# Assembling Model Terms for Smoothing Spline ANOVA Models Assemble numerical functions for calculating model terms in a smoothing spline ANOVA model. ```r mkterm(mf, type) ``` ## Arguments - `mf`: Model frame of the model formula. - `type`: List specifying the type of spline for each variable. ## Background Tensor-product splines are constructed based on the model formula and the marginal reproducing kernels, as described in Gu (2002, Sec. 2.4). The marginal variables can be factors, numerical vectors, and numerical matrices, as specified in the details section. One-way ANOVA decompositions are built in the supported marginal constructions, in which one has the constant, a "nonparametric contrast," and possibly also a "parametric contrast." To the "nonparametric contrast" there corresponds a reproducing kernal `rk`, and to a "parametric contrast" there corresponds a set of null space basis `phi`. The reproducing kernels and null space basis on the product domain can be constructed from the marginal `rk` and `phi` in a systematic manner. The marginal one-way ANOVA structures induce a multi-way ANOVA structure on the product domain, with model terms consisting of unpenalized "parametric contrasts" and/or penalized "nonparametric contrasts." One only needs to construct `rk`'s and `phi`'s associated with the model terms implied by the model formula. ## Details For a factor `x`, `type$x` is ignored; `mkrk.ordinal` is used if `is.ordered(x)==TRUE` and `mkrk.nominal` is used otherwise. Factors with 3 or more levels are penalized. For a numerical vector `x`, `type$x` is of the form `type.x` for `type.x`=`"cubic"`, `"linear"`, or of the form `list(type.x, range)` for `type.x`=`"per"`, `"cubic.per"`, `"linear.per"`, `"cubic"`, `"linear"`; `"per"` is short for `"cubic.per"`. See `mkfun.poly` for the functions used. For `type.x` missing, the default is `"cubic"`. For `range` missing with `type.x`=`"cubic"`, `"linear"`, the default is `c(min(x),max(x))+c(-1,1)*(max(x)-mimn(x))*.05`. For a numerical matrix `x`, `type$x` is of the form `type.x` or `list(type.x, order)` for `type.x`=`"tp"`, `"sphere"`, or of the form `list("tp",list(order=order,mesh=mesh,weight=weight))`. See `mkfun.tp` for the functions used. For `type.x` missing, the default is `"tp"`. For `order` missing, the default is `2`. For `mesh` and `weight` missing with `type.x`=`"tp"` and `order` given, the defaults are `mesh`=`x` and `weight`=`1`. For a numerical vector or numerical matrix `x`, one may also use `type$x` of the form `list("custom",list(nphi=nphi,mkphi=mkphi,mkrk=mkrk,env=env))`; `nphi` is the null space dimension **excluding the constant**, and `mkphi` is ignored if `nphi`=0. See examples below. This feature allows the use of other marginal constructions; one may modify `mkphi.cubic` or `mkphi.tp.p` for `mkphi` and modify `mkrk.cubic` or `mkrk.sphere` for `mkrk`. ## Note For a numerical vector `x` in `ssden`, the default `range` is `domain$x`. For a numerical matrix `x` with `type.x`=`"sphere"`, it is assumed that `dim(x)[2]==2`, `x[,1]` between [-90,90] the latitude in degrees, and `x[,2]` between [-180,180] the longitude in degrees. For backward compatibility , one may set `type="cubic"`, `"linear"`, or `"tp"`, but then the default parameters can not be overridden; the type is simply duplicated for each variable. ## Returns A list object with an element `labels` containing the labels of all model terms. For each of the model terms, there is an element holding the numerical functions for calculating the unpenalized and penalized parts within the term. ## References Gu, C. (2013), **Smoothing Spline ANOVA Models (2nd Ed)**. New York: Springer-Verlag. ## Examples ```r ## cubic marginals x1 <- rnorm(100); x2 <- rnorm(100); y <- 3+5*sin(x1-2*x2)+rnorm(x1) fit <- ssanova0(y~x1*x2) ## the same fit fit1 <- ssanova0(y~x1*x2,type=list(x1="cubic")) ## the same fit one more time par <- list(nphi=1,mkphi=mkphi.cubic,mkrk=mkrk.cubic, env=c(min(x2),max(x2))+c(-1,1)*(max(x2)-min(x2))*.05) fit2 <- ssanova0(y~x1*x2,type=list(x2=list("custom",par))) ## Clean up ## Not run: rm(x1,x2,y,fit,fit1,par,fit2) ## cubic and thin-plate marginals x1 <- rnorm(100); x2 <- matrix(rnorm(200),100,2) y <- 3+5*sin(x1-2*x2[,1]*x2[,2])+rnorm(x1) fit <- ssanova0(y~x1*x2) ## the same fit fit1 <- ssanova0(y~x1*x2,type=list(x2="tp")) ## the same fit one more time mkphi.tp1 <- function(x) mkphi.tp(x$dm,x$ord,x$mesh,x$wt) mkrk.tp1 <- function(x) mkrk.tp(x$dm,x$ord,x$mesh,x$wt) env <- list(dm=2,ord=2,mesh=x2,wt=1) par <- list(nphi=2,mkphi=mkphi.tp1,mkrk=mkrk.tp1,env=env) fit2 <- ssanova0(y~x1*x2,type=list(x2=list("custom",par))) ## Clean up ## Not run: rm(x1,x2,y,fit,fit1,mkphi.tp1,mkrk.tp1,env,par,fit2) ```

mkterm function