Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets

Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components. For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper. We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles. Quaternions are isomorphic to an algebra of structured 4x4 real matrices. This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms. A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance. Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs. Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces

Seeking Order in Chaos: A Definition of Culture

Since its identification as a unique field of research, the modern study of culture has become very popular. Its analytical-interpretive power has earned it a place of honor among the natural and social sciences, and the humanities. Despite its central status, however, the term “culture” itself has not yet found an accepted, customary definition. The absence of such a definition compels scholars of culture to search for ways to explore the discipline they are engaged in. This situation literally blocks knowledge of who we are as human beings and how we live, and muddles research goals and methodologies. This article aims to deal with this drawback. Taking as its starting point Freud’s basic definition of culture as the “total achievements and institutions, which moved us away from our animal-like ancestors,” and organizing this “total” under three analytical categories of taste, value, and control — it suggests a coherent definition of culture that encompasses most of the existing ones, while embracing them under the “rule” of “radical simplification,” conceived by Robert Darnton in hard-to-define cases. This may offer a better understanding of culture as an all-encompassing human phenomenon, and a a more effective means for selecting the appropriate methodologies needed for the analysis of relevant questions

DSS SUCCESS: MEASUREMENT AND FACILITATION

Decision Support Systems (DSS) represent an ever increasing portion of the investment in computer-based systems in organizations. Unlike earlier systems which aimed to replace existing clerical processes with faster, more efficient clerical processes, DSS attempt to extend and expand the capabilities of organizational decision makers. This fundamental difference in purpose between DSS and clerical systems causes our existing notions about system success to be inadequate. This paper explores the issue of DSS success, asking what it is, how it can be measured, and what can be done to facilitate it.Information Systems Working Papers Serie