gam {mgcv}  R Documentation 
Fits a generalized additive model (GAM) to
data. The degree of smoothness of model terms is estimated as part of
fitting; isotropic or scale invariant smooths of any number of variables are
available as model terms; confidence/credible intervals are readily
available for any quantity predicted using a fitted model; gam
is
extendable: i.e. users can add smooths.
Smooth terms are represented using penalized regression splines (or similar smoothers)
with smoothing parameters selected by GCV/UBRE or by regression splines with
fixed degrees of freedom (mixtures of the two are
permitted). Multidimensional smooths are available using penalized thin plate
regression splines (isotropic) or tensor product splines (when an isotropic smooth is inappropriate).
For more on specifying models see gam.models
. For more on model
selection see gam.selection
. For faster fits use the "cr"
bases for smooth terms, te
smooths for smooths of several
variables, and performance iteration for smoothing parameter estimation (see
gam.method
). For large datasets see warnings.
gam()
is not a clone of what SPLUS provides: the major
differences are (i) that by default estimation of the
degree of smoothness of model terms is part of model fitting, (ii) a
Bayesian approach to variance estimation is employed that makes for easier
confidence interval calculation (with good coverage probabilites) and (iii) the
facilities for incorporating smooths of more than one variable are
different: specifically there are no lo
smooths, but instead (a) s
terms can have more than one argument, implying an isotropic smooth and (b) te
smooths are
provided as an effective means for modelling smooth interactions of any
number of variables via scale invariant tensor product smooths. If you want
a clone of what SPLUS provides use gam from package gam
.
gam(formula,family=gaussian(),data=list(),weights=NULL,subset=NULL, na.action,offset=NULL,control=gam.control(),method=gam.method(), scale=0,knots=NULL,sp=NULL,min.sp=NULL,H=NULL,gamma=1, fit=TRUE,G=NULL,...)
formula 
A GAM formula (see also gam.models ). This is exactly like the formula for a
GLM except that smooth terms can be added to the right hand side of the
formula (and a formula of the form y ~ . is not allowed).
Smooth terms are specified by expressions of the form: s(var1,var2,...,k=12,fx=FALSE,bs="tp",by=a.var) where var1 ,
var2 , etc. are the covariates which the smooth
is a function of and k is the dimension of the basis used to
represent the smooth term. If k is not
specified then k=10*3^(d1) is used where d is the number
of covariates for this term. fx is used to indicate whether or
not this term has a fixed number of degrees of freedom (fx=FALSE
to select d.f. by GCV/UBRE). bs indicates the basis to use for the smooth: for a full
list see s , but note that the default "tp" , while
it possesses nice optimality properties is slow and memory hungry for very
large datasets (but see examples for how to get around this). by can be used to specify a variable by which
the smooth should be multiplied. For example gam(y~z+s(x,by=z))
would specify a model E(y)=f(x)z where
f(.) is a smooth function (the formula is
y~x+s(x,by=z) rather than y~s(x,by=z) because
the smooths are always set up to sum to zero over the covariate
values). The by option is particularly useful for models in
which different functions of the same variable are required for
each level of a factor and for `variable parameter models': see s .
An alternative for specifying smooths of more than one covariate is e.g.: te(x,z,bs=c("tp","tp"),m=c(2,3),k=c(5,10)) which would specify a tensor product
smooth of the two covariates x and z constructed from marginal t.p.r.s. bases
of dimension 5 and 10 with marginal penalties of order 2 and 3. Any combination of basis types is
possible, as is any number of covariates.
Formulae can involve nested or ``overlapping'' terms such as y~s(x)+s(z)+s(x,z) or y~s(x,z)+s(z,v) : see
gam.side for further details and examples. 
family 
This is a family object specifying the distribution and link to use in
fitting etc. See glm and family for more
details. The negative binomial families provided by the MASS library
can be used, with or without known theta parameter: see
gam.neg.bin for details.

