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tionally inherit spatial structure from the model’s formulation. Scientifically, a special case of (11) are principal component decompositions or the closely related truncated Karhunen– L` oeve expansions, which are referred to as empirical orthogonal functions (EOFs) in climate science. EOFs are particularly useful expansions for teleconnection because these trans- formations meaningfully characterize phenomena that impact global climate (Ashok et al., 2007). 

## 2.4 Inference 

While inference for the RESP model (3) can use standard hierarchical Bayesian modeling techniques, the Bayesian framework provides crucial intuition and interpretation for esti- mates of teleconnection effects (8) and (12). Full description of model priors and compu- tational techniques for inference are discussed in Supplement A ?? . The Gaussian process assumption and separable covariance (6) for the vector of teleconnection coefficients α $^{∗}$( s ) with associated covariance matrix R ∗ defined in Section 2.2 imply the normally-distributed prior α $^{∗}$( s ) | R ∗ ∼ N ( 0 , R $^{∗}$). Gaussian process assumptions for the RESP model’s spatial correlation also imply the likelihood for the vector of responses observed at n$_{t}$ timepoints Y ( s ) = [ Y ( s , t$_{1}$ ) , . . . , Y ( s , t$_{n}$$_{t}$ )] T ∈ R n$_{t}$ is 

<img src='content_image/324311.jpg'>

with σ 2 s = C$_{w}$ { ( s , t ) , ( s , t ) } and matrices of local covariates X ( s ) = $^{[}$x ( s , t ) T ] t$_{n}$$_{t}$ t = t$_{1}$ ∈ R n$_{t}$ × p and reduced-rank remote covariates Z ∗ ∈ R k × $^{n$_{t}$}$. The matrix Z ∗ is comprised of column vectors z ∗ t = R ∗ − $^{1}$c ∗ $^{T}$z$_{t}$ ∈ R k built from remote covariate vectors z$_{t}$ = [ z ( r$_{j}$ , t )] n$_{r}$ j =1 ∈ R $^{n$_{r}$}$. Our formulation of the spatial correlation (5) implies the scalar σ 2 s is constant across time; non-stationary extensions are discussed in Section 4. Standard Bayesian linear regression 

