Sven-Oliver Troschke

Matrix representation of quaternions

Abstract We establish that there are a total of 48 distinct ordered sets of three 4×4 (skew-symmetric) signed permutation matrices which will serve as the basis of an algebra of quaternions.

Quaternions: further contributions to a matrix oriented approach

Abstract By representing quaternions as four-dimensional vectors and the multiplication of quaternions as matrix-by-vector product we employ a matrix oriented approach to the topic. We investigate properties of the fundamental real matrix associated with a quaternion and examine a frequently considered quaternion equation, from which the n th power of a quaternion can be determined.

On semi-orthogonality and a special class of matrices

Abstract The concept of semi-orthogonality of two complex vectors is introduced. As a consequence, a generalization of the class of orthogonal projectors is investigated.

On properties of 3 X 3 semi-magic squares

In this paper we consider 3 X 3 semi-magic squares understood as 3 X 3 matrices whose rows and columns add up tothe same constant. We identify the most general form of such matrices, calculate their Moore-Penrose inverse, their rank and their eigenvalues.

A Simulation Study To Compare Various Covariance Adjustment Techniques

A common procedure when combining two multivariate unbiased estimates (or forecasts) is the covariance adjustment technique (CAT). Here the optimal combination weights depend on the covariance structure of the estimators. In practical applications, however, this covariance structure is hardly ever known and, thus, has to be estimated. An effect of this drawback may be that the theoretically best method is no longer the best. In a simulation study (using normally distributed data) three different variants of CAT are compared with respect to their accuracy. These variants are different in the portion of the covariance structure that is estimated. We characterize which variant is appropriate in different situations and quantify the gains and losses that occur.

A selective procedure for combining forecasts

If there are various forecasts for the same random variable, it is common practice to combine these forecasts in order to obtain a better forecast. But an important question is how to perform the combination, especially if the system under investigation is subject to structural changes and consequently the best combination method is not the same all of the time. This paper presents a data driven approach, which for each point of time selects a combination technique from a given set of combination techniques. Properties and limitations of this selection procedure are investigated using simulated data from normal distributions.

SOME REMARKS ON PARTIAL ORDERINGS OF NONNEGATIVE DEFINITE MATRICES

Abstract Baksalary and Pukelsheim (1990) investigated partial orderings of nonnegative definite matrices. Some additional remarks are given in this note.

Regression Approach to the Linear Combination of Multivariate Forecasts

In Troschke (2002) the author introduces a linear approach to the scalar mean square error optimal combination of forecasts for a vector random variable. In this paper it is shown how the optimal combination parameters can be obtained with the help of linear regression. Thus the application of these combination methods to empirical data is facilitated. An empirical example illustrating the performance of the new methods is given. These methods are compared to the classical univariate treatment of the respective variables.

Regression approach to the linear plus quadratic combination of forecasts

In Troschke and Trenkler (2000) the authors introduce linear plus quadratic approaches to the mean square error optimal combination of forecasts for a scalar random variable. In this paper it is shown how the optimal combination parameters can be obtained with the help of linear regression. Thus numerical considerations as well as application of linear plus quadratic combination to empirical data are facilitated. First results on the comparison of the new methods to the classical linear approaches are given. It is found that there are situations where the linear plus quadratic approaches may be employed bene cially, but further investigations have to be carried out.