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\name{NonLinStatistics}
\alias{NonLinStatistics}
\alias{mutualPlot}
\alias{falsennPlot}
\alias{recurrencePlot}
\alias{separationPlot}
\alias{lyapunovPlot}
\title{Chaotic Time Series Statistics}
\description{
A collection and description of functions to
investigate the chaotic behavior of time series
processes.
\cr
Functions to Analyse Chaotic Time Series:
\tabular{ll}{
\code{mutualPlot} \tab Returns mutual information, \cr
\code{falsennPlot} \tab returns false nearest neigbours, \cr
\code{recurrencePlot} \tab returns a recurrence plot, \cr
\code{separationPlot} \tab returns a space-time separation plot, \cr
\code{lyapunovPlot} \tab computes maximum lyapunov exponent. }
}
\usage{
mutualPlot(x, partitions = 16, lag.max = 20, doplot = TRUE, \dots)
falsennPlot(x, m, d, t, rt = 10, eps = NULL, doplot = TRUE, \dots)
recurrencePlot(x, m, d, end.time, eps, nt = 10, doplot = TRUE, \dots)
separationPlot(x, m, d, mdt, idt = 1, doplot = TRUE, \dots)
lyapunovPlot(x, m, d, t, ref, s, eps, k = 1, doplot = TRUE, \dots)
}
\arguments{
\item{d}{
an integer value setting the value of the time delay.
}
\item{eps}{
[falsennPlot] - \cr
a numeric value setting the value of the neighbour diameter.
If NULL, which is the default value, then the value will be
automatically setted to \code{eps=sd(x)/10}.
\cr
[lyapunovPlot] - \cr
the radius where to find nearest neighbours.
\cr
[recurrencePlot] - \cr
the neighbourhood threshold.
}
\item{doplot}{
a logical flag. Should a plot be displayed?
}
\item{end.time}{
[recurrencePlot] - \cr
ending time as number of observations.
}
\item{idt}{
[separationPlot] - \cr
an integer value setting the number of observation steps in
each iterations. By default 1.
}
\item{k}{
[lyapunovPlot] - \cr
an integer setting th enumber of considered neighbours.
By default 1.
}
\item{lag.max}{
[mutualPlot] - \cr
an integer value setting the number of maximum lags, by
default 20.
}
\item{m}{
[*Plot] - \cr
an integer value setting the value of the maximum embedding
dimension.
}
\item{mdt}{
[separationPlot] - \cr
an integer value setting the number of iterations.
}
\item{nt}{
[recurrencePlot] - \cr
observations in each step which will be plotted, by default 10.
Increasing \code{nt} reduces number of points plotted which
is usefule especially with highly sampled data.
}
\item{rt}{
[falsennPlot] - \cr
an integer value setting the value for the escape factor. By
default 10.
}
\item{partitions}{
[mutualPlot] - \cr
an integer value setting the number of bins, by default 16.
}
\item{ref}{
[lyapunovPlot] - \cr
the number of points to take into account.
}
\item{s}{
[lyapunovPlot] - \cr
the iterations along which follow the neighbours of each point.
}
\item{t}{
[*Plot] - \cr
an integer value setting the value for the Theiler window.
}
\item{x}{
[*Plot] - \cr
a numeric vector, or an object either of class 'ts' or
of class 'timeSeries'.
}
\item{\dots}{
arguments to be passed.
}
}
\details{
\bold{Phase Space Representation:}
\cr\cr
The function \code{mutualPlot} estimates and plots the mutual
information index of a given time series for a specified number
of lags. The joint probability distribution function is estimated
with a simple bi-dimensional density histogram.
\cr
The function \code{falsennPlot} uses the Method of false nearest
neighbours to help deciding the optimal embedding dimension.
\cr
\bold{Non-Stationarity:}
\cr\cr
The funcdtion \code{recurrencePlot} creates a recurrence plot as
proposed by Eckmann et al. [1987].
\cr
The function \code{separationPlot} creates a space-time separation
plot qs introduced by Provenzale et al. [1992]. It plots the
probability that two points in the reconstructed phase-space have
distance smaller than epsilon in function of epsilon and of the
time between the points, as iso-lines at levels 10, 20, ..., 100
percent levels. The plot can be used to decide the Theiler time
window.
\cr
\bold{Lyapunov Exponents:}
\cr\cr
The function \code{lyapunovPlot} evaluates and plots the largest
Lyapunov exponent of a dynamic system from a univariate time series.
The estimate of the Lyapunov exponent uses the algorithm of Kantz.
In addition, the function computes the regression coefficients of
a user specified segment of the sequence given as input.
\cr
\bold{Dimensions and Entropies:}
\cr\cr
The function \code{C2} computes the sample correlation integral on
the provided time series for the specified length scale and
Theiler window. It uses a naiv algorithm: simply returns the
fraction of points pairs nearer than eps. It is prefarable to use
the function \code{d2}, which takes roughly the same time, but
computes the correlation sum for multiple length scales and
embedding dimensions at once.
\cr
The function \code{d2} computes the sample correlation integral
over given length scales \code{neps} for embedding dimensions
\code{1:m} for a given Theiler window. The slope of the linear
segment in the log-log plot gives an estimate of the correlation
dimension.
}
\references{
Brock, W.A., Dechert W.D., Sheinkman J.A. (1987);
\emph{A Test of Independence Based on the Correlation
Dimension},
SSRI no. 8702, Department of Economics, University of
Wisconsin, Madison.
Eckmann J.P., Oliffson Kamphorst S., Ruelle D. (1987),
\emph{Recurrence plots of dynamical systems},
Europhys. Letters 4, 973.
Hegger R., Kantz H., Schreiber T. (1999);
\emph{Practical implementation of nonlinear time series
methods: The TISEAN package},
CHAOS 9, 413--435.
Kennel M.B., Brown R., Abarbanel H.D.I. (1992);
\emph{Determining embedding dimension for phase-space
reconstruction using a geometrical construction},
Phys. Rev. A45, 3403.
Rosenstein M.T., Collins J.J., De Luca C.J. (1993);
\emph{A practical method for calculating largest Lyapunov
exponents from small data sets},
Physica D 65, 117.
}
\seealso{
\code{RandomInnovations}.
}
\author{
Diethelm Wuertz for the Rmetrics \R-port.
}
\examples{
## mutualPlot -
mutualPlot(logisticSim(1000))
## recurrencePlot -
lorentz = lorentzSim(
times = seq(0, 40, by = 0.01),
parms = c(sigma = 16, r = 45.92, b = 4),
start = c(-14, -13, 47),
doplot = FALSE)
recurrencePlot(lorentz[, 2], m = 3, d = 2, end.time = 800, eps = 3,
nt = 5, pch = '.', cex = 2)
}
\keyword{models}
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