null.space.dimension {mgcv} | R Documentation |

## The basis of the space of un-penalized functions for a TPRS

### Description

The thin plate spline penalties give zero penalty to some
functions. The space of these functions is spanned by a set of
polynomial terms. `null.space.dimension`

finds the dimension of this space, `M`

, given
the number of covariates that the smoother is a function of, `d`

,
and the order of the smoothing penalty, `m`

. If `m`

does not
satisfy `2m>d`

then the smallest possible dimension
for the null space is found given `d`

and the requirement that
the smooth should be visually smooth.

### Usage

```
null.space.dimension(d,m)
```

### Arguments

`d` |
is a positive integer - the number of variables of which the t.p.s. is a function. |

`m` |
a non-negative integer giving the order of the penalty functional, or signalling that the default order should be used. |

### Details

Thin plate splines are only visually smooth if the order of the
wiggliness penalty, `m`

, satisfies `2m > d+1`

. If `2m<d+1`

then this routine finds the smallest
`m`

giving visual smoothness
for the given `d`

, otherwise the supplied `m`

is used. The null space dimension is given by:

`M=(m+d-1)!/(d!(m-1)!)`

which is the value returned.

### Value

An integer (array), the null space dimension
`M`

.

### Author(s)

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

### References

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

https://www.maths.ed.ac.uk/~swood34/

### See Also

### Examples

```
require(mgcv)
null.space.dimension(2,0)
```

*mgcv*version 1.9-0 Index]