hurstBlock              package:fractal              R Documentation

_H_u_r_s_t _c_o_e_f_f_i_c_i_e_n_t _e_s_t_i_m_a_t_i_o_n _i_n _t_h_e _t_i_m_e _d_o_m_a_i_n

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

     Function to estimate the Hurst parameter H of a long memory time
     series by one of several methods as specified in input. These
     methods all work directly with the sample values of the time
     series (not the spectrum).

     _a_g_g_a_b_s The series is partitioned into m groups. Within each group,
          the first absolute moment about the mean of the entire series
          is evaluated. A measure of the variability of this statistic
          between groups is calculated. The number of groups, m, is
          increased and the process is repeated. The observed
          variability changes with increasing m in a way related by
          theory to the Hurst parameter H of the input series. For the
          methods used here, a log-log plot of variability versus
          number of groups is, ideally, linear, with a slope related to
          H, so H can be determined by linear regression.

     _a_g_g_v_a_r The series is partitioned into m groups. Within each group,
          the variance (relative to the mean of the entire series) is
          evaluated. A measure of the variability of this statistic
          between groups is calculated. The number of groups, m, is
          increased and the process is repeated. The observed
          variability changes with increasing m in a way related by
          theory to the Hurst parameter H of the input series. For the
          methods used here, a log-log plot of variability versus
          number of groups is, ideally, linear, with a slope related to
          H, so H can be determined by linear regression.

     _d_i_f_f_v_a_r The series is partitioned into m groups. Within each
          group, the variance, relative to the mean of the entire
          series, is evaluated. The first difference of the variances
          is then evaluated. A measure of the variability of this
          statistic between groups is calculated. The number of groups,
          m, is increased and the process is repeated. The observed
          variability changes with increasing m in a way related by
          theory to the Hurst parameter H of the input series. For the
          methods used here, a log-log plot of variability versus
          number of groups is, ideally, linear, with a slope related to
          H, so H can be determined by linear regression.

     _h_i_g_u_c_h_i The series is assumed to have the character of a noise,
          not a motion. The series is partitioned into m groups. The
          cumulative sums of the series are evaluated to convert the
          series from a noise to a motion. Absolute differences of the
          cumulative sums between groups are analyzed to estimate the
          fractal dimension of the path. The number of groups, m, is
          increased and the process is repeated. The result changes
          with increasing m in a way related by Higuchi's theory to the
          Hurst parameter H of the input series. A log-log plot of the
          statistic versus number of groups is, ideally, linear, with a
          slope related to H, so H can be determined by linear
          regression. .in -5 

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

     hurstBlock(x, method="aggAbs", scale.min=8, scale.max=NULL,
         scale.ratio=2, weight=function(x) rep(1,length(x)), fit=lm)

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

       x: a vector containing a uniformly-sampled real-valued time
          series.

     fit: a function representing the linear regression scheme to use
          in fitting the resulting statistics (on a log-log scale).
          Supported functions are: 'lm', 'lmsreg', and 'ltsreg'. See
          the on-line help documentation for each of these for more
          information: in R, these are found in the 'MASS' package
          while in S-PLUS they are indigenous and found in the 'splus'
          database. Default: 'lm'.

  method: a character string indicating the method to be used to
          estimate the Hurst coefficient (H). Choices are:

          '"_a_g_g_a_b_s"' Absolute Values of the Aggregated Series

          '"_a_g_g_V_a_r"' Aggregated Variance Method

          '"_d_i_f_f_v_a_r"' Differenced Variance Method

          '"_h_i_g_u_c_h_i"' Higuchi's Method .in -5

               Default: '"aggabs"'.

scale.max: an integer denoting the maximum scale (block size) to use in
          partitioning the series. Default: 'length(x)'.

scale.min: an integer denoting the minimum scale (block size) to use in
          partitioning the series. Default: '8'.

scale.ratio: ratio of successive scales to use in partitioning the
          data. For example, if 'scale.min=8' and 'scale.ratio=2', the
          first scale will be 8, the second scale 16, the third scale
          32, and so on. Default: '2'.

  weight: a function with a single required variable ('x') used to
          weight the resulting statistics ('x') for each scale during
          linear regression. Currently, only supported when 'fit=lm'.
          Default: 'function(x) rep(1,length(x))'.

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

     an object of class 'fractalBlock'.

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

     T. Higuchi (1988), Approach to an irregular time series on the
     basis of the fractal theory, _Physica D_, *31*, 277-283.

     M.S. Taqqu, V. Teverovsky, and W. Willinger, Estimators for Long-
     Range Dependence: an Empirical Study (1995), _Fractals_, *3*, pp.
     785-798.

     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.

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

     'fractalBlock', 'hurstSpec', 'lm'.

_E_x_a_m_p_l_e_s:

     ## create test series 
     set.seed(100)
     x <- rnorm(1024)
     walk <- cumsum(x)

     ## calculate the Hurst coefficient of a random 
     ## walk series using various techniques 
     methods <- c("aggabs","aggvar","diffvar","higuchi")
     z <- lapply(methods, function(method, walk){
         hurstBlock(ifelse1(method=="higuchi",diff(walk),walk), method=method)
     },walk=walk )
     names(z) <- methods

     ## plot results 
     old.plt <- splitplot(2,2,1)
     for (i in 1:4){
         if (i > 1)
            splitplot(2,2,i)
         plot(z[[i]], key=FALSE)
         mtext(paste(attr(z[[i]],"stat.name"), round(as.numeric(z[[i]]),3), sep=", H="),
            line=0.5, adj=1)
     }
     par(old.plt)

