# Analysing data with spatial dependence

## Kriging example

Exponential decay is used ($e^{-\lambda D}$). Try different values for $\lambda$ (0, 0.1, 0.5). Try modifying it to half-Normal ($e^{-(\lambda D)^{2}}$).

The models is defined as:

where $\mathbf{\mu}$ is the mean vector for the Multivariate Normal (MVN) distribution, and $\mathbf{\Sigma}$ is the variance-covariance matrix with a spatial dependence structure. $\mathbf{\Sigma}$ is defined as $\sigma^{2} exp(-\lambda \mathbf{D})$, where $\mathbf{D}$ is an $n$-by-$n$ spatial distance matrix (we use Euclidean distances).

set.seed(2345)
library(MASS)
mu <- 5 # global mean
sigma_sq <- 1 # global variance
lambda <- 0.5

## set up an m x m square lattice
m <- 10
xy <- expand.grid(x=seq_len(m), y=seq_len(m))
n <- nrow(xy)
D <- as.matrix(dist(xy))

Sigma <- sigma_sq * exp(-lambda*D)

Y <- mvrnorm(1, rep(mu, n), Sigma)

op <- par(mfrow = c(2, 2))
image(seq_len(n), seq_len(n), D, main = "D", ylab="i", xlab="i")
image(seq_len(n), seq_len(n), Sigma, main="Sigma", ylab="i", xlab="i")
Distance <- seq(0, m * sqrt(2), by = 0.1)
plot(Distance, exp(-lambda*Distance), type = "l")
image(seq_len(m), seq_len(m), matrix(Y, m, m), main = "Y", ylab="y", xlab="x")
par(op)


Bayesian analysis:

library(dclone)
model <- custommodel("model {
for (i in 1:n) {
for (j in 1:n) {
Sigma[i,j] <- sigma_sq * exp(-lambda*D[i,j])
}
mu_vec[i] <- mu
}
Y[1:n] ~ dmnorm(mu_vec, invSigma)
invSigma <- inverse(Sigma)
log_sigma ~ dnorm(0, 0.001)
sigma_sq <- exp(log_sigma)^2
mu ~ dnorm(0, 0.1)
lambda ~ dgamma(1, 0.1)
}")
dat <- list(Y = Y, n = n, D = D)
fit <- jags.fit(data = dat, params = c("mu", "sigma_sq","lambda"),
model = model, n.iter = 1000)
summary(fit)


DC: replicating the whole experiment $K$ times (i.e. clones are independent, and identical in terms of within-clone dependence structure)

library(dclone)
model <- custommodel("model {
for (k in 1:K) {
for (i in 1:n) {
for (j in 1:n) {
Sigma[i,j,k] <- sigma_sq * exp(-lambda*D[i,j])
}
mu_vec[i,k] <- mu # mu_vec does not really require cloning
}
Y[1:n,k] ~ dmnorm(mu_vec[1:n,k], invSigma[1:n,1:n,k])
invSigma[1:n,1:n,k] <- inverse(Sigma[1:n,1:n,k])
}
log_sigma ~ dnorm(0, 0.001)
sigma_sq <- exp(log_sigma)^2
mu ~ dnorm(0, 0.1)
lambda ~ dgamma(1, 0.1)
}")
dat <- list(Y = dcdim(data.matrix(Y)), n = n, D = D, K=1)
dcfit <- dc.fit(data = dat, params = c("mu", "sigma_sq","lambda"),
model = model, n.iter = 1000,
n.clones=c(1,2),
multiply="K", unchanged=c("n", "D"))
summary(dcfit)
dcdiag(dcfit)


Inverse Wishart prior, $\sigma^{2}$ and $\lambda$ is hard to recover (it requires post processing the posterior estimates, i.e. monitor the whole invSigma matrix or its inverse):

library(dclone)
model <- custommodel("model {
for (i in 1:n) {
mu_vec[i] <- mu
}
Y[1:n] ~ dmnorm(mu_vec, invSigma)
invSigma[1:n,1:n] ~ dwish(R[1:n,1:n], n)
mu ~ dnorm(0, 0.1)
}")
dat <- list(Y = Y, n = n, R = diag(1, n, n))
fit <- jags.fit(data = dat, params = "mu",
model = model, n.iter = 1000)
summary(fit)


DC for inverse Wishart parametrization:

library(dclone)
model <- custommodel("model {
for (i in 1:n) {
mu_vec[i] <- mu
}
for (k in 1:K) {
Y[1:n,k] ~ dmnorm(mu_vec[1:n], invSigma[1:n,1:n,k])
invSigma[1:n,1:n,k] ~ dwish(R[1:n,1:n], n)
}
mu ~ dnorm(0, 0.1)
}")
dat <- list(Y = dcdim(data.matrix(Y)), n = n, R = diag(1, n, n), K=1)
dcfit <- dc.fit(data = dat, params = "mu",
model = model, n.iter = 1000,
n.clones=c(1,2),
multiply="K", unchanged=c("n", "R"))
summary(dcfit)
dcdiag(dcfit)


Correlation ($\rho$) based parametrization. Try different values for $\rho$.

