Space-time bivariate model in Ngme2

This is a demonstration of using ngme2 package to fit a space-time bivariate model. Before use this model, please make sure you are familiar with the space-time (tensor product) model and bivariate model separately (Please see Ngme2 space-time model and Ngme2 bivariate model for more details).

This model is a combination of the previous two, we use the bivariate model as the second part of the tensor product model.

Simulation

library(ngme2)

## This is ngme2 of version 0.6.0
## - See our homepage: https://davidbolin.github.io/ngme2 for more details.

First we specify the two indices for the space-time model. We have 20 time points and 40 mesh locations for space.

time_len = 20
half_bv_len = 50

# specify indices for 2 parts
time_idx = rep(1:time_len, each=half_bv_len*2)
bv_idx = rep(c(1:half_bv_len,1:half_bv_len), time_len)

# group to indicate two different fields for bivariate model
group = rep(rep(c("f1", "f2"), each=half_bv_len),time_len)

Next, we specify the bivariate model using a list, here we consider a bivariate model with two AR(1) processes.

bv_list <- list(
  model="bv",
  sub_models = list(
    f1 = list(
      model="ar1",
      rho=0.5
    ),
    f2 = list(
      model="ar1",
      rho=-0.3
    )
  )
)

Then we specify the full model using the f function, which is a tensor product model with the first part being an AR(1) process and the second part being the bivariate model.

f_model <- f(
  map=list(
    time_idx,
    bv_idx
  ),
  model="tp",
  first = list(
    model="ar1", rho=0.6
  ),
  second = bv_list,
  group = group,
  noise=noise_nig(
    mu=-2,
    sigma=1,
    nu=2
  )
)

# Simulate the latent fields
W = simulate(f_model)[[1]]

# Adding measurement noise
Y = W + rnorm(length(W), sd=0.2)

Estimation

Now we turn to estimate the model using simulated data.

bv_list_2 <- list(
  model="bv",
  sub_models = list(
    f1 = list(
      model="ar1"
    ),
    f2 = list(
      model="ar1"
    )
  )
)
nn = length(Y)

fit0 = ngme(
  Y ~ 0 + f(1:nn, model="rw1") +
    f(map=list(
      time_idx, bv_idx
    ),
    model="tp",
    first = list(model="ar1"),
    second = bv_list_2,
    noise=noise_nig()
  ),
  control_opt = control_opt(
    iterations = 10
  ),
  data=data.frame(Y=Y),
  group = group
)

## Starting estimation... 
## 
## Starting posterior sampling... 
## Posterior sampling done! 
## Note:
##       1. Use ngme_post_samples(..) to access the posterior samples.
##       2. Use ngme_result(..) to access different latent models.

fit0

## *** Ngme object ***
## 
## Fixed effects: 
##   None
## 
## Models: 
## $field1
##   Model type: Random walk (order 1)
##       No parameter.
##   Noise type: NORMAL
##   Noise parameters: 
##       sigma = 0.75
## 
## $field2
##   Model type: Tensor product
##       first: AR(1)
##           rho = 0.24
##       second: Bivariate model (non-Gaussian noise)
##           theta = -0.0628
##           rho = 0.0516
##           f1: AR(1)
##               rho = 0.24
##           f2: AR(1)
##               rho = -0.227
##   Noise type: NIG
##   Noise parameters: 
##       mu = -0.471
##       sigma = 1.48
##       nu = 0.7
## 
## Measurement noise: 
##   Noise type: NORMAL
##   Noise parameters: 
##       sigma = 1.22
## 
## 
## Number of replicates is  1

traceplot(fit0, "field1")

Here we can see, we get the simulated parameter correctly.