data 
A data frame containing the model response variable and
covariates required by the formula. By default the variables are taken
from environment(formula) : typically the environment from
which gam is called. 
weights 
prior weights on the data. 
subset 
an optional vector specifying a subset of observations to be used in the fitting process. 
na.action 
a function which indicates what should happen when the data contain `NA's. The default is set by the `na.action' setting of `options', and is `na.fail' if that is unset. The ``factoryfresh'' default is `na.omit'. 
offset 
Can be used to supply a model offset for use in fitting. Note
that this offset will always be completely ignored when predicting, unlike an offset
included in formula : this conforms to the behaviour of
lm and glm . 
control 
A list of fit control parameters returned by
gam.control . 
method 
A list controlling the fitting methods used. This can make a big
difference to computational speed, and, in some cases, reliability of
convergence: see gam.method for details. 
scale 
If this is zero then GCV is used for all distributions
except Poisson and binomial where UBRE is used with scale parameter
assumed to be 1. If this is greater than 1 it is assumed to be the scale
parameter/variance and UBRE is used: to use the negative binomial in this case
theta must be known. If scale is negative GCV
is always used, which means that the scale parameter will be estimated by GCV and the Pearson
estimator, or in the case of the negative binomial theta will be estimated
in order to force the GCV/Pearson scale estimate to unity (if this is possible). For binomial models in
particular, it is probably worth comparing UBRE and GCV results; for ``overdispersed Poisson'' GCV is
probably more appropriate than UBRE. 
knots 
this is an optional list containing user specified knot values to be used for basis construction.
For the cr and cc bases the user simply supplies the knots to be used, and there must be the same number as the basis
dimension, k , for the smooth concerned. For the tp basis knots has two uses. Firstly, for large datasets
the calculation of the tp basis can be timeconsuming. The user can retain most of the advantages of the t.p.r.s.
approach by supplying a reduced set of covariate values from which to obtain the basis 
typically the number of covariate values used will be substantially
smaller than the number of data, and substantially larger than the basis dimension, k . The second possibility
is to avoid the eigendecomposition used to find the t.p.r.s. basis altogether and simply use
the basis implied by the chosen knots: this will happen if the number of knots supplied matches the
basis dimension, k . For a given basis dimension the second option is
faster, but gives poorer results (and the user must be quite careful in choosing knot locations).
Different terms can use different
numbers of knots, unless they share a covariate.

sp 
A vector of smoothing parameters for each term can be provided here.
Smoothing parameters must
be supplied in the order that the smooth terms appear in the model
formula. With fit method "magic" (see gam.control
and magic ) then negative elements indicate that the
parameter should be estimated, and hence a mixture of fixed and estimated
parameters is possible. With fit method "mgcv" , if sp is
supplied then all its elements must be positive. Note that fx=TRUE
in a smooth term overrides what is supplied here effectively setting the
smoothing parameter to zero. 
min.sp 
for fit method "magic" only, lower bounds can be
supplied for the smoothing parameters. Note that if this option is used then
the smoothing parameters sp , in the returned object, will need to be added to
what is supplied here to get the actual smoothing parameters. Lower bounds on the smoothing
parameters can sometimes help stabilize otherwise divergent PIRLS iterations. 
H 
With fit method "magic" a user supplied fixed quadratic
penalty on the parameters of the
GAM can be supplied, with this as its coefficient matrix. A common use of this term is
to add a ridge penalty to the parameters of the GAM in circumstances in which the model
is close to unidentifiable on the scale of the linear predictor, but perfectly well
defined on the response scale. 
gamma 
It is sometimes useful to inflate the model degrees of
freedom in the GCV or UBRE score by a constant multiplier. This allows
such a multiplier to be supplied if fit method is "magic" . 
fit 
If this argument is TRUE then gam sets up the model and fits it, but if it is
FALSE then the model is set up and an object G is returned which is the output from
gam.setup plus some extra items required to complete the GAM fitting process. 
G 
Usually NULL , but may contain the object returned by a previous call to gam with
fit=FALSE , in which case all other arguments are ignored except for gamma , family , control and fit . 