Same model as above, but now $\mathbf{\Sigma}$ is defined as $\sigma^{2} \rho^{\mathbf{D}})$.

set.seed(2345)
library(MASS)
mu <- 5
sigma_sq <- 1
rho <- 0.8

## set up an m x m square lattice
m <- 10
xy <- expand.grid(x=seq_len(m), y=seq_len(m))
n <- nrow(xy)
D <- as.matrix(dist(xy))

Sigma <- sigma_sq * rho^D

Y <- mvrnorm(1, rep(mu, n), Sigma)

op <- par(mfrow = c(2, 2))
image(seq_len(n), seq_len(n), D, main = "D", ylab="i", xlab="i")
image(seq_len(n), seq_len(n), Sigma, main="Sigma", ylab="i", xlab="i")
Distance <- seq(0, m * sqrt(2), by = 0.1)
plot(Distance, rho^Distance, type = "l")
image(seq_len(m), seq_len(m), matrix(Y, m, m), main = "Y", ylab="y", xlab="x")
par(op)


Bayesian analysis:

library(dclone)
model <- custommodel("model {
for (i in 1:n) {
for (j in 1:n) {
Sigma[i,j] <- sigma_sq * rho^D[i,j]
}
mu_vec[i] <- mu
}
Y[1:n] ~ dmnorm(mu_vec, invSigma)
invSigma <- inverse(Sigma)
log_sigma ~ dnorm(0, 0.001)
sigma_sq <- exp(log_sigma)^2
mu ~ dnorm(0, 0.1)
rho ~ dunif(0, 0.999)
}")
dat <- list(Y = Y, n = n, D = D)
fit <- jags.fit(data = dat, params = c("mu", "sigma_sq","rho"),
model = model, n.iter = 1000)
summary(fit)


DC:

model <- custommodel("model {
for (k in 1:K) {
for (i in 1:n) {
for (j in 1:n) {
Sigma[i,j,k] <- sigma_sq * rho^D[i,j]
}
mu_vec[i,k] <- mu # mu_vec does not really require cloning
}
Y[1:n,k] ~ dmnorm(mu_vec[1:n,k], invSigma[1:n,1:n,k])
invSigma[1:n,1:n,k] <- inverse(Sigma[1:n,1:n,k])
}
log_sigma ~ dnorm(0, 0.001)
sigma_sq <- exp(log_sigma)^2
mu ~ dnorm(0, 0.1)
rho ~ dunif(0, 0.999)
}")
dat <- list(Y = dcdim(data.matrix(Y)), n = n, D = D, K=1)
dcfit <- dc.fit(data = dat, params = c("mu", "sigma_sq","rho"),
model = model, n.iter = 1000,
n.clones=c(1,2),
multiply="K", unchanged=c("n", "D"))
summary(dcfit)
dcdiag(dcfit)


## Cluster sampling design, Poisson-Lognormal GLMM

We assume that points within clusters are dependent (i.e. close to each other to assume high dependence), but clusters are independent (i.e. far apart to assume independence). We assume each cluster has same number of points for simplicity, but number of points within cluster can vary.

The model for any given cluster is defined as: $(Y_{ij} \mid \lambda_i) \sim Poisson(\lambda_{i})$, $i = 1, 2, \ldots, n$, $j = 1, 2, \ldots, m$, $log(\lambda_i) = \alpha_i + \mu$, and $\mathbf{\alpha_i} \sim MVN(\mathbf{0}, \mathbf{\Sigma})$.

$\mathbf{\Sigma}$ is the $m$-by-$m$ variance covariance matrix for cluster $i$. Diagonal elements are defined as $\sigma^{2}$, off-diagonal elements are defined as $\sigma^{2} \rho$.

set.seed(1234)
library(MASS)
## total sample size is n x m
m <- 5 # number of points in a cluster
n <- 25 # number of clusters
mu <- 1.6 # global mean on log scale
sigma_sq <- 1 # global variance
rho <- 0.8

## variance-covariance matrix for a single cluster
Sigma1 <- sigma_sq * diag(1, m, m) + sigma_sq * rho * (1 - diag(1, m, m))
## the full variance covariance matrix is 'block-diagonal'
## which means it is filled with zeros across clusters
alpha <- matrix(0, n, m)
for (i in 1:n) {
alpha[i,] <- mvrnorm(1, rep(0, m), Sigma1)
}
Y <- rpois(n*m, exp(as.numeric(alpha) + mu))
dim(Y) <- dim(alpha)
Y
boxplot(t(Y), xlab="Clusters", ylab="Y", col="tomato")
points(1:n, rowMeans(Y), pch=4, col=4)


