| magic {mgcv} | R Documentation |
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.
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))
y |
is the response data vector. |
X |
is the model matrix (more columns than rows are allowed). |
sp |
is the array of smoothing parameters. The vector |
S |
is a list of of penalty matrices. |
off |
is an array indicating the first parameter in the parameter vector that is
penalized by the penalty involving |
L |
is a matrix mapping |
lsp0 |
If |
rank |
is an array specifying the ranks of the penalties. This is useful, but not essential, for forming square roots of the penalty matrices. |
H |
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. |
C |
is the optional matrix specifying any linear equality constraints on the fitting problem. If b is the parameter vector then the parameters are forced to satisfy Cb=0. |
w |
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 |
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 |
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:
|
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 |
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. |
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:
min ||W(Xb-y)||^2 + b'Hb + theta_1 b'S_1 b + theta_2 b'S_2 b + . . .
possibly subject to constraints Cb=0. X is a design matrix, b a parameter vector, y a data vector, W a weight matrix, S_i a positive semi-definite matrix of coefficients defining the ith penalty with associated smoothing parameter theta_i, H is the positive semi-definite offset penalty matrix and C a matrix of coefficients defining any linear equality constraints on the problem. X need not be of full column rank.
The theta_i are chosen to minimize either the GCV score:
V_g = n ||W(y-Ay)||^2/[tr(I - g A)]^2
or the UBRE score:
V_u =||W(y-Ay||^2/n - 2 s tr(I - g A)/n + s
where g is gamma the inflation factor for degrees of freedom (usually set to 1) and s
is scale, the scale parameter. 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 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 b parameters are easily found.)
The method is coded in C with matrix factorizations performed using LINPACK and LAPACK routines.
The function returns a list with the following items:
b |
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. |
sp |
an array of the estimated smoothing parameters. |
sp.full |
an array of the smoothing parameters that actually multiply the elements of
|
rV |
a factored form of the parameter covariance matrix. The (Bayesian) covariance
matrix of the parametes |
gcv.info |
is a list of information about the performance of the method with the following elements:
|
Note that some further useful quantities can be obtained using magic.post.proc.
Simon N. Wood simon.wood@r-project.org
Wood, S.N. (2004) Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Amer. Statist. Ass. 99:673-686
http://www.maths.bath.ac.uk/~sw283/
## 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")