Covariance Inference and Decompositions for Tensor Datasets

A collection of functions for Kronecker structured covariance estimation and testing under the array normal model. For estimation, maximum likelihood and Bayesian equivariant estimation procedures are implemented. For testing, a likelihood ratio testing procedure is available. This package also contains additional functions for manipulating and decomposing tensor data sets. This work was partially supported by NSF grant DMS-1505136. Details of the methods are described in Gerard and Hoff (2015) and Gerard and Hoff (2016) .

BuildStatus AppVeyor BuildStatus CRANVersion License: GPLv3


This package contains a collection of functions for statistical analysis with tensor(array)-variate data sets. In particular, tensr has the following features:

  • Basic functions for handling arrays, such as vectorization, matrix unfolding, and multilinear multiplication.
  • Functions for calculating (Tucker) tensor decompositions, such as the incredible higher-order LQ decomposition (incredible HOLQ), the incredible singular value decomposition (ISVD), the incredible higher-order polar decomposition (IHOP), the higher-order singular value decomposition (HOSVD), and the low multilinear rank approximation using the higher-order orthogonal iteration (HOOI).
  • Perform likelihood inference in mean-zero Kronecker structured covariance models, such as
    • Derive the maximum likelihood estimates of the covariance matrices under the array normal model,
    • Run a likelihood ratio test in separable covariance models, and
    • Calculate AIC’s and BIC’s for separable covariance models.
  • Run a Gibbs sampler to draw from the posterior distribution of the Kronecker structured covariance matrix in the array normal model. This posterior is with respect to a (non-informative) prior induced by the right Haar measure over a product group of lower triangular matrices acting on the space of Kronecker structured covariance matrices. For any invariant loss function, any Bayes rule with respect to this prior will be the uniformly minimum risk equivariant estimator (UMREE) with respect to that loss.
  • Calculate the UMREE under a multiway generalization of Stein’s loss. This estimator dominates the maximum likelihood estimator uniformly over the entire parameter space of Kronecker structured covariance matrices.
  • Calculate a (randomized) orthogonally invariant estimator of the Kronecker structured covariance matrix. This estimator dominates the UMREE under the product group of lower triangular matrices.

This package is also published on CRAN.

Vignettes are available on Equivariant Inference and Likelihood Inference.


To install from CRAN, run in R:


To install the latest version from Github, run in R:



Gerard, D., & Hoff, P. (2016). A higher-order LQ decomposition for separable covariance models. Linear Algebra and its Applications, 505, 57-84. [Link to LAA] [Link to arXiv]

Gerard, D., & Hoff, P. (2015). Equivariant minimax dominators of the MLE in the array normal model. Journal of Multivariate Analysis, 137, 32-49. [Link to JMVA] [Link to arXiv]


Reference manual

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1.0.1 by David Gerard, 2 years ago

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Browse source code at

Authors: David Gerard [aut, cre] , Peter Hoff [aut]

Documentation:   PDF Manual  

GPL-3 license

Imports assertthat

Suggests knitr, rmarkdown, covr, testthat

Imported by TULIP, catch.

See at CRAN