Smooth Data with a Positive Function using a Roughness Penalty
Smooth Data with a Positive Function using a Roughness Penalty
DESCRIPTION:
This function smooths data from a single curve or function using a
roughness penalty. Unlike function
smooth. basis, however, the function is constrained to be
positive. The function is defined by a functional data object
Wfdobj, which defines the logarithm of the positive function.
The roughness of Wfdobj, and hence of the positive smoothing
function, is definable in a wide variety of ways using either
derivatives or a linear differential operator.
USAGE:
posfd(y, argvals, Wfdobj, Lfd=3, lambda=0, conv=0.0001, iterlim=20,
dbglev=1)
REQUIRED ARGUMENTS:
- y
-
A vector of observations on a single curve.
- argvals
-
A vector of argument values corresponding to y.
- Wfdobj
-
A functional data object defining a single univariate function.
OPTIONAL ARGUMENTS:
- Lfd
-
Either a nonnegative integer or a linear differential operator object.
If present, the derivative or the value of applying the operator is
evaluated rather than the functions themselves.
- lambda
-
A nonnegative value controlling the amount of roughness in the
data.
- conv
-
A criterion for convergence of the iterations.
- iterlim
-
A limit on the number of iterations.
- dbglev
-
Either 0, 1, or 2. This controls the amount information printed out on
each iteration, with 0 implying no output, 1 intermediate output level,
and 2 full output.
VALUE:
A list containing:
- Wfdobj
-
A functional data object defining function W(x) that that
optimizes the fit to the data of the monotone function that it defines.
- Flist
- A list containing results for the final converged solution:
- f
- The optimal function value being minimized.
- grad
- The gradient vector at the optimal solution.
- norm
- The norm of the gradient vector at the optimal solution.
- iternum
- The number of iterations.
- iterhist
- A iternum+1 by 5 matrix containing the iteration
history.
DETAILS:
The computation of the positive function requires iterative
optimization of the fitting criterion, and is therefore more
computationally intensive than unconstrained smoothing. For this
reason, only one curve at a time is allowed to be smoothed.
The computational problem of estimating a positive function is very
similar to that for estimating a density function, except that there is
no normalizing constraint. For this reason, this function has much in
common with function densityfd.
To plot the positive function or to evaluate it, evaluate
Wfdobj, and then
exponentiate the resulting vector.
REFERENCES: