Hess_rho_mult: Hessian matrix of LCV and LAML wrt rho (log smoothing parameters). Version for multiplicative decomposition : relative mortality ratio model

Description

Hessian matrix of LCV and LAML wrt rho (log smoothing parameters). Version for multiplicative decomposition : relative mortality ratio model

Usage

Hess_rho_mult(
  X_GL,
  X_GL_Q,
  GL_temp,
  haz_GL,
  deriv2_rho_beta,
  deriv_rho_beta,
  weights,
  tm,
  nb_smooth,
  p,
  n_legendre,
  deriv_rho_inv_Hess_beta,
  deriv_rho_Hess_unpen_beta,
  S_list,
  minus_eigen_inv_Hess_beta,
  temp_LAML,
  temp_LAML2,
  Vp,
  S_beta,
  beta,
  inverse_new_S,
  X,
  X_Q,
  event,
  expected,
  Ve,
  deriv_rho_Ve,
  mat_temp,
  deriv_mat_temp,
  eigen_mat_temp,
  method
)

Value

Hessian matrix of LCV or LAML wrt rho

Arguments

X_GL: list of matrices (length(X.GL)=n.legendre) for Gauss-Legendre quadrature
X_GL_Q: list of transformed matrices from X_GL in order to calculate only the diagonal of the fourth derivative of the likelihood
GL_temp: list of vectors used to make intermediate calculations and save computation time
haz_GL: list of all the matrix-vector multiplications X.GL[[i]]%*%beta for Gauss Legendre integration in order to save computation time
deriv2_rho_beta: second derivatives of beta wrt rho (implicit differentiation)
deriv_rho_beta: firt derivatives of beta wrt rho (implicit differentiation)
weights: vector of weights for Gauss-Legendre integration on [-1;1]
tm: vector of midpoints times for Gauss-Legendre integration; tm = 0.5*(t1 - t0)
nb_smooth: number of smoothing parameters
p: number of regression parameters
n_legendre: number of nodes for Gauss-Legendre quadrature
deriv_rho_inv_Hess_beta: list of first derivatives of Vp wrt rho
deriv_rho_Hess_unpen_beta: list of first derivatives of Hessian of unpenalized log likelihood wrt rho
S_list: List of all the rescaled penalty matrices multiplied by their associated smoothing parameters
minus_eigen_inv_Hess_beta: vector of eigenvalues of Vp
temp_LAML: temporary matrix used when method="LAML" to save computation time
temp_LAML2: temporary matrix used when method="LAML" to save computation time
Vp: Bayesian covariance matrix
S_beta: List such that S_beta[[i]]=S_list[[i]]%*%beta
beta: vector of estimated regression parameters
inverse_new_S: inverse of the penalty matrix
X: design matrix for the model
X_Q: transformed design matrix in order to calculate only the diagonal of the fourth derivative of the likelihood
event: vector of right-censoring indicators
expected: vector of expected hazard rates
Ve: frequentist covariance matrix
deriv_rho_Ve: list of derivatives of Ve wrt rho
mat_temp: temporary matrix used when method="LCV" to save computation time
deriv_mat_temp: list of derivatives of mat_temp wrt rho
eigen_mat_temp: vector of eigenvalues of mat_temp
method: criterion used to select the smoothing parameters. Should be "LAML" or "LCV"; default is "LAML"