ARRAY CALCULUS WITH APPLICATIONS IN STATISTICS (APL, MATRIX, MULTIVARIATE ANALYSIS, COMPUTING, TENSOR)

A significant portion of modern statistics and other quantitative sciences is concerned with linearity. Matrix theory is a basic tool used in linear analysis. But matrices have only two indices--one index for rows and one for columns. With only two indices, it often is awkward to represent multivariate data in matrix form. This has led to the use of awkward, specialized methods such as stringing out data and using partitioned matrices, incidence matrices and Kronecker products. Moreover, there has not been any generally agreed-upon unified theory for multivariate derivatives. Such a theory would provide consistent definitions and useful formulas for computing the first and higher order derivatives of a scalar, vector or matrix of functions with respect to a scalar, vector or matrix of independent variables.;The array mathematics is applied to a variety of problems in statistics. Three new results are obtained: (1) New, efficient methods for computer calculation of sample multivariate means and covariances are presented. (2) A characterization of the commutation matrix is provided. (3) For the full factorial model of any order, explicit formulas are given for the cell means and for the mean of any set of cells that can be defined in terms of levels of fixed factors.;An historical basis for the array concept is provided. Suggestions for further research are given.;To try to resolve these linearity problems due to limitations of matrix theory, a theory of arrays with any fixed, finite number of indices is developed. The development includes the arithmetic and differential calculus of arrays. A technique for displaying arrays is provided.