Temporal distributions

Temporal smoothing with car.normal

In one dimension, the intrinsic Gaussian CAR distribution reduces to a Gaussian random walk (see e.g.Fahrmeir and Lang, 2001). Assume we have a set of temporally correlated random effects θ_t, t=1,..., T (where T is the number of equally-spaced time points). In the simplest case of a random walk of order 1, RW(1), we may write

θ_t | θ_-t ~ Normal ( θ_t+1, φ ) for t = 1
~ Normal ( (θ_t-1 + θ_t+1)/2, φ / 2 ) for t = 2, ...., T-1
~ Normal ( θ_t-1, φ ) for t = T

where θ_-t denotes all elements of θexcept the θ_t. This is equivalent to specifying

θ_t | θ_-t ~ Normal ( Σ_k C_tk θ_k, φ M_tt) for t = 1, ..., T

where C_tk = W_tk / W_t+, W_t+ = Σ_k W_tk and W_tk = 1 if k = (t-1) or (t+1) and 0 otherwise; M_tt = 1/W_t+. Hence the RW(1) prior may be fitted using the car.normal distribution in WinBUGS, with appropriate specification of the weight and adjacency matrices, and num vector (see the Air Pollution Example)

A second order random walk prior is defined as

θ_t | θ_-t ~ Normal ( 2θ_t+1 - θ_t+2, φ ) for t = 1
~ Normal ( (2θ_t-1 + 4θ_t+1 - θ_t+2)/5, φ / 5 ) for t = 2
~ Normal ( (-θ_t-2 + 4θ_t-1 + 4θ_t+1 - θ_t+2)/6, φ / 6 ) for t = 3, ...., T-2
~ Normal ( (-θ_t-2 + 4θ_t-1 + 2θ_t+1)/5, φ / 5 ) for t = T-1
~ Normal ( -θ_t-2+ 2θ_t-1, φ ) for t = T

Again this is equivalent to specifying

θ_t | θ_-t ~ Normal ( Σ_k C_tk θ_k, φ M_tt) for t = 1, ..., T

where C_tk is defined as above, but this time with W_tk = -1 if k = (t-2) or (t+2), W_tk = 4 if k = (t-1) or (t+1) and t in (3, T-2), W_tk = 2 if k = (t-1) or (t+1) and t = 2 or T-1, W_tk = 0 otherwise; M_tt = 1/W_t+.

Note that if the observed time points are not equally spaced, it is necessary to include missing values (NA) for the intermediate time points (see the Air Pollution Example).