fit.dnetwork {PartialNetwork}

Fitting Network Distribution using ARD.

Description

fit.dnetwork computes the network distribution using the simulations from the posterior distribution of the ARD network formation model. The linking probabilities are also computed for individuals without ARD. The degrees and the gregariousness of the individuals without ARD are computed from the sample with ARD using a k-nearest neighbors method.

Usage

fit.dnetwork(
  object,
  X = NULL,
  obsARD = NULL,
  m = NULL,
  burnin = NULL,
  print = TRUE
)

Arguments

object

estim.ARD object returned by mcmcARD.

X

(required when ARD are available for a sample of individuals) is a matrix of variables describing individuals with ARD and those without ARD. This matrix will be used to compute distance between individuals in the k-nearest neighbors approach. This could be the matrix of traits (see details).

obsARD

logical vector of length nrow(X) (number of individuals with and without ARD), where the i-th entry equal to TRUE if the i-th individual in X has ARD and FALSE otherwise. If missing, obsARD = rep(c(TRUE, FALSE), n1, n2), where n1 is the number of individuals with ARD (see details).

m

number of neighbors used to compute the gregariousness and the degree for individuals without ARD (default value is 1).

burnin

number of simulations from the posterior distribution used as burn-in. The network distribution will be computed used the simulation from the iteration burnin + 1.

print

logical; if TRUE, the progression will be printed in the console.

Details

The order of individuals provided through the arguments traitARD and ARD (when calling the function mcmcARD) should fit the order of individuals in X and obsARD. Especially, the i-th row of X[obsARD,] should correspond to the i-th row in traitARD or ARD.

Value

A list consisting of:

dnetwork

posterior mean of the network distribution.

degree

posterior mean of the degree.

nu

posterior mean of the gregariousness, nu.

Examples

set.seed(123)
# GENERATE DATA
# Sample size
N  <- 500 
n  <- 300

# ARD parameters
genzeta <- 1
mu      <- -1.35
sigma   <- 0.37
K       <- 12    # number of traits
P       <- 3     # Sphere dimension 

# Generate z (spherical coordinates)
genz    <- rvMF(N,rep(0,P))

# Genetate nu  from a Normal distribution with parameters mu and sigma (The gregariousness)
gennu   <- rnorm(N,mu,sigma)

# compute degrees
gend <- N*exp(gennu)*exp(mu+0.5*sigma^2)*exp(logCpvMF(P,0) - logCpvMF(P,genzeta))

# Link probabilities
Probabilities <- sim.dnetwork(gennu,gend,genzeta,genz) 

# Adjacency matrix
G <- sim.network(Probabilities)

# Generate vk, the trait location
genv <- rvMF(K,rep(0,P))

# set fixed some vk  distant
genv[1,] <- c(1,0,0)
genv[2,] <- c(0,1,0)
genv[3,] <- c(0,0,1)

# eta, the intensity parameter
geneta   <-abs(rnorm(K,2,1))

# Build traits matrix
densityatz      <- matrix(0,N,K)
for(k in 1:K){
  densityatz[,k] <- dvMF(genz,genv[k,]*geneta[k])
}

trait       <- matrix(0,N,K)
NK          <- floor(runif(K, 0.8, 0.95)*colSums(densityatz)/apply(densityatz, 2, max)) 
for (k in 1:K) {
  trait[,k]       <- rbinom(N, 1, NK[k]*densityatz[,k]/sum(densityatz[,k]))
}

# print a percentage of people having a trait
colSums(trait)*100/N

# Build ARD
ARD         <- G %*% trait

# generate b
genb        <- numeric(K)
for(k in 1:K){
  genb[k]   <- sum(G[,trait[,k]==1])/sum(G)
}

############ ARD Posterior distribution ################### 
# EXAMPLE 1: ARD observed for the entire population
# initialization 
d0     <- exp(rnorm(N)); b0 <- exp(rnorm(K)); eta0 <- rep(1,K);
zeta0  <- 1; z0 <- matrix(rvMF(N,rep(0,P)),N); v0 <- matrix(rvMF(K,rep(0,P)),K)

# We need to fix some of the vk and bk for identification (see Breza et al. (2020) for details).
vfixcolumn      <- 1:6
bfixcolumn      <- c(3, 5)
b0[bfixcolumn]  <- genb[bfixcolumn]
v0[vfixcolumn,] <- genv[vfixcolumn,]

start  <- list("z" = z0, "v" = v0, "d" = d0, "b" = b0, "eta" = eta0, "zeta" = zeta0)
# MCMC ARD
out    <- mcmcARD(Y = ARD, traitARD = trait, start = start, fixv = vfixcolumn,
                  consb = bfixcolumn, iteration = 5000)
                  
# fit network distribution
dist   <- fit.dnetwork(out)

plot(rowSums(dist$dnetwork), gend)
abline(0, 1, col = "red")

# EXAMPLE 2: ARD observed for a sample of the population
# observed sample
selectARD   <- sort(sample(1:N, n, FALSE))
traitard    <- trait[selectARD,]
ARD         <- ARD[selectARD,]
logicalARD  <- (1:N) %in% selectARD

# initianalization 
d0     <- exp(rnorm(n)); b0 <- exp(rnorm(K)); eta0 <- rep(1,K);
zeta0  <- 1; z0 <- matrix(rvMF(n,rep(0,P)),n); v0 <- matrix(rvMF(K,rep(0,P)),K)

# We need to fix some of the vk and bk for identification (see Breza et al. (2020) for details).
vfixcolumn      <- 1:6
bfixcolumn      <- c(3, 5)
b0[bfixcolumn]  <- genb[bfixcolumn]
v0[vfixcolumn,] <- genv[vfixcolumn,]

start  <- list("z" = z0, "v" = v0, "d" = d0, "b" = b0, "eta" = eta0, "zeta" = zeta0)
# MCMC ARD
out    <- mcmcARD(Y = ARD, traitARD = traitard, start = start, fixv = vfixcolumn,
                  consb = bfixcolumn, iteration = 5000)
                  
# fit network distribution
dist   <- fit.dnetwork(out, X = trait, obsARD = logicalARD, m = 1)

library(ggplot2)
ggplot(data.frame("etimated.degree" = dist$degree,
                  "true.degree"     = gend,
                  "observed"        = ifelse(logicalARD, TRUE, FALSE)),
       aes(x = etimated.degree, y = true.degree, colour = observed)) +
  geom_point()