HE Plots for Multivariate Linear Models

Multivariate analysis of variance (MANOVA) extends the ideas and methods of univariate ANOVA in simple and straightforward ways. But the familiar graphical methods typically used for univariate ANOVA are inadequate for showing how measures in a multivariate response vary with each other, and how their means vary with explanatory factors. Similarly, the graphical methods commonly used in multiple regression are not widely available or used in multivariate multiple regression (MMRA). We describe a variety of graphical methods for multiple-response (MANOVA and MMRA) data aimed at understanding what is being tested in a multivariate test, and how factor/predictor effects are expressed across multiple response measures.

In particular, we describe and illustrate: (a) Data ellipses and biplots for multivariate data; (b) HE plots, showing the hypothesis and error covariance matrices for a given pair of responses, and a given effect; (c) HE plot matrices, showing all pairwise HE plots; and (d) reduced-rank analogs of HE plots, showing all observations, group means, and their relations to the response variables. All of these methods are implemented in a collection of easily used SAS macro programs.     

Existence of Matrix-Valued Multiresolution Analysis-Based Matrix-Valued Tight Wavelet Frames

In this article, we introduce and study the matrix-valued tight wavelet frames for analyzing matrix-valued signal based on matrix-valued multiresolution analysis (MMRA). We put our emphasis on the existence of the MMRA-based matrix-valued tight wavelet frames by establishing the correspondence with their the unitary extension principle (UEP). Here in particular we introduce the square brackets product and the quasi-interpolatory operator, which makes the certificating process for UEP become relatively simple. Some interesting byproducts, such as features on the quasi-interpolatory operator 𝒫_n in the matrix-valued function space case, are the critical foundation for our main work.