likelihood.smooth.spline {aroma.light} | R Documentation |
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(g|x) = (y-g(x))'W(y-g(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.
## S3 method for class 'smooth.spline': likelihood(object, x=NULL, y=NULL, w=NULL, base=exp(1), rel.tol=.Machine$double.eps^(1/8), ...)
object |
The smooth.spline object. |
x, y |
The x and y values for which the (weighted) likelihood will
be calculated. If x is of type xy.coords any value of
argument y will be omitted. If x==NULL , the x and y values
of the smoothing spline will be used. |
w |
The weights for which the (weighted) likelihood will be
calculated. If NULL , weights equal to one are assumed. |
base |
The base of the logarithm of the likelihood. If NULL ,
the non-logged likelihood is returned. |
rel.tol |
The relative tolerance used in the call to
integrate . |
... |
Not used. |
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
.
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*roughness . |
lambda |
the smoothing parameter |
roughness |
the value of the roughness functional given the specific smoothing spline and the range of data |
Henrik Bengtsson (http://www.braju.com/R/)
smooth.spline
and robustSmoothSpline
().
# 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 non-unique x[1:5] <- x[1] # Non-unique 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)