R: Generalized additive models with integrated smoothness estimation

gam {mgcv} | R Documentation

## Generalized additive models with integrated smoothness estimation

### Description

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 and `te` smooths for smooths of several variables.

`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`.

### 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 ~ .` 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 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 ``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 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 ``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` 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 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. Negative elements indicate that the parameter should be estimated, and hence a mixture of fixed and estimated parameters is possible. However, when routine `mgcv` is used as the underlying smoothness estimation method (not the default), then all elements of `sp` must be positive, if it is supplied. Note that `fx=TRUE` in a smooth term over-rides what is supplied here effectively setting the smoothing parameter to zero.
`min.sp` | 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. This option cannot be used with `mgcv` as the undelying smoothness selection routine (but it is not the default).
`H` | 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. This option cannot be used with `mgcv` as the undelying smoothness selection routine (but it is not the default).
`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 (not used if underlying fit routine is non-default `mgcv`).
`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` containing what would be required to fit is returned is returned. See argument `G`.
`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`, `method` and `fit`.
`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 to be estimated, but those for fixed s.p.s are not used); `scale` is the typical scale of the GCV/UBRE function, for passing to the outer optimizer.
`...` | further arguments for passing on e.g. to `gam.fit` (such as `mustart`).

### 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~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. A better behaved alternative is GACV (again see `gam.method`). Smoothing parameters are chosen to minimize the GCV or UBRE/AIC 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 (P-IRLS) (see e.g. Wood 2000). Smoothing parameter selection is integrated in one of two ways. (i) `Performance iteration' uses the fact that at each P-IRLS iteration a penalized weighted least squares problem is solved, and the smoothing parameters of that problem can estimated by GCV or UBRE. Eventually, in most cases, both model parameter estimates and smoothing parameter estimates converge. (ii) 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 the P-IRLS iteration has to be differentiated, to facilitate optimization, and `gam.fit3` is used in place of `gam.fit`. The default is the second method, outer iteration.

Several 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. See `choose.k` for information on checking the basis dimension choise.

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. For this reason the default behaviour is to randomly subsample `max.knots` unique data locations if there are more than `max.knots` such, and to use the sub-sample for basis construction. The sampling is always done with the same random seed to ensure repeatability (does not reset R RNG). `max.knots` is 3000, by default. Both seed and `max.knots` can be modified using the `xt` argument to `s`. Alternatively the user can supply knots from which to construct a basis.

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 default underlying fitting methods are given in Wood (2004 and 2008). Some alternative methods are discussed in Wood (2000 and 2006).

### 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 non-default fit method 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.

### Author(s)

Simon N. Wood simon.wood@r-project.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).

### References

Key References on this implementation:

Wood, S.N. (2004) Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Amer. Statist. Ass. 99:673-686. [Default method for additive case (but no longer for generalized)]

Wood, S.N. (2008) Fast stable direct fitting and smoothness selection for generalized additive models. J.R.Statist.Soc.B 70(2): — [Default method for generalized additive model case]

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

Wood, S.N. (2006a) Low rank scale invariant tensor product smooths for generalized additive mixed models. Biometrics 62(4):1025-1036

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

Wood, S.N. (2006c) On confidence intervals for generalized additive models based on penalized regression splines. Australian and New Zealand Journal of Statistics. 48(4): 445-464.

Wood, S.N. (2000) Modelling and Smoothing Parameter Estimation with Multiple Quadratic Penalties. J.R.Statist.Soc.B 62(2):413-428 [The original paper, but no longer the default methods.]

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

http://www.maths.bath.ac.uk/~sw283/

### 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 with user defined 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")
    # varying the default large dataset behaviour via `xt'
    b7 <- gam(y~s(x,z,k=50,xt=list(max.knots=1000,seed=2)))
    vis.gam(b7)

[Package mgcv version 1.3-29 Index]