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(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.

Usage

## S3 method for class 'smooth.spline'
likelihood(object, x=NULL, y=NULL, w=NULL, base=exp(1),
  rel.tol=.Machine$double.eps^(1/8), ...)

Arguments

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:

Author(s)

Henrik Bengtsson

See Also

smooth.spline and robustSmoothSpline().

Examples

# 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)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(aroma.light)
aroma.light v3.2.0 (2016-01-06) successfully loaded. See ?aroma.light for help.
> png(filename="/home/ddbj/snapshot/RGM3/R_BC/result/aroma.light/likelihood.smooth.spline.Rd_%03d_medium.png", width=480, height=480)
> ### Name: likelihood.smooth.spline
> ### Title: Calculate the log likelihood of a smoothing spline given the
> ###   data
> ### Aliases: likelihood.smooth.spline
> ### Keywords: methods smooth internal
> 
> ### ** Examples
> 
> # 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")
Log likelihood with unique x values:
> print(l)
Likelihood of smoothing spline: -293865.1 
 Log base: 2.718282 
 Weighted residuals sum of square: 293865.2 
 Penalty: -0.1210083 
 Smoothing parameter lambda: 0.0009257147 
 Roughness score: 130.7187 
> 
> # 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");

Log likelihood of the *spline* data set:
> print(l)
Likelihood of smoothing spline: -293228.5 
 Log base: 2.718282 
 Weighted residuals sum of square: 293228.6 
 Penalty: -0.1210751 
 Smoothing parameter lambda: 0.0009261969 
 Roughness score: 130.7229 
> 
> # 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");

Log likelihood of the *original* data set:
> print(l)
Likelihood of smoothing spline: -293864.2 
 Log base: 2.718282 
 Weighted residuals sum of square: 293864.3 
 Penalty: -0.1210758 
 Smoothing parameter lambda: 0.0009261969 
 Roughness score: 130.7236 
> 
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> 
> dev.off()
null device 
          1 
>