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manipulate the integral in (3) shows that the reduced rank approximation can be interpreted as inducing transformed covariates z $^{∗}$( r $^{∗}$, t ) via 

<img src='content_image/320680.jpg'>

where z ∗ $^{(}$r ∗ $_{j}$, t ) = ∫ D$_{Z}$ z ( r , t ) h $^{(}$r , r ∗ j $^{)}$d r . The z ∗ $^{(}$r ∗ $_{j}$, t ) and α $^{(}$s , r ∗ j ) may be collected into the covariate vector z ( t ) and spatially varying effects θ ( s ) in (1). We remark that the RESP model differs from standard spatially varying coefficient models in that the z ∗ $^{(}$r ∗ $_{j}$, t $^{)}$represent induced—rather than a priori—covariates, and the α $^{(}$s , r ∗ j ) inherit spatial structure from the model’s formulation. 

Scientifically, the predictive process approach to addressing multicollinearity in the re- mote covariates reduces the remote covariates z ( r , t ), r ∈ D$_{Z}$ at each time point to k spatially-averaged indices z $^{∗}$( r $^{∗}$, t ) centered at r ∗ for r ∗ ∈ { r ∗ $_{1}$, . . . , r ∗ $_{k}$} . This manipulation is fairly generic and should be applicable to all predictive process models. For teleconnection, this manipulation connects the RESP model to one set of standard teleconnection method- ologies in which teleconnection effects are measured with respect to ocean indices based on spatial averages of remote covariates (Ashok et al., 2007; Towler et al., 2016). 

## 2.3 Spatial basis function transformation of remote coefficients 

The RESP model (3) is also related to another set of standard teleconnection methodologies in which teleconnection effects are measured with respect to complex ocean indices such as empirical orthogonal functions (Montroy, 1997; Ting & Wang, 1997). Spatial basis functions provide a means to reparameterize the RESP model and show it can identify and leverage known teleconnections with complex patterns. We use the following reparameterization of the teleconnection effects α ( s , r ) to discuss teleconnection between Pacific Ocean sea surface 

