Spatial model in greta


#1

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


#2

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!


#3

@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)

#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!


#5

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