... 
further arguments for
passing on e.g. to gam.fit 
A generalized additive model (GAM) is a generalized linear model (GLM) in which the linear predictor is given by a user specified sum of smooth functions of the covariates plus a conventional parametric component of the linear predictor. A simple example is:
log(E(y_i))=f_1(x_1i)+f_2(x_2i)
where the (independent) response variables y_i~Poi, and f_1 and f_2 are smooth functions of covariates x_1 and x_2. The log is an example of a link function.
If absolutely any smooth functions were allowed in model fitting then maximum likelihood estimation of such models would invariably result in complex overfitting estimates of f_1 and f_2. For this reason the models are usually fit by penalized likelihood maximization, in which the model (negative log) likelihood is modified by the addition of a penalty for each smooth function, penalizing its `wiggliness'. To control the tradeoff between penalizing wiggliness and penalizing badness of fit each penalty is multiplied by an associated smoothing parameter: how to estimate these parameters, and how to practically represent the smooth functions are the main statistical questions introduced by moving from GLMs to GAMs.
The mgcv
implementation of gam
represents the smooth functions using
penalized regression splines, and by default uses basis functions for these splines that
are designed to be optimal, given the number basis functions used. The smooth terms can be
functions of any number of covariates and the user has some control over how smoothness of
the functions is measured.
gam
in mgcv
solves the smoothing parameter estimation problem by using the
Generalized Cross Validation (GCV) criterion
n D/(n  DoF)^2
or an UnBiased Risk Estimator (UBRE )criterion
D/n + 2 s DoF / n s
where D is the deviance, n the number of data, s
the scale parameter and
DoF the effective degrees of freedom of the model. Notice that UBRE is effectively
just AIC rescaled, but is only used when s is known. It is also
possible to replace D by the Pearson statistic (see gam.method
),
but this can lead to over smoothing. Smoothing parameters are chosen to
minimize the GCV or UBRE score for the model, and the main computational challenge solved
by the mgcv
package is to do this efficiently and reliably. Various
alternative numerical methods are provided: see gam.method
.
Broadly gam
works by first constructing basis functions and one or more quadratic penalty
coefficient matrices for each smooth term in the model formula, obtaining a model matrix for
the strictly parametric part of the model formula, and combining these to obtain a
complete model matrix (/design matrix) and a set of penalty matrices for the smooth terms.
Some linear identifiability constraints are also obtained at this point. The model is
fit using gam.fit
, a modification of glm.fit
. The GAM
penalized likelihood maximization problem is solved by penalized Iteratively
Reweighted Least Squares (IRLS) (see e.g. Wood 2000). At each iteration a penalized
weighted least squares problem is solved, and the smoothing parameters of that problem are
estimated by GCV or UBRE. Eventually both model parameter estimates and smoothing
parameter estimates converge. Alternatively the PIRLS scheme is iterated to
convergence for each trial set of smoothing parameters, and GCV or UBRE scores
are only evaluated on convergence  optimization is then `outer' to the PIRLS
loop: in this case extra derivatives have to be carried along with the PIRLS
iteration, to facilitate optimization, and gam.fit2
is used.
Five alternative basispenalty types are built in for representing model
smooths, but alternatives can easily be added (see smooth.construct
which
uses psplines to illustrate how to add new smooths).
The built in alternatives for univariate smooths terms are: a conventional penalized
cubic regression spline basis, parameterized in terms of the function values at the knots;
a cyclic cubic spline with a similar parameterization and thin plate regression splines.
The cubic spline bases are computationally very efficient, but require `knot' locations to be
chosen (automatically by default). The thin plate regression splines are optimal low rank
smooths which do not have knots, but are more computationally costly to set
up. Smooths of several variables can be represented using thin plate
regression splines, or tensor products of any available basis
including user defined bases (tensor product penalties are obtained automatically form
the marginal basis penalties). The t.p.r.s. basis is isotropic, so if this is not appropriate tensor
product terms should be used. Tensor product smooths have one penalty and smoothing parameter per marginal
basis, which means that the relative scaling of covariates is essentially determined automatically by GCV/UBRE.
