te                   package:mgcv                   R Documentation

_D_e_f_i_n_e _t_e_n_s_o_r _p_r_o_d_u_c_t _s_m_o_o_t_h_s _i_n _G_A_M _f_o_r_m_u_l_a_e

_D_e_s_c_r_i_p_t_i_o_n:

     Function used in definition of tensor product smooth terms within
     'gam' model formulae. The function does not evaluate a smooth - it
     exists purely to help set up a model using tensor product  based
     smooths.

_U_s_a_g_e:

     te(..., k=NA,bs="cr",m=0,d=NA,by=NA,fx=FALSE,mp=TRUE)

_A_r_g_u_m_e_n_t_s:

     ...: a list of variables that are the covariates that this smooth
          is a function of.

       k: the dimension(s) of the bases used to represent the smooth
          term. If not supplied then set to '5^d'. If supplied as a
          single number then this  basis dimension is used for each
          basis. If supplied as an array then the elements are the
          dimensions of the component (marginal) bases of the tensor
          product.

      bs: array (or single character string) specifying the type for
          each  marginal basis. '"cr"' for cubic regression spline;
          '"cc"' for periodic/cyclic  cubic regression spline; '"tp"'
          for thin plate regression spline. User defined bases can 
          also be used here (see 'smooth.construct' for an example). If
          only one  identifier is given then this is used for all
          bases.

       m: The order of the penalty for each t.p.r.s. term (e.g. 2 for
          normal cubic spline penalty with 2nd derivatives). If a
          single number is given  then it is used for all terms. '0'
          autoinitializes. 'm' is ignored for the  '"cr"' and '"cc"'
          bases.

       d: array of marginal basis dimensions. For example if you want a
          smooth for 3 covariates  made up of a tensor product of a 2
          dimensional t.p.r.s. basis and a 1-dimensional basis, then 
          set 'd=c(2,1)'.

      by: specifies a covariate by which the whole smooth term is to be
          multiplied. This is particularly useful for creating models
          in which a smooth interacts with a factor: in this case the
          'by' variable would usually be the dummy variable coding one
          level of the factor. See the examples below.

      fx: indicates whether the term is a fixed d.f. regression spline
          ('TRUE') or a penalized regression spline ('FALSE').

      mp: 'TRUE' to use multiple penalties for the smooth. 'FALSE' to
          use only  a single penalty: single penalties are not
          recommended - they tend to allow only rather  wiggly models.

_D_e_t_a_i_l_s:

     Smooths of several covariates can be constructed from tensor
     products of the bases used to represent smooths of one (or
     sometimes more) of the covariates. To do this `marginal' bases are
     produced with associated model matrices and penalty matrices, and
     these are then combined in the manner described in
     'tensor.prod.model.matrix' and 'tensor.prod.penalties', to produce
      a single model matrix for the smooth, but multiple penalties (one
     for each marginal basis). The basis dimension  of the whole smooth
     is the product of the basis dimensions of the marginal smooths.

     An option for operating with a single penalty (The Kronecker
     product of the marginal penalties) is provided, but  it is rarely
     of practical use: the penalty is typically so rank deficient that
     even the smoothest resulting model  will have rather high
     estimated degrees of freedom. 

     Tensor product smooths are especially useful for representing
     functions of covariates measured in different units,  although
     they are typically not quite as nicely behaved as t.p.r.s. smooths
     for well scaled covariates.

     The function does not evaluate the variable arguments.

_V_a_l_u_e:

     A class 'tensor.smooth.spec' object defining a tensor product
     smooth to be turned into a basis and penalties by the
     'smooth.construct.tensor.smooth.spec' function. 

     The returned object contains the following items:

  margin: A list of 'smooth.spec' objects of the type returned by 's', 
          defining the basis from which the tensor product smooth is
          constructed.

    term: An array of text strings giving the names of the covariates
          that  the term is a function of.

      by: is the name of any 'by' variable as text ('"NA"' for none).

      fx: logical array with element for each penalty of the term
          (tensor product smooths have multiple penalties). 'TRUE' if
          the penalty is to  be ignored, 'FALSE', otherwise. 

 p.order: The order of the t.p.r.s. penalty, or 0 for auto-selection of
          the penalty order.

full.call: Text for pasting into a string to be converted to a gam
          formula, which has the values of function options given
          explicitly - this is useful for constructing a fully expanded
          gam formula which can be used without needing access to any
          variables that may have been used to define k, fx, bs or m in
          the original call. i.e. this is text which when parsed and
          evaluated generates a call to 's()' with all the options
          spelled out explicitly.

   label: A suitable text label for this smooth term.

     dim: The dimension of the smoother - i.e. the number of covariates
          that it is a function of.

_A_u_t_h_o_r(_s):

     Simon N. Wood simon@stats.gla.ac.uk

_S_e_e _A_l_s_o:

     's','gam','gamm'

_E_x_a_m_p_l_e_s:

     # following shows how tensor pruduct deals nicely with 
     # badly scaled covariates (range of x 5% of range of z )
     test1<-function(x,z,sx=0.3,sz=0.4)  
     { x<-x*20
       (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)/20;z<-runif(n);
     xs<-seq(0,1,length=30)/20;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)
     f <- test1(x,z)
     y <- f + rnorm(n)*0.2
     b1<-gam(y~s(x,z))
     persp(xs,zs,truth);title("truth")
     vis.gam(b1);title("t.p.r.s")
     b2<-gam(y~te(x,z))
     vis.gam(b2);title("tensor product")
     b3<-gam(y~te(x,z,bs=c("tp","tp")))
     vis.gam(b3);title("tensor product")
     par(old.par)

     test2<-function(u,v,w,sv=0.3,sw=0.4)  
     { ((pi**sv*sw)*(1.2*exp(-(v-0.2)^2/sv^2-(w-0.3)^2/sw^2)+
       0.8*exp(-(v-0.7)^2/sv^2-(w-0.8)^2/sw^2)))*(u-0.5)^2*20
     }
     n <- 500
     v <- runif(n);w<-runif(n);u<-runif(n)
     f <- test2(u,v,w)
     y <- f + rnorm(n)*0.2
     # tensor product of a 2-d thin plate regression spline and 1-d cr spline
     b <- gam(y~te(v,w,u,k=c(30,5),d=c(2,1),bs=c("tp","cr")))
     op <- par(mfrow=c(2,2))
     vis.gam(b,cond=list(u=0),color="heat",zlim=c(-0.2,3.5))
     vis.gam(b,cond=list(u=.33),color="heat",zlim=c(-0.2,3.5))
     vis.gam(b,cond=list(u=.67),color="heat",zlim=c(-0.2,3.5))
     vis.gam(b,cond=list(u=1),color="heat",zlim=c(-0.2,3.5))
     par(op)

