dminML.LA.ridgeGLM {squeezy}

Partial derivatives of -log(ML) of ridge penalised GLMs

Description

Returns the partial derivatives (w.r.t. 'loglambdas') of the minus log Laplace approximation (LA) of the marginal likelihood of ridge penalised generalised linear models. Note: currently only implemented for linear and logistic regression.

Usage

dminML.LA.ridgeGLM(loglambdas, XXblocks, Y, sigmasq = 1, 
                  Xunpen = NULL, intrcpt = TRUE, model, minlam = 0, 
                  opt.sigma = FALSE)

Arguments

loglambdas

Logarithm of the ridge penalties as returned by ecpc or squeezy; Gx1 vector.

XXblocks

List of sample covariance matrices X_g %*% t(X_g) for groups g = 1,..., G.

Y

Response data; n-dimensional vector (n: number of samples) for linear and logistic outcomes.

sigmasq

(linear model only) Noise level (Y~N(X*beta,sd=sqrt(sigmasq))).

Xunpen

Unpenalised variables; nxp_1-dimensional matrix for p_1 unpenalised variables.

intrcpt

Should an intercept be included? Set to TRUE by default.

model

Type of model for the response; linear or logistic.

minlam

Minimum value of lambda that is added to exp(loglambdas); set to 0 as default.

opt.sigma

(linear model only) TRUE/FALSE if log(sigmasq) is given as first argument of loglambdas for optimisation purposes

Value

Partial derivatives of the Laplace approximation of the minus log marginal likelihood to the model parameters 'loglambdas';

For opt.sigma=FALSE: Gx1-dimensional vector for the G log(group ridge penalties).

For opt.sigma=TRUE (linear model only): (G+1)x1-dimensional vector for the partial derivative to log(sigmasq) (first element) and for the G log(group ridge penalties).

Examples

#Simulate toy data
n<-100
p<-300
X <- matrix(rnorm(n*p),n,p)
Y <- rnorm(n)
groupset <- list(1:(p/2),(p/2+1):p)
sigmahat <- 2
alpha <- 0.5
tauMR <- c(0.01,0.005)

XXblocks <- lapply(groupset, function(x)X[,x]%*%t(X[,x]))

#compute partial derivatives of the minus log marginal likelihood to the penalties only
dminML.LA.ridgeGLM(loglambdas = log(sigmahat/tauMR), 
                  XXblocks, Y, sigmasq = sigmahat, 
                  model="linear",opt.sigma=FALSE)

#additionally, compute the partial derivative to the linear regression noise parameter sigma^2
dminML.LA.ridgeGLM(loglambdas = log(c(sigmahat,sigmahat/tauMR)), 
                   XXblocks, Y, sigmasq = sigmahat, 
                   model="linear",opt.sigma=TRUE)