magic: Stable Multiple Smoothing Parameter Estimation by GCV or UBRE

Description

Function to efficiently estimate smoothing parameters in generalized ridge regression problems 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 the smoothing parameters for each penalty (one penalty may have a smoothing parameter fixed at 1).

For maximal numerical stability the method is based on orthogonal decomposition methods, and attempts to deal with numerical rank deficiency gracefully using a truncated singular value decomposition approach.

Usage

magic(y,X,sp,S,off,L=NULL,lsp0=NULL,rank=NULL,H=NULL,C=NULL,
      w=NULL,gamma=1,scale=1,gcv=TRUE,ridge.parameter=NULL,
      control=list(tol=1e-6,step.half=25,rank.tol=
      .Machine$double.eps^0.5),extra.rss=0,n.score=length(y),nthreads=1)

Arguments

is the response data vector.

is the model matrix (more columns than rows are allowed).

is the array of smoothing parameters. The vector L%*%log(sp) + lsp0 contains the logs of the smoothing parameters that actually multiply the penalty matrices stored in S (L is taken as the identity if NULL). Any sp values that are negative are autoinitialized, otherwise they are taken as supplying starting values. A supplied starting value will be reset to a default starting value if the gradient of the GCV/UBRE score is too small at the supplied value.

is a list of of penalty matrices. S[[i]] is the ith penalty matrix, but note that it is not stored as a full matrix, but rather as the smallest square matrix including all the non-zero elements of the penalty matrix. Element 1,1 of S[[i]] occupies element off[i], off[i] of the ith penalty matrix. Each S[[i]] must be positive semi-definite. Set to list() if there are no smoothing parameters to be estimated.

off

is an array indicating the first parameter in the parameter vector that is penalized by the penalty involving S[[i]].

is a matrix mapping log(sp) to the log smoothing parameters that actually multiply the penalties defined by the elemts of S. Taken as the identity, if NULL. See above under sp.

lsp0

If L is not NULL this is a vector of constants in the linear transformation from log(sp) to the actual log smoothing parameters. So the logs of the smoothing parameters multiplying the S[[i]] are given by L%*%log(sp) + lsp0. Taken as 0 if NULL.

rank

is an array specifying the ranks of the penalties. This is useful, but not essential, for forming square roots of the penalty matrices.

is the optional offset penalty - i.e. a penalty with a smoothing parameter fixed at 1. This is useful for allowing regularization of the estimation process, fixed smoothing penalties etc.

is the optional matrix specifying any linear equality constraints on the fitting problem. If $\bf b$ is the parameter vector then the parameters are forced to satisfy $ {\bf Cb} = {\bf 0} $.

the regression weights. If this is a matrix then it is taken as being the square root of the inverse of the covariance matrix of y, specifically $ {\bf V}_y^{-1} = {\bf w}^\prime{\bf w}$. If w is an array then it is taken as the diagonal of this matrix, or simply the weight for each element of y. See below for an example using this.

gamma

is an inflation factor for the model degrees of freedom in the GCV or UBRE score.

scale

is the scale parameter for use with UBRE.

gcv

should be set to TRUE if GCV is to be used, FALSE for UBRE.

ridge.parameter

It is sometimes useful to apply a ridge penalty to the fitting problem, penalizing the parameters in the constrained space directly. Setting this parameter to a value greater than zero will cause such a penalty to be used, with the magnitude given by the parameter value.

control

is a list of iteration control constants with the following elements:

tol: The tolerance to use in judging convergence.
step.half: If a trial step fails then the method tries halving it up to a maximum of step.half times.
rank.tol: is a constant used to test for numerical rank deficiency of the problem. Basically any singular value less than rank_tol multiplied by the largest singular value of the problem is set to zero.

extra.rss

is a constant to be added to the residual sum of squares (squared norm) term in the calculation of the GCV, UBRE and scale parameter estimate. In conjuction with n.score, this is useful for certain methods for dealing with very large data sets.

n.score

number to use as the number of data in GCV/UBRE score calculation: usually the actual number of data, but there are methods for dealing with very large datasets that change this.

nthreads

magic can make use of multiple threads if this is set to >1.

Value

The function returns a list with the following items:

The best fit parameters given the estimated smoothing parameters.

scale

the estimated (GCV) or supplied (UBRE) scale parameter.

score

the minimized GCV or UBRE score.

an array of the estimated smoothing parameters.

sp.full

an array of the smoothing parameters that actually multiply the elements of S (same as sp if L was NULL). This is exp(L%*%log(sp)).

a factored form of the parameter covariance matrix. The (Bayesian) covariance matrix of the parametes b is given by rV%*%t(rV)*scale.

gcv.info

is a list of information about the performance of the method with the following elements:

full.rank: The apparent rank of the problem: number of parameters less number of equality constraints.
rank: The estimated actual rank of the problem (at the final iteration of the method).
fully.converged: is TRUE if the method converged by satisfying the convergence criteria, and FALSE if it coverged by failing to decrease the score along the search direction.
hess.pos.def: is TRUE if the hessian of the UBRE or GCV score was positive definite at convergence.
iter: is the number of Newton/Steepest descent iterations taken.
score.calls: is the number of times that the GCV/UBRE score had to be evaluated.
rms.grad: is the root mean square of the gradient of the UBRE/GCV score w.r.t. the smoothing parameters.
R: The factor R from the QR decomposition of the weighted model matrix. This is un-pivoted so that column order corresponds to X. So it may not be upper triangular.

