RandomFields (version 2.0.71)

hurst: Hurst coefficient

Description

The function estimates the Hurst coefficient of a process

Usage

hurst(x, y = NULL, z = NULL, data,
      gridtriple = FALSE, sort = TRUE,
      block.sequ = unique(round(exp(seq(log(min(3000, dim[1] / 5)), log(dim[1]),
                          len=min(100, dim[1]))))),
      fft.m = c(1, min(1000, (fft.len - 1) / 10)),
      fft.max.length = Inf, method=c("dfa", "fft", "var"),
      mode=c("plot", "interactive"), pch=16, cex=0.2, cex.main=0.85,
      PrintLevel=RFparameters()$Print,height=3.5,...)

Arguments

x

matrix of coordinates, or vector of x coordinates

y

vector of y coordinates

z

vector of z coordinates

data

the data

gridtriple

logical. Only relevant if grid=TRUE. If gridtriple=TRUE then x, y, and z are of the form c(start,end,step); if gridtriple=FALSE then x,

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 is 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': [object Object],[object Object],[object Object] Usually only one mode is given. Two modes may make sense in the combination c("plot", "interactive") in which case

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, see regression; 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 parameters

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

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)

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 nucleotidesPhys. Rev. E49, 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.Fractals3, 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 {F}ast {F}ourier {T}ransform for the estimation of power spectra: a method based on time averaging over short, modified periodogramsIEEE Trans. Audio Electroacoustics15, 70-73.

See Also

CovarianceFct, fractal.dim

Examples

x <- runif(1000)
if (interactive()) {
   h <- hurst(1:length(x), data=x)
} else {
   h <- hurst(1:length(x), data=x, mode = "nographics")
}