The t.p.r.s. basis and cubic regression spline bases are both available with
either conventional `wiggliness penalties' or penalties augmented with a
shrinkage component: the conventional penalties treat some space of functions
as `completely smooth' and do not penalize such functions at all; the
penalties with extra shrinkage will zero a term altogether for high enough
smoothing parameters: gam.selection
has an example of the use of
such terms.
For any basis the user specifies the dimension of the basis for each smooth term. The dimension of the basis is one more than the maximum degrees of freedom that the term can have, but usually the term will be fitted by penalized maximum likelihood estimation and the actual degrees of freedom will be chosen by GCV. However, the user can choose to fix the degrees of freedom of a term, in which case the actual degrees of freedom will be one less than the basis dimension.
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 (2003). 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. The t.p.r.s. basis can become expensive to calculate for large datasets. In this case the user can supply a reduced set of knots to use in basis construction (see knots, in the argument list), or use tensor products of cheaper bases.
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. The values at neighbouring knots
are connected by sections of cubic polynomial constrained to be
continuous up to and including second derivative at the knots. The resulting curve
is 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.
Details of the underlying fitting methods are given in Wood (2000, 2004a).
If fit == FALSE
the function returns a list G
of items needed to
fit a GAM, but doesn't actually fit it.
Otherwise the function returns an object of class "gam"
as described in gamObject
.
If fit method "mgcv"
is selected, the code does not check for rank deficiency of the
model matrix that may result from lack of identifiability between the
parametric and smooth components of the model.
You must have more unique combinations of covariates than the model has total parameters. (Total parameters is sum of basis dimensions plus sum of nonspline 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.
With large datasets (more than a few thousand data) the "tp"
basis gets very slow to use: use the knots
argument as discussed above and
shown in the examples. Alternatively, for 1d smooths you can use the "cr"
basis and
for multidimensional smooths use te
smooths.
For data with many zeroes clustered together in the covariate space it is quite easy to set up
GAMs which suffer from identifiability problems, particularly when using Poisson or binomial
families. The problem is that with e.g. log or logit links, mean value zero corresponds to
an infinite range on the linear predictor scale. Some regularization is possible in such cases: see
gam.control
for details.
Simon N. Wood simon.wood@rproject.org
Front end design inspired by the S function of the same name based on the work of Hastie and Tibshirani (1990). Underlying methods owe much to the work of Wahba (e.g. 1990) and Gu (e.g. 2002).
Key References on this implementation:
Wood, S.N. (2000) Modelling and Smoothing Parameter Estimation with Multiple Quadratic Penalties. J.R.Statist.Soc.B 62(2):413428
Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95114
Wood, S.N. (2004a) Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Amer. Statist. Ass. 99:673686
Wood, S.N. (2004b) On confidence intervals for GAMs based on penalized regression splines. Technical Report 0412 Department of Statistics, University of Glasgow.
Wood, S.N. (2004c) Low rank scale invariant tensor product smooths for generalized additive mixed models. Technical Report 0413 Department of Statistics, University of Glasgow.
Key Reference on GAMs and related models:
Hastie (1993) in Chambers and Hastie (1993) Statistical Models in S. Chapman and Hall.
Hastie and Tibshirani (1990) Generalized Additive Models. Chapman and Hall.
Wahba (1990) Spline Models of Observational Data. SIAM
Background References:
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. 12:383398
Gu (2002) Smoothing Spline ANOVA Models, Springer.
