| mgcv {mgcv} | R Documentation |
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 to iteratively adjust a set of relative smoothing parameters for each penalty. To ensure that the overall level of smoothing is optimal, and to guard against trapping by local minima, a highly efficient global minimisation with respect to one overall smoothing parameter is also made at each iteration.
For a listing of all routines in the mgcv package type:
library(help="mgcv")
mgcv(y,X,sp,S,off,C=NULL,w=rep(1,length(y)),H=NULL,
scale=1,gcv=TRUE,control=mgcv.control())
y |
The response data vector. |
X |
The design matrix for the problem, note that ncol(X)
must give the number of model parameters, while nrow(X)
should give the number of data. |
sp |
An array of smoothing parameters. If control$fixed==TRUE then these are taken as being the
smoothing parameters. Otherwise any positive values are assumed to be initial estimates and negative values to
signal auto-initialization. |
S |
A list of penalty matrices. Only the smallest square block containing all non-zero matrix
elements is actually stored, and off[i] indicates the element of the parameter vector that
S[[i]][1,1] relates to. |
off |
Offset values indicating where in the overall parameter a particular stored penalty starts operating.
For example if p is the model parameter vector and k=nrow(S[[i]])-1, then the ith penalty is given by t(p[off[i]:(off[i]+k)])%*%S[[i]]%*%p[off[i]:(off[i]+k)]. |
C |
Matrix containing any linear equality constraints on the problem (i.e. C in Cp=0). |
w |
A vector of weights for the data (often proportional to the
reciprocal of the standard deviation of y). |
H |
A single fixed penalty matrix to be used in place of the multiple
penalty matrices in S. mgcv cannot mix fixed and estimated penalties. |
scale |
This is the known scale parameter/error variance to use with UBRE.
Note that it is assumed that the variance of y_i is
given by scale/w_i. |
gcv |
If gcv is TRUE then smoothing parameters are estimated by GCV,
otherwise UBRE is used. |
control |
A list of control options returned by mgcv.control. |
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 (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.
An object is returned with the following elements:
b |
The best fit parameters given the estimated smoothing parameters. |
scale |
The estimated or supplied scale parameter/error variance. |
score |
The UBRE or GCV score. |
sp |
The estimated (or supplied) smoothing parameters (lambda_i/rho) |
Vb |
Estimated covariance matrix of model parameters. |
hat |
diagonal of the hat/influence matrix. |
edf |
array of estimated degrees of freedom for each parameter. |
info |
A list of convergence diagnostics, with the following elements:
|
The method may not behave well with near column rank deficient X especially in contexts where the weights vary wildly.
Simon N. Wood simon.wood@r-project.org
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, S.N. (2000) Modelling and Smoothing Parameter Estimation with Multiple Quadratic Penalties. J.R.Statist.Soc.B 62(2):413-428
http://www.stats.gla.ac.uk/~simon/
library(help="mgcv") # listing of all routines set.seed(1);n<-400;sig2<-4 x0 <- runif(n, 0, 1);x1 <- runif(n, 0, 1) x2 <- runif(n, 0, 1);x3 <- runif(n, 0, 1) f <- 2 * sin(pi * x0) f <- f + exp(2 * x1) - 3.75887 f <- f+0.2*x2^11*(10*(1-x2))^6+10*(10*x2)^3*(1-x2)^10-1.396 e <- rnorm(n, 0, sqrt(sig2)) y <- f + e # set up additive model G<-gam(y~s(x0)+s(x1)+s(x2)+s(x3),fit=FALSE) # fit using mgcv mgfit<-mgcv(G$y,G$X,G$sp,G$S,G$off,C=G$C)