mgcv                  package:mgcv                  R Documentation

_M_u_l_t_i_p_l_e _S_m_o_o_t_h_i_n_g _P_a_r_a_m_e_t_e_r _E_s_t_i_m_a_t_i_o_n _b_y _G_C_V _o_r _U_B_R_E

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

     Function to efficiently estimate smoothing parameters in
     Generalized Ridge Regression Problem with multiple (quadratic)
     penalties, by GCV  or UBRE. The function uses Newton's method in
     multi-dimensions, backed up by steepest descent.

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

     mgcv(M)

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

       M: is the single argument to `mgcv' and should be a list with
          the elements listed below:

     M$y: The response data vector.

     M$w: A vector of weights for the data (often proportional to the 
          reciprocal of the variance). 

     M$X: The design matrix for the problem, note that `ncol(M$X)' must
          give the number of model parameters, while `nrow(M$X)' 
          should give the number of data.

     M$C: Matrix containing any linear equality constraints  on the
          problem (i.e. C in Cp=0).

     M$S: A one dimensional array containing the non-zero elements of
          the penalty matrices. Let `start<-sum(M$df[1:(k-1)]^2)' or
          `start<-0' if `k==1'. Then penalty matrix `k' has
          `M$S[start+i+M$df[i]*(j-1)' on its ith row and jth column.
          The kth full penalty matrix is constructed by placing the
          matrix just derived into an appropriate matrix of zeroes with
          it's 1,1 element at `M$off[k],M$off[k]' 

   M$off: Offset values locating the elements of `M$S[]' in the correct
          location within each penalty coefficient matrix.

   M$fix: `M$fix[i]' is FALSE if the corresponding smoothing parameter 
          is to be estimated, or TRUE if it is to be fixed at zero.

    M$df: `M$df[i]' gives the number of rows and columns of `M$S[i]'
          that are non-zero.

  M$sig2: This serves 2 purposes. If it is strictly positive then UBRE
          is  used with sig2 taken as the known error variance. If it
          is  non-positive  then GCV is used (and an estimate of the
          error variance  returned). Note that it is assumed that the
          variance of y_i is  given by `sig2'/w_i.

    M$sp: An array of initial smoothing parameter estimates if any of 
          these is not strictly positive then automatic generation of
          initial values is used.

M$conv.tol: The convergence tolerence to use for the multiple smoothing
          parameter search. 1e-6 is usually ok.

M$max.half: Each step of the GCV minimisation is repeatedly halved if
          it fails to improve the GCV score. This is the maximum number
          of halvings to try at each step. 15 is usually enough.

_D_e_t_a_i_l_s:

     This is documentation for the code implementing the method
     described in section  4 of  Wood (2000) . The method is a
     computationally efficient means of applying GCV to  the  problem
     of smoothing parameter selection in generalized ridge regression
     problems  of  the form:

 minimise || W^0.5 (Xp-y) ||^2 rho +  lambda_1 p'S_1 p + lambda_1 p'S_2 p + . . .

     possibly subject to constraints Cp=0.  X is a design matrix, p a
     parameter vector,  y a data vector, W a diagonal weight matrix,
     S_i a positive semi-definite matrix  of coefficients defining the
     ith penalty and C a matrix of coefficients  defining any linear
     equality constraints on the problem. The smoothing parameters are
     the lambda_i but there is an overall smoothing parameter rho as
     well. Note that X must be of full column rank, at least when
     projected  into the null space of any equality constraints.  

     The method operates by alternating very efficient direct searches
     for  rho with Newton or steepest descent updates of the logs of
     the lambda_i.  Because the GCV/UBRE scores are flat w.r.t. very
     large or very small lambda_i,  it's important to get good starting
     parameters, and to be careful not to step into a flat region of
     the smoothing parameter space. For this reason the algorithm
     rescales any Newton step that  would result in a log(lambda_i)
     change of more than 5. Newton steps are only used if the Hessian
     of the GCV/UBRE is postive definite, otherwise steepest descent is
     used. Similarly steepest  descent is used if the Newton step has
     to be contracted too far (indicating that the quadratic model 
     underlying Newton is poor). All initial steepest descent steps are
     scaled so that their largest component is 1. However a step is
     calculated, it is never expanded if it is successful (to avoid
     flat portions of the objective),  but steps are successively
     halved if they do not decrease the GCV/UBRE score, until they do,
     or the direction is deemed to have  failed. `M$conv' provides some
     convergence diagnostics.

