gam() is not a clone of what Splus provides.
Smooth terms are represented using penalized regression splines
with smoothing parameters selected by GCV or by regression splines with
fixed degrees of freedom (mixtures of the two are
permitted). Multi-dimensional smooths are available using penalized thin plate
regression splines, but the user must make sure that covariates are sensibly scaled
relative to each other when using such terms. For a general overview see
Wood (2001).gam(formula,family=gaussian(),data=list(),weights=NULL,control=gam.control,scale=0)y ~ . is not allowed).
Smooth terms are specified by expressions of the form:
gam was called is
searched for the variables specified in the formula.gam.control, with five user controllable elements:
maxit controls maximum iterations in
gam.fit, convergence tolerance in gam.fit is controlled by epsilon"gam" which has the following elements:xp[i,] are the locations for the ith smooth.call object containing the call to gam() that produced
this gam object (useful for constructing model frames).g above - i.e. the leading diagonal of the Hessian.}
TRUE if the second smoothing parameter guess improved the GCV/UBRE score. (Please report examples
where this is FALSE)}
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.}You must have more unique combinations of covariates than the model has total parameters. (Total parameters is sum of basis dimensions plus sum of non-spline terms less the number of spline terms).
Automatic smoothing parameter selection is not likely to work well when fitting models to very few response data.
Relative scaling of covariates to a multi-dimensional smooth term has an affect on the results: make sure that relative scalings are sensible. For example, measuring one spatial co-ordinate in millimetres and the other in lightyears will usually produce poor results.
With large datasets (more than a few thousand data) the "tp"
basis gets very slow to use. In this case use "cr" for 1-d
smooths. If you need to use multi-dimensional terms with large datasets
and find gam too slow, please let me know - and I'll up the
priority for fixing this!
Thin plate regression splines are constructed by starting with the
basis for a full thin plate spline and then truncating this basis in
an optimal manner, to obtain a low rank smoother. Details are given in
Wood (MS submitted). One key advantage of the approach is that it avoids
the knot placement problems of conventional regression spline
modelling, but it also has the advantage that smooths of lower rank
are nested within smooths of higher rank, so that it is legitimate to
use conventional hypothesis testing methods to compare models based on
pure regression splines.
In the case of the cubic regression spline basis, knots of the spline are placed evenly
throughout the covariate values to which the term refers: For
example, if fitting 101 data with an 11 knot spline of x then
there would be a knot at every 10th (ordered) x value. The
parameterization used represents the spline in terms of its
values at the knots. Connection of these values at neighbouring knots
by sections of cubic polynomial constrainted to join at the knots so as to be
continuous up to and including second derivative yields a natural cubic
spline through the values at the knots (given two extra conditions specifying
that the second derivative of the curve should be zero at the two end
knots). This parameterization gives the parameters a nice interpretability.
Given a basis for each smooth term, it easy to obtain a wiggliness
penalty for each, and to construct a penalized likelihood, which
balances the fit of the overall model against it's
complexity. This consists of the log likelihood for the model minus a
sum of wiggliness penalties (one for each smooth) each multiplied by a
smoothing parameter. The smoothing parameters control the trade-off
between fit and smoothness.
So, the gam fitting problem has become a penalized glm fitting problem, which can be fitted using a
slight modification of glm.fit : gam.fit. The penalized
glm approach also allows smoothing parameters for all smooth terms to
be selected simultaneously by GCV or UBRE. This is achieved as
part of fitting by calling mgcv within gam.fit.
Details of the GCV/UBRE minimization method are given in Wood (2000).
Green and Silverman (1994) Nonparametric Regression and Generalized Linear Models. Chapman and Hall. 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
Wood (2001) mgcv:GAMs and Generalized Ridge Regression for R. R News 1(2):20-25 Wood (MS submitted) Thin Plate Regression Splines
Wahba (1990) Spline Models of Observational Data. SIAM
s predict.gam plot.gamlibrary(mgcv)
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)
pi <- asin(1) * 2
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(abs(sig2)))
y <- f + e
b<-gam(y~s(x0)+s(x1)+s(x2)+s(x3))
plot(b,pages=1)
# now a GAM with 3df regression spline term & 2 penalized terms
b1<-gam(y~s(x0,k=4,fx=TRUE,bs="tp")+s(x1,k=12)+s(x2,15))
plot(b1,pages=1)
# now fit a 2-d term to x0,x1
b3<-gam(y~s(x0,x1)+s(x2)+s(x3))
par(mfrow=c(2,2))
plot(b3)
par(mfrow=c(1,1))
# now simulate poisson data
g<-exp(f/5)
y<-rpois(rep(1,n),g)
b2<-gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=poisson)
plot(b2,pages=1)
# negative binomial data
set.seed(1)
y<-rnbinom(g,size=2,mu=g)
b3<-gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=neg.binom)
plot(b3,pages=1)
# and a pretty 2-d smoothing example....
test1<-function(x,z,sx=0.3,sz=0.4)
{ (pi**sx*sz)*(1.2*exp(-(x-0.2)^2/sx^2-(z-0.3)^2/sz^2)+
0.8*exp(-(x-0.7)^2/sx^2-(z-0.8)^2/sz^2))
}
n<-500
old.par<-par(mfrow=c(2,2))
x<-runif(n);z<-runif(n);
xs<-seq(0,1,length=30);zs<-seq(0,1,length=30)
pr<-data.frame(x=rep(xs,30),z=rep(zs,rep(30,30)))
truth<-matrix(test1(pr$x,pr$z),30,30)
contour(xs,zs,truth)
y<-test1(x,z)+rnorm(n)*0.1
b4<-gam(y~s(x,z))
fit1<-matrix(predict.gam(b4,pr,se=FALSE),30,30)
contour(xs,zs,fit1)
persp(xs,zs,truth)
persp(b4)
par(old.par)Run the code above in your browser using DataLab