# R: Define tensor product smooths in GAM formulae

te {mgcv} | R Documentation

## Define tensor product smooths in GAM formulae

### Description

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.

### Usage

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

### Arguments

`...` | 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. See `choose.k` for further information.

`bs`

array (or single character string) specifying the type for each marginal basis. `"cr"` for cubic regression spline; `"cs"` for cubic regression spline with shrinkage; `"cc"` for periodic/cyclic cubic regression spline; `"tp"` for thin plate regression spline; `"ts"` for t.p.r.s. with extra shrinkage. 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.
`np` | `TRUE` to use the `normal parameterization' for a tensor product smooth. This parameterization represents any 1-d marginal smooths using a parameterization where the parameters are function values at `knots' spread evenly through the data. The parameterization makes the penalties easily interpretable, however it can reduce numerical stability.
`xt` | Either a single object, providing any extra information to be passed to each marginal basis constructor, or a list of such objects, one for each marginal basis.

### Details

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.

Note also that GAMs constructed from lower rank tensor product smooths are nested within GAMs constructed from higher rank tensor product smooths if the same marginal bases are used in both cases (the marginal smooths themselves are just special cases of tensor product smooths.)

The `normal parameterization' (`np=TRUE`) re-parameterizes the marginal smooths of a tensor product smooth so that the parameters are function values at a set of points spread evenly through the range of values of the covariate of the smooth. This means that the penalty of the tensor product associated with any particular covariate direction can be interpreted as the penalty of the appropriate marginal smooth applied in that direction and averaged over the smooth. Currently this is only done for marginals of a single variable. This parameterization can reduce numerical stability for when used with marginal smooths other than `"cc"`, `"cr"` and `"cs"`: if this causes problems, set `np=FALSE`.

The function does not evaluate the variable arguments.

### Value

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.
`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.
`mp` | `TRUE` is multiple penalties are to be used (default).
`np` | `TRUE` to re-parameterize 1-D marginal smooths in terms of function values (defualt).

### Author(s)

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

### References

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

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

### See Also

`s`,`gam`,`gamm`

### Examples

    # 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)

[Package mgcv version 1.3-27 Index]