| D2ss {sfsmisc} | R Documentation |
Compute the numerical first or 2nd derivatives of f() given
observations (x[i], y ~= f(x[i])).
D1tr is the trivial discrete first derivative
using simple difference ratios, whereas D1ss and D2ss
use cubic smoothing splines (see smooth.spline)
to estimate first or second derivatives, respectively.
D2ss first uses smooth.spline for the first derivative
f'() and then applies the same to the predicted values
f'^(t[i]) (where t[i] are the values of
xout) to find f''^(t[i]).
D1tr(y, x = 1) D1ss(x, y, xout = x, spar.offset = 0.1384, spl.spar=NULL) D2ss(x, y, xout = x, spar.offset = 0.1384, spl.spar=NULL)
x,y |
numeric vectors of same length, supposedly from a model
y ~ f(x). For D1tr(), x can have length one
and then gets the meaning of h = Delta x. |
xout |
abscissa values at which to evaluate the derivatives. |
spar.offset |
numeric fudge added to the smoothing parameter(s),
see spl.par below. Note that the current default is there
for historical reasons only, and we often would recommend to use
spar.offset = 0 instead. |
spl.spar |
direct smoothing parameter(s) for smooth.spline.
If it is NULL (as per default), the smoothing parameter used
will be spar.offset + sp$spar, where sp$spar is the GCV
estimated smoothing parameter for both smooths, see
smooth.spline. |
It is well known that for derivative estimation, the optimal smoothing
parameter is larger (more smoothing needed) than for the function itself.
spar.offset is really just a fudge offset added to the
smoothing parameters. Note that in R's implementation of
smooth.spline, spar is really on the
logλ scale.
D1tr() and D1ss() return a numeric vector of the length
of y or xout, respectively.
D2ss() returns a list with components
x |
the abscissae values (= xout) at which the
derivative(s) are evaluated. |
y |
estimated values of f''(x_i). |
spl.spar |
numeric vector of length 2, contain the spar
arguments to the two smooth.spline calls. |
spar.offset |
as specified on input (maybe rep()eated to length 2). |
Martin Maechler, in 1992 (for S).
D1D2 which directly uses the 2nd derivative of
the smoothing spline; smooth.spline.
## First Derivative --- spar.off = 0 ok "asymptotically" (?)
set.seed(330)
mult.fig(12)
for(i in 1:12) {
x <- runif(500, 0,10); y <- sin(x) + rnorm(500)/4
f1 <- D1ss(x=x,y=y, spar.off=0.0)
plot(x,f1, ylim = range(c(-1,1,f1)))
curve(cos(x), col=3, add= TRUE)
}
set.seed(8840)
x <- runif(100, 0,10)
y <- sin(x) + rnorm(100)/4
op <- par(mfrow = c(2,1))
plot(x,y)
lines(ss <- smooth.spline(x,y), col = 4)
str(ss[c("df", "spar")])
xx <- seq(0,10, len=201)
plot(xx, -sin(xx), type = 'l', ylim = c(-1.5,1.5))
title(expression("Estimating f''() : " * frac(d^2,dx^2) * sin(x) == -sin(x)))
offs <- c(0.05, 0.1, 0.1348, 0.2)
i <- 1
for(off in offs) {
d12 <- D2ss(x,y, spar.offset = off)
lines(d12, col = i <- i+1)
}
legend(2,1.6, c("true : -sin(x)",paste("sp.off. = ", format(offs))), lwd=1,
col = 1:(1+length(offs)), cex = 0.8, bg = NA)
par(op)