Calculate the log likelihood of a smoothing spline given the data
Description
Calculate the (log) likelihood of a spline given the data used to fit the spline, g. The likelihood consists of two main parts: 1) (weighted) residuals sum of squares, and 2) a penalty term. The penalty term consists of a smoothing parameter lambda and a roughness measure of the spline J(g) = \int g''(t) dt. Hence, the overall log likelihood is
\log L(gx) = (yg(x))'W(yg(x)) + λ J(g)
In addition to the overall likelihood, all its seperate components are also returned.
Note: when fitting a smooth spline with (x,y) values where the
x's are not unique, smooth.spline
will replace
such (x,y)'s with a new pair (x,y') where y' is a
reweighted average on the original y's. It is important to
be aware of this. In such cases, the resulting smooth.spline
object does not contain all (x,y)'s and therefore this
function will not calculate the weighted residuals sum of square on
the original data set, but on the data set with unique x's.
See examples below how to calculate the likelihood for the spline with
the original data.
Usage
1 2 3 
Arguments
object 
The smooth.spline object. 
x, y 
The x and y values for which the (weighted) likelihood will
be calculated. If 
w 
The weights for which the (weighted) likelihood will be
calculated. If 
base 
The base of the logarithm of the likelihood. If 
rel.tol 
The relative tolerance used in the call to

... 
Not used. 
Details
The roughness penalty for the smoothing spline, g, fitted from data in the interval [a,b] is defined as
J(g) = \int_a^b g''(t) dt
which is the same as
J(g) = g'(b)  g'(a)
The latter is calculated internally by using
predict.smooth.spline
.
Value
Returns the overall (log) likelihood of class
SmoothSplineLikelihood
, a class with the following attributes:
wrss 
the (weighted) residual sum of square 
penalty 
the penalty which is equal to 
lambda 
the smoothing parameter 
roughness 
the value of the roughness functional given the specific smoothing spline and the range of data 
Author(s)
Henrik Bengtsson
See Also
smooth.spline
and robustSmoothSpline
().
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  # Define f(x)
f < expression(0.1*x^4 + 1*x^3 + 2*x^2 + x + 10*sin(2*x))
# Simulate data from this function in the range [a,b]
a < 2; b < 5
x < seq(a, b, length=3000)
y < eval(f)
# Add some noise to the data
y < y + rnorm(length(y), 0, 10)
# Plot the function and its second derivative
plot(x,y, type="l", lwd=4)
# Fit a cubic smoothing spline and plot it
g < smooth.spline(x,y, df=16)
lines(g, col="yellow", lwd=2, lty=2)
# Calculating the (log) likelihood of the fitted spline
l < likelihood(g)
cat("Log likelihood with unique x values:\n")
print(l)
# Note that this is not the same as the log likelihood of the
# data on the fitted spline iff the x values are nonunique
x[1:5] < x[1] # Nonunique x values
g < smooth.spline(x,y, df=16)
l < likelihood(g)
cat("\nLog likelihood of the *spline* data set:\n");
print(l)
# In cases with non unique x values one has to proceed as
# below if one want to get the log likelihood for the original
# data.
l < likelihood(g, x=x, y=y)
cat("\nLog likelihood of the *original* data set:\n");
print(l)
