A symbolic wrapper to indicate a smooth term in a formala argument to gam

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`...` |
the unspecified |

`span` |
the number of observations in a neighborhood. This is the smoothing
parameter for a |

`degree` |
the degree of local polynomial to be fit; currently
restricted to be |

`x` |
for |

`y` |
a response variable passed to |

`w` |
weights |

`ncols` |
for |

`xeval` |
If this argument is present, then |

A smoother in gam separates out the parametric part of the fit
from the non-parametric part. For local regression, the parametric
part of the fit is specified by the particular polynomial being fit
locally. The workhorse function `gam.lo`

fits the local
polynomial, then strips off this parametric part. All the parametric
pieces from all the terms in the additive model are fit
simultaneously in one operation for each loop of the backfitting
algorithm.

`lo`

returns a numeric matrix. The simplest case is when there is a
single argument to `lo`

and `degree=1`

; a one-column matrix is
returned, consisting of a normalized version of the vector. If
`degree=2`

in this case, a two-column matrix is returned, consisting
of a degree-2 polynomial basis. Similarly, if there are
two arguments, or the single argument is a two-column matrix, either a
two-column matrix is returned if `degree=1`

, or a five-column matrix
consisting of powers and products up to degree `2`

. Any dimensional
argument is allowed, but typically one or two vectors are used in
practice.

The matrix is endowed with a number of attributes; the matrix itself is
used in the construction of the model matrix, while the attributes are
needed for the backfitting algorithms `general.wam`

(weighted additive
model) or `lo.wam`

(currently not implemented). Local-linear curve
or surface fits reproduce linear responses, while local-quadratic fits
reproduce quadratic curves or surfaces. These parts of the `loess`

fit are computed exactly together with the other parametric linear parts

When two or more smoothing variables are given, the user should make
sure they are in a commensurable scale; `lo()`

does no
normalization. This can make a difference, since `lo()`

uses a
spherical (isotropic) neighborhood when establishing the nearest neighbors.

Note that `lo`

itself does no smoothing; it simply sets things up
for `gam`

; `gam.lo`

does the actual smoothing.
of the model.

One important attribute is named `call`

. For example, `lo(x)`

has a call component
`gam.lo(data[["lo(x)"]], z, w, span = 0.5, degree = 1, ncols = 1)`

.
This is an expression that gets evaluated repeatedly in `general.wam`

(the backfitting algorithm).

`gam.lo`

returns an object with components

`residuals` |
The residuals from the smooth fit. Note that the
smoother removes the parametric part of the fit (using a linear fit
with the columns in |

`nl.df` |
the nonlinear degrees of freedom |

`var` |
the pointwise variance for the nonlinear fit |

When `gam.lo`

is evaluated with an `xeval`

argument, it returns a
matrix of predictions.

Written by Trevor Hastie, following closely the design in the "Generalized Additive Models" chapter (Hastie, 1992) in Chambers and Hastie (1992).

Hastie, T. J. (1992)
*Generalized additive models.*
Chapter 7 of *Statistical Models in S*
eds J. M. Chambers and T. J. Hastie, Wadsworth \& Brooks/Cole.

Hastie, T. and Tibshirani, R. (1990)
*Generalized Additive Models.*
London: Chapman and Hall.

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