

   CCaallccuullaatteess llooccaall LLyyaappuunnoovv eexxppoonneennttss ffoorr pplloottttiinngg..

        lle(jac, model=1, nprod=c(5, 10, 20, 40, 80), skip, statevector=F,
        lags=NA)

   AArrgguummeennttss::

        jac: Jacobian matrix or a nnreg fit.

      model: Model number of fit used to calculate Jacobians.

      nprod: Vector of LLE products of Jacobians.

       skip: Columns of Jacobian matrix to skip in calculating
             LLEs.  For example, skip the columns associated
             with forcing functions.

   statevector: If false, a time-delay reconstruction model is
             assumed and a Jacobian matrix n by d is expected,
             where n is the length of the time series and d is
             the dimension of the state space.  If true, a
             state space vector model is assumed and a Jacobian
             matrix n by d^2 is expected.

       lags: Lagged time values used in the Jacobian matrix.

   VVaalluuee::

      local: Matrix of LLEs with columns corresponding to the
             LLEs of the nprod values.

      nprod: Vector of LLE products of Jacobians.

        glb: Global Lyapunov exponent.

      model: Model number used to calculate Jacobians.

   RReeffeerreenncceess::

        S. Ellner, D.W. Nychka, and A.R. Gallant. 1992.  LENNS,
        a program  to  estimate  the  dominant  Lyapunov expo-
        nent of noisy nonlinear systems from time series  data.
        Institute of  Statistics  Mimeo Series #2235, Statis-
        tics Department, North Carolina State University,
        Raleigh, NC 27695-8203.

   SSeeee AAllssoo::

        make.lle

   EExxaammpplleess::

        make.lags(rossler.state[1:200,1],c(1,2,3)) -> data.r  # create
        # 3-d time delay vector model of the x variable of rossler system.
        nnreg(data.r$x,data.r$y,5,5) -> fit # fit time series model using nnreg.
        jac<- predict(fit, derivative=1)
        lle(jac) -> rossler.lle  # LLEs of Rossler data
        summary(lle)
        plot(rossler.lle)  # plot LLEs

        # here is an easier way
        nlar( rossler[1:200], lags=1:3, method="nnreg", k1=5)-> ou
        lle( out) -> rossler.lle

