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## - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
# Copyright (C) 2012 - 2023 Reza Mohammadi |
# |
# This file is part of BDgraph package. |
# |
# BDgraph is free software: you can redistribute it and/or modify it under |
# the terms of the GNU General Public License as published by the Free |
# Software Foundation; see <https://cran.r-project.org/web/licenses/GPL-3>.|
# |
# Maintainer: Reza Mohammadi <a.mohammadi@uva.nl> |
## - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
# Bayesian parameter estimation for discrete Weibull regression
## - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
bdw.reg = function( data, formula = NA, iter = 5000, burnin = NULL,
dist.q = dnorm, dist.beta = dnorm,
par.q = c( 0, 1 ), par.beta = c( 0, 1 ), par.pi = c( 1, 1 ),
initial.q = NULL, initial.beta = NULL, initial.pi = NULL,
ZI = FALSE, scale.proposal = NULL, adapt = TRUE, print = TRUE )
{
if( is.null( burnin ) ) burnin = round( iter * 0.75 )
if( !is.vector( data ) )
{
mf = stats::model.frame( formula, data = data )
y = stats::model.response( mf, "numeric" )
# x = as.matrix( mf )
# x[ , 1 ] = 1
x = stats::model.matrix(formula, data = data)
}else{
y = data
x = matrix( 1, ncol = 1, nrow = length( y ) )
}
n = length( y )
ncol_x = ncol( x )
ind_y0 = ifelse( y == 0, 1, 0 )
#setting initial parameters
if( is.null( initial.q ) )
{
theta.q = rep( 0, times = ncol_x )
q = 1 - sum( y == 0 ) / n
if( q == 1 ) q = 0.999
theta.q[ 1 ] = log( q / ( 1 - q ) )
}else{
theta.q = initial.q
}
if( is.null( initial.beta ) )
{
theta.beta = rep( 0, times = ncol_x )
}else{
theta.beta = initial.beta
}
if( is.null( scale.proposal ) ) scale.proposal = 2.38 ^ 2 / ( 2 * ncol_x )
if( !ZI ){
pii = 1
z = rep( 1, n )
}else{
pii = ifelse( is.null( initial.pi ), sum( y != 0 ) / n + sum( y == 0 ) / ( n * 2 ), initial.pi )
# z = ifelse( y == 0, 0, 1 )
prob_z = ifelse( y != 0, 1, pii )
z = stats::rbinom( n = n, size = 1, prob = prob_z )
}
fit = stats::optim( par = c( theta.q, theta.beta ),
fn = BDgraph::log_post_cond_dw,
x = x, y = y, z = z, dist.q = dist.q, par.q = par.q, dist.beta = dist.beta, par.beta = par.beta,
control = list( "fnscale" = -1, maxit = 10000 ), hessian = TRUE )
#Initial values: initial posterior mode
theta.q = fit $ par[ 1 : ncol_x ]
theta.beta = fit $ par[ ( ncol_x + 1 ) : ( 2 * ncol_x ) ]
para = c( theta.q, theta.beta )
if( ZI ) para = c( para, pii )
#Setting the covariance matrix of the proposal distribution for the regression coefficients
fisher_info_inv = BDgraph::near_positive_definite( -fit $ hessian[ 1 : ( 2 * ncol_x ), 1 : ( 2 * ncol_x ) ] )
if( !isSymmetric( fisher_info_inv ) || isTRUE( class( try( solve( fisher_info_inv ), silent = TRUE ) ) == "try-error" ) )
{
# sigma_proposal = diag( 0.5, length( para ) )
sigma_proposal = diag( 0.5, 2 * ncol_x )
}else{
fisher_info = solve( fisher_info_inv )
sigma_proposal = scale.proposal * fisher_info
}
sample = matrix( nrow = iter + 1, ncol = length( para ) )
sample[ 1, ] = para
count_accept = 0
for( i_mcmc in 1:iter )
{
if( ( print == TRUE ) && ( i_mcmc %% round( iter / 100 ) == 0 ) )
{
info = paste( round( i_mcmc / iter * 100 ), '% done, Acceptance rate = ',
round( count_accept / iter * 100, 2 ),'%' )
cat( '\r ', info, rep( ' ', 20 ) )
}
theta.q = sample[ i_mcmc, 1 : ncol_x ]
theta.beta = sample[ i_mcmc, ( ncol_x + 1 ) : ( 2 * ncol_x ) ]
if( ZI )
{
pii = sample[ i_mcmc, ncol( sample ) ]
dens = BDgraph::ddweibull_reg( x, y, theta.q, theta.beta )
prob_z0 = ind_y0 * ( 1 - pii )
prob_z1 = dens * pii
prob_z = prob_z1 / ( prob_z1 + prob_z0 )
prob_z[ is.na( prob_z ) ] = 1
#Gibbs sampling for z: sample from posterior distribution given current values of theta.q and theta.beta
z = stats::rbinom( n = n, size = 1, prob = prob_z )
#Gibbs sampling pi: sample from z
n_z0 = sum( z == 0 )
n_z1 = sum( z == 1 )
### Sample new pi from posterior distribution
pii = stats::rbeta( n = 1, shape1 = par.pi[ 1 ] + n_z1, shape2 = par.pi[ 2 ] + n_z0 )
