R: Multiple Smoothing Parameter Estimation by GCV or UBRE

mgcv {mgcv}

R Documentation

Multiple Smoothing Parameter Estimation by GCV or UBRE

Description

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.

The other functions in the mgcv package are listed under `See Also'.

Usage

mgcv(M)

Arguments

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 standard deviation of M$y).

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$m

The number of penalty terms that are to be applied.

M$off

Offset values locating the elements of M$S[] in the correct location within each penalty coefficient matrix. indexing starts at zero.

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 on return it will contain an estimate of the error variance). 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. If M$fixed.sp is non-zero then these parameters will used as known smoothing parameters.

M$fixed.sp

Non-zero to indicate that all smoothing parameters are supplied as known, and are not to be estimated.

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.

M$min.edf

The minimum estimated degrees of freedom for the model: set to less than or equal to 0 to ignore. Supplying this quantity helps to maintain numerical stability in cases in which the best model is actually very smooth.

M$target.edf

Set this to negative for it to be ignored. If it is positive then this is taken as the target edf to use for the model - if the GCV/UBRE function has multiple minima w.r.t. the overall smoothing parameter then the minimum closest to the target edf will be selected. This feature is primarily useful when selecting smoothing parameters as part of iterative least squares methods - the GCV/UBRE score can develop spurious minima in these cases that are not representative of the problem being approximated.

Details

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.

Value

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$Vp`	Estimated covariance matrix of model parameters.
`M$gcv.ubre`	The minimized GCV/UBRE score.
`M$conv`	A list of convergence diagnostics, with the following elements: edf Array of whole model estimated degrees of freedom. hat Array of elements from leading diagonal of `hat' matrix (`influence' matrix). Same length as `M$y`. score Array of ubre/gcv scores at the edfs for the final set of relatinve smoothing parameters. 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! iter the number of iterations taken. in.ok `TRUE` if the second smoothing parameter guess improved the GCV/UBRE score. (Please report examples where this is `FALSE`) step.fail `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.

WARNING

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

Author(s)

Simon N. Wood simon@stats.gla.ac.uk

References

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

Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114

http://www.stats.gla.ac.uk/~simon/

Examples

set.seed(0)    
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)
# create list for passing to GAMsetup
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),by=0,by.exists=c(FALSE,FALSE,FALSE,FALSE),
          p.order=c(0,0,0,0), x = x,n.knots=rep(0,4),fit.method="mgcv") 
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$max.half<-15
H$min.edf<-5;H$fixed.sp<-0
H <- mgcv(H)  # select smoothing parameters and fit model

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