Hello, I’m wondering if is possible to implement spatial models in greta e.g. Conditional Autoregressive (CAR) prior? Thanks in advance.
In STAN: http://mc-stan.org/users/documentation/case-studies/mbjoseph-CARStan.html
In BUGS: https://www.mrc-bsu.cam.ac.uk/wp-content/uploads/geobugs12manual.pdf
Hi Hemingway, this should be very possible!
Below is my first attempt at an example in Greta (based on the non-sparse example in the Stan document)
library(greta) # version from GitHub
W <- A # adjacency matrix
scaled_x <- c(scale(x))
X <- model.matrix(~scaled_x)
D <- solve(diag(rowSums(W)))
# Model data
X <- as_data(X)
y <- as_data(O)
W <- as_data(W) # adjacency matrix
log_offset <- as_data(log(E))
D <- as_data(D)
sigma <- tau * (D - alpha * W)
# priors
alpha <- uniform(0, 1)
beta <- normal(0, 1)
tau <- gamma(2, 2)
phi <- multivariate_normal(0, sigma)
distribution(y) <- poisson(exp(X %*% beta + phi + log_offset))
model <- model(beta, phi, alpha, tau)
draws <- mcmc(model, n_samples = 1000, warmup = 1000, chains = 4)
There may well be bugs however as I don’t have any data to run it on and am not at all an expert on such models. Hopefully this gives you the general idea though!
If you have any further questions just ask.
All the best!
@Voltemand, Thanks for the prompt answer.
I was wondering what’s the issue with:
phi <- multivariate_normal(0, sigma)
Error: the dimension of this distribution must be at least 2 but was 1.
Thanks.
Codes:
N <- 56
# observed
O <- c( 9, 39, 11, 9, 15, 8, 26, 7, 6, 20,
13, 5, 3, 8, 17, 9, 2, 7, 9, 7,
16, 31, 11, 7, 19, 15, 7, 10, 16, 11,
5, 3, 7, 8, 11, 9, 11, 8, 6, 4,
10, 8, 2, 6, 19, 3, 2, 3, 28, 6,
1, 1, 1, 1, 0, 0)
# expected
E <- c( 1.4, 8.7, 3.0, 2.5, 4.3, 2.4, 8.1, 2.3, 2.0, 6.6,
4.4, 1.8, 1.1, 3.3, 7.8, 4.6, 1.1, 4.2, 5.5, 4.4,
10.5,22.7, 8.8, 5.6,15.5,12.5, 6.0, 9.0,14.4,10.2,
4.8, 2.9, 7.0, 8.5,12.3,10.1,12.7, 9.4, 7.2, 5.3,
18.8,15.8, 4.3,14.6,50.7, 8.2, 5.6, 9.3,88.7,19.6,
3.4, 3.6, 5.7, 7.0, 4.2, 1.8)
# covariate
x <- c(16,16,10,24,10,24,10, 7, 7,16,
7,16,10,24, 7,16,10, 7, 7,10,
7,16,10, 7, 1, 1, 7, 7,10,10,
7,24,10, 7, 7, 0,10, 1,16, 0,
1,16,16, 0, 1, 7, 1, 1, 0, 1,
1, 0, 1, 1,16,10)
# adjacency matrix
A <- structure(c(0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
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0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0), .Dim = c(N, N))
W <- A # adjacency matrix
scaled_x <- c(scale(x)) # scaling the predictor
X <- model.matrix(~scaled_x)
D <- solve(diag(rowSums(W)))
# Model data
X <- as_data(X)
y <- as_data(O)
W <- as_data(W) # adjacency matrix
log_offset <- as_data(log(E))
D <- as_data(D)
alpha <- uniform(0, 1)
beta <- normal(0, 1)
tau <- gamma(2, 2)
sigma <- tau * (D - alpha * W)
# priors
phi <- multivariate_normal(0, sigma)
distribution(y) <- poisson(exp(X %*% beta + phi + log_offset))
model <- model(beta, phi, alpha, tau)
draws <- mcmc(model, n_samples = 1000, warmup = 1000, chains = 4)No worries!
The problem with that line was that the mean argument I had passed (0) did not have the right dimension (I had mistakenly assumed it would implicitly vectorise).
I have also cleaned up a few other small mistakes in the model I noticed after running it on your data. The model now is:
W <- A # adjacency matrix
scaled_x <- c(scale(x)) # scaling the predictor
X <- model.matrix(~ scaled_x)
D <- diag(rowSums(W))
X <- as_data(X)
y <- as_data(O)
W <- as_data(W)
log_offset <- as_data(log(E))
D <- as_data(D)
alpha <- uniform(0, 1)
beta <- normal(0, 1, dim = c(2, 1))
tau <- gamma(2, 2)
mu <- t(zeros(N))
sigma <- tau * (D - alpha * W)
phi <- t(multivariate_normal(mu, solve(sigma)))
distribution(y) <- poisson(exp(X %*% beta + phi + log_offset))
model <- model(beta, phi, alpha, tau)
draws <- mcmc(model, n_samples = 1000, warmup = 1000, chains = 4, one_by_one = TRUE)
I would be very interested if you could compare the results you get from this model with what you get from the model same model implemented in stan.
All the best!
Thank you so much for clarification. It works really well and the parameter estimates seem to match Stan.
