This document provides verification test results for normal, lognormal, and uniform distributions that are used in Sandia's Latin Hypercube Sampling (LHS) software. The purpose of this testing is to verify that the sample values being generated in LHS are distributed according to the desired distribution types. The testing of distribution correctness is done by examining summary statistics, graphical comparisons using quantile-quantile plots, and format statistical tests such as the Chisquare test, the Kolmogorov-Smirnov test, and the Anderson-Darling test. The overall results from the testing indicate that the generation of normal, lognormal, and uniform distributions in LHS is acceptable.
We propose two new multilinear operators for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions. The first operator, which we call the Tucker operator, is shorthand for performing an n-mode matrix multiplication for every mode of a given tensor and can be employed to concisely express the Tucker decomposition. The second operator, which we call the Kruskal operator, is shorthand for the sum of the outer-products of the columns of N matrices and allows a divorce from a matricized representation and a very concise expression of the PARAFAC decomposition. We explore the properties of the Tucker and Kruskal operators independently of the related decompositions. Additionally, we provide a review of the matrix and tensor operations that are frequently used in the context of tensor decompositions.