Heinz Neudecker

An approach ton-mode components analysis

As an extension of Lastovickas four-mode components analysis ann-mode components analysis is developed. Using a convenient notation, both a canonical and a least squares solution are derived. The relation between both solutions and their computational aspects are discussed.

BLOCK KRONECKER PRODUCTS AND THE VECB OPERATOR

This paper is concerned with two generalizations of the Kronecker product and two related generalizations of the vec operator. It is demonstrated that they pairwise match two different kinds of matrix partition, viz. the balanced and unbalanced ones. Relevant properties are supplied and proved. A related concept, the so-called tilde transform of a balanced block matrix, is also studied. The results are illustrated with various statistical applications of the five concepts studied.

Some results on commutation matrices, with statistical applications

The commutation matrix Pmn changes the order of multiplication of a Kronecker matrix product. The vec operator stacks columns of a matrix one under another in a single column. It is possible to express the vec of a Kronecker matrix product in terms of a Kronecker product of vecs of matrices. The commutation matrix plays an important role here. “Super-vec-operators” like vec A ⊗ vec A vec (A ⊗ A), and vec{(A ⊗ A)Pnn} are very convenient. Several of their properties are being studied. Both the traditional commutation matrix and vec operator and the newer concepts developed from these are applied to multivariate statistical and related problems.

Experiments with Mixtures: Optimal Allocations for Becker's Models

Extending Scheffé’s simplex-centroid design for experiments with mixtures, we introduce aweighted simplex-centroid design for a class of mixture models. Becker’s homogeneous functions of degree one belong to this class. By applying optimal design theory, we obtainA-, D- andI-optimal allocations of observations for Becker’s models.

A matrix trace inequality

Abstract In a short note Y. Yang [2] solved a problem set by R. Bellman [1]. It is not difficult to prove a slightly relaxed version of Yangs result. We shall use a different method.

Fourth-Order Properties of Normally Distributed Random Matrices

Abstract Three results are given involving a normally distributed matrix X , namely (1) the expectation of X ′ AXCX ′ BX , (2) the covariance of vec X ′ AX and vec X ′ BX , and (3) the expectation of X ⊗ X ⊗ X ⊗ X .

Numerical solutions of the algebraic matrix Riccati equation.

Abstract The linear-quadratic control model is one of the most widely used control models in both empirical and theoretical economic modeling. In order to obtain the equilibrium solution of this control model, the so-called algebraic matrix Riccati equation has to be solved. In this note we present a numerical solution method for solving this equation. Our method solves the Riccati equation as a multidimensional fixed-point problem. By establishing the analytical derivative of the Riccati equation we have been able to construct a very efficient Newton-type solution method with quadratic convergence properties. Our method is an extension for the Newton-Raphson method described in Kwakernaak and Sivan (1972) and does not require any special conditions on the transition matrix as in the nonrecursive method of Vaughan (1970).

A direct derivation of the exact Fisher information matrix of Gaussian vector state space models

This paper deals with a direct derivation of Fishers information matrix of vector state space models for the general case, by which is meant the establishment of the matrix as a whole and not element by element. The method to be used is matrix differentiation, see [4]. We assume the model to be Gaussian and use the negative logarithm of the likelihood function as used in the definition of Fishers information. In a related paper Klein et al. [3] establish the information matrix by assembling its elements as derived in the literature [2,5,6] and for an approximation of the Hessian of the log-likelihood function one can refer to [7,8].

On second-order and fourth-order moments of jointly distributed random matrices: a survey

Abstract The study is concerned with second-order and fourth-order moments of jointly distributed random matrices. When distributional properties are required, normality is adopted. Some of the results can also be applied to elliptical or Wishart distributions. The developments are entirely algebraic. Full use is made of the Kronecker product, (repeated) vectorization, commutation matrices and related items. There are a few references in the main text but many additional references to other work, in which the same or kindred results (obtained by other methods, procedures or concepts) can be found, have been included in the References.

A V-optimal design for Scheffé's polynomial model

In experiments with mixtures, the so-called weighted simplex-centroid design is applied to obtain V-optimal allocation of observations which is shown to be an optimal design over the entire simplex by using the equivalence theorem.