2.3.1
BYMar model
The model proposed by Martínez-Beneito et al. (2008) will be named BYMar be- cause it models the random region effect using the convolution model ofBesag et al.
(1991) (called in short BYM) and the temporal correlation is induced through a first order autoregressive structure. The model is briefly described next.
log ri1 = η + α1+ (1 − ρ2)−1/2(θi1+ φi1), i = 1, . . . , n, θi1 ∼ N (0, τθ−1), i = 1, . . . , n, φ1 = (φ11, . . . , φn1) 0 ∼ N (0, [τφRφ]−), (iCAR model) (2.4)
where η quantifies the global log-risk, α1models the mean deviation of the risk in the
first period from the mean level of all of them, Rφ is the n × n spatial neighborhood
2.3 Autoregressive and moving average models 33 log rit = η + αt+ ρ(log ri,t−1− η − αt−1) + θit+ φit,
θit ∼ N (0, τθ−1), φt= (φ1t, . . . , φnt) 0 ∼ N (0, [τφRφ]−), α = (α1, . . . , αt) 0 ∼ N (0, [ταRα]−), (2.5)
where Rα is the structure matrix of a random walk of order one. The temporal
dependence introduced inEquation (2.5)makes the relative risks in each region and time point depend not only on their neighbors relative risks in the same time point, but also on the relative risks in previous periods. The dependence in each region among different time points has been defined as a first order autoregressive time series. Finally, spatial dependence has been introduced using a BYM prior.
In Equation (2.4), the term (1 − ρ2)−1/2 is introduced in order to make the
variance-covariance matrix of log r1 equal to the stationary covariance matrix of the
series {log rt}∞t=1. Then,
(log r1, log r2, . . . , log rT)
0
| . . . ∼ N ((η · 1T + α) ⊗ 1n, Λ ⊗ Σ),
where (log r1, log r2, . . . , log rT)
0
| . . . denotes the log-relative risk distribution given all the parameters in the different hierarchies of the model, Σ is the covariance matrix of θ·t+ φ·t for any period t and Λ denotes the correlation matrix of a first-
order autoregressive time series of length T . In contrast to CAR models and the
Knorr-Held (2000) proposal in particular, Λ depends now on one parameter, ρ,
what makes this matrix more flexible than those used in the other proposals. With respect to prior distributions, a Unif(0, 1) prior was used for ρ, while Unif(0, 5) prior distributions were used for all the standard deviations (square root inverse of precision parameters τθ, τψ and τα) involved in this model. The upper limit of this
distribution was considered as a vague choice since it is referred to variables defined in a logarithmic scale (log-risks). Finally, an improper flat prior distribution was used for η. See Martínez-Beneito et al.(2008) for more details about this model.
2.3.2
STMARS model
The STMARS model is the spatio-temporal extension of the Spatial Moving Average Risk Smoothing (SMARS) model proposed in Botella-Rocamora et al. (2013). The SMARS model can be considered as an alternative to CAR-based processes. If CAR models are the equivalent in the spatial domain to the temporal integrated and autoregressive processes, then SMARS can be considered as the spatial equivalent to moving average processes in time. The spatio-temporal extension is defined as in the BYMar model using an autoregressive temporal structure of first order. In the
34 Evaluation of models for the detection of high-risk areas STMARS model, the log-risks are modeled as
log rit = η + αt+ ω0ψit+ ω1 X i0∼1i ψi0 t+ . . . + ωm X i0∼mi ψi0 t,
where η represents the mean of the log-risks for the first period and αtthe differential
risk of the period t with regards to the first (with α1 = 0). A RW1 structure
is considered for these differential risks, i.e., αt ∼ N (αt−1, τα−1) for t = 2, . . . , T .
The set i0 ∼k i denotes all the geographical units being k-th order neighbors of
region i (those with the shortest path between both regions having k edges) and ω = (ω0, . . . , ωm)
0
is a vector weighting the contribution of the neighbors of differing orders of region i. The maximum order of the neighbors having an effect on the log-relative risks, m, is also a variable estimated by the model. As mentioned above, temporal dependence is achieved using a first-order autoregressive process on the latent effects ψit, such that
ψi1 ∼ N (0, τψ−1),
ψit ∼ N (ρ · ψi,(t−1), (1 − ρ2)/τψ) t = 2, . . . , T.
The set of k-th order neighbors defined for this model, as for the SMARS model, will form a kind of irregular disc centered on region i and a radius defined by k. In general, if any region is a k-th order neighbor of region i, it will be also a k0-th order neighbor for any k0 > k. The zeroth-order neighbor of region i will be considered as its own region i. This makes neighbors of lower order to be more influential than those of higher order. Additional details about this model model can be seen in
Botella-Rocamora (2010, Chapter 4).
A Unif(0, 1) was used for ρ. Improper flat prior distributions were used for η and σψ = 1/
√
τψ, in the latter case restricted to the positive real line. As the scale
of ψ is controlled by τψ, the restriction
Pm
j=0ωj = 0 is imposed on ω. Thus, a flat
Direchlet(1n+1) is used as prior distribution for ω. Finally, P (m) ∝ (m!)−1was used
as prior distribution for m. A deeper reasoning on the use of these priors is given in Botella-Rocamora et al. (2013).