O'Sullivan, Yandall and Raynor (1986) Automatic smoothing of regression functions in generalized linear models. J. Am. Statist.Ass. 81:96103
Wood (2001) mgcv:GAMs and Generalized Ridge Regression for R. R News 1(2):2025
Wood and Augustin (2002) GAMs with integrated model selection using penalized regression splines and applications to environmental modelling. Ecological Modelling 157:157177
http://www.stats.gla.ac.uk/~simon/
gamObject
, gam.models
, s
, predict.gam
,
plot.gam
, summary.gam
, gam.side
,
gam.selection
,mgcv
, gam.control
gam.check
, gam.neg.bin
, magic
,vis.gam
library(mgcv) set.seed(0) n<400 sig<2 x0 < runif(n, 0, 1) x1 < runif(n, 0, 1) x2 < runif(n, 0, 1) x3 < runif(n, 0, 1) f0 < function(x) 2 * sin(pi * x) f1 < function(x) exp(2 * x) f2 < function(x) 0.2*x^11*(10*(1x))^6+10*(10*x)^3*(1x)^10 f3 < function(x) 0*x f < f0(x0) + f1(x1) + f2(x2) e < rnorm(n, 0, sig) y < f + e b<gam(y~s(x0)+s(x1)+s(x2)+s(x3)) summary(b) plot(b,pages=1,residuals=TRUE) # same fit in two parts ..... G<gam(y~s(x0)+s(x1)+s(x2)+s(x3),fit=FALSE) b<gam(G=G) # an extra ridge penalty (useful with convergence problems) .... bp<gam(y~s(x0)+s(x1)+s(x2)+s(x3),H=diag(0.5,37)) print(b);print(bp);rm(bp) # set the smoothing parameter for the first term, estimate rest ... bp<gam(y~s(x0)+s(x1)+s(x2)+s(x3),sp=c(0.01,1,1,1)) plot(bp,pages=1);rm(bp) # set lower bounds on smoothing parameters .... bp<gam(y~s(x0)+s(x1)+s(x2)+s(x3),min.sp=c(0.001,0.01,0,10)) print(b);print(bp);rm(bp) # now a GAM with 3df regression spline term & 2 penalized terms b0<gam(y~s(x0,k=4,fx=TRUE,bs="tp")+s(x1,k=12)+s(x2,k=15)) plot(b0,pages=1) # now fit a 2d term to x0,x1 b1<gam(y~s(x0,x1)+s(x2)+s(x3)) par(mfrow=c(2,2)) plot(b1) par(mfrow=c(1,1)) # now simulate poisson data g<exp(f/4) y<rpois(rep(1,n),g) b2<gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=poisson) plot(b2,pages=1) # repeat fit using performance iteration gm < gam.method(gam="perf.magic") b3<gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=poisson,method=gm) plot(b3,pages=1) # a binary example g < (f5)/3 g < binomial()$linkinv(g) y < rbinom(g,1,g) lr.fit < gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=binomial) ## plot model components with truth overlaid in red op < par(mfrow=c(2,2)) for (k in 1:4) { plot(lr.fit,residuals=TRUE,select=k) xx < sort(eval(parse(text=paste("x",k1,sep="")))) ff < eval(parse(text=paste("f",k1,"(xx)",sep=""))) lines(xx,(ffmean(ff))/3,col=2) } par(op) anova(lr.fit) lr.fit1 < gam(y~s(x0)+s(x1)+s(x2),family=binomial) lr.fit2 < gam(y~s(x1)+s(x2),family=binomial) AIC(lr.fit,lr.fit1,lr.fit2) # and a pretty 2d smoothing example.... test1<function(x,z,sx=0.3,sz=0.4) { (pi**sx*sz)*(1.2*exp((x0.2)^2/sx^2(z0.3)^2/sz^2)+ 0.8*exp((x0.7)^2/sx^2(z0.8)^2/sz^2)) } n<500 old.par<par(mfrow=c(2,2)) x<runif(n);z<runif(n); xs<seq(0,1,length=30);zs<seq(0,1,length=30) pr<data.frame(x=rep(xs,30),z=rep(zs,rep(30,30))) truth<matrix(test1(pr$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) vis.gam(b4) par(old.par) # very large dataset example using knots n<10000 x<runif(n);z<runif(n); y<test1(x,z)+rnorm(n) ind<sample(1:n,1000,replace=FALSE) b5<gam(y~s(x,z,k=50),knots=list(x=x[ind],z=z[ind])) vis.gam(b5) # and a pure "knot based" spline of the same data b6<gam(y~s(x,z,k=100),knots=list(x= rep((1:100.5)/10,10), z=rep((1:100.5)/10,rep(10,10)))) vis.gam(b6,color="heat")