lmModel               package:fractal               R Documentation

_C_o_n_s_t_r_u_c_t_o_r _f_u_n_c_t_i_o_n _f_o_r _o_b_j_e_c_t_s _o_f _c_l_a_s_s "_l_m_M_o_d_e_l"

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

     Packs the parameters defining a specfied stochastic fractal time
     series model into a list an returns the result.

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

     lmModel(model, variance.=1.0, delta=0.45,
         alpha=-0.9, HG=0.95, HB=0.95,
         innovations.var=NULL, Cs=NULL,
         bterms=10, dterms=10, M=100)

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

   model: a character string defining the model type. Choices are

          '"_p_p_l"' Pure power law (PPL) process. A process {X(t)} is a
               PPL process if its SDF is given by

                       S(X,f) = Cs * |f|^alpha

               where Cs > 0. The innovations variance for this process
               is given by Cs * exp(-alpha*(log(2)+1)) (this is the
               variance of the best linear predictor of the process
               given its infinite past).

          '"_f_d_p"' Fractionally differenced (FD) process. A process
               {X(t)} is a FD process if its SDF is given by

    S(X,f) = sigma^2 / (2 * sin(pi*f) )^(2 * delta) for |f| <= 1/2

               where sigma^2 is the innovations variance, and delta is
               the FD parameter. Thus, an FD model is completely
               defined by the innovations variance and FD parameter.

          '"_f_g_n"' Fractional Gaussian noise (FGN) process. An FGN
               process {X(t)} is a _stationary_ Gaussian process if its
               ACVF is given by

 s(X,k) = var{X}/2 *(|k + 1|^(2*Hg) - 2*|k|^(2*Hg) + |k + 1|^(2*Hg))

               where var{X} > 0 is the variance of the process, while
               Hg is the so-called Hurst coefficient. The coefficient
               Hg is sometimes called the self-similarity parameter for
               a FGN process and is usually designated in the
               literature as simply H.

          '"_d_f_b_m"' Discrete Fractional Brownian Motion. i.e.,
               regularly-spaced samples from a FBM process that is
               defined over the entire real axis. .in -5 

      Cs: pure power law constant. If supplied, this argument is used
          to compute 'variance' and 'innovations.var'. If not supplied
          and 'innovations.var' is supplied, then 'Cs' and 'variance'
          are determined from the 'innovations.var'. Default: 'NULL'.

      HB: the Hurst coefficient for a DFBM process. Default: '0.95'.

      HG: the Hurst coefficient for an FGN process. Default: '0.95'.

       M: sets the number of terms used in the Euler-Maclaurin
          summation for calculating the SDF of an FGN process and DFBM
          process. The default value should be adequate for all values
          of the Hurst coefficient. Default: '100'.

   alpha: power law exponent for a PPL model. Default: '-0.9'.

  bterms: an integer used to control the number of primary terms
          cumulatively summed in computing an ACVS for a PPL process.
          Default: '10'.

   delta: the FD parameter. Default: '0.45'.

  dterms: an integer used to control the number of secondary terms
          cumulatively summed in computing an ACVS for a PPL process.
          Default: '10'.

innovations.var: innovations variance for an FD or PPL model. If
          supplied, this argument is used to compute 'variance' and,
          for a PPL model, 'Cs'. If not supplied and 'Cs' is supplied
          for a PPL model, then 'Cs' determines 'innovations.var'. If
          not supplied and 'Cs' is also not supplied for a PPL model or
          if not supplied for an FD model, then 'variance' determines
          'innovations.var'. Default: 'NULL'.

variance.: the process variance with a default of unity. If 'cs' or
          'innovations.var' is specified, this parameter is set in
          agreement with those. If the process is nonstationary but has
          stationary differences, i.e., incrementally stationary, then
          the process variance is taken to be the variance of the
          stationary process that is formed by appropriately
          differencing the nonstationary process.

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

     an object of class 'lmModel' containing a list of model
     parameters.

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

     D. Percival and A. Walden (2000), _Wavelet Methods for Time Series
     Analysis_, Cambridge University Press, Chapter 7.

     J. Beran (1994), _Statistics for Long-Memory Processes_, Chapman
     and Hall, Chapter 2.

     D. Percival and A. Walden (1993), _Spectral Analysis for Physical
     Applications_, Cambridge University Press, 1993, Chapter 9.

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

     'lmACF', 'lmSDF', 'lmSimulate', 'lmConvert', 'lmConfidence',
     'FDWhittle'.

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

     lmModel("ppl", alpha=-2.0)
     lmModel("fdp", delta=0.45, innov=1.3)
     lmModel("fgn", HG=0.98)
     lmModel("dfbm", HB=0.35)

