ChaoticTimeSeries {fSeries}R Documentation

Chaotic Time Series Modelling

Description

A collection and description of functions to investigate the chaotic behavior of time series processes. Included are functions to simulate different types of chaotic time series maps.

Chaotic Time Series Maps:

tentSim Simulates data from the Tent Map,
henonSim simulates data from the Henon Map,
ikedaSim simulates data from the Ikeda Map,
logisticSim simulates data from the Logistic Map,
lorentzSim simulates data from the Lorentz Map,
roesslerSim simulates data from the Roessler Map.

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

tentSim(n = 1000, n.skip = 100, parms = c(a = 2), start = runif(1), 
    doplot = FALSE)
henonSim(n = 1000, n.skip = 100, parms = c(a = 1.4, b = 0.3), 
    start = runif(2), doplot = FALSE)
ikedaSim(n = 1000, n.skip = 100, parms = c(a = 0.4, b = 6.0, c = 0.9), 
    start = runif(2), doplot = FALSE)
logisticSim(n = 1000, n.skip = 100, parms = c(r = 4), start = runif(1), 
    doplot = FALSE)
lorentzSim(times = seq(0, 40, by = 0.01), parms = c(sigma = 16, r = 45.92, 
    b = 4), start = c(-14, -13, 47), doplot = TRUE, ...)
roesslerSim(times = seq(0, 100, by = 0.01), parms = c(a = 0.2, b = 0.2, c = 8.0),
    start = c(-1.894, -9.920, 0.0250), doplot = TRUE, ...)
    
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 [*Plot] -
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? [*Plot] -
By default TRUE.
[*Sim] -
By default TRUE.
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.
n, n.skip [henonSim][ikedaSim][logisticSim] -
the number of chaotic time series points to be generated and the number of initial values to be skipped from the series.
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.
parms the named parameter vector characterizing the chaotic map.
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.
start the vector of start values to initiate the chaotic map.
t [*Plot] -
an integer value setting the value for the Theiler window.
times [lorentzSim][roesslerSim] -
the sequence of time series points at which to generate the map.
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.

Value

[*Sim] -
All functions return invisible a vector of time series data.

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.

See Also

RandomInnovations.

Examples

## SOURCE("fSeries.035A-ChaoticTimeSeries")

## bdsTest -
   # iid Time Series:
   par(mfrow = c(3, 1))
   x = rnorm(100)
   plot(x, type = "l", main = "iid Time Series")
   bdsTest(x, m = 3)
   # Non Identically Distributed Time Series:
   x = c(rnorm(50), runif(50))
   plot(x, type = "l", main = "Non-iid Time Series")
   bdsTest(x, m = 3)  
   # Non Independent Innovations from Quadratic Map:
   x = rep(0.2, 100)
   for (i in 2:100) x[i] = 4*(1-x[i-1])*x[i-1]
   plot(x, type = "l", main = "Quadratic Map")
   bdsTest(x, m = 3)
   
## tnnTest -
   # Time Series Non-linear in "mean" regression 
   par(mfrow = c(2, 1))
   n = 1000
   x = runif(1000, -1, 1)  
   tnnTest(x)
   # Generate time series which is nonlinear in "mean"
   x[1] = 0.0
   for (i in (2:n)) {
     x[i] = 0.4*x[i-1] + tanh(x[i-1]) + rnorm (1, sd = 0.5) }
   plot(x, main = "Teraesvirta Test", type = "l")
   tnnTest(x)
   
## wnnTest -
   # Time Series Non-Linear in "mean" Regression
   par(mfrow = c(2, 1))
   n = 1000
   x = runif(1000, -1, 1)
   wnnTest(x)
   # Generate time series which is nonlinear in "mean"
   x[1] = 0.0
   for (i in (2:n)) {
     x[i] = 0.4*x[i-1] + tanh(x[i-1]) + rnorm (1, sd = 0.5) }
   plot(x, main = "White Test", type = "l")
   wnnTest(x)                     

[Package fSeries version 240.10068 Index]