LongRangeDependence         package:fSeries         R Documentation

_L_o_n_g _R_a_n_g_e _D_e_p_e_n_d_e_n_c_e _M_o_d_e_l_l_i_n_g

_D_e_s_c_r_i_p_t_i_o_n:

     A collection and description of functions  to investigate the long
     range dependence or  long memory behavior of an univariate time 
     series process. Included are functions to  simulate fractional
     Gaussian noise and  fractional ARMA processes, functions to model 
     the true autocorrelations and the spectrum of  these processes,
     and functions to compute the  Hurst exponent by several different
     methods. 

     The Functions and methods are:

     Functions to simulate long memory time series processes:

       'fnmSim'     Simulates fractio nal Brownian motion,
       '- mvn'      from the numerical approximation of the stochastic integral,
       '- chol'     from the Choleski's decomposition of the covariance matrix,
       '- lev'      using the method of Levinson,
       '- circ'     using the method of Wood and Chan,
       '- wave'     using the wavelet synthesis,
       'fgnSim'     Simulates fractional Gaussian noise,
       '- beran'    using the method of Beran,
       '- durbin'   using the method Durbin and Levinson,
       '- paxson'   using the method of Paxson,
       'farimaSim'  simulates FARIMA time series processes.

     Functions to model the true autocorrelation function and spectrum:

       'fgnTrueacf'     Returns true FGN covariances,
       'fgnTruefft'     returns true FGN fast Fourier transform,
       'farimaTrueacf'  returns true FARIMA covariances,
       'farimaTruefft'  returns true FARIMA fast Fourier transform.

     Functions to estimate the Hurst exponent:

       'aggvarFit'   Aggregated variance method,
       'diffvarFit'  Differenced aggregated variance method,
       'absvalFit'   aggregated absolute value (moment) method,
       'higuchiFit'  Higuchi's or fractal dimension method,
       'pengFit'     Peng's or variance of residuals method,
       'rsFit'       R/S Rescaled Range Statistic method,
       'perFit'      periodogram method,
       'boxperFit'   boxed (modified) periodogram method,
       'whittleFit'  Whittle estimator.

     Function for the wavelet estimator:

       'waveletFit'  wavelet estimator.

_U_s_a_g_e:

     fbmSim(n = 100, H = 0.7, method = c("mvn", "chol", "lev", "circ", "wave"),
         waveJ = 7, doplot = TRUE, fgn = FALSE)
     fgnSim(n = 1000, H = 0.7, method = c("beran", "durbin", "paxson"))
     farimaSim(n = 1000, model = list(ar = c(0.5, -0.5), d = 0.3, ma = 0.1),
         method = c("freq", "time"), ...) 
         
     fgnTrueacf(n, H) 
     fgnTruefft(n, H) 
     farimaTrueacf(n, H) 
     farimaTruefft(n, H)
        
     aggvarFit(x, levels = 50, minnpts = 3, cut.off = 10^c(0.7, 2.5), 
         doplot = FALSE, trace = FALSE, title = NULL, description = NULL)    
     diffvarFit(x, levels = 50, minnpts = 3, cut.off = 10^c(0.7, 2.5), 
         doplot = FALSE, trace = FALSE, title = NULL, description = NULL) 
     absvalFit(x, levels = 50, minnpts = 3, cut.off = 10^c(0.7, 2.5), moment = 1, 
         doplot = FALSE, trace = FALSE, title = NULL, description = NULL) 
     higuchiFit(x, levels = 50, minnpts = 2, cut.off = 10^c(0.7, 2.5), 
         doplot = FALSE, trace = FALSE, title = NULL, description = NULL)
     pengFit(x, levels = 50, minnpts = 3, cut.off = 10^c(0.7, 2.5), 
         method = c("mean", "median"), 
         doplot = FALSE, trace = FALSE, title = NULL, description = NULL)
     rsFit(x, levels = 50, minnpts = 3, cut.off = 10^c(0.7, 2.5), 
         doplot = FALSE, trace = FALSE, title = NULL, description = NULL)
     perFit(x, cut.off = 0.1, method = c("per", "cumper"), 
         doplot = FALSE, title = NULL, description = NULL)
     boxperFit(x, nbox = 100, cut.off = 0.10, 
         doplot = FALSE, trace = FALSE, title = NULL, description = NULL)      
     whittleFit(x, order = c(1, 1), subseries = 1, method = c("fgn", "farma"), 
         trace = FALSE, spec = FALSE, title = NULL, description = NULL)
       
     waveletFit(x, length = NULL, order = 2, octave = c(2, 8), 
         doplot = FALSE, title = NULL, description = NULL)
             
     show.fHURST(object)

