gam                   package:mgcv                   R Documentation

_G_e_n_e_r_a_l_i_z_e_d _a_d_d_i_t_i_v_e _m_o_d_e_l_s _w_i_t_h _i_n_t_e_g_r_a_t_e_d _s_m_o_o_t_h_n_e_s_s _e_s_t_i_m_a_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

     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'.

_U_s_a_g_e:

     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,...)

_A_r_g_u_m_e_n_t_s:

 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
          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 ``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.
          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
          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 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"'.

     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', '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').

_D_e_t_a_i_l_s:

     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. 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).

_V_a_l_u_e:

     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'.

_W_A_R_N_I_N_G_S:

     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.

_A_u_t_h_o_r(_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).

_R_e_f_e_r_e_n_c_e_s:

     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

     <URL: http://www.maths.bath.ac.uk/~sw283/>

_S_e_e _A_l_s_o:

     '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'

_E_x_a_m_p_l_e_s:

     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")

