# Define tensor product smooths or tensor product interactions in GAM formulae Functions used for the definition of tensor product smooths and interactions within `gam` model formulae. `te` produces a full tensor product smooth, while `ti` produces a tensor product interaction, appropriate when the main effects (and any lower interactions) are also present. The functions do not evaluate the smooth - they exists purely to help set up a model using tensor product based smooths. Designed to construct tensor products from any marginal smooths with a basis-penalty representation (with the restriction that each marginal smooth must have only one penalty). ```r te(..., k=NA,bs="cr",m=NA,d=NA,by=NA,fx=FALSE, np=TRUE,xt=NULL,id=NULL,sp=NULL,pc=NULL) ti(..., k=NA,bs="cr",m=NA,d=NA,by=NA,fx=FALSE, np=TRUE,xt=NULL,id=NULL,sp=NULL,mc=NULL,pc=NULL) ``` ## Arguments - `...`: a list of variables that are the covariates that this smooth is a function of. Transformations whose form depends on the values of the data are best avoided here: e.g. `te(log(x),z)` is fine, but `te(I(x/sd(x)),z)` is not (see `predict.gam`). - `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. See `smooth.terms` for details and full list. User defined bases can also be used here (see `smooth.construct` for an example). If only one basis code is given then this is used for all bases. - `m`: The order of the spline and its penalty (for smooth classes that use this) for each term. If a single number is given then it is used for all terms. A vector can be used to supply a different `m` for each margin. For marginals that take vector `m` (e.g. `p.spline` and `Duchon.spline`), then a list can be supplied, with a vector element for each margin. `NA` autoinitializes. `m` is ignored by some bases (e.g. `"cr"`). - `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)`. Incompatibilities between built in basis types and dimension will be resolved by resetting the basis type. - `by`: a numeric or factor variable of the same dimension as each covariate. In the numeric vector case the elements multiply the smooth evaluated at the corresponding covariate values (a `varying coefficient model' results). In the factor case causes a replicate of the smooth to be produced for each factor level. See `gam.models` for further details. May also be a matrix if covariates are matrices: in this case implements linear functional of a smooth (see `gam.models` and `linear.functional.terms` for details). - `fx`: indicates whether the term is a fixed d.f. regression spline (`TRUE`) or a penalized regression spline (`FALSE`). - `np`: `TRUE` to use the `normal parameterization' for a tensor product smooth. This represents any 1-d marginal smooths via parameters that are function values at `knots', spread evenly through the data. The parameterization makes the penalties easily interpretable, however it can reduce numerical stability in some cases. - `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. - `id`: A label or integer identifying this term in order to link its smoothing parameters to others of the same type. If two or more smooth terms have the same `id` then they will have the same smoothing paramsters, and, by default, the same bases (first occurance defines basis type, but data from all terms used in basis construction). - `sp`: any supplied smoothing parameters for this term. Must be an array of the same length as the number of penalties for this smooth. Positive or zero elements are taken as fixed smoothing parameters. Negative elements signal auto-initialization. Over-rides values supplied in `sp` argument to `gam`. Ignored by `gamm`. - `mc`: For `ti` smooths you can specify which marginals should have centering constraints applied, by supplying 0/1 or `FALSE`/`TRUE` values for each marginal in this vector. By default all marginals are constrained, which is what is appropriate for, e.g., functional ANOVA models. Note that `'ti'` only applies constraints to the marginals, so if you turn off all marginal constraints the term will have no identifiability constraints. Only use this if you really understand how marginal constraints work. - `pc`: If not `NULL`, signals a point constraint: the smooth should pass through zero at the point given here (as a vector or list with names corresponding to the smooth names). Never ignored if supplied. See `identifiability`. ## 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. 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. It is sometimes useful to investigate smooth models with a main-effects + interactions structure, for example $$ f_1(x) + f_2(z) + f_3(x,z)f_1(x) + f_2(z) + f_3(x,z) $$ This functional ANOVA decomposition is supported by `ti` terms, which produce tensor product interactions from which the main effects have been excluded, under the assumption that they will be included separately. For example the `~ ti(x) + ti(z) + ti(x,z)` would produce the above main effects + interaction structure. This is much better than attempting the same thing with `s`or `te` terms representing the interactions (although mgcv does not forbid it). Technically `ti` terms are very simple: they simply construct tensor product bases from marginal smooths to which identifiability constraints (usually sum-to-zero) have already been applied: correct nesting is then automatic (as with all interactions in a GLM framework). See Wood (2017, section 5.6.3). 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 when used with marginal smooths other than `"cc"`, `"cr"` and `"cs"`: if this causes problems, set `np=FALSE`. Note that tensor product smooths should not be centred (have identifiability constraints imposed) if any marginals would not need centering. The constructor for tensor product smooths ensures that this happens. The function does not evaluate the variable arguments. ## Returns 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). - **id**: the `id` argument supplied to `te`. - **sp**: the `sp` argument supplied to `te`. - **inter**: `TRUE` if the term was generated by `ti`, `FALSE` otherwise. - **mc**: the argument `mc` supplied to `ti`. ## Author(s) Simon N. Wood simon.wood@r-project.org ## References Wood, S.N. (2006) Low rank scale invariant tensor product smooths for generalized additive mixed models. Biometrics 62(4):1025-1036 tools:::Rd_expr_doi("10.1111/j.1541-0420.2006.00574.x") Wood S.N. (2017) Generalized Additive Models: An Introduction with R (2nd edition). Chapman and Hall/CRC Press. tools:::Rd_expr_doi("10.1201/9781315370279") [https://www.maths.ed.ac.uk/~swood34/](https://www.maths.ed.ac.uk/~swood34/) ## See Also `s`,`gam`,`gamm`, `smooth.construct.tensor.smooth.spec` ## Examples ```r # following shows how tensor pruduct deals nicely with # badly scaled covariates (range of x 5% of range of z ) require(mgcv) 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~ ti(x) + ti(z) + ti(x,z)) vis.gam(b3);title("tensor anova") ## now illustrate partial ANOVA decomp... vis.gam(b3);title("full anova") b4 <- gam(y~ ti(x) + ti(x,z,mc=c(0,1))) ## note z constrained! vis.gam(b4);title("partial anova") plot(b4) par(old.par) ## now with a multivariate marginal.... 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 2D Duchon spline and 1D cr spline m <- list(c(1,.5),rep(0,0)) ## example of list form of m b <- gam(y~te(v,w,u,k=c(30,5),d=c(2,1),bs=c("ds","cr"),m=m)) plot(b) ```

te function