_A_r_g_u_m_e_n_t_s:

 cut.off: [*Fit] - 
           a numeric vector with the lower and upper cut-off  points
          for the estimation. They should be chosen  to define a linear
          range. The default values are  c(0.7, 2.5), i.e. 10\^0.7 and
          10\^2.5, respectively. 

description: [*Fit] - 
           a character string which allows for a brief description. 

  doplot: [*Fit] - 
           a logical flag, by default FALSE. Should a plot be
          displayed? 

     fgn: [fbmSim] - 
           a logical flag, if 'FALSE', the functions returns a FBM 
          series otherwise a FGN series. 

       H: [fgnSim] - 
           the Hurst exponent, a numeric value between 0.5 and 1, by
          default 0.7. 

  length: [waveletFit] - 
           the length of data to be used, must be power of 2. If set to
          'NULL', the previous power will be used. 

  levels: [*Fit] - 
           the number of aggregation levels or number of blocks from 
          which the variances or moments are computed. 

  method: [fbmSim] - 
           the method how to generate the FBM time series sequence, one
          of the following five character strings: '"mvn"',  '"chol"',
          '"lev"', '"circ"', or '"wave"'. [fgnSim] - 
           the method how to generate the FGN time series sequence, one
          of the following three character strings: '"beran"', 
          '"durbin"', or '"paxson"'. 
           [farimaSim] - 
           the method how to generate the time series sequence, one of
          the following tow character strings: '"freq"', or '"time"'. 
           [pengFit] - 
           a string naming the method how to do the averaging, either
          calculating the '"mean"' or the '"median"'.  
           [perFit] - 
           a string naming the method how to fit the data, either using
          the peridogram itself '"per"', or using the cumulated
          periodogram '"cumper"'.  
           [whittleFit] - 
           a string naming the underlying time series process to  be
          estimated, either '"fgn"' for FGN processes, or '"farima"'for
          FARIMA models. 

 minnpts: [*Fit] - 
           the minimum number of points or blocksize to be used to 
          estimate the variance or moments at any aggregation level. 

   model: a list with model parameters 'ar', 'ma' and 'd'. 'ar' is a
          numeric vector giving the AR coefficients,  'd' is an integer
          value giving the degree of differencing, and 'ma' is a
          numeric vector giving the MA coefficients. Thus the order of
          the time series process is FARMA(p, d, q) with 'p=length(ar)'
          and 'q=length(ma)'. 'd' is a fractional value for FARMA
          models. By default an  FARMA(2, d, 1) model with coefficients
          'ar=c(0.5, -0.5)', 'ma=0.1', and 'd=0.3' will be generated. 

  moment: [absvalHurst] - 
           an integer value, by default 1 which denotes absolute
          values.  For values larger than one this argument determines
          what  absolute moment should be calculated. 

       n: [fgnSim][farimaSim] - 
           number of data points to be simulated, a numeric value,  by
          default 1000. 

    nbox: [boxperFit] - 
           is the number of boxes to divide the data into. A numeric
          value, by default 100. 

  object: an object of class 'fHurst'. 

  octave: [waveletFit] - 
           beginning and ending octave for estimation. An integer
          vector with two elements. By default 'c(2, 8)'. If the upper
          value is too large, it will be replaced by the maximum
          allowed value. 

   order: [waveletFit] - 
           the order of the wavelet. An integer value, by default  '2'. 

    spec: [whittleFit] - 
           Should the periodogram be returned? A logical flag, by
          default  FALSE. 

subseries: [whittleFit] - 
           allows optionally to subdivide the series into subseries. A
          numeric value, by default 1. 

   title: a character string which allows for a project title. 

   trace: a logical value, by defaul FALSE. Should the estimation 
          process be traced? 

   waveJ: [fbmSim] - 
           an integer parameter for the simulation of FBM using the
          wavelet  method. 

       x: [*Fit] - 
           the numeric vector of data, an object of class 'timeSeries',
          or any other object which can be transofrmed into a numeric
          vector by the function 'as.vector'. 

