derivative.expAD: Work Function for 'smoothSurvReg', currently nowhere used

In smoothSurv: Survival Regression with Smoothed Error Distribution

Work Function for 'smoothSurvReg', currently nowhere used

Description

Function to compute derivatives of exp(a) w.r.t. exp(d) where d stands for a shorter vector of 'a' G-spline coefficients.

Usage

derivative.expAD(knots, sdspline, last.three, all = TRUE)

Arguments

knots	A vector of G-spline knots \mu.
sdspline	Standard deviation \sigma_0 of the basis G-spline.
last.three	Indeces of the three 'a' G-spline coefficients which are expressed as a function of the remaining (g-3) 'a' G-spline coefficients such that the three constraints are satisfied. This must be a vector of length 3 with three different numbers from 1:length(mu). Consequently, a[last.three[1]] <- 0.
all	If TRUE, matrix (g - 2) x g (there is one zero column) is returned. If FALSE, matrix (g - 2) x 2 is returned. The first row is always an intercept. See details.

Details

To satisfy the three constraints

\sum_{j=1}^g c_j = 1,

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

\sum_{j=1}^g c_j \mu_j^2 = 1 - \sigma_0^2

imposed on the G-spline we use the following parametrization:

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

The constraints can be solved such that a[last.three[1]] = 0 and a[last.three[2:3]] are expressed as a function of a[-last.three] in the following way:

a_{k} = \log\Bigl\{\omega_{0,k} + \sum_{j\neq last.three}\omega_{j,k}\exp(a_j)\Bigr\},% \qquad k = last.three[2], last.three[3],

where \omega coefficients are a function of knots and G-spline standard deviation. If we denote d the vector a[-last.three] this function computes derivatives of \exp(a) w.r.t. \exp(d) together with the intercept term used to compute \exp(a) from \exp(d). This is actually a matrix of \omega coefficients. If we denote it as \Omega then if all == TRUE

\exp(a) = \Omega_{1,\cdot}^T + \Omega_{-1,\cdot}^T\exp(d)

and if all == FALSE

\exp(a[last.three[2:3]]) = \Omega_{1,\cdot}^T + \Omega_{-1,\cdot}^T\exp(d).

Value

A matrix with \omega coefficients.

Note

WARNING: This function is primarily used inside smoothSurvReg. Consequently, it has very few error checks on its input arguments.

Author(s)

Arnošt Komárek [e-mail]

smoothSurv documentation built on June 22, 2024, 11:11 a.m.