smooth.spline: Fit a Smoothing Spline

Description

Fits a cubic smoothing spline to the supplied data.

Usage

smooth.spline(x, y = NULL, w = NULL, df, spar = NULL, lambda = NULL, cv = FALSE,
              all.knots = FALSE, nknots = .nknots.smspl,
              keep.data = TRUE, df.offset = 0, penalty = 1,
              control.spar = list(), tol = 1e-6 * IQR(x), keep.stuff = FALSE)

Arguments

a vector giving the values of the predictor variable, or a list or a two-column matrix specifying x and y.

responses. If y is missing or NULL, the responses are assumed to be specified by x, with x the index vector.

optional vector of weights of the same length as x; defaults to all 1.

the desired equivalent number of degrees of freedom (trace of the smoother matrix). Must be in $(1,n_x]$, $n_x$ the number of unique x values, see below.

spar

smoothing parameter, typically (but not necessarily) in $(0,1]$. When spar is specified, the coefficient $\lambda$ of the integral of the squared second derivative in the fit (penalized log likelihood) criterion is a monotone function of spar, see the details below. Alternatively lambda may be specified instead of the scale free spar=$s$.

lambda

if desired, the internal (design-dependent) smoothing parameter $\lambda$ can be specified instead of spar. This may be desirable for resampling algorithms such as cross validation or the bootstrap.

ordinary leave-one-out (TRUE) or ‘generalized’ cross-validation (GCV) when FALSE; is used for smoothing parameter computation only when both spar and df are not specified; it is used however to determine cv.crit in the result. Setting it to NA for speedup skips the evaluation of leverages and any score.

all.knots

if TRUE, all distinct points in x are used as knots. If FALSE (default), a subset of x[] is used, specifically x[j] where the nknots indices are evenly spaced in 1:n, see also the next argument nknots.

Alternatively, a strictly increasing numeric vector specifying “all the knots” to be used; must be rescaled to $[0, 1]$ already such that it corresponds to the ans $ fit$knots sequence returned, not repeating the boundary knots.

nknots

integer or function giving the number of knots to use when all.knots = FALSE. If a function (as by default), the number of knots is nknots(nx). By default for $n_x > 49$ this is less than $n_x$, the number of unique x values, see the Note.

keep.data

logical specifying if the input data should be kept in the result. If TRUE (as per default), fitted values and residuals are available from the result.

df.offset

allows the degrees of freedom to be increased by df.offset in the GCV criterion.

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 spar is computed, i.e., missing or NULL, see below.

Note that this is partly experimental and may change with general spar computation improvements!

low:: lower bound for spar; defaults to -1.5 (used to implicitly default to 0 in R versions earlier than 1.4).
high:: upper bound for spar; defaults to +1.5.
tol:: the absolute precision (tolerance) used; defaults to 1e-4 (formerly 1e-3).
eps:: the relative precision used; defaults to 2e-8 (formerly 0.00244).
trace:: logical indicating if iterations should be traced.
maxit:: integer giving the maximal number of iterations; defaults to 500.

Note that spar is only searched for in the interval $[low, high]$.

tol

a tolerance for same-ness or uniqueness of the x values. The values are binned into bins of size tol and values which fall into the same bin are regarded as the same. Must be strictly positive (and finite).

keep.stuff

an experimental logical indicating if the result should keep extras from the internal computations. Should allow to reconstruct the $X$ matrix and more.

Value

An object of class "smooth.spline" with components

the distinct x values in increasing order, see the ‘Details’ above.

the fitted values corresponding to x.

the weights used at the unique values of x.

yin

the y values used at the unique y values.

tol

the tol argument (whose default depends on x).

data

only if keep.data = TRUE: itself a list with components x, y and w of the same length. These are the original $(x_i,y_i,w_i), i = 1, \dots, n$, values where data$x may have repeated values and hence be longer than the above x component; see details.

lev

(when cv was not NA) leverages, the diagonal values of the smoother matrix.

cv.crit

cross-validation score, ‘generalized’ or true, depending on cv. The CV score is often called “PRESS” (and labeled on print()), for ‘PREdiction Sum of Squares’.

pen.crit

the penalized criterion, a non-negative number; simply the (weighted) residual sum of squares (RSS), sum(.$w * residuals(.)^2) .

