# NonLinStatistics: Chaotic Time Series Statistics In fNonlinear: Rmetrics - Nonlinear and Chaotic Time Series Modelling

## Description

A collection and description of functions to investigate the chaotic behavior of time series processes.

Functions to Analyse Chaotic Time Series:

 `mutualPlot` Returns mutual information, `falsennPlot` returns false nearest neigbours, `recurrencePlot` returns a recurrence plot, `separationPlot` returns a space-time separation plot, `lyapunovPlot` computes maximum lyapunov exponent.

## Usage

 ```1 2 3 4 5 6``` ``` mutualPlot(x, partitions = 16, lag.max = 20, doplot = TRUE, ...) falsennPlot(x, m, d, t, rt = 10, eps = NULL, doplot = TRUE, ...) recurrencePlot(x, m, d, end.time, eps, nt = 10, doplot = TRUE, ...) separationPlot(x, m, d, mdt, idt = 1, doplot = TRUE, ...) lyapunovPlot(x, m, d, t, ref, s, eps, k = 1, doplot = TRUE, ...) ```

## Arguments

 `d` an integer value setting the value of the time delay. `eps` [falsennPlot] - 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 `eps=sd(x)/10`. [lyapunovPlot] - the radius where to find nearest neighbours. [recurrencePlot] - the neighbourhood threshold. `doplot` a logical flag. Should a plot be displayed? `end.time` [recurrencePlot] - ending time as number of observations. `idt` [separationPlot] - an integer value setting the number of observation steps in each iterations. By default 1. `k` [lyapunovPlot] - an integer setting th enumber of considered neighbours. By default 1. `lag.max` [mutualPlot] - an integer value setting the number of maximum lags, by default 20. `m` [*Plot] - an integer value setting the value of the maximum embedding dimension. `mdt` [separationPlot] - an integer value setting the number of iterations. `nt` [recurrencePlot] - observations in each step which will be plotted, by default 10. Increasing `nt` reduces number of points plotted which is usefule especially with highly sampled data. `rt` [falsennPlot] - an integer value setting the value for the escape factor. By default 10. `partitions` [mutualPlot] - an integer value setting the number of bins, by default 16. `ref` [lyapunovPlot] - the number of points to take into account. `s` [lyapunovPlot] - the iterations along which follow the neighbours of each point. `t` [*Plot] - an integer value setting the value for the Theiler window. `x` [*Plot] - a numeric vector, or an object either of class 'ts' or of class 'timeSeries'. `...` arguments to be passed.

## Details

Phase Space Representation:

The function `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.
The function `falsennPlot` uses the Method of false nearest neighbours to help deciding the optimal embedding dimension.

Non-Stationarity:

The funcdtion `recurrencePlot` creates a recurrence plot as proposed by Eckmann et al. [1987].
The function `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.

Lyapunov Exponents:

The function `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.

Dimensions and Entropies:

The function `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 `d2`, which takes roughly the same time, but computes the correlation sum for multiple length scales and embedding dimensions at once.
The function `d2` computes the sample correlation integral over given length scales `neps` for embedding dimensions `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.

## Author(s)

Diethelm Wuertz for the Rmetrics R-port.

## References

Brock, W.A., Dechert W.D., Sheinkman J.A. (1987); 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), Recurrence plots of dynamical systems, Europhys. Letters 4, 973.

Hegger R., Kantz H., Schreiber T. (1999); Practical implementation of nonlinear time series methods: The TISEAN package, CHAOS 9, 413–435.

Kennel M.B., Brown R., Abarbanel H.D.I. (1992); 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); A practical method for calculating largest Lyapunov exponents from small data sets, Physica D 65, 117.

`RandomInnovations`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```## 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) ```

### Example output

```Loading required package: timeDate

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