corrDim               package:fractal               R Documentation

_C_o_r_r_e_l_a_t_i_o_n _d_i_m_e_n_s_i_o_n

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

     Estimates the correlation dimension by forming a delay embedding
     of a time series, calculating correlation summation curves (one
     per embedding dimension), and subsequently fitting the slopes of
     these curves on a log-log scale using a robust linear regression
     model. If the slopes converge at a given embedding dimension E,
     then E is the correct embedding dimension and the (convergent)
     slope value is an estimate of the correlation dimension for the
     data.

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

     corrDim(x, dimension=5,
         tlag=timeLag(x, method="acfdecor"), olag=0, resolution=2)

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

       x: a vector containing a uniformly-sampled real-valued time
          series or a matrix containing an embedding with each column
          representing a different coordinate. If the latter, the
          'dimension' input is set to the number of columns and the
          'tlag' input is ignored.

dimension: the maximal embedding dimension. Default: '5'.

    olag: the number of points along the trajectory of the current
          point that must be exceeded in order for another point in the
          phase space to be considered a neighbor candidate. This
          argument is used to help attenuate temporal correlation in
          the the embedding which can lead to spuriously low
          correlation dimension estimates. The orbital lag must be
          positive or zero. Default: 'length(x)/10' or '500', whichever
          is smaller.

resolution: an integer representing the spatial resolution factor. A
          value of P increases the number of effective scales by a
          factor of P at a cost of raising the L-infinity norm to the
          Pth power. For example, setting the resolution to 2 will
          double the number of scales while imposing and additional
          multiplication operation. The resolution must exceed unity.
          Default: '2'.

    tlag: the time delay between coordinates. Default: 'timeLag(x,
          method="acfdecor")', the decorrelation time of the
          autocorrelation function.

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

     To estimate the correlation dimension, correlation summation
     curves must be generated and subsequently fit with a robust linear
     regression model to obtain the slopes of these curves on a log-log
     plot. The dimension at which these slope estimates (appear to)
     converge reveals the proper embedding dimension for the data and
     the slope at this (and higher) embedding dimensions is an estimate
     of the correlation dimension. The function used to fit the
     correlation summation curves is 'lmsreg' which fits a robust
     linear model to the data using the method of least median of
     squares regression. See the on-line help documentation for help on
     the 'lmsreg' function: in R, 'lmsreg' is found in the 'MASS'
     package while in S-PLUS it is indigenous and appears in the
     'splus' database.

     The correlation summation at scale eps for a given embedding
     dimension is defined as

 C2(eps)=2 / (N - gamma)/(N - gamma - 1) * sum{i=1:N}sum{j=i+1+gamma:N} H(eps - || Xi - Xj ||),

     where H is the Heavyside function

                H(x)=0 if x <= 0 and H(x)=1 for x > 1,


     and Xi is the ith point of a collection of 'N' points in the phase
     space. The parameter gamma is the orbital lag.

     The algorithm used to calculate the correlation summation is made
     computationally efficient by using:

     _1 The L-infinity norm to calculate the distance between neighbors
          in the phase space as opposed to (say) the L2 norm which
          involves taking computationally intense square root and power
          of two operations. The L-infinity norm of the distance
          between two points in the phase space is the absolute value
          of the maximal difference between any of the points'
          respective coordinates, i.e. if X ={z1, z2, z3} then ||X||
          sub infinity=max{i}(zi).

     _2 Bitwise masking and shift operations to reveal the radix-2
          exponent of the L-infinity norm. This direct means of
          obtaining the exponent immediately yields the associated
          scale of the distance between neighbors in the phase space
          while avoiding costly log operations. The bitwise mask and
          shift factors are based on the IEEE standard 754 for binary
          floating-point arithmetic. Initial tests are performed in the
          code to verify that the current machine follows this
          standard.

     _3 a computationally efficient routine to calculate the resulting
          value of a float raised to a positive integer power.
          Specifically, the L-infinity norm is raised to an integer
          power ('p') to effectively increase the spatial resolution by
          a factor of 'p'. .in -5

          The correlation summation curves C2(E,eps) where 'E' is the
          embedding dimension and eps is the scale, the correlation
          dimension curves D2(E,eps) can be calculated by

           D2(E,eps)=log2 (C2(E,2*eps) / C2(E,eps / 2)) / 2

          This formulation is used to help suppress numerical
          instabilities that are present in other numerical derivative
          schemes such as a first order difference.

          As a caveat to the user, the slope estimates of the
          correlation summation curves will typically display a fair
          amount of variability and the range of scales over which the
          slopes are approximately linear may be small. Inasmuch, the
          correlation dimension estimate should always be interpretted
          as a subjective summary statistic, even when the original
          times series is representative of a truly noise-free chaotic
          response.

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

     an object of class 'chaoticInvariant'.

_S_3 _M_E_T_H_O_D_S:


     _e_d_a._p_l_o_t plots an extended data analysis plot, which graphically
          summarizes the process of obtaining a correlation dimension
          estimate. A time history, phase plane embeddding, correlation
          summation curves, and the slopes of correlation summation
          curves as a function of scale are plotted.

     _p_l_o_t plots the correlation summation curves on a log-log scale.
          The following options may be used to adjust the plot
          components:

          _t_y_p_e Character string denoting the type of data to be
               plotted. The '"stat"' option plots the correlation
               summation curves while the '"dstat"' option plots a
               3-point estimate of the derivatives of the correlation
               summation curves. The '"slope"' option plots the
               estimated slope of the correlation summation curves as a
               function of embedding dimension. Default: '"stat"'.

          _f_i_t Logical flag. If 'TRUE', a regression line is overlaid
               for each curve. Default: 'TRUE'.

          _g_r_i_d Logical flag. If 'TRUE', a grid is overlaid on the plot.
               Default: 'TRUE'.

          _l_e_g_e_n_d Logical flag. If 'TRUE', a legend of the estimated
               slopes as a function of embedding dimension is
               displayed. Default: 'TRUE'.

          ... Additional plot arguments (set internally by the 'par'
               function). .in -5


          _p_r_i_n_t prints a qualitiative summary of the results.

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

     Peter Grassberger and Itamar Procaccia (1983), Measuring the
     strangeness of strange attractors, _Physica D_, *9*, 189-208.

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

     Peter Grassberger and Itamar Procaccia (1983), Characterization of
     strange attractors, _Physical Review Letters_, *50*(5), 346-349.

     Rousseeuw, P. J. (1984). Least median of squares regression.
     _Journal of the American Statistical Association_, *79*, 871-88.

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

     'infoDim', 'embedSeries', 'timeLag', 'chaoticInvariant',
     'lyapunov', 'poincareMap', 'spaceTime', 'findNeighbors',
     'determinism'.

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

     ## calculate the correlation dimension estimates 
     ## for chaotic beam data using a delay 
     ## embedding for dimensions 1 through 10, a 
     ## orbital lag of 10, and a spatial resolution 
     ## of 4. 
     beam.d2 <- corrDim(beamchaos, olag=10, dim=10, res=4)

     ## print a summary of the results 
     print(beam.d2)

     ## plot the correlation summation curves 
     plot(beam.d2, fit=FALSE, legend=FALSE)

     ## plot an extended data analysis plot 
     eda.plot(beam.d2)

