This function estimates the information dimension by forming a delay embedding of a time series, calculating related statistical 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 information dimension for the data.
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a vector containing a uniformly-sampled real-valued time series.
the maximal embedding dimension. Default:
let p=k/N for 0 < p <= 1 be the mass density where N
is the number of points in the embedding and k is the number of neighbors found
near an arbitrary reference point in the embedding. The
the metric used to define the distance between
points in the embedding. Choices are limited to
the number of points to create in developing the density vector.
For a given reference point in the phase space, the density is defined by the
relation p=k / N where k is the number of neighbors in the phase space and N
is the total number of points in the embedding. To obtain the informaiton dimension
statistics, the density is varied logarithmically from 1/N to 1.0.
the number of reference points to use in forming the information dimension
statistic. This argument directly specifies the number of equi-dense neighborhoods to average in
forming the average neighborhood radius statistic. As with the
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:
the time delay between coordinates. Default: the decorrelation time of the autocorrelation function.
The information dimension (D1) is one of an infinite number of fractal dimensions of a chaotic system. For generalized fractal dimension estimations, correlation integral moments are determined as an average of the contents of neighbohoods in the phase space of equal radius eps. Using this approach. the information dimension for a given embedding dimension E is estimated via D1(E)=< ln(p) > / ln(eps) in the limit as eps approaches zero, where eps is the radius of an E-dimensional hypersphere, p is the density (also known as the mass fraction), and < ln(p) > is the average Shannon information needed to specify an arbitrary point in the phase space with accuracy eps.
Alternatively, the neighborhoods can be constructed with variable radii but with constant density. The scaling behavior of the average radii of these neighborhoods as a function of density is then used to estimate the fractal dimensions. In this function, we use this constant density approach to calculate the statistics for estimating the information dimension.
For single variable time series, the phase space is approximated with a delay embedding and D1(E) is thus estimated over statistics gathered for dimensions 1,...,E. For chaotic systems, these statistics will ‘saturate’ at a finite embedding dimension, revealing both the (estimated) information dimension and an appropriate embedding dimension for the system. A linear regression scheme should be to estimate the D1(E) using the statistics returned by this function.
an object of class
plots an extended data analysis plot, which graphically summarizes the process of obtaining a information dimension estimate. A time history, phase plane embeddding, information dimension curves, and the slopes of information dimension curves as a function of scale are plotted.
plots the information dimension curves on a log-log scale. The following options may be used to adjust the plot components:
Character string denoting the type of data to be plotted. The
"stat"option plots the information dimension curves while the
"dstat"option plots a 3-point estimate of the derivatives of the information dimension curves. The
"slope"option plots the estimated slope of the information dimension curves as a function of embedding dimension. Default:
Logical flag. If
TRUE, a regression line is overlaid for each curve. Default:
Logical flag. If
TRUE, a grid is overlaid on the plot. Default:
Logical flag. If
TRUE, a legend of the estimated slopes as a function of embedding dimension is displayed. Default:
Additional plot arguments (set internally by the
prints a qualitiative summary of the results.
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.
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## calculate the information dimension estimates ## for chaotic beam data using a delay ## embedding for dimensions 1 through 10 beam.d1 <- infoDim(beamchaos, dim=10) ## print a summary of the results print(beam.d1) ## plot the information dimension curves without ## regression lines plot(beam.d1, fit=FALSE, legend=FALSE) ## plot an extended data analysis plot eda.plot(beam.d1)
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