Revision 68994912262e9c2ae98bff7968c23cd4cc8b9e8b authored by Martin Schlather on 24 August 2008, 00:00 UTC, committed by Gabor Csardi on 24 August 2008, 00:00 UTC
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hurst.Rd
\name{hurst}
\alias{hurst}
\alias{Hurst}
\alias{long memory dependence}
\title{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(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,...) 
}
\arguments{
  \item{x}{matrix of coordinates, or vector of x coordinates}
  \item{y}{vector of y coordinates}
  \item{z}{vector of z coordinates}
  \item{data}{the data}
%  \item{grid}{logical; determines whether the vectors \code{x},
%    \code{y}, and \code{z} should be
%    interpreted as a grid definition, see Details.  \code{grid}
%    does not apply for \code{T}.}
  \item{gridtriple}{logical. Only relevant if \code{grid=TRUE}.
    If \code{gridtriple=TRUE}
    then \code{x}, \code{y}, and \code{z} are of the
    form \code{c(start,end,step)}; if
    \code{gridtriple=FALSE} then \code{x}, \code{y}, and \code{z}
    must be vectors of ascending values
  }
  \item{sort}{logical.  If \code{TRUE} then the coordinates are permuted
    such that the largest grid length is in \code{x}-direction; this is
    of interest for algorithms that slice higher dimensional fields
    into one-dimensional sections.
  }
  \item{block.sequ}{ascending sequences of block lengths for which the
    detrended fluctuation analysis and the variance method is
    performed.}
  \item{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.}
  \item{fft.max.length}{if the number of points in \code{x}-direction is
    larger than \code{fft.max.length} then the segments of length
    \code{fft.max.length} are considered, shifted by
    \code{fft.max.length/2} (WOSA-estimator).}
  \item{method}{list of implemented methods to calculate the Hurst parameter; see Details}
  \item{mode}{character. A vector with components
    'nographics', 'plot', or 'interactive': %; see the Details.
    \describe{
      \item{'nographics'}{no graphical output}
      \item{'plot'}{the regression line is plotted}
      \item{'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.
  }
  \item{pch}{vector or scalar; sign by which data are plotted.}
  \item{cex}{vector or scalar; size of \code{pch}.}
  \item{cex.main}{font size for title in regression plot, see
    \command{\link{regression}}; only used if mode includes 'plot'
    or 'interactive'}
  \item{PrintLevel}{integer.  If \code{PrintLevel} is 0 or 1
    nothing is printed. 
    If \code{PrintLevel=2} warnings and the regression results
    are given.  If \code{PrintLevel>2} tracing information is given.
  }
  \item{height}{height of the graphics window}
  \item{...}{graphical parameters}
}
\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:
  \itemize{
    \item detrended fluctuation analysis (dfa)
    \item aggregated variation (var)
    \item periodogram or WOSA estimator (fft)
  }
}
\value{
  The function returns a list with elements
  \code{dfa}, \code{varmeth}, \code{fft} corresponding to
  the three methods given in the Details.
  
  Each of the elements is itself a list that contains the
  following elements.
  
  \item{x}{the x-coordinates used for the regression fit}
  
  \item{y}{the y-coordinates used for the regression fit}
  
  \item{regr}{the coefficients of the \command{\link[stats]{lm}}}
  
  \item{sm}{smoothed curve through the (x,y) points}
    
  \item{x.u}{\code{NULL} or the restricted x-coordinates given
    by the user in the interactive plot}

  \item{y.u}{\code{NULL} or y-coordinates according to \code{x.u}}

  \item{regr.u}{\code{NULL} or the coefficients of 
    \command{\link[stats]{lm}} for \code{x.u} and \code{y.u}}

  \item{H}{the Hurst coefficient}

  \item{H.u}{\code{NULL} or the Hurst coefficient corresponding to the
      user's regression line}
}
\references{
%  Overviews:
%  \itemize{
%    \item 
%  }

  detrended fluctuation analysis
  \itemize{
    \item Peng, C.K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley,
    H.E. and Goldberger, A.L. (1994)
    Mosaic organization of DNA nucleotides
    \emph{Phys. Rev. E} \bold{49}, 1685-1689
  }

  aggregated variation
  \itemize{
    \item 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. \emph{A Practical Guide to Heavy Tails, Statistical
      Techniques an Applications.} Boston: Birkhaeuser

    \item
    Taqqu, M.S. and  Teverovsky, V. and  Willinger, W. (1995)
    Estimators for long-range dependence: an empirical study.
    \emph{Fractals} \bold{3}, 785-798
  }

  periodogram
  \itemize{
    \item Percival, D.B. and Walden, A.T. (1993)
    \emph{Spectral Analysis for Physical Applications: Multitaper and
      Conventional Univariate Techniques}, Cambridge: Cambridge
    University Press.
    
    \item 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 \emph{IEEE Trans. Audio
	Electroacoustics} \bold{15}, 70-73.  
  }
}
\author{Martin Schlather, \email{martin.schlather@math.uni-goettingen.de}
  \url{http://www.stochastik.math.uni-goettingen.de/institute}}
\seealso{
  \command{\link{CovarianceFct}}, \command{\link{fractal.dim}}
}
%\examples{
%}
\keyword{ spatial }%-- one or more ...






%  LocalWords:  hurst gridtriple sequ exp len fft Inf dfa var pch cex WOSA regr
%  LocalWords:  PrintLevel RFparameters periodogram nographics itemize varmeth
%  LocalWords:  lm sm Schlather url ealso CovarianceFct
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