derivative.cc3: 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 differe

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.

Value

A matrix with $\omega$ coefficients.

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