A symbolic wrapper to indicate a smooth term in a formala argument to gam
gam.lo(
x,
y,
w = rep(1, length(y)),
span = 0.5,
degree = 1,
ncols = p,
xeval = x
)lo(..., span = 0.5, degree = 1)
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
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 x), so these
residual represent the nonlinear part of the fit.
the nonlinear degrees of freedom
the pointwise variance for the nonlinear fit
When gam.lo is evaluated with an xeval argument, it returns a
matrix of predictions.
for gam.lo, the appropriate basis of polynomials generated
from the arguments to lo. These are also the variables that receive
linear coefficients in the GAM fit.
a response variable passed to gam.lo during backfitting
weights
the number of observations in a neighborhood. This is the
smoothing parameter for a loess fit. If specified, the full argument
name span must be written.
the degree of local polynomial to be fit; currently restricted
to be 1 or 2. If specified, the full argument name
degree must be written.
for gam.lo the number of columns in x used as the
smoothing inputs to local regression. For example, if degree=2, then
x has two columns defining a degree-2 polynomial basis. Both are
needed for the parameteric part of the fit, but ncol=1 telling the
local regression routine that the first column is the actually smoothing
variable.
If this argument is present, then gam.lo produces a
prediction at xeval.
the unspecified ...{} can be a comma-separated list of
numeric vectors, numeric matrix, or expressions that evaluate to either of
these. If it is a list of vectors, they must all have the same length.
Written by Trevor Hastie, following closely the design in the "Generalized Additive Models" chapter (Hastie, 1992) in Chambers and Hastie (1992).
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.
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.
y ~ Age + lo(Start)
# fit Start using a loess smooth with a (default) span of 0.5.
y ~ lo(Age) + lo(Start, Number)
y ~ lo(Age, span=0.3) # the argument name span cannot be abbreviated.
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