| gam {mgcv} | R Documentation |
Fits the specified generalized additive model (GAM) to
data. gam() is not a clone of what Splus provides.
Smooth terms are represented using penalized regression splines
with smoothing parameters selected by GCV/UBRE 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). For more on specifying models see gam.models. For more on model
selection see gam.selection.
gam(formula,family=gaussian(),data=list(),weights=NULL,subset=NULL,
na.action,control=gam.control(),scale=0,knots=NULL,sp=NULL,
min.sp=NULL,H=NULL,gamma=1,...)
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^(d-1) 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 muber of degrees of freedom (fx=FALSE
to select d.f. by GCV/UBRE). bs indicates the basis to use, with
"cr" indicating cubic regression spline, and "tp"
indicating thin plate regression spline: "cr" can only be used
with 1-d smooths. 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 in GAMs
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: see s.
For backwards compatibility the formula may also include terms like s(x,12|f), which specifies a regression spline which is not to be penalized
and has 12 knots, or s(x,z,25) indicating a rank 25 penalized
t.p.r.s. In such cases arguements k, fx and bs are
ignored if supplied and a one dimensional term will always use a cubic
regression spline basis. Note that a term of the form s(x) will
result in a term with a "tp" basis.
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.conditions 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 ``factory-fresh'' default is `na.omit'. |
control |
A list of fit control parameters returned by
gam.control. |
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 ``over-dispersed 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 basis 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 time-consuming. 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 eigen-decomposition 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 over-rides what is supplied here exectively 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 P-IRLS 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 un-identifiable 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". |
... |
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 (idependent) 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 or an Un-Biased Risk Estimator criterion
(UBRE) which is in practice an approximation to AIC. 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. Two alternative
numerical methods are provided, see mgcv, magic and
gam.control.
Broadly gam works by first constructing basis functions and a quadratic penalty
coefficient matrix 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.
The fitting approach just described, in which the smoothing parameters are estimated for
each approximating linear model of the IRLS process was suggested by Chong Gu (see, e.g.
Gu 2002), and is very computationally efficient. However, because the approach neglects
the dependence of the iterative weights on the smoothing parameters, it is usually
possible to find smoothing parameters which actually yield slightly lower GCV/UBRE score
estimates than those resulting from this `performance iteration'. gam therefore also
allows the user to `improve' the smoothing parameter estimates, by using O'Sullivan's
(1986) suggested method, in which for each trial set of smoothing parameters the IRLS is
iterated to convergance before the UBRE/GCV score is evaluated. This requires much less
efficient minimisation of the power iteration based on nlm, and is
therefore quite slow.
Two alternative bases are available for representing model terms. Univariate smooth terms can be represented using conventional cubic regression splines - which are very efficient computationally - or thin plate regression splines. Multivariate terms must be represented using thin plate regression splines. For either 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).
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 constrainted 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 "mgcv" GCV/UBRE minimization method are given in Wood (2000):
the basis of the approach
is to alternate efficient global optimization with respect to one overall smoothing
parameter with Newton updates of a set of relative smoothing parameters for each smooth term. The Newton updates are
backed up by steepest descent, since the GCV/UBRE score functions are not positive definite everywhere.
