tensr, Because tensor
was Already Taken

Description
This package contains a collection of functions for statistical
analysis with tensor(array)-variate data sets.
Let \(X\) be a multidimensional
array (also called a tensor) of \(K\)
dimensions. This package provides a series of functions to perform
statistical inference when \(\text{vec}(X)
\sim N(0,\Sigma)\), where \(\Sigma\) is assumed to be Kronecker
structured. That is, \(\Sigma\) is the
Kronecker product of \(K\) covariance
matrices, each of which has the interpretation of being the covariance
of \(X\) along its \(k\)th mode, or dimension.
Pay particular attention to the zero mean assumption. That is, you
need to de-mean your data prior to applying these functions. If you have
more than one sample, \(X_i\) for \(i = 1,\ldots,n\), then you can concatenate
these tensors along a \((K+1)\)th mode
to form a new tensor \(Y\) and apply
the demean_tensor()
function to \(Y\) which will return a tensor that
satisfies the mean-zero assumption.
Details of the methods may be found in Gerard & Hoff (2015) and
Gerard & Hoff (2016). 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.
Installation
To install from CRAN, run in R
:
install.packages("tensr")
To install the latest version from Github, run in R
:
## install.packages("pak")
pak::pak("github::dcgerard/tensr")
References
Gerard, D., & Hoff, P. (2016). A higher-order LQ decomposition
for separable covariance models. Linear Algebra and its
Applications, 505, 57-84. doi: 10.1016/j.laa.2016.04.033
Gerard, D., & Hoff, P. (2015). Equivariant minimax dominators of
the MLE in the array normal model. Journal of Multivariate
Analysis, 137, 32-49. doi: 10.1016/j.jmva.2015.01.020