infDim: Information dimension
In nonlinearAnalysis: R package for nonlinear time series analysis

Description Usage Arguments Details Value Author(s) References See Also Examples

Functions for estimating the information dimension of a dynamical system from 1-dimensional time series using Takens' vectors

  infDim(time.series, embedding.dim = 2, time.lag = 1,
    min.fixed.mass, max.fixed.mass,
    number.fixed.mass.points = 10, radius,
    increasing.radius.factor = sqrt(2),
    number.boxes = NULL, number.reference.vectors,
    theiler.window = 1, kMax = 100, do.plot = TRUE)

  getFixedMass(x)

  getLogRadius(x)

  ## S3 method for class 'infDim'
 estimate(x, regression.range = NULL,
    do.plot = TRUE, ...)

  ## S3 method for class 'infDim'
 plot(x, ...)

`time.series`	The original time series from which the information dimension will be estimated.
`embedding.dim`	Integer denoting the dimension in which we shall embed the time.series (see `buildTakens`).
`time.lag`	Integer denoting the number of time steps that will be use to construct the Takens' vectors (see `buildTakens`).
`min.fixed.mass`	Minimum percentage of the total points that the algorithm shall use for the estimation.
`max.fixed.mass`	Maximum percentage of the total points that the algorithm shall use for the estimation.
`number.fixed.mass.points`	The number of different fixed mass fractions between min.fixed.mass and max.fixed.mass that the algorithm will use for estimation.
`radius`	Initial radius for searching neighbour points in the phase space. Ideally, it should be small enough so that the fixed mass contained in this radius is slightly greater than the min.fixed.mass. However, whereas the radius is not too large (so that the performance decreases) the choice is not critical.
`increasing.radius.factor`	Numeric value. If no enough neighbours are found within radius, the radius is increased by a factor increasing.radius.factor until succesful. Default: sqrt(2) = 1.414214.
`number.boxes`	Number of boxes that will be used in the box assisted algorithm (see neighbourSearch).
`number.reference.vectors`	Number of reference points that the routine will try to use, saving computation time.
`theiler.window`	Integer denoting the Theiler window: Two Takens' vectors must be separated by more than theiler.window time steps in order to be considered neighbours. By using a Theiler window, we exclude temporally correlated vectors from our estimations.
`kMax`	Maximum number of neighbours used for achieving p with all the points from the time series (see Details). Default: 100.
`do.plot`	Logical value. If TRUE (default value), a plot of the correlation sum is shown.
`x`	A infDim object.
`regression.range`	Vector with 2 components denoting the range where the function will perform linear regression.
`...`	Additional parameters.

The information dimension is a particular case of the generalized correlation dimension when setting the order q = 1. It is possible to demonstrate that the information dimension D1 may be defined as: D1=lim{r->0} <ln p(r)>/ln(r). Here, p(r) is the probability of finding a neighbour in a neighbourhood of size r and <> is the mean value. Thus, the information dimension specifies how the average Shannon information scales with the radius r.

In order to estimate D1, the algorithm looks for the scaling behaviour of the the average radius that contains a given portion (a "fixed-mass") of the total points in the phase space. By performing a linear regression of ln p Vs ln <r> (being p the fixed-mass of the total points), an estimate of D1 is obtained.

The algorithm also introduces a variation of p for achieving a better performance: for small values of p, all the points in the time series (N) are considered for obtaining p=n/N. Above a maximum number of neighbours kMax, the algorithm obtains p by decreasing the number of points considerd from the time series M<N. Thus p = kMax/M.

Even with these improvements, the calculations for the information dimension are heavier than those needed for the correlation dimension. Thus, this method only explores one embedding dimension. However, the user should run the method for different embedding dimensions for checking if D1 saturates.

A infDim object that consist of a list with two components: log.radius and fixed.mass. log.radius contains the average log10(radius) in which the fixed.mass can be found.

The getFixedMass function returns the fixed mass vector used in the information dimension algorithm

The getLogRadius function returns average log(radius) computed on the information dimension algorithm

The 'estimate' function estimates the information dimension of the 'infDim' object by performing a linear regression over the fixed mass' range specified in 'regression.range'. If do.plot is TRUE, a graphic of the regression over the data is shown.

The 'plot' function shows two graphics of the information dimension estimate: a graphic of <log10(radius)> Vs fixed mass and a graphic of the local slopes of the information dimension Vs the fixed mass, both in a semi-log scale.

Constantino A. Garcia

H. Kantz and T. Schreiber: Nonlinear Time series Analysis (Cambridge university press)

corrDim.

## Not run: 
s = sinaiMap(a=0.3,n.sample=5000,start=c(0.23489,0.8923),do.plot=FALSE)
inf.dim = infDim(time.series = s$x, embedding.dim = 2, time.lag = 1,
                       min.fixed.mass=0.01, max.fixed.mass=0.03,
                       number.fixed.mass.points=1000, radius =0.1,
                       increasing.radius.factor = sqrt(2), number.boxes=100,
                       number.reference.vectors=100, theiler.window = 200,
                       kMax = 100,do.plot=FALSE)
estimate(inf.dim)
## End(Not run)