Fit a Smoothing Spline
Description
Fits a cubic smoothing spline to the supplied data.
Usage
1 2 3 4 
Arguments
x 
a vector giving the values of the predictor variable, or a list or a twocolumn matrix specifying x and y. 
y 
responses. If 
w 
optional vector of weights of the same length as 
df 
the desired equivalent number of degrees of freedom (trace of the smoother matrix). Must be in (1,nx], nx the number of unique x values, see below. 
spar 
smoothing parameter, typically (but not necessarily) in
(0,1]. The coefficient λ of the integral of the
squared second derivative in the fit (penalized log likelihood)
criterion is a monotone function of 
cv 
ordinary ( 
all.knots 
if 
nknots 
integer or 
keep.data 
logical specifying if the input data should be kept
in the result. If 
df.offset 
allows the degrees of freedom to be increased by

penalty 
the coefficient of the penalty for degrees of freedom in the GCV criterion. 
control.spar 
optional list with named components controlling the
root finding when the smoothing parameter Note that this is partly experimental and may change with general spar computation improvements!
Note that 
tol 
a tolerance for sameness or uniqueness of the 
Details
Neither x
nor y
are allowed to containing missing or
infinite values.
The x
vector should contain at least four distinct values.
‘Distinct’ here is controlled by tol
: values which are
regarded as the same are replaced by the first of their values and the
corresponding y
and w
are pooled accordingly.
The computational λ used (as a function of \code{spar}) is λ = r * 256^(3*spar  1) where r = tr(X' W X) / tr(Σ), Σ is the matrix given by Σ[i,j] = Integral B''[i](t) B''[j](t) dt, X is given by X[i,j] = B[j](x[i]), W is the diagonal matrix of weights (scaled such that its trace is n, the original number of observations) and B[k](.) is the kth Bspline.
Note that with these definitions, f_i = f(x_i), and the Bspline basis representation f = X c (i.e., c is the vector of spline coefficients), the penalized log likelihood is L = (y  f)' W (y  f) + λ c' Σ c, and hence c is the solution of the (ridge regression) (X' W X + λ Σ) c = X' W y.
If spar
is missing or NULL
, the value of df
is used to
determine the degree of smoothing. If both are missing, leaveoneout
crossvalidation (ordinary or ‘generalized’ as determined by
cv
) is used to determine λ.
Note that from the above relation,
spar
is spar = s0 + 0.0601 * log(λ),
which is intentionally different from the SPLUS implementation
of smooth.spline
(where spar
is proportional to
λ). In R's (log λ) scale, it makes more
sense to vary spar
linearly.
Note however that currently the results may become very unreliable
for spar
values smaller than about 1 or 2. The same may
happen for values larger than 2 or so. Don't think of setting
spar
or the controls low
and high
outside such a
safe range, unless you know what you are doing!
The ‘generalized’ crossvalidation method will work correctly when
there are duplicated points in x
. However, it is ambiguous what
leaveoneout crossvalidation means with duplicated points, and the
internal code uses an approximation that involves leaving out groups
of duplicated points. cv = TRUE
is best avoided in that case.
Value
An object of class "smooth.spline"
with components
x 
the distinct 
y 
the fitted values corresponding to 
w 
the weights used at the unique values of 
yin 
the y values used at the unique 
data 
only if 
lev 
(when 
cv.crit 
crossvalidation score, ‘generalized’ or true, depending
on 
pen.crit 
penalized criterion 
crit 
the criterion value minimized in the underlying

df 
equivalent degrees of freedom used. Note that (currently)
this value may become quite imprecise when the true 
spar 
the value of 
lambda 
the value of λ corresponding to 
iparms 
named integer(3) vector where 
fit 
list for use by

call 
the matched call. 
Note
The number of unique x
values, nx, are
determined by the tol
argument, equivalently to
1  nx < length(x)  sum(duplicated( round((x  mean(x)) / tol) ))