}
#Random Metropolis-Hasting for theta.q and theta.beta (given Z and pi)
#Log-posterior at current parameters
log_posterior_t = BDgraph::log_post_cond_dw( par = c( theta.q, theta.beta ), par.q = par.q, par.beta = par.beta,
dist.q = dist.q, dist.beta = dist.beta, x = x, y = y, z = z )
#Adaptive MH
if( adapt )
if( i_mcmc >= 100 && i_mcmc %% 100 == 0 )
sigma_proposal = BDgraph::near_positive_definite( stats::cov( sample[ ( i_mcmc - 99 ) : i_mcmc, 1 : ( 2 * ncol_x ) ] ) )
if( !isSymmetric( sigma_proposal ) || isTRUE( class( try( chol.default( sigma_proposal ), silent = TRUE ) ) == "try-error" ) )
sigma_proposal = diag( 0.5, ncol( fisher_info ) )
b1 = BDgraph::rmvnorm( n = 1, mean = rep( 0, 2 * ncol_x ), sigma = sigma_proposal )
#New proposed values of theta.q and theta.beta
value_proposal = sample[ i_mcmc, 1 : ( 2 * ncol_x ) ] + as.vector( b1 )
#Log-posterior at proposed parameters (given the same Z and pi)
log_posterior_proposal = BDgraph::log_post_cond_dw( par = value_proposal,
par.q = par.q, par.beta = par.beta,
dist.q = dist.q, dist.beta = dist.beta,
x = x, y = y, z = z )
log_alpha = min( 0, log_posterior_proposal - log_posterior_t )
if( log( stats::runif( 1 ) ) <= log_alpha )
{
sample[ i_mcmc + 1, 1 : ( 2 * ncol_x ) ] = value_proposal
count_accept = count_accept + 1
}else
sample[ i_mcmc + 1, 1 : ( 2 * ncol_x ) ] = sample[ i_mcmc, 1 : ( 2 * ncol_x ) ]
if( ZI )
sample[ i_mcmc + 1, ncol( sample ) ] = pii
}
cat( "\n" )
sample = sample[ ( burnin + 1 ) : ( iter + 1 ), ]
sample.mean = apply( sample, 2, mean )
theta.q = sample.mean[ 1 : ncol_x ]
theta.beta = sample.mean[ ( ncol_x + 1 ) : ( 2 * ncol_x ) ]
if( ZI )
pii = sample.mean[ 2 * ncol_x + 1 ]
q_reg = 1 / ( 1 + exp( - x %*% theta.q ) )
beta_reg = exp( x %*% theta.beta )
if( ncol_x == 1 )
{
q_reg = q_reg[ 1 ]
beta_reg = beta_reg[ 1 ]
}
return( list( sample = sample, q.est = q_reg, beta.est = beta_reg,
pi.est = pii, accept.rate = count_accept / iter ) )
}
## - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
# Discrete Weibull density based on regression
## - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
ddweibull_reg = function( x, y, theta.q, theta.beta )
{
q_reg = 1 / ( 1 + exp( - x %*% theta.q ) )
beta_reg = exp( x %*% theta.beta )
density = q_reg ^ ( y ^ beta_reg ) - q_reg ^ ( ( y + 1 ) ^ beta_reg )
return( density )
}
## - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
## Posterior of q and beta given Y & Z: P( X, Y | Z ) P( Z ) P( q ) P( beta )
## - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
log_post_cond_dw = function( par, x, y, z, dist.q, par.q, dist.beta, par.beta )
{
ncol_x = ncol( x )
theta.q = par[ 1 : ncol_x ]
theta.beta = par[ ( ncol_x + 1 ) : ( 2 * ncol( x ) ) ]
dens_dw_z1 = BDgraph::ddweibull_reg( x = as.matrix( x[ z == 1, ] ), y = y[ z == 1 ],
theta.q = theta.q, theta.beta = theta.beta )
# log_lik = sum( log( dens_dw_z1[ dens_dw_z1 != 0 ] ) )
dens_dw_z1[ dens_dw_z1 == 0 ] = .Machine $ double.xmin
log_lik = sum( log( dens_dw_z1 ) )
log_prior_q = sum( dist.q( theta.q, par.q[ 1 ], par.q[ 2 ], log = TRUE ) )
log_prior_b = sum( dist.beta( theta.beta, par.beta[ 1 ], par.beta[ 2 ], log = TRUE ) )
log_post = log_lik + log_prior_q + log_prior_b
return( log_post )
}
## - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
near_positive_definite = function( m )
{
# Copyright 2003-05 Korbinian Strimmer
# Rank, condition, and positive definiteness of a matrix; Method by Higham 1988
d = ncol( m )
eigen_m = eigen( m )
eigen_vectors = eigen_m $ vectors
eigen_values = eigen_m $ values
delta = 2 * d * max( abs( eigen_values ) ) * .Machine $ double.eps
# factor two is just to make sure the resulting
# matrix passes all numerical tests of positive definiteness
tau = pmax( 0, delta - eigen_values )
dm = eigen_vectors %*% diag( tau, d ) %*% t( eigen_vectors )
return( m + dm )
}
## - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |
|