     The method is coded in `C' and is intended to be portable. It
     should be  noted that seriously ill conditioned problems (i.e.
     with close to column rank  deficiency in the design matrix) may
     cause problems, especially if weights vary  wildly between
     observations.

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

     The function returns a list of the same form as the input list
     with the  following elements added/modified: 

     M$p: The best fit parameters given the estimated smoothing
          parameters.

    M$sp: The estimated smoothing parameters  (lambda_i/rho)

  M$sig2: The estimated (GCV) or supplied (UBRE) error variance.

   M$edf: The estimated degrees of freedom associated with each
          penalized term. If p_i is the vector of parameters penalized
          by the ith penalty then we can write p_i = P_i y, and the edf
          for the ith term is tr(XP_i)... while this is sensible in the
          non-overlapping penalty case (e.g. GAMs) it makes alot less
          sense if penalties overlap!

    M$Vp: Estimated covariance matrix of model parameters.

  M$conv: A list of convergence diagnostics, with the following
          elements:

     _g the gradient of the GCV/UBRE score w.r.t. the smoothing
          parameters at termination.

     _h the second derivatives corresponding to `g' above - i.e. the
          leading diagonal of the Hessian.

     _e the eigen-values of the Hessian. These should all be
          non-negative!

     _i_t_e_r the number of iterations taken.

     _i_n._o_k `TRUE' if the second smoothing parameter guess improved the
          GCV/UBRE score. (Please report examples  where this is
          `FALSE')

     _s_t_e_p._f_a_i_l `TRUE' if the algorithm terminated by failing to improve
          the GCV/UBRE score rather than by "converging".  Not
          necessarily a problem, but check the above derivative
          information quite carefully. 

_W_A_R_N_I_N_G:

     The method may not behave well with near column rank deficient X
     especially in contexts where the weights vary wildly.

_A_u_t_h_o_r(_s):

     Simon N. Wood snw@st-and.ac.uk

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

     Gu and Wahba (1991) Minimizing GCV/GML scores with multiple
     smoothing parameters via the Newton method. SIAM J. Sci. Statist.
     Comput. 12:383-398

     Wood (2000) Modelling and Smoothing Parameter Estimation  with
     Multiple  Quadratic Penalties. JRSSB 62(2):413-428

     <URL: http://www.ruwpa.st-and.ac.uk/simon.html>

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

     `GAMsetup' `gam'

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

      # This example modified from routine SANtest()

         n<-100 # number of observations to simulate
         x <- runif(4 * n, 0, 1) # simulate covariates
         x <- array(x, dim = c(4, n)) # put into array for passing to GAMsetup
         pi <- asin(1) * 2  # begin simulating some data
         y <- 2 * sin(pi * x[1, ])
         y <- y + exp(2 * x[2, ]) - 3.75887
         y <- y + 0.2 * x[3, ]^11 * (10 * (1 - x[3, ]))^6 + 10 * (10 *
             x[3, ])^3 * (1 - x[3, ])^10 - 1.396
         sig2<- -1    # set magnitude of variance
         e <- rnorm(n, 0, sqrt(abs(sig2)))
         y <- y + e          # simulated data
         w <- matrix(1, n, 1) # weight matrix
         par(mfrow = c(2, 2)) # scatter plots of simulated data   
         plot(x[1, ], y)
         plot(x[2, ], y)
         plot(x[3, ], y)
         plot(x[4, ], y)
         G <- list(m = 4, n = n, nsdf = 0, df = c(15, 15, 15, 15),dim=c(1,1,1,1),s.type=c(0,0,0,0), 
            p.order=c(0,0,0,0), x = x) # creat list for passing to GAMsetup
         H <- GAMsetup(G)
         H$y <- y    # add data to H
         H$sig2 <- sig2  # add variance (signalling GCV use in this case) to H
         H$w <- w       # add weights to H
         H$sp<-array(-1,H$m)
         H$fix<-array(FALSE,H$m)
         H$conv.tol<-1e-6;H$max.half<-15
         H <- mgcv(H)  # select smoothing parameters and fit model