all.knots = FALSE
and nknots = .nknots.smspl
,
entails using only O(nx ^ 0.2)
knots instead of nx for nx > 49. This cuts
speed and memory requirements, but not drastically anymore since R
version 1.5.1 where it is only O(nk) + O(n) where
nk is the number of knots.
In this case where not all unique x
values are
used as knots, the result is not a smoothing spline in the strict
sense, but very close unless a small smoothing parameter (or large
df
) is used.
Author(s)
R implementation by B. D. Ripley and Martin Maechler
(spar/lambda
, etc).
Source
This function is based on code in the GAMFIT
Fortran program by
T. Hastie and R. Tibshirani (originally taken from
http://lib.stat.cmu.edu/general/, which is now defunct but
mirrors may still be available),
which makes use of spline code by Finbarr O'Sullivan. Its design
parallels the smooth.spline
function of Chambers & Hastie (1992).
References
Chambers, J. M. and Hastie, T. J. (1992) Statistical Models in S, Wadsworth & Brooks/Cole.
Green, P. J. and Silverman, B. W. (1994) Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman and Hall.
Hastie, T. J. and Tibshirani, R. J. (1990) Generalized Additive Models. Chapman and Hall.
See Also
predict.smooth.spline
for evaluating the spline
and its derivatives.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56  require(graphics)
attach(cars)
plot(speed, dist, main = "data(cars) & smoothing splines")
cars.spl < smooth.spline(speed, dist)
(cars.spl)
## This example has duplicate points, so avoid cv = TRUE
lines(cars.spl, col = "blue")
lines(smooth.spline(speed, dist, df = 10), lty = 2, col = "red")
legend(5,120,c(paste("default [C.V.] => df =",round(cars.spl$df,1)),
"s( * , df = 10)"), col = c("blue","red"), lty = 1:2,
bg = 'bisque')
detach()
## Residual (Tukey Anscombe) plot:
plot(residuals(cars.spl) ~ fitted(cars.spl))
abline(h = 0, col = "gray")
## consistency check:
stopifnot(all.equal(cars$dist,
fitted(cars.spl) + residuals(cars.spl)))
## Visualize the behavior of .nknots.smspl()
nKnots < Vectorize(.nknots.smspl) ; c.. < adjustcolor("gray20",.5)
curve(nKnots, 1, 250, n=250)
abline(0,1, lty=2, col=c..); text(90,90,"y = x", col=c.., adj=.25)
abline(h=100,lty=2); abline(v=200, lty=2)
n < c(1:799, seq(800, 3490, by=10), seq(3500, 10000, by = 50))
plot(n, nKnots(n), type="l", main = "Vectorize(.nknots.smspl) (n)")
abline(0,1, lty=2, col=c..); text(180,180,"y = x", col=c..)
n0 < c(50, 200, 800, 3200); c0 < adjustcolor("blue3", .5)
lines(n0, nKnots(n0), type="h", col=c0)
axis(1, at=n0, line=2, col.ticks=c0, col=NA, col.axis=c0)
axis(4, at=.nknots.smspl(10000), line=.5, col=c..,col.axis=c.., las=1)
## artificial example
y18 < c(1:3, 5, 4, 7:3, 2*(2:5), rep(10, 4))
xx < seq(1, length(y18), len = 201)
(s2 < smooth.spline(y18)) # GCV
(s02 < smooth.spline(y18, spar = 0.2))
(s02. < smooth.spline(y18, spar = 0.2, cv = NA))
plot(y18, main = deparse(s2$call), col.main = 2)
lines(s2, col = "gray"); lines(predict(s2, xx), col = 2)
lines(predict(s02, xx), col = 3); mtext(deparse(s02$call), col = 3)
## The following shows the problematic behavior of 'spar' searching:
(s2 < smooth.spline(y18, control =
list(trace = TRUE, tol = 1e6, low = 1.5)))
(s2m < smooth.spline(y18, cv = TRUE, control =
list(trace = TRUE, tol = 1e6, low = 1.5)))
## both above do quite similarly (Df = 8.5 + 0.2)

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