crit

the criterion value minimized in the underlying .Fortran routine sslvrg. When df has been specified, the criterion is $3 + (tr(S_\lambda) - df)^2$, where the $3 +$ is there for numerical (and historical) reasons.

equivalent degrees of freedom used. Note that (currently) this value may become quite imprecise when the true df is between and 1 and 2.

spar

the value of spar computed or given, unless it has been given as c(lambda = *), when it set to NA here.

ratio

(when spar above is not NA), the value $r$, the ratio of two matrix traces.

lambda

the value of $\lambda$ corresponding to spar, see the details above.

iparms

named integer(3) vector where ..$ipars["iter"] gives number of spar computing iterations used.

auxMat

experimental; when keep.stuff was true, a “flat” numeric vector containing parts of the internal computations.

fit

list for use by predict.smooth.spline, with components

knot:: the knot sequence (including the repeated boundary knots), scaled into $[0, 1]$ (via min and range).
nk:: number of coefficients or number of ‘proper’ knots plus 2.
coef:: coefficients for the spline basis used.
min, range:: numbers giving the corresponding quantities of x.

call

the matched call.

method(class = "smooth.spline") shows a hatvalues() method based on the lev vector above.

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.

Unless lambda has been specified instead of spar, the computational $\lambda$ used (as a function of $s=spar$) is $\lambda = r * 256^{3 s - 1}$ where $r = tr(X' W X) / tr(\Sigma)$, $\Sigma$ is the matrix given by $\Sigma_{ij} = \int B_i''(t) B_j''(t) dt$, $X$ is given by $X_{ij} = B_j(x_i)$, $W$ is the diagonal matrix of weights (scaled such that its trace is $n$, the original number of observations) and $B_k(.)$ is the $k$-th B-spline.

Note that with these definitions, $f_i = f(x_i)$, and the B-spline basis representation $f = X c$ (i.e., $c$ is the vector of spline coefficients), the penalized log likelihood is $L = (y - f)' W (y - f) + \lambda c' \Sigma c$, and hence $c$ is the solution of the (ridge regression) $(X' W X + \lambda \Sigma) c = X' W y$.

If spar and lambda are missing or NULL, the value of df is used to determine the degree of smoothing. If df is missing as well, leave-one-out cross-validation (ordinary or ‘generalized’ as determined by cv) is used to determine $\lambda$.

Note that from the above relation, spar is $s = s0 + 0.0601 * \bold{\log}\lambda$, which is intentionally different from the S-PLUS implementation of smooth.spline (where spar is proportional to $\lambda$). In R's ($\log \lambda$) 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! Similarly, specifying lambda instead of spar is delicate, notably as the range of “safe” values for lambda is not scale-invariant and hence entirely data dependent.

The ‘generalized’ cross-validation method GCV will work correctly when there are duplicated points in x. However, it is ambiguous what leave-one-out cross-validation 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.

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.

Examples

Run this code

# NOT RUN {
require(graphics)
plot(dist ~ speed, data = cars, main = "data(cars)  &  smoothing splines")
cars.spl <- with(cars, smooth.spline(speed, dist))
cars.spl
## This example has duplicate points, so avoid cv = TRUE
# }
# NOT RUN {
lines(cars.spl, col = "blue")
ss10 <- smooth.spline(cars[,"speed"], cars[,"dist"], df = 10)
lines(ss10, 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')


## Residual (Tukey Anscombe) plot:
plot(residuals(cars.spl) ~ fitted(cars.spl))
abline(h = 0, col = "gray")

## consistency check:
stopifnot(all.equal(cars$dist,
                    fitted(cars.spl) + residuals(cars.spl)))
## The chosen inner knots in original x-scale :
with(cars.spl$fit, min + range * knot[-c(1:3, nk+1 +1:3)]) # == unique(cars$speed)

## 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)

## Specifying 'lambda' instead of usual spar :
(s2. <- smooth.spline(y18, lambda = s2$lambda, tol = s2$tol))

# }
# NOT RUN {
## The following shows the problematic behavior of 'spar' searching:
(s2  <- smooth.spline(y18, control =
                      list(trace = TRUE, tol = 1e-6, low = -1.5)))
(s2m <- smooth.spline(y18, cv = TRUE, control =
                      list(trace = TRUE, tol = 1e-6, low = -1.5)))
## both above do quite similarly (Df = 8.5 +- 0.2)
# }

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