minPenalty {smoothSurv}

Minimize the penalty term under the two (mean and variance) constraints

Description

This function minimizes

\frac{1}{2}\sum_{j=m+1}^g \Bigl(\Delta^m a_j\Bigr)^2

with respect to a_1,\dots, a_g under the constraints

\sum_{j=1}^g c_j \mu_j = 0

and

\sum_{j=1}^g c_j (\mu_j^2 + \sigma_0^2) = 1,

where

c_j = \frac{\exp(a_j)}{\sum_{l=1}^g}\exp(a_l)

with one of a's fixed to zero.

Note that the minimum is always zero. We are thus mainly interested in the point where the minimum is reached.

Usage

minPenalty(knots = NULL, dist.range = c(-6, 6), by.knots = 0.3, sdspline = NULL,
    difforder = 3, init.c,
    maxiter = 200, rel.tolerance = 1e-10, toler.chol = 1e-15, toler.eigen = 1e-3,
    maxhalf = 10, debug = 0, info = TRUE)

Arguments

knots

A vector of knots \mu_1,\dots,\mu_g.

dist.range

Approximate minimal and maximal knot. If not given by knots the knots are determined as c(seq(0, dist.range[2], by = by.knots), seq(0, dist.range[1], by = -by.knots)). The sequence of knots is sorted and multiple entries are removed.

by.knots

The distance between the two knots used when building a vector of knots if these are not given by knots.

sdspline

Standard deviation \sigma_0^2 of the basis G-spline (here it appeares only in the variance constraint). If not given it is determined as 2/3 times the maximal distance between the two knots. If sdspline >= 1 it is changed to 0.9 to be able to satisfy the constraints.

difforder

The order of the finite difference used in the penalty term.

init.c

Optional vector of the initial values for the G-spline coefficients c, all values must lie between 0 and 1 and must sum up to 1.

maxiter

Maximum number of Newton-Raphson iterations.

rel.tolerance

(Relative) tolerance to declare the convergence. For this function, the convergence is declared if absolute value of the penalty is lower than rel.tolerance and if both constraints are satisfied up to rel.tolerance.

toler.chol

Tolerance to declare Cholesky decomposition singular.

toler.eigen

Tolerance to declare an eigen value of a matrix to be zero.

maxhalf

Maximum number of step-halving steps if updated estimate leads to a decrease of the objective function.

debug

If non-zero print debugging information.

info

If TRUE information concerning the iteration process is printed during the computation to the standard output.

Value

A list with the components “spline”, “penalty”, “warning”, “fail”.

spline

A data frame with columns named “Knot”, “SD basis”, “c coef.” and “a coef.” which gives the optimal values of c_1,\dots, c_g and a_1,\dots, a_g in the latter two columns. This data.frame can be further worked out using the function eval.Gspline.

penalty

The value of the penalty term when declaring convergence.

warning

Possible warnings concerning the convergence.

fail

Failure indicator. It is zero if everything went OK.

Author(s)

Arnošt Komárek [e-mail]

Examples

optimum <- minPenalty(knots=seq(-4.2, 4.2, by = 0.3), sdspline=0.2, difforder=3)
where <- optimum$spline
print(where)
show <- eval.Gspline(where, seq(-4.2, 4.2, by=0.05))
plot(show, type="l", bty="n", lwd=2)