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We develop a statistical theory for spatial kernel restricted canonical correlation analysis (skrCCA), characterizing its association matrices as kernel-weighted averages of cross-covariances and deriving exact null variances under independence. Recovery and stability bounds are established in two- and multiblock settings, revealing how spatial weighting and within-block dependence jointly influence effective information for association estimation. This work clarifies the interpretation of fitted spatial association axes and provides conditions for their reliable recovery.
Spatial association axes estimated by skrCCA are shown to be kernel-weighted averages of cross-covariances, with recovery reliability depending on population separation and inter-block connection strength.
Relating observations of different categories across space requires methods that accommodate distinct sampling locations and spatial dependence. Spatial kernel restricted canonical correlation analysis (skrCCA) uses spatial weights to identify associated feature combinations across data categories. Here, we develop a statistical theory of its covariance form after feature reduction. Conditional on the observed locations and under a mean-zero model, the pairwise association matrices target kernel-weighted averages of cross-covariances. For prespecified directions, we derive the exact null variance under independence between blocks. We establish recovery and stability bounds in two- and multiblock settings under explicit population models, and show that exact cyclic updates have convergent objective values. The variance characterization shows how spatial weighting and within-block dependence jointly determine the effective information available for estimating association. The recovery bounds separate statistical and optimization error, relating stability to population separation and the strength of connections between blocks. These results clarify what fitted spatial association axes estimate and provide conditions under which they can be recovered reliably.