derivative.cc3: Work Function for 'smoothSurvReg', currently nowhere used
In smoothSurv: Survival Regression with Smoothed Error Distribution

derivative.cc3

R Documentation

Work Function for 'smoothSurvReg', currently nowhere used

Description

Function to compute derivatives of all 'c' G-spline coefficients with respect to chosen (g - 3) coefficients such that the whole vector of g 'c' coefficients satisfies the constraints.

Usage

derivative.cc3(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 'c' G-spline coefficients which are expressed as a function of the remaining (g - 3) 'c' 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(knots)`.
`all`	If `TRUE`, matrix (g - 2) x g (there is one zero column) is returned. If `FALSE`, matrix (g - 2) x 3 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 can express the three 'c' coefficients as a function of the remaining g - 3 'c' coefficients in the following way.

c_{k} = \omega_{0,k} + \sum_{j\neq last.three}\omega_{j,k} c_j, % \qquad k \in last.three,

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

c = \Omega_{1,\cdot}^T + \Omega_{-1,\cdot}^T d

and if all == FALSE

c[last.three] = \Omega_{1,\cdot}^T + \Omega_{-1,\cdot}^T d.