Envelope regression is a crucial part of multivariate linear modeling, leveraging separation into material and immaterial parts to provide parsimonious data reduction. Rekabdarkolaee et al. (2020) construct spatial envelope, a novel multivariate Gaussian process, extended by May et al. (2022) to linear coregionalization envelope. Our proposal (TEASE) combines spatial envelope and full-scale basis graphical lasso (LeDuc et al., 2025), leveraging targeted graphical elastic net (Kovács et al., 2021) for immaterial precision matrices. Material part is a multivariate multiresolution Gaussian Markov random field (Kleiber et al., 2019; Caringi and Secchi, 2026). Enhanced envelope penalty (Kwon and Zou, 2025) regularizes regression coefficients. Demonstrations are on climate data.

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