nnregCI               package:funfits               R Documentation

_F_i_n_d_s _a _c_o_n_f_i_d_e_n_c_e _s_e_t _o_f _p_a_r_a_m_e_t_e_r_s _f_o_r _a _n_e_u_r_a_l _n_e_t _f_i_t.

_D_e_s_c_r_i_p_t_i_o_n:

     The joint parameter confidence set for a neural net fit is all the
      neural net parameter sets (theta) such that  S(theta) <=
     S(theta^hat)*[1+(p/n-p)*F(p,n-p,alpha)], where S(theta) is the
     residual sum of squares, theta^hat is the least-  squares estimate
     of theta, p is the number of parameters of the model  and n is the
     number of data points. For the F distribution, alpha is the
     probability level.

     The program finds parameter sets which satisfy the above
     inequality. The value of cut1 is
     RMSE(theta^hat)*sqrt([1+(p/n-p)*F(p,n-p,alpha)]). The value of
     cut2 is .8*cut1. Approximately 20% of the fits will have a  RMSE
     of cut1 and the remaining 80% will be uniform between
     RMSE(theta^hat) and cut1. This distribution of parameter sets is
     to make sure that the  parameter sets cover the confidence region.
     The actual value of cut2 is  used only as a check for the covering
     of the confidence region. The returned  component summary has a
     count of the fits between cut1 and cut2 and also  below cut2.

     Parameters of the model are estimated by nonlinear least squares.
     The parameter space has a large number of local minimum so the
     strategy is to generate "many" parameter sets at random and
     iterate these starts with a minimization algorithm. The two
     function parameters ntries and ngrid are used in generating the
     many starting parameter sets for nonlinear least squares. Ngrind
     is the number of cubes growing geometrically over a range of
     magnitude of parameters. Ntries is the number of parameter sets
     generated at random by a uniform distribution in each cube. The
     best parameter set ( out the Ntries ) in each cube is used as the
     start of a coarse optimization.  Npol of these coarse fits are
     selected for further refinement by a minimization with smaller
     tolerance.

     The target RMS for a fit is generated as described above. The
     parameter  sets for the confidence sets are generated in the
     polishing stage and in groups of the optional argument npol. The
     file nnregCI.cut contains  information about the polished fits.
     The 7th column is target RMSE value  the 8th column is the
     difference between target RMSE and the root finder's  RMSE. The
     9th column is the value of cut1 and the 10th column is the value 
     of cut2.

_U_s_a_g_e:

     nnregCI(fit, model=fit$best.model, ngrind=250, ntries=100, npol=20, 
     clevel=0.95, cut1=NA, cut2=NA, nfits=500, tol1=1e-06, tol2=1e-09, 
     itmax1=250, itmax2=10000, fdata, fout="nnci.out", seed)

_A_r_g_u_m_e_n_t_s:

     fit: A nnreg object. 

   model: Model number used in finding joint parameter confidence set.
          Default is the  best model based on GCV(2). 

  ngrind: Number of coarse optimizations. 

  ntries: Number of random starting values for each coarse
          optimization. 

    npol: Number of coarse fits improved, i.e polish, using smaller
          minimization tolerance. 

  clevel: Confidence level used in finding joint parameter confidence
          set. Default is the 0.95 level. 

    cut1: RMSE value corresponding to the clevel confidence level. 

    cut2: RMSE value corresponding to 80% of the RMSE value
          corresponding to the clevel confidence level. 

   nfits: Number of fits (parameter sets) found in the confidence set.
          Maximum is 500. 

    tol1: Minimization tolerance for coarse optimizations. 

    tol2: Minimization tolerance for polish optimizations. 

  itmax1: Maximum number of iterations performed in the minimization
          routine for coarse optimizations. 

  itmax2: Maximum number of iterations performed in the minimization
          routine for polish optimizations. 

   fdata: Temporary UNIX file name for the data. 

    fout: Temporary UNIX file name for the output. 

    seed: Seed used in generating the random parameter starts. 

_V_a_l_u_e:

     Object of class nnreg. The component model is a list of the
     parameters for each fitted model. Each component model is of class
     netfit. 

   model: Component model of class netfit. Includes a list of the
          dimension of the x matrix, the number of hidden units used in
          the model, the mean of each column of the x matrix, the mean
          of the y values, the standard deviation of each column of the
          x matrix, the standard deviation of the y values, the number
          of parameters in the model and the parameters of model.  

 summary: Partial Fortan program output. Summary of the nnreg fit.
          Includes a  summary of the specified number of fitted values. 

    call: Call to the function. 

       x: Matrix of independent variables. 

       y: Vector of dependent variables. 

       n: Number of observations or length of y. 

   nfits: Number of fits (parameter sets) found in the confidence set. 

    seed: Seed used in generating the random parameter starts. 

_S_i_d_e _E_f_f_e_c_t_s:

     This function does the bulk of the computation using a stand-alone
     FORTRAN program running in the UNIX shell. This operation is
     transparent to the user. For large problems the input files can be
     setup using this function and the fitting program can be run
     separately in the background.

_R_e_f_e_r_e_n_c_e_s:

     B.A. Bailey, S. Ellner, D.W. Nychka. 1996. Chaos with Confidence:
     Asymptotics  and Applications of Local Lyapunov Exponents.
     Proceedings on Nonlinear Dynamics and Time Series, Building a
     Bridge Between the Natural and Statistical Sciences. Fields
     Institute Communications.

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

     D.W. Nychka, S. Ellner, D. McCaffrey, and A.R. Gallant. 1992.
     Finding Chaos in Noisy Systems. J. R. Statist. Soc. B 54:399-426.

_S_e_e _A_l_s_o:

     predict.nnreg, predict.netfit, plot.nnreg, summary.nnreg,
     print.nnreg

_E_x_a_m_p_l_e_s:

     nnreg(ozone$x,ozone$y,1,2) -> fit # fitting a surface to ozone 
     # measurements, from 1 to 2 hidden units

     nnregCI(fit) -> fit.ci # finds 500 fits in the .95 confidence set based
     # on the best model from the above fit