     ...: arguments to be passed. 

_D_e_t_a_i_l_s:

     *I: Functions to Simulate Long Memory Processes:* 

     _Fractional Gaussian Noise:_ 
      The function 'fgnSim' simulates a series of fractional Gaussian
     noise, FGN. FGN provides a parsimonious model for  stationary
     increments of a self-similar process parameterised  by the Hurst
     exponent H and variance. Fractional Gaussian noise  with H < 0.5
     demonstrates negatively autocorrelated or  anti-persistent
     behaviour, and FGN with H > 0.5 demonstrates  1/f , long memory or
     persistent behaviour, and the special  case. The case H = 0.5
     corresponds to the classical Gaussian  white noise. One can select
     from three different  methods. The first generator named '"beran"'
     uses the fast Fourier transform to generate the series based on 
     SPLUS code written originally by J. Beran [1994]. The second
     generator named '"durbin"' produces a FGN series by  using the
     Durbin-Levinson coefficients. The algorithm was reimplemented in
     pure S based on the C source code written by V. Teverovsky [199x].
     The third generator named  '"paxson"' was proposed by V. Paxson
     [199x], this approaximate method is a very fast and requires low
     storage.  However, the algorithm reveals some weakness in the
     method which was discussed by D.A. Rolls [2001]. 

     _Fractional ARIMA Processes:_ 
      The function 'farimaSim' is a generator for fractional ARIMA time
     series processes. A Gaussian FARIMA(0,d,0) series can be created,
     where _d _ is related to the the Hurst  exponent _H_ through
     _d=H-0.5_. This is a particular  case of the more general Gaussian
     FARIMA(p,d,q) process which  follows the same asymptotic relations
     for their autocovariance  and the spectral density as do the
     Gaussian FARIMA(0,d,0)  processes. Two different generators are
     implement in S. The  first named '"freq"' works in the frequence
     domain and  generates the series from the fast Fourier transform
     based on  SPLUS code written originally by J. Beran [1994]. The
     second  method creates the series in the time domain, therefore
     named  '"time"'. The algorithm was reimplemented in pure S based 
     on the Fortran source code from the R's 'fracdiff' package 
     originally written by C. Fraley [1991]. Details for the algorithm 
     are given in J Haslett and A.E. Raftery [1989].  

     *II: Functions to Model True Correlations and Spectrum:* 

     The functions 'fgnTrueacf' and 'farimaTrueacf' return the true
     covariances of an FGN and Gaussian FARIMA(0,d,0) time  series
     process. The functions 'fgnTruefft' and 'farimaTruefft' return the
     true fast Fourier transform of an FGN and Gaussian FARIMA(0,d,0) 
     time series process. The R functions are implemented from SPlus
     code written by J. Beran [1994]. 

     *III: Functions to Estimate the Hurst Exponent:* 

     These are 9 functions as described by M.S. Taqqu, V. Teverovsky,
     and W. Willinger [1995] to estimate the self similarity parameter 
     and/or the intensity of long-range dependence in a time series. 

     _Aggregated Variance Method:_ 
      The function 'aggvarFit' computes the Hurst exponent from  the
     variance of an aggregated FGN or FARIMA time series process.  The
     original time series is divided into blocks of size 'm'.  Then the
     sample variance within each block is computed. The slope 
     'beta=2*H-2' from the least square fit of the logarithm of  the
     sample variances versus the logarithm of the block sizes  provides
     an estimate for the Hurst exponent 'H'.  

     _Differenced Aggregated Variance Method:_ 
      To distinguish jumps and slowly decaying trends which are two
     types of non-stationary, from long-range dependence, the function 
     'diffvarFit' differences the sample variances of successive 
     blocks. The slope 'beta=2*H-2' from the least square fit of  the
     logarithm of the differenced sample variances versus the 
     logarithm of the block sizes provides an estimate for the Hurst 
     exponent 'H'.  

