# gam

##### Generalized additive models with integrated smoothness estimation

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). Multi-dimensional 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 S-PLUS 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 S-PLUS provides use gam from package `gam`

.

- Keywords
- models, regression, smooth

##### Usage

```
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,in.out,...)
```

##### Arguments

- 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 ~ .`

i - 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 b - 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'.
- 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 bi
- 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, < - 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 - 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 - 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 circumst - 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 - 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`

,`in.out`

,`control`

, - in.out
- optional list for initializing outer iteration. If supplied
then this must contain two elements:
`sp`

should be an array of initialization values for all smoothing parameters (there must be a value for all smoothing parameters, whether fixed or - ...
- further arguments for
passing on e.g. to
`gam.fit`

##### Details

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 $y_i \sim {\rm Poi }$, 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 Un-Biased 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 P-IRLS 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 P-IRLS
loop: in this case extra derivatives have to be carried along with the P-IRLS
iteration, to facilitate optimization, and `gam.fit2`

is used.

Five alternative basis-penalty types are built in for representing model
smooths, but alternatives can easily be added (see `smooth.construct`

which
uses p-splines 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, 2006b).

##### Value

- 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`

.

##### WARNINGS

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 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.

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 and
for multi-dimensional 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.

##### References

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):413-428

Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114

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

Wood, S.N. (2004b) On confidence intervals for GAMs based on penalized regression splines. Technical Report 04-12 Department of Statistics, University of Glasgow.

Wood, S.N. (2006a) Low rank scale invariant tensor product smooths for generalized additive mixed models. Biometrics

Wood S.N. (2006b) Generalized Additive Models: An Introduction with R. CRC Press.

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. 12:383-398

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. 81:96-103

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

##### See Also

`mgcv-package`

, `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`

##### Examples

```
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*(1-x))^6+10*(10*x)^3*(1-x)^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 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/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 <- (f-5)/3
g <- binomial()$linkinv(g)
y <- rbinom(g,1,g)
lr.fit <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=binomial)
## plot model components with truth overlaid in red
op <- par(mfrow=c(2,2))
for (k in 1:4) {
plot(lr.fit,residuals=TRUE,select=k)
xx <- sort(eval(parse(text=paste("x",k-1,sep=""))))
ff <- eval(parse(text=paste("f",k-1,"(xx)",sep="")))
lines(xx,(ff-mean(ff))/3,col=2)
}
par(op)
anova(lr.fit)
lr.fit1 <- gam(y~s(x0)+s(x1)+s(x2),family=binomial)
lr.fit2 <- gam(y~s(x1)+s(x2),family=binomial)
AIC(lr.fit,lr.fit1,lr.fit2)
# and a pretty 2-d smoothing example....
test1<-function(x,z,sx=0.3,sz=0.4)
{ (pi**sx*sz)*(1.2*exp(-(x-0.2)^2/sx^2-(z-0.3)^2/sz^2)+
0.8*exp(-(x-0.7)^2/sx^2-(z-0.8)^2/sz^2))
}
n<-500
old.par<-par(mfrow=c(2,2))
x<-runif(n);z<-runif(n);
xs<-seq(0,1,length=30);zs<-seq(0,1,length=30)
pr<-data.frame(x=rep(xs,30),z=rep(zs,rep(30,30)))
truth<-matrix(test1(pr$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")
```

*Documentation reproduced from package mgcv, version 1.3-22, License: GPL version 2 or later*