# Information dimension

### Description

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.

### Usage

1 2 3 4 |

### Arguments

`x` |
a vector containing a uniformly-sampled real-valued time series. |

`dimension` |
the maximal embedding dimension. Default: |

`max.neighbors` |
let |

`metric` |
the metric used to define the distance between
points in the embedding. Choices are limited to |

`n.density` |
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 |

`n.reference` |
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 |

`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: |

`tlag` |
the time delay between coordinates. Default: the decorrelation time of the autocorrelation function. |

### Details

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.

### Value

an object of class `chaoticInvariant`

.

### S3 METHODS

- eda.plot
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.

- plot
plots the information dimension curves on a log-log scale. The following options may be used to adjust the plot components:

- type
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:`"stat"`

.- fit
Logical flag. If

`TRUE`

, a regression line is overlaid for each curve. Default:`TRUE`

.- grid
Logical flag. If

`TRUE`

, a grid is overlaid on the plot. Default:`TRUE`

.- legend
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).

prints a qualitiative summary of the results.

### References

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.

### See Also

`corrDim`

, `embedSeries`

, `timeLag`

, `chaoticInvariant`

, `lyapunov`

, `poincareMap`

, `spaceTime`

, `findNeighbors`

, `determinism`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
## 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)
``` |