[Show abstract][Hide abstract] 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.
Linear Algebra and its Applications 01/2003; 362:251–255. · 0.97 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper is concerned with the fitting of non-linear relationships such as the logistic and Gompertz functions. Although Gauss had proposed the now standard Gauss–Newton procedure for this purpose in 1809, and it was strongly championed by Schultz in 1930, this procedure did not come into common use until modern computing equipment was introduced in the 1960s. In its stead a variety of virtually arbitrary procedures were employed. These arbitrary procedures are still used when the practitioner requires preliminary estimates of the parameters of a given non-linear function
Journal of the Royal Statistical Society Series D (The Statistician) 01/2002; 47(1):137 - 147.
[Show abstract][Hide abstract] ABSTRACT: We describe the geometrical representation of allocations of quantities of goods, votes or probabilities between two or more persons, parties or strategies. We are particularly concerned with the representation of the time-varying allocation of votes between three political parties and with the time-invariant allocation of probabilities between the three strategies available to one of the participants in some matrix games. Copyright 2001 by Blackwell Publishers Ltd and The Victoria University of Manchester
Manchester School 01/2001; 69(4):477-80. · 0.26 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we investigate the algebraic relationships between some of the more familiar estimation and testing procedures employed in multivariate econometrics and the principal components and continuum regression techniques of multivariate statistics.
Linear Algebra and its Applications 01/1999; 289(1):121-126. · 0.97 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper is concerned with the historical development of a traditional procedure for determining appropriate values for the parameters defining a linear relationship. This traditional procedure is variously known as the minimax absolute residual, Chebyshev, or L∞-norm procedure. Besides being of considerable interest in its own right as one of the earliest objective methods for estimating the parameters of such relationships, this procedure is also closely related to Rousseeuw's least median of squared residuals and to the least sum of absolute residuals or L1-norm procedures.The minimax absolute residual procedure was first proposed by Laplace in 1786 and developed over the next 40 years by de Prony, Cauchy, Fourier, and Laplace himself. More recent contributions to this traditional literature include those of de la Vallée Poussin and Stiefel.Nowadays, the minimax absolute residual procedure is usually implemented as the solution of a primal or dual linear programming problem. It therefore comes as no surprise to discover that some of the more prominent features of such problems, including early variants of the simplex algorithm are to be found in these contributions.In this paper we re-examine some of the conclusions reached by Grattan-Guinness (1970), Franksen (1985) and Grattan-Guinness (1994) and suggest several amendments to their findings. In particular, we establish the nature of de Prony's geometrical fitting procedure and trace the origins of Fourier's prototype of the simplex algorithm.
[Show abstract][Hide abstract] ABSTRACT: We briefly outline the origins of formal matrix theory in the 1870s and discuss Aitken's role in the dissemination of matrix methods in the 1940s with particular reference to the subject area of statistics and economics.
Linear Algebra and its Applications 01/1997; · 0.97 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We outline the history of some of the concepts and techniques of linear algebra which are intimately connected with the development of the method of least squares and related fitting procedures. Our study concentrates on contributions made during the early years of the nineteenth century, but it is not entirely restricted to this period.
Linear Algebra and its Applications 01/1996; · 0.97 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we generalise Rousseeuw's least median squared residual and minimum volume ellipsoid criteria and obtain a suitable criterion for fitting a q-dimensional hyperplane. This new criterion includes Rousseeuw's criteria as special cases. We also outline the corresponding criteria for fitting two or more q-dimensional hyperplanes.