CVEK (version 0.1-2)

estimate_ridge: Estimating a Single Model

Description

Estimating projection matrices and parameter estimates for a single model.

Usage

estimate_ridge(
  Y,
  X,
  K,
  lambda,
  compute_kernel_terms = TRUE,
  converge_thres = 1e-04
)

Arguments

Y

(matrix, n*1) The vector of response variable.

X

(matrix, n*d_fix) The fixed effect matrix.

K

(list of matrices) A nested list of kernel term matrices, corresponding to each kernel term specified in the formula for a base kernel function in kern_func_list.

lambda

(numeric) A numeric string specifying the range of tuning parameter to be chosen. The lower limit of lambda must be above 0.

compute_kernel_terms

(logic) Whether to computing effect for each individual terms. If FALSE then only compute the overall effect.

converge_thres

(numeric) The convergence threshold for computing kernel terms.

Value

beta

(matrix, d_fixed*1) Fixed effect estimates.

alpha

(matrix, n*k_terms) Kernel effect estimates for each kernel term.

kern_term_mat

(matrix, n*k_terms) Kernel effect for each kernel term.

A_list

(list of length k_terms) Projection matrices for each kernel term.

proj_matrix

(list of length 4) Estimated projection matrices, combined across kernel terms.

Details

For a single model, we can calculate the output of gaussian process regression, the solution is given by $$\hat{\beta}=[X^T(K+\lambda I)^{-1}X]^{-1}X^T(K+\lambda I)^{-1}y$$ $$\hat{\alpha}=(K+\lambda I)^{-1}(y-\hat{\beta}X)$$.

References

Andreas Buja, Trevor Hastie, and Robert Tibshirani. (1989) Linear Smoothers and Additive Models. Ann. Statist. Volume 17, Number 2, 453-510.