysis of Random Fields

## Description

The function estimates the Hurst coefficient of a process

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```RFhurst(x, y = NULL, z = NULL, data, sort = TRUE, block.sequ = unique(round(exp(seq(log(min(3000, dimen/5)), log(dimen), len = min(100, dimen))))), fft.m = c(1, min(1000, (fft.len - 1)/10)), fft.max.length = Inf, method = c("dfa", "fft", "var"), mode = if (interactive ()) c("plot", "interactive") else "nographics", pch = 16, cex = 0.2, cex.main = 0.85, printlevel = RFoptions()\$basic\$printlevel, height = 3.5, ...) ```

## Arguments

 `x` \argX `y,z` \argYz `data` the data
 `sort` logical. If `TRUE` then the coordinates are permuted such that the largest grid length is in `x`-direction; this is of interest for algorithms that slice higher dimensional fields into one-dimensional sections. `block.sequ` ascending sequences of block lengths for which the detrended fluctuation analysis and the variance method are performed. `fft.m` vector of 2 integers; lower and upper endpoint of indices for the frequency which are used in the calculation of the regression line for the periodogram near the origin. `fft.max.length` if the number of points in `x`-direction is larger than `fft.max.length` then the segments of length `fft.max.length` are considered, shifted by `fft.max.length/2` (WOSA-estimator). `method` list of implemented methods to calculate the Hurst parameter; see Details `mode` character. A vector with components `'nographics'`, `'plot'` or `'interactive'`: `'nographics'`no graphical output `'plot'`the regression line is plotted `'interactive'`the regression domain can be chosen interactively Usually only one mode is given. Two modes may make sense in the combination c("plot", "interactive") in which case all the results are plotted first, and then the interactive mode is called. In the interactive mode, the regression domain is chosen by two mouse clicks with the left mouse; a right mouse click leaves the plot. `pch` vector or scalar; sign by which data are plotted. `cex` vector or scalar; size of `pch`. `cex.main` font size for title in regression plot; only used if mode includes `'plot'` or `'interactive'` `printlevel` integer. If `printlevel` is 0 or 1 nothing is printed. If `printlevel=2` warnings and the regression results are given. If `printlevel>2` tracing information is given. `height` height of the graphics window `...` graphical arguments

## Details

The function is still in development. Several functionalities do not exist - see the code itself for the current stage.

The function calculates the Hurst coefficient by various methods:

• detrended fluctuation analysis (dfa)

• aggregated variation (var)

• periodogram or WOSA estimator (fft)

## Value

The function returns a list with elements `dfa`, `varmeth`, `fft` corresponding to the three methods given in the Details.

Each of the elements is itself a list that contains the following elements.

 `x` the x-coordinates used for the regression fit `y` the y-coordinates used for the regression fit `regr` the coefficients of the `lm`. `sm` smoothed curve through the (x,y) points `x.u` `NULL` or the restricted x-coordinates given by the user in the interactive plot `y.u` `NULL` or y-coordinates according to `x.u` `regr.u` `NULL` or the coefficients of `lm` for `x.u` and `y.u` `H` the Hurst coefficient `H.u` `NULL` or the Hurst coefficient corresponding to the user's regression line

## References

detrended fluctuation analysis

• Peng, C.K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E. and Goldberger, A.L. (1994) Mosaic organization of DNA nucleotides Phys. Rev. E 49, 1685-1689

aggregated variation

• Taqqu, M.S. and Teverovsky, V. (1998) On estimating the intensity of long range dependence in finite and infinite variance time series. In: Adler, R.J., Feldman, R.E., and Taqqu, M.S. A Practical Guide to Heavy Tails, Statistical Techniques an Applications. Boston: Birkhaeuser

• Taqqu, M.S. and Teverovsky, V. and Willinger, W. (1995) Estimators for long-range dependence: an empirical study. Fractals 3, 785-798

periodogram

• Percival, D.B. and Walden, A.T. (1993) Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques, Cambridge: Cambridge University Press.

• Welch, P.D. (1967) The use of Fast Fourier Transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms IEEE Trans. Audio Electroacoustics 15, 70-73.

`RMmodel`, `RFfractaldim`
 ```1 2 3 4 5 6``` ```RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set ## RFoptions(seed=NA) to make them all random again x <- runif(1000) h <- RFhurst(1:length(x), data=x) ```