Note that some further useful quantities can be obtained using magic.post.proc.

Details

The method is a computationally efficient means of applying GCV or UBRE (often approximately AIC) to the problem of smoothing parameter selection in generalized ridge regression problems of the form: $$ minimise~ \| { \bf W} ({ \bf Xb - y} ) \|^2 + {\bf b}^\prime {\bf Hb} + \sum_{i=1}^m \theta_i {\bf b^\prime S}_i{\bf b} $$ possibly subject to constraints $ {\bf Cb}={\bf 0}$. $ {\bf X}$ is a design matrix, $\bf b$ a parameter vector, $\bf y$ a data vector, $\bf W$ a weight matrix, $ {\bf S}_i$ a positive semi-definite matrix of coefficients defining the ith penalty with associated smoothing parameter $\theta_i$, $\bf H$ is the positive semi-definite offset penalty matrix and $\bf C$ a matrix of coefficients defining any linear equality constraints on the problem. $ {\bf X}$ need not be of full column rank.

The $\theta_i$ are chosen to minimize either the GCV score:

$$V_g = \frac{n\|{\bf W}({\bf y} - {\bf Ay})\|^2}{[tr({\bf I} - \gamma {\bf A})]^2}$$

or the UBRE score:

$$V_u=\|{\bf W}({\bf y}-{\bf Ay})\|^2/n-2 \phi tr({\bf I}-\gamma {\bf A})/n + \phi$$

where $\gamma$ is gamma the inflation factor for degrees of freedom (usually set to 1) and $\phi$ is scale, the scale parameter. $\bf A$ is the hat matrix (influence matrix) for the fitting problem (i.e the matrix mapping data to fitted values). Dependence of the scores on the smoothing parameters is through $\bf A$.

The method operates by Newton or steepest descent updates of the logs of the $\theta_i$. A key aspect of the method is stable and economical calculation of the first and second derivatives of the scores w.r.t. the log smoothing parameters. Because the GCV/UBRE scores are flat w.r.t. very large or very small $\theta_i$, 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 $log(\theta_i)$ 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. (Given the smoothing parameters the optimal $\bf b$ parameters are easily found.)

The method is coded in C with matrix factorizations performed using LINPACK and LAPACK routines.

References

Wood, S.N. (2004) Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Amer. Statist. Ass. 99:673-686

https://www.maths.ed.ac.uk/~swood34/

Examples

Run this code

# NOT RUN {
## Use `magic' for a standard additive model fit ... 
   library(mgcv)
   set.seed(1);n <- 200;sig <- 1
   dat <- gamSim(1,n=n,scale=sig)
   k <- 30
## set up additive model
   G <- gam(y~s(x0,k=k)+s(x1,k=k)+s(x2,k=k)+s(x3,k=k),fit=FALSE,data=dat)
## fit using magic (and gam default tolerance)
   mgfit <- magic(G$y,G$X,G$sp,G$S,G$off,rank=G$rank,
                  control=list(tol=1e-7,step.half=15))
## and fit using gam as consistency check
   b <- gam(G=G)
   mgfit$sp;b$sp  # compare smoothing parameter estimates
   edf <- magic.post.proc(G$X,mgfit,G$w)$edf # get e.d.f. per param
   range(edf-b$edf)  # compare

## p>n example... fit model to first 100 data only, so more
## params than data...

   mgfit <- magic(G$y[1:100],G$X[1:100,],G$sp,G$S,G$off,rank=G$rank)
   edf <- magic.post.proc(G$X[1:100,],mgfit,G$w[1:100])$edf

## constrain first two smooths to have identical smoothing parameters
   L <- diag(3);L <- rbind(L[1,],L)
   mgfit <- magic(G$y,G$X,rep(-1,3),G$S,G$off,L=L,rank=G$rank,C=G$C)

## Now a correlated data example ... 
    library(nlme)
## simulate truth
    set.seed(1);n<-400;sig<-2
    x <- 0:(n-1)/(n-1)
    f <- 0.2*x^11*(10*(1-x))^6+10*(10*x)^3*(1-x)^10
## produce scaled covariance matrix for AR1 errors...
    V <- corMatrix(Initialize(corAR1(.6),data.frame(x=x)))
    Cv <- chol(V)  # t(Cv)%*%Cv=V
## Simulate AR1 errors ...
    e <- t(Cv)%*%rnorm(n,0,sig) # so cov(e) = V * sig^2
## Observe truth + AR1 errors
    y <- f + e 
## GAM ignoring correlation
    par(mfrow=c(1,2))
    b <- gam(y~s(x,k=20))
    plot(b);lines(x,f-mean(f),col=2);title("Ignoring correlation")
## Fit smooth, taking account of *known* correlation...
    w <- solve(t(Cv)) # V^{-1} = w'w
    ## Use `gam' to set up model for fitting...
    G <- gam(y~s(x,k=20),fit=FALSE)
    ## fit using magic, with weight *matrix*
    mgfit <- magic(G$y,G$X,G$sp,G$S,G$off,rank=G$rank,C=G$C,w=w)
## Modify previous gam object using new fit, for plotting...    
    mg.stuff <- magic.post.proc(G$X,mgfit,w)
    b$edf <- mg.stuff$edf;b$Vp <- mg.stuff$Vb
    b$coefficients <- mgfit$b 
    plot(b);lines(x,f-mean(f),col=2);title("Known correlation")
# }

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