This package provides a collection of functions for likelihood and equivariant inference for covariance matrices under the array normal model. Also included are functions for calculating tensor decompositions that are related to likelihood inference in the array normal model.
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
vec(X) \sim N(0,Σ),
where Σ is assumed to be Kronecker structured. That is, Σ is the Kronecker product of K covariance matrices, each of which has the interpretation of being the covariance of X along its kth 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,…,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.
amprod k-mode product.
anorm_cd Array normal conditional distributions.
array_bic_aic Calculate the AIC and BIC.
arrIndices Array indices.
atrans Tucker product.
collapse_mode Collapse multiple modes into one mode.
convert_cov Convert the output from
component covariance matrices.
demean_tensor Demeans array data.
equi_mcmc Gibbs sampler using an invariant prior.
fnorm Frobenius norm of an array.
get_equi_bayes Get the Bayes rule under multiway Stein's
get_isvd Calculate the incredible SVD (ISVD).
holq Calculate the incredible higher-order LQ decomposition
hooi Calculate the higher-order orthogonal iteration (HOOI).
hosvd Calculate the (truncated) higher-order SVD (HOSVD).
Kom Commutation matrix.
ihop The incredible higher-order polar decomposition (IHOP).
ldan Log-likelihood of array normal model.
listprod Element-wise matrix products between two lists.
lq LQ decomposition.
lrt_null_dist_dim_same Draw from null distribution of
likelihood ratio test statistic.
lrt_stat Calculate the likelihood ratio test statistic.
mat Unfold a matrix.
mhalf The symmetric square root of a positive definite
mle_from_holq Get MLE from output of
multi_stein_loss Calculate multiway Stein's loss from square
multi_stein_loss_cov Calculate multiway Stein's loss from
component covariance matrices.
multiway_takemura Calculate a truncated multiway Takemura
polar The left polar decomposition.
qr2 QR Decomposition.
random_ortho Generate a list of orthogonal matrices drawn
from Haar distribution.
rmirror_wishart Sample from the mirror-Wishart distribution.
sample_sig Update for total variation parameter in
sample_right_wishart Gibbs update of
start_ident Get list of identity matrices.
start_resids Sample covariance matrices for each mode.
tsum Tucker sum.
tr Trace of a matrix.
trim Truncates small numbers to 0.
Gerard, D., & Hoff, P. (2016). A higher-order LQ decomposition for separable covariance models. Linear Algebra and its Applications, 505, 57-84. https://doi.org/10.1016/j.laa.2016.04.033 http://arxiv.org/pdf/1410.1094v1.pdf
Gerard, D., & Hoff, P. (2015). Equivariant minimax dominators of the MLE in the array normal model. Journal of Multivariate Analysis, 137, 32-49. https://doi.org/10.1016/j.jmva.2015.01.020 http://arxiv.org/pdf/1408.0424.pdf
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