REMOTE EFFECTS SPATIAL PROCESS MODELS 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). 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}$$_{t}$ )] T ∈ R n$_{t}$ is with σ 2 s = C$_{w}$ { ( s , t ) , ( s , t ) } and matrices of local covariates X ( s ) = $^{[}$x ( s , t ) T ] t$_{n}$$_{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 Y ( s ) | α $^{∗}$( s ) , β , R $^{∗}$, c $^{∗}$, σ 2 s ∼ N $^{(}$X ( s ) β + Z ∗ $^{T}$α $^{∗}$( s ) , σ 2 $_{s}$I$_{n}$$_{t}$ ) (14) 13 2.4 Inference