# s

##### Defining smooths in GAM formulae

Function used in definition of smooth terms within
`gam`

model formulae. The function does not evaluate a (spline)
smooth - it exists purely to help set up a model using spline based smooths.

- Keywords
- models, regression, smooth

##### Usage

`s(..., k=-1,fx=FALSE,bs="tp",m=NA,by=NA,xt=NULL,id=NULL,sp=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.

`s(log(x))`

is fine, but`s(I(x/sd(x)))`

is not (see`predict.gam`

).- k
the dimension of the basis used to represent the smooth term. The default depends on the number of variables that the smooth is a function of.

`k`

should not be less than the dimension of the null space of the penalty for the term (see`null.space.dimension`

), but will be reset if it is. See`choose.k`

for further information.- fx
indicates whether the term is a fixed d.f. regression spline (

`TRUE`

) or a penalized regression spline (`FALSE`

).- bs
a two letter character string indicating the (penalized) smoothing basis to use. (eg

`"tp"`

for thin plate regression spline,`"cr"`

for cubic regression spline). see`smooth.terms`

for an over view of what is available.- m
The order of the penalty for this term (e.g. 2 for normal cubic spline penalty with 2nd derivatives when using default t.p.r.s basis).

`NA`

signals autoinitialization. Only some smooth classes use this. The`"ps"`

class can use a 2 item array giving the basis and penalty order separately.- 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). For the numeric

`by`

variable case the resulting smooth is not usually subject to a centering constraint (so the`by variable`

should not be added as an additional main effect). In the factor`by`

variable case a replicate of the smooth is produced for each factor level (these smooths will be centered, so the factor usually needs to be added as a main effect as well). See`gam.models`

for further details. A`by`

variable 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).- xt
Any extra information required to set up a particular basis. Used e.g. to set large data set handling behaviour for

`"tp"`

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 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). An`id`

with a factor`by`

variable causes the smooths at each factor level to have the same smoothing parameter.- 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`

.- 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

The function does not evaluate the variable arguments. To use this function to specify use of
your own smooths, note the relationships between the inputs and the output object and see the example
in `smooth.construct`

.

##### Value

A class `xx.smooth.spec`

object, where `xx`

is a basis identifying code given by
the `bs`

argument of `s`

. These `smooth.spec`

objects define smooths and are turned into
bases and penalties by `smooth.construct`

method functions.

The returned object contains the following items:

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

The dimension of the basis used to represent the smooth.

TRUE if the term is to be treated as a pure regression spline (with fixed degrees of freedom); FALSE if it is to be treated as a penalized regression spline

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

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

is the name of any `by`

variable as text (`"NA"`

for none).

A suitable text label for this smooth term.

The object passed in as argument `xt`

.

An identifying label or number for the smooth, linking it to other
smooths. Defaults to `NULL`

for no linkage.

array of smoothing parameters for the term (negative for
auto-estimation). Defaults to `NULL`

.

##### References

Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114

Wood S.N. (2017) Generalized Additive Models: An Introduction with R (2nd edition). Chapman and Hall/CRC Press.

##### See Also

##### Examples

```
# NOT RUN {
# example utilising `by' variables
library(mgcv)
set.seed(0)
n<-200;sig2<-4
x1 <- runif(n, 0, 1);x2 <- runif(n, 0, 1);x3 <- runif(n, 0, 1)
fac<-c(rep(1,n/2),rep(2,n/2)) # create factor
fac.1<-rep(0,n)+(fac==1);fac.2<-1-fac.1 # and dummy variables
fac<-as.factor(fac)
f1 <- exp(2 * x1) - 3.75887
f2 <- 0.2 * x1^11 * (10 * (1 - x1))^6 + 10 * (10 * x1)^3 * (1 - x1)^10
f<-f1*fac.1+f2*fac.2+x2
e <- rnorm(n, 0, sqrt(abs(sig2)))
y <- f + e
# NOTE: smooths will be centered, so need to include fac in model....
b<-gam(y~fac+s(x1,by=fac)+x2)
plot(b,pages=1)
# }
```

*Documentation reproduced from package mgcv, version 1.8-31, License: GPL (>= 2)*