The function returns an object of class "gam" which has the following elements:
boundary |
did parameters end up at boundary of parameter space? |
by |
a 2-d array of by variables (i.e. covariates that multiply a
smooth term) by[i,j] is the jth value for the ith by
variable. There are only as many rows of this array as there are
by variables in the model (often 0). The rownames of by
give the by variable names. |
by.exists |
an array of logicals: by.exists[i] is
TRUE if the ith smooth has a by variable associated with
it, FALSE otherwise. |
call |
the matched call (allows update to be used with gam objects, for example). |
coefficients |
the coefficients of the fitted model. Parametric coefficients are first, followed by coefficients for each spline term in turn. |
converged |
indicates whether or not the iterative fitting method converged. |
covariate.shift |
covariates get shifted so that they are centred around zero - this is by how much. |
deviance |
(unpenalized) |
df |
The maximum degrees of freedom for each smooth term plus one. In the "cr" basis case
this is the number of knots used. For the "tp" basis it is the rank of the smooth (one greater than the
maximum degrees of freedom because of the centering constraint). |
df.null |
null degrees of freedom. |
dim |
number of covariates of which term is a function. |
edf |
estimated degrees of freedom for each smooth. |
family |
family object specifying distribution and link used. |
fit.method |
The underlying multiple GCV/UBRE method used: "magic"
for the new more stable method, "mgcv" for the Wood (2000) method. |
fitted.values |
fitted model predictions of expected value for each datum. |
formula |
the model formula. |
full.formula |
the model formula with each smooth term fully expanded and with option arguments given explicitly (i.e. not with reference to other variables) - useful for later prediction from the model. |
gcv.ubre |
The minimized GCV or UBRE score. |
gcv.used |
TRUE if GCV used for smoothing parameter selection, FALSE if UBRE used. |
hat |
array of elements from the leading diagonal of the `hat' (or `influence') matrix. Same length as response data vector. |
iter |
number of iterations of P-IRLS taken to get convergence. |
linear.predictor |
fitted model prediction of link function of expected value for each datum. |
mgcv.conv |
A list of covergence diagnostics relating to smoothing
parameter estimation. Differs for method "magic" and "mgcv". Here is
the "mgcv" version:
g above - i.e. the leading diagonal of the Hessian.TRUE if the second smoothing parameter guess improved the GCV/UBRE score.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.In the case of "magic" the items are:
TRUE is multiple GCV/UBRE converged by meeting
convergence criteria. FALSE if method stopped with a steepest descent step
failure. |
min.edf |
Minimum possible degrees of freedom for whole model. |
model |
model frame containing all variables needed in original model fit. |
nsdf |
number of parametric, non-smooth, model terms including the intercept. |
null.deviance |
deviance for single parameter model. |
p.order |
the order of the penalty used for each term. 0 signals auto-selection. |
prior.weights |
prior weights on observations. |
residuals |
the deviance residuals for the fitted model. |
sig2 |
estimated or supplied variance/scale parameter. |
sp |
smoothing parameter for each smooth. |
s.type |
type of spline basis used: 0 for conventional cubic regression spline, 1 for t.p.r.s. |
UZ |
array storing the matrices for transforming from t.p.r.s. basis to equivalent t.p.s. basis - see GAMsetup for details of how the matrices are packed in this array. |
Vp |
estimated covariance matrix for parameter estimators. |
weights |
final weights used in IRLS iteration. |
xp |
knot locations for each cubic regression spline based smooth.
xp[i,]+covariate.shift[i] are the locations for the ith smooth. |
Xu |
The set of unique covariate locations used to define t.p.s. from which t.p.r.s. basis was derived. Again see GAMsetup for details of the packing algorithm. |
xu.length |
The number of unique covariate combinations in the data. |
y |
response data. |
The "mgcv" code does not check for rank defficiency 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 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: use the knots argument as discussed above and
shown in the examples. Alternatively, for 1-d smooths you can use the "cr" basis.
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@stats.gla.ac.uk
Key References:
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
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. 12:383-398
Gu (2002) Smoothing Spline ANOVA Models, Springer.
Hastie and Tibshirani (1990) Generalized Additive Models. Chapman and Hall.
O'Sullivan, Yandall and Raynor (1986) Automatic smoothing of regression functions in generalized linear models. J. Am. Statist.Ass. 81:96-103
Wahba (1990) Spline Models of Observational Data. SIAM
Wood (2001) mgcv:GAMs and Generalized Ridge Regression for R. R News 1(2):20-25
Wood and Augustin (2002) GAMs with integrated model selection using penalized regression splines and applications to environmental modelling. Ecological Modelling 157:157-177
http://www.stats.gla.ac.uk/~simon/
gam.models, s, predict.gam,
plot.gam, summary.gam, gam.side.conditions,
gam.selection,mgcv, gam.control
gam.check, gam.neg.bin, magic,vis.gam
library(mgcv)
set.seed(0)
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))
summary(b)
plot(b,pages=1)
# an extra ridge penalty (useful with convergence problems) ....
bp<-gam(y~s(x0)+s(x1)+s(x2)+s(x3),H=diag(0.5,41))
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(x2,15))
plot(b0,pages=1)
# now fit a 2-d 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/5)
y<-rpois(rep(1,n),g)
b2<-gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=poisson)
plot(b2,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)
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((1:10-0.5)/10,10),
z=rep((1:10-0.5)/10,rep(10,10))))
vis.gam(b6,color="heat")