     _Aggregated Absolute Value/Moment Method:_ 
      The function 'absvalFit' computes the Hurst exponent from the
     moments 'moment=M' of absolute values of an aggregated  FGN or
     FARIMA time series process. The first moment 'M=1' coincides with
     the absolute value method, and the second moment  'M=2' with the
     aggregated variance method. Again, the slope  'beta=M*(H-1)' of
     the regression line of the logarithm of  the statistic versus the
     logarithm of the block sizes provides  an estimate for the Hurst
     exponent 'H'.  

     _Higuchi or Fractal Dimension Method:_ 
      The function 'higuchiFit' implements a technique which is  very
     similar to the absolute value method. Instead of blocks a  sliding
     window is used to compute the aggregated series. The  function
     involves the calculation the calculation of the length  of a path
     and, in principle, finding its fractal Dimension 'D'.  The slope
     'D=2-H' from the least square fit of the logarithm  of the
     expected path lengths versus the logarithm of the block  (window)
     sizes provides an estimate for the Hurst exponent 'H'.  

     _Peng or Variance of Residuals Method:_ 
      The function 'pengFit' uses the method described by peng. In
     Peng's variance of residuals method the series is also divided
     into blocks of size 'm'. Within each block the cumulated sums are
     computed up to 't' and a least-squares line  'a+b*t' is fitted to
     the cumulated sums. Then the sample  variance of the residuals is
     computed which is proportional to 'm^(2*H)'. The '"mean"' or
     '"median"' are computed over the blocks. The slope 'beta=2*H' from
     the  least square provides an estimate for the Hurst exponent 'H'.       

     _The R/S Method:_ 
      The function 'rsFit' implements the algorithm named  _rescaled
     range analysis_ which is dicussed for example  in detail by B.
     Mandelbrot and Wallis [199x], B. Mandelbrot [199x]  and B.
     Mandelbrot and M.S. Taqqu [199x]. 

     _The Periodogram Method:_ 
      The function 'perFit' estimates the Hurst exponent from the
     periodogram. In the finite variance case, the periodogram is an
     estimator of the spectral density of the time series. A series
     with long range dependence will show a spectral density with a
     lower law behavior in the frequency. Thus, we expect that a
     log-log plot of the periodogram versus frequency will display a
     straight line, and the slopw can be computed as _1-2H_. In
     practice one uses only the lowest 10% of the frequencies,  since
     the power law behavior holds only for frequencies close to zero.
     Varying this cut off may provide additional information. Plotting
     _H_ versus the cut off, one should select that cut off where the
     curve flattens out to estimate 'H'. This approach can be selected
     by the argument 'method="per"'. Alternatively we can select
     'method="cumper"'. In this case, instead of using the periodgram
     itself, the cmulative periodgram will be investigated. The slope
     of the double logarithmic fit is given by _2-2H_. More details can
     be found in the work of J. Geweke and S. Porter-Hudak [1983] and
     in Taqqu [?]. 

     _The Boxed or Modified Periodogram Method:_ 
      The function 'boxperFit' is a modification of the periodogram
     method. The algorithm devides the frequency axis into
     logarithmically equally spaced boxes and averages the periodogram
     values corresponding  to the frequencies inside the box. 

     _The Whittle Estimator:_ 
      The function 'whittleFit' performs also a periodogram analysis.
     The algorithm is based on the minimization of a likelihood
     function defined in the frequency domain. For FGN and
     FARIMA(0,d,0) processes the parameter _H_ or _d_ is the unknown
     parameter which minimizes the function. This approach also allows
     to compute confidence intervals. Unlike the previous eight
     estimators the Whittle estimator is not a graphical method, it
     just returns the values of _H_ or _d_ together with their
     confidence intervals. The function allows also to investigate
     FARIMA(p,d,q) models, then the parameter set to be optimized is
     enlarged by the AR and MA coefficients. It is worth to remark,
     that the empirical series is required to be a  Gaussian process
     and that the underlying form must be specified.  

