poincareMap             package:fractal             R Documentation

_C_r_e_a_t_e _a _P_o_i_n_c_a_r_e _m_a_p

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

     Create a map using the extrema of a scalar time series.

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

     poincareMap(x, extrema="min", denoise=FALSE)

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

       x: a vector holding a scalar time series.

 denoise: a logical value. If 'TRUE', the data is first denoised via
          waveshrink prior to analysis. Default: 'FALSE'.

 extrema: the type of extrema desired. May be "min" for minima, "max"
          for maxima, or "all" for both maxima and minima. Default:
          '"min"'.

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

     This function finds the extrema of a scalar time series to form a
     map. The time series is assumed to be a uniform sampling of s(t),
     where s(t) is a (possibly noisy) measurement from a deterministic
     non-linear system. It is known that s'(t), s''(t), ... are
     legitimate coordinate vectors in the phase space. Hence the
     hyperplane given by s'(t)=0 may be used as a Poincare surface of
     section. The intersections with this plane are exactly the extrema
     of the time series. The time series minima (or maxima) are the
     interesections in a given direction and form a map that may be
     used to estimate invariants, e.g., correlation dimension and
     Lyapunov exponents, of the underlying non-linear system.

     The algorithm used to create a Poincare map is as follows.

     _1 The first and second derivatives of the resulting series are
          approximated via the continuous wavelet transform (CWT) using
          the first derivative of a Gaussian as a mother wavelet filter
          (see references for details).

     _2 The locations of the local extrema are then estimated using the
          standard first and second derivative tests on the CWT
          coefficients at a single and appropriate scale (an
          appropriate scale is one that is large enough to smooth out
          noisy components but not so large as to the oversmooth the
          data).

     _3 The extrema locations are then fit with a quadratic
          interpolation scheme to estimate the amplitude of the extrema
          using the original time series. .in -5 

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

     a list where the first element ('location') is a vector containing
     the temporal locations of the extrema values, with respect to
     sample numbers 1,...N, where N is the length of the original time
     series. The second element ('amplitude') is a vector containing
     the extrema amplitudes.

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

     Holger Kantz and Thomas Schreiber, _Nonlinear Time Series
     Analysis_, Cambridge University Press, 1997.

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

     'embedSeries', 'corrDim', 'infoDim'.

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

     ## Using the third coordinate (\eqn{z} state) of a 
     ## chaotic Lorenz system, form a discrete map 
     ## using the series maxima. Embed the resulting 
     ## extrema in a 2-dimensional delay embedding 
     ## (with delay=1 for a map). The resulting plot 
     ## reveals a tent map structure common to 
     ## Poincare sections of chaotic flows. 
     z <- poincareMap(lorenz[,3], extrema="max")
     z <- embedSeries(z$amplitude, tlag=1, dimension=2)
     plot(z, pch=1, cex=1)

