lyapunov.spec {DChaos}

Estimates the Lyapunov exponent spectrum

Description

This function estimates the Lyapunov exponent spectrum through the QR decomposition procedure based on the partial derivatives computed by the jacobian.net function.

Usage

lyapunov.spec(
  data,
  blocking = c("BOOT", "NOVER", "EQS", "FULL"),
  B = 1000,
  doplot = TRUE
)

Arguments

data

should be a jacobian object containing the partial derivatives computed by the jacobian.net function.

blocking

a character denoting the blocking method chosen for figuring out the Lyapunov exponent spectrum through the QR decomposition procedure. Available options are FULL if the user considers the full sample, NOVER if the user considers the non-overlapping sample, EQS if the user considers the equally spaced sample or BOOT if the user considers the bootstrap sample (Default BOOT).

B

a non-negative integer denoting the number of bootstrap iterations (Default 1000).

doplot

a logical value denoting if the user wants to draw a plot TRUE or not FALSE. If it is TRUE the evolution of the Lyapunov exponent values are represented for the whole period considering the blocking method chosen by the user. It shows as many graphs as embedding dimensions have been considered (Default TRUE).

Value

This function returns several objects considering the parameter set selected by the user. The Lyapunov exponent spectrum considering the QR decomposition procedure by each blocking method are estimated. It also contains some useful information about the estimated jacobian, the best-fitted feed-forward single hidden layer neural net model, the best set of weights found, the fitted values, the residuals obtained, the best embedding parameters set chosen, the sample size or the block length considered by each blocking method. This function provides the standard error, the z test value and the p-value for testing the null hypothesis H0: \lambda_k > 0 for k = 1,2,3, \ldots, m (full spectrum). Reject the null hypothesis $H_0$ means lack of chaotic behaviour. That is, the data-generating process does not have a chaotic attractor because of it does not show the property of sensitivity to initial conditions.

Note

The DChaos package provides several ways to figure out robustly the neural net estimator of the k-th Lyapunov exponent. On the one hand if the R users have previously obtained the partial derivatives from the jacobian.net function they can apply directly the function lyapunov.spec which estimates the Lyapunov exponent spectrum taking into account the QR decomposition procedure. They can also use the function lyapunov.max which estimates only the largest Lyapunov exponent considering the Norma-2 procedure. Hence the DChaos package allows the R users to choose between two different procedures to obtain the neural net estimator of the k-th Lyapunov exponent and four ways of subsampling by blocks: full sample, non-overlapping sample, equally spaced sample and bootstrap sample. The blocking methods what they do is to split the time-series data into several blocks by estimating a Lyapunov exponent for each subsample. If the R users choose the non-overlapping sample (blocking = "NOVER"), the equally spaced sample (blocking = "EQS") or the bootstrap sample (blocking = "BOOT") the mean and median values of the Lyapunov exponent for each block or subsample are saved. By default we recommend using the median value as it is more robust to the presence of outliers. Notice that the parameter B will only be considered if the R users choose the bootstrap blocking method.

Author(s)

Julio E. Sandubete, Lorenzo Escot

References

Ellner, S., Gallant, A., McCaffrey, D., Nychka, D. 1991 Convergence rates and data requirements for jacobian-based estimates of lyapunov exponents from data. Physics Letters A 153(6):357-363.

McCaffrey, D.F., Ellner, S., Gallant, A.R., Nychka, D.W. 1992 Estimating the lyapunov exponent of a chaotic system with nonparametric regression. Journal of the American Statistical Association 87(419):682-695.

Nychka, D., Ellner, S., Gallant, A.R., McCaffrey, D. 1992 Finding chaos in noisy systems. Journal of the Royal Statistical Society 54(2):399-426.

Whang, Y.J., Linton, O. 1999 The asymptotic distribution of nonparametric estimates of the lyapunov exponent for stochastic time series. Journal of Econometrics 91(1):1-42.

Shintani, M., Linton, O. 2004 Nonparametric neural network estimation of Lyapunov exponents and a direct test for chaos. Journal of Econometrics 120(1):1-33.

Examples

## set.seed(34)
## Simulates time-series data from the Logistic map with chaos
## ts        <- DChaos::logistic.sim(n=1000, a=4)
## show(head(ts, 5))

## Provides the Lyapunov exponent spectrum by the QR decomposition procedure considering the
## bootstrap blocking method from the best-fitted neural net model and the partial
## derivatives showed in the jacobian.net example.
## jacobian <- DChaos::jacobian.net(data=ts, m=3:3, lag=1:1, timelapse="FIXED", h=2:10)
## summary(jacobian)
## exponent <- DChaos::lyapunov.spec(data=jacobian, blocking="BOOT", B=100, doplot=FALSE)
## summary(exponent)