hurst              package:RandomFields              R Documentation

_H_u_r_s_t _c_o_e_f_f_i_c_i_e_n_t

_D_e_s_c_r_i_p_t_i_o_n:

     The function estimates the Hurst coefficient of a process

_U_s_a_g_e:

     hurst(x, y = NULL, z = NULL, data,
           gridtriple = FALSE, sort = TRUE,
           block.sequ = unique(round(exp(seq(log(3000), 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,...) 

_A_r_g_u_m_e_n_t_s:

       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', 'y',
          and 'z' must be vectors of ascending values 

    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': 

          '_n_o_g_r_a_p_h_i_c_s' no graphical output

          '_p_l_o_t' the regression line is plotted

          '_i_n_t_e_r_a_c_t_i_v_e' 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, 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

_D_e_t_a_i_l_s:

     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)

_V_a_l_u_e:

     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 'lsfit'

      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  'lsfit' for 'x.u' and 'y.u'

       H: the Hurst coefficient

     H.u: 'NULL' or the Hurst coefficient corresponding to the user's
          regression line

_A_u_t_h_o_r(_s):

     Martin Schlather, martin.schlather@cu.lu <URL:
     http://www.cu.lu/~schlathe>

_R_e_f_e_r_e_n_c_e_s:

     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 {F}ast {F}ourier {T}ransform
           for the estimation of power spectra: a method based on time
           averaging over short, modified periodograms _IEEE Trans.
           Audio Electroacoustics_ *15*, 70-73.  

_S_e_e _A_l_s_o:

     'CovarianceFct', 'fractal.dim'