Output from greta:
Mean SD Naive SE Time-series SE
beta[1,1] 0.01060 0.25262 0.003994 0.022356
beta[2,1] 0.27469 0.09395 0.001485 0.002317
alpha 0.93200 0.06492 0.001026 0.001904
tau 1.64185 0.48640 0.007691 0.012095
And from Stan:
mean se_mean sd 2.5% 25%
beta[1] -0.01792750 0.013038796 0.28861331 -0.631593939 -0.17496830
beta[2] 0.27432038 0.001415310 0.09442595 0.087170821 0.21124791
alpha 0.93210676 0.001040790 0.06480737 0.759120479 0.91085712
tau 1.63267268 0.006709894 0.49358056 0.849935240 1.27696174
Benchmark:
greta: Time difference of 2.937537 mins
Stan: Time difference of 4.895678 mins
@Voltemand and @Hemingway is it possible to fit a similar model, but when multiple rows of observations are nested within the same location?
Assuming that the spatial effect (the phi’s) are constant across the multiple observations per locations (no spatial-effect), you could just pass a vector of location id (named id_loc below) corresponding to the rows of the W matrix to the linear predictor, something like:
distribution(y) <- poisson(exp(X %*% beta + phi[id_loc] + log_offset))
Hope this helps
Thanks for clarifying. I did exactly the same but it is nice to get some confirmation from folks here 
I read somewhere else that one could have a slightly more complex spatial error structure that takes among-site (co)variance and within-site variance but it is a bit out of my capacity currently. Will read more…
I have been trying CAR models on my data but due to the large numbers of locations (> 1000) the computation are long, I guess due to the multivariate_normal call.
So I decided to have a go at defining an exact sparse equivalent based on Max Joseph Stan example. To do this I assume that one needs to define a new distribution (sparse_car) to sample the phi’s from, I have little knowledge in tensorflow but I decided to have a shot:
library(greta)
library(R6)
# get greta internals
distribution_node <- .internals$nodes$node_classes$distribution_node
as.greta_array <- .internals$greta_arrays$as.greta_array
check_dims <- .internals$utils$checks$check_dims
distrib <- .internals$nodes$constructors$distrib
fl <- .internals$utils$misc$fl
tf_sum <- greta:::tf_sum # oops, looks like this one didn't make it into .internals!
sparse_car_distribution <- R6Class(
"sparse_car_distribution",
inherit = distribution_node,
public = list(
initialize = function(tau, alpha,
W_sparse, D_sparse,
lambda, dim) {
tau <- as.greta_array(tau)
alpha <- as.greta_array(alpha)
W_sparse <- as.greta_array(W_sparse)
D_sparse <- as.greta_array(D_sparse)
lambda <- as.greta_array(lambda)
dim <- check_dims(tau, alpha, target_dim = dim)
super$initialize("sparse_car", dim)
self$add_parameter(tau, "tau")
self$add_parameter(alpha, "alpha")
self$add_parameter(W_sparse, "W_s")
self$add_parameter(D_sparse, "D_s")
self$add_parameter(lambda, "lbd")
},
tf_distrib = function(parameters, dag) {
alpha <- parameters$alpha
tau <- parameters$tau
W_sparse <- parameters$W_s
D_sparse <- parameters$D_s
lambda <- parameters$lbd
log_prob <- function(x) {
nloc <- length(lambda)
npair <- nrow(W_sparse)
phit_d <- x * D_sparse
phit_w <- rep(0, 4)
for(i in 1:3){
phit_w[W_sparse[i, 1]] <- phit_w[W_sparse[i, 1]] +
x[W_sparse[i, 2]]
phit_w[W_sparse[i, 2]] <- phit_w[W_sparse[i, 2]] +
x[W_sparse[i, 1]]
}
ldet_terms <- log(fl(1) - alpha * lambda)
res <- fl(0.5) * (nloc * log(tau) + tf_sum(ldet_terms) -
tau * (phit_d * phi - alpha *
(phit_w * phi)))
}
list(log_prob = log_prob, cdf = NULL, log_cdf = NULL)
},
tf_cdf_function = NULL,
tf_log_cdf_function = NULL
)
)
sparse_car <- function (tau, alpha,
W_sparse, D_sparse,
lambda, dim = NULL) {
distrib("sparse_car", tau, alpha,
W_sparse, D_sparse,
lambda, dim)
}
And below some toy example just to try sampling the phi’s:
# a simple example with 4 locations
loc <- data.frame(x=runif(4),
y=runif(4))
d_mat <- matrix(0, ncol = 4, nrow=4)
d_mat[lower.tri(d_mat)][dist(loc) < 0.4] <- 1
d_mat[upper.tri(d_mat)] <- t(d_mat)[upper.tri(d_mat)]
# make W_sparse, D_sparse and lambda
nloc <- nrow(loc)
npair <- sum(d_mat) / 2
counter <- 1
W_sparse <- matrix(0, nrow= npair, ncol = 2)
for (i in 1:(nloc - 1)) {
for (j in (i + 1):nloc) {
if (d_mat[i, j] == 1) {
W_sparse[counter, 1] = i
W_sparse[counter, 2] = j
counter = counter + 1
}
}
}
# number of neighbour per site
D_sparse <- rowSums(d_mat)
# compute eigenvalues
invsqrtD <- base::diag(1 / sqrt(D_sparse), nrow = nloc, ncol = nloc)
lambda <- base::eigen(t(invsqrtD) %*% d_mat %*% invsqrtD)$values
# the parameter
alpha <- uniform(0, 1)
tau <- gamma(2, 2)
phi <- sparse_car(tau, alpha, W_sparse, D_sparse, lambda, dim = 4)
mm <- model(alpha, tau)
dd <- mcmc(mm)
Which errors at the stage when phit_w is created. So if someone has some ideas on how to debug this that would be helpful.