Bayesian analysis:

library(dclone)
model <- custommodel("model {
for (i in 1:n) {
for (j in 1:m) {
Y[i,j] ~ dpois(lambda[i,j])
lambda[i,j] <- exp(alpha[i,j] + mu)
}
alpha[i,1:m] ~ dmnorm(zeros, invSigma)
}
Sigma <- sigma_sq * R + sigma_sq * rho * (1 - R)
invSigma <- inverse(Sigma)
log_sigma ~ dnorm(0, 0.001)
sigma_sq <- exp(log_sigma)^2
mu ~ dnorm(0, 0.1)
rho ~ dunif(0, 0.999)
}")
dat <- list(Y = Y, n = n, m = m, R = diag(1, m, m),
zeros = rep(0, m))
fit <- jags.fit(data = dat, params = c("mu", "sigma_sq","rho"),
model = model, n.iter = 1000)
summary(fit)


DC: cloning the whole dat set which means more independent clusters (i.e. not increased levels of replication within cluster)

library(dclone)
model <- custommodel("model {
for (k in 1:K) {
for (i in 1:n) {
for (j in 1:m) {
Y[i,j,k] ~ dpois(lambda[i,j,k])
lambda[i,j,k] <- exp(alpha[i,j,k] + mu)
}
alpha[i,1:m,k] ~ dmnorm(zeros, invSigma)
}
}
Sigma <- sigma_sq * R + sigma_sq * rho * (1 - R)
invSigma <- inverse(Sigma)
log_sigma ~ dnorm(0, 0.001)
sigma_sq <- exp(log_sigma)^2
mu ~ dnorm(0, 0.1)
rho ~ dunif(0, 0.999)
}")
dat <- list(Y = dcdim(array(Y, c(dim(Y), 1))), n = n, m = m, R = diag(1, m, m),
zeros = rep(0, m), K=1)
dcfit <- dc.fit(data = dat, params = c("mu", "sigma_sq","rho"),
model = model, n.iter = 1000,
n.clones=c(1,2),
unchanged = c("n", "m", "R", "zeros"), multiply="K")
summary(dcfit)
dcdiag(dcfit)


## Binomial GLMM for clustered data

set.seed(1234)
library(MASS)
## total sample size is n x m
m <- 5 # number of points in a cluster
n <- 25 # number of clusters
mu <- 0 # global mean on logit scale
sigma_sq <- 1 # global variance
rho <- 0.8

## variance-covariance matrix for a single cluster
Sigma1 <- sigma_sq * diag(1, m, m) + sigma_sq * rho * (1 - diag(1, m, m))
## the full variance covariance matrix is 'block-diagonal'
## which means it is filled with zeros across clusters
alpha <- matrix(0, n, m)
for (i in 1:n) {
alpha[i,] <- mvrnorm(1, rep(0, m), Sigma1)
}
Y <- rbinom(n*m, 1, plogis(as.numeric(alpha) + mu))
dim(Y) <- dim(alpha)
Y
boxplot(t(Y), xlab="Clusters", ylab="Y", col="tomato")
points(1:n, rowMeans(Y), pch=4, col=4)


Bayesian analysis:

library(dclone)
model <- custommodel("model {
for (i in 1:n) {
for (j in 1:m) {
Y[i,j] ~ dbern(p[i,j])
p[i,j] <- ilogit(alpha[i,j] + mu)
}
alpha[i,1:m] ~ dmnorm(zeros, invSigma)
}
Sigma <- sigma_sq * R + sigma_sq * rho * (1 - R)
invSigma <- inverse(Sigma)
log_sigma ~ dnorm(0, 0.001)
sigma_sq <- exp(log_sigma)^2
mu ~ dnorm(0, 0.1)
rho ~ dunif(0, 0.999)
}")
dat <- list(Y = Y, n = n, m = m, R = diag(1, m, m),
zeros = rep(0, m))
fit <- jags.fit(data = dat, params = c("mu", "sigma_sq","rho"),
model = model, n.iter = 1000)
summary(fit)


DC:

library(dclone)
model <- custommodel("model {
for (k in 1:K) {
for (i in 1:n) {
for (j in 1:m) {
Y[i,j,k] ~ dbern(p[i,j,k])
p[i,j,k] <- ilogit(alpha[i,j,k] + mu)
}
alpha[i,1:m,k] ~ dmnorm(zeros, invSigma)
}
}
Sigma <- sigma_sq * R + sigma_sq * rho * (1 - R)
invSigma <- inverse(Sigma)
log_sigma ~ dnorm(0, 0.001)
sigma_sq <- exp(log_sigma)^2
mu ~ dnorm(0, 0.1)
rho ~ dunif(0, 0.999)
}")
dat <- list(Y = dcdim(array(Y, c(dim(Y), 1))), n = n, m = m, R = diag(1, m, m),
zeros = rep(0, m), K=1)
dcfit <- dc.fit(data = dat, params = c("mu", "sigma_sq","rho"),
model = model, n.iter = 1000,
n.clones=c(1,2),
unchanged = c("n", "m", "R", "zeros"), multiply="K")
summary(dcfit)
dcdiag(dcfit)