     The original functions were written by V. Teverovsky and W.
     Willinger for SPLUS calling internal functions written in C. The
     software can  be found on M. Taqqu's home page:
      _http://math.bu.edu/people/murad/_  
      In addition the Whittle estimator uses SPlus functions written 
     by J. Beran. They can be found in the appendix of his book or on 
     the StatLib server:
      _http://lib.stat.cmu.edu/S/_ 
      Note, all nine R functions and internal utility functions are 
     reimplemented entirely in S. 

     *IV: Functions to perform a Wavelet Analysis:* 

     The function 'waveletFit' computes the Discrete Wavelet 
     Transform, averages the squares of the coefficients of the
     transform,  and then performs a linear regression on the logarithm
     of the  average, versus the log of the scale parameter of the
     transform.  The result should be directly proportional to 'H'
     providing an estimate for the Hurst exponent.  

     *V: Statistical Tests:* 

     The ...

_V_a_l_u_e:

     'fgnSim' and 'farimaSim' return a numeric vector of length  'n',
     the FGN or FARIMA series. 

     'fgnTrueacf(n, H)', 'fgnTruefft(n, H)',  'farimaTrueacf(n, H)',
     and 'farimaTruefft(n, H)' return the true covariance and the true
     spectrum of the FGN or FARIMA time series process. 

     '*Fit' returns an S4 object of class 'fHURST' with the following
     slots:

   @call: the function call.       

 @method: a character string with the selected method string.       

  @hurst: a list with at least one element, the Hurst exponent named
          'H'. Optional values may be the value of the fitted  slope
          'beta', or information from the fit.       

@parameters: a list with a varying number of elements describing the
          input parameters from the argument list.       

   @data: a list holding the input data. 

    @fit: a list object with all information of the fit. 

   @plot: a list object which holds information to create a plot of the
          fit. 

  @title: a character string with the name of the test. 

@description: a character string with a brief description of the test. 


     'waveletFit' 

_A_u_t_h_o_r(_s):

     V. Paxson, code as listed in the Appendix of his paper 1995, 
      J. Beran, ported by Maechler, code as listed in the Appendix of
     his Book, 
      M.S. Taqqu et al. for the S-Plus and C code concerned with the
     Hurst exponent, 
      C. Fraley for the FARIMA simulation code, 
      Guy Nason for the functions from the R package 'wavethresh', 
      Diethelm Wuertz for the Rmetrics R-port.

_R_e_f_e_r_e_n_c_e_s:

     Beran J. (1992); _Statistics for Long-Memory Processes_, Chapman
     and Hall, New York, 1994.

     Haslett J., Raftery A.E. (1989); _Space-Time Modelling with
     Long-Memory Dependence:  Assessing Ireland's Wind Power Resource_,
     Applied Statistics 38, pp. 1-50.

     Paxson V. (1995);  _Fast Approximation of Self-Similar Network
     Traffic_, Technical report, LBL-36750/UC-405, Berkeley, and
     Computer Communcation Review27, p.5-18, 1997.

     Rolls D.A. (2001); _Improved Fast Approximate Synthesis of
     Fractional Gaussian Noise_, Thesis, Department of Mathematics and
     Statistics, Queen's University at Kingston, Kingston, Ontario,
     Canada, 5 pages.

     Taqqu M., et al. _Hurst Exponent_, Several Preprints.

_E_x_a_m_p_l_e_s:

     ## SOURCE("fSeries.33A-LongRangeDependence")

     ## fgnSim -
        par(mfrow = c(3, 1), cex = 0.75)  
        # Beran's Method:
        plot(fgnSim(n = 200, H = 0.75), type = "l",  
          ylim = c(-3, 3), xlab = "time", ylab = "x(t)", main = "Beran")
        # Durbin's Method:
        plot(fgnSim(n = 200, H = 0.75, method = "durbin"), type = "l",
          ylim = c(-3, 3), xlab = "time", ylab = "x(t)", main = "Durbin")
        # Paxson's Method:
        plot(fgnSim(n = 200, H = 0.75, method = "paxson"), type = "l",
          ylim = c(-3, 3), xlab = "time", ylab = "x(t)", main = "Paxson")

