Doubly-Unlinked Regression for Dependent Data

Shuffled regression concerns settings in which covariates and responses are observed without their correct pairing. In dependent-data problems, a second form of missing correspondence can arise when responses are also detached from the latent temporal, spatial, or geometric domain that induces their dependence structure. We study regression under this joint loss of correspondence and, to our knowledge, provide the first systematic treatment of this setting. Specifically, we consider a doubly-unlinked regression model in which both the covariate-response link and the response-domain link are unknown, represented by two latent permutation matrices, while dependence is induced by an unobserved stochastic process. This framework unifies shuffled regression and latent-domain permutation models within a common dependent-data setting. We characterize signal-to-noise regimes governing recovery of the regression parameter and the latent permutations, and show that consistent estimation of the regression coefficient can be achieved under strictly weaker conditions than exact permutation recovery. To address the combinatorial difficulty of inference, we develop REPAIR, a variational Bayes method based on a block-structured permutation model that captures localized scrambling while substantially reducing computational complexity. Simulations and an applied example illustrate the empirical behavior of REPAIR and support the theoretical results.