gam {mgcv}R Documentation

Generalized Additive Models using penalized regression splines and GCV

Description

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 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).

Usage

 gam(formula,family=gaussian(),data=list(),weights=NULL,control=gam.control,scale=0)

Arguments

formula A GAM formula. 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") 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.
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.
family This is a family object specifying the distribution and link to use in fitting etc. See glm and family for more details. Where the family is neg.binom then a negative binomial family is used based on the implementation in the MASS library. In this case, if the value of theta is not given, a version of glm.nb :gam.nbut is used to estimate theta iteratively, starting from a Poisson distribution. This extra layer of iteration slows down fitting.
data A data frame containing the model response variable and covariates required by the formula. If this is missing then the frame from which gam was called is searched for the variables specified in the formula.
weights prior weights on the data.
control A list as returned by gam.control, with five user controllable elements: maxit controls maximum iterations in gam.fit, convergence tolerance in gam.fit is controlled by epsilon and the third item is trace. The smoothing parameter selection method is controlled by two further items: mgcv.tol controls the convergence tolerance to use in smoothing parameter estimation, while mgcv.half.max controls the maximum number of step halvings to try in each optimization step if the step fails to reduce the GCV score.
scale If this is zero then GCV is used for all distributions except Poisson, binomial and negative 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. If scale is negative GCV is always used (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.)

Details

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 (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).

Value

The function returns an object of class "gam" which has the following elements:

coefficients the coefficients of the fitted model. Parametric coefficients are first, followed by coefficients for each spline term in turn.
residuals the deviance residuals for the fitted model.
fitted.values fitted model predictions of expected value for each datum.
family family object specifying distribution and link used.
linear.predictor fitted model prediction of link function of expected value for each datum.
deviance (unpenalized)
null.deviance deviance for single parameter model.
df.null null degrees of freedom
iter number of iterations of IRLS taken to get convergence.
weights final weights used in IRLS iteration.
prior.weights prior weights on observations.
df.null number of data
y response data.
converged indicates whether or not the iterative fitting method converged.
sig2 estimated or supplied variance/scale parameter.
edf estimated degrees of freedom for each smooth.
boundary did parameters end up at boundary of parameter space?
sp smoothing parameter for each smooth.
df number of knots for each smooth (one more than maximum degrees of freedom).
nsdf number of parametric, non-smooth, model terms including the intercept.
Vp estimated covariance matrix for parameter estimators.
covariate.shift covariates get shifted so that they are centred around zero - this is by how much.
xp knot locations for each cubic regression spline based smooth. xp[i,] are the locations for the ith smooth.
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.
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.
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.
x parametric design matrix columns (including intercept) followed by the data that form arguments of the smooths.
s.type type of spline basis used: 0 for conventional cubic regression spline, 1 for t.p.r.s.
p.order the order of the penalty used for each term. 0 signals auto-selection.
dim number of covariates of which term is a function
call a mode call object containing the call to gam() that produced this gam object (useful for constructing model frames).
mgcv.conv A list of smoothing parameter convergence diagnostics, with the following elements (irrelevant for models with only one smoothing parameter to estimate):
g
the gradient of the GCV/UBRE score w.r.t. the smoothing parameters at termination.
h
the second derivatives corresponding to g above - i.e. the leading diagonal of the Hessian.
e
the eigen-values of the Hessian. These should all be non-negative!
iter
the number of iterations taken.
in.ok
TRUE if the second smoothing parameter guess improved the GCV/UBRE score. (Please report examples where this is FALSE)
step.fail
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.

WARNINGS

The code does not check for rank defficiency of the model matrix -it will likely just fail instead!

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!

Author(s)

Simon N. Wood snw@st-and.ac.uk

References

Hastie and Tibshirani (1990) Generalized Additive Models. Chapman and Hall.

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

http://www.ruwpa.st-and.ac.uk/simon.html

See Also

s predict.gam plot.gam

Examples

library(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)

[Package Contents]