Efficient computational algorithms for analysis of linear models using the M-P inverse

Abstract

The generalized inverse plays a significant role in the analysis of linear models. Especially the Moore-Penrose inverse is very important. In this thesis we derive a procedure for analyzing the unbalanced model using the Moore-Penrose inverse based on the results of the corresponding balanced model. For a given unbalanced linear modely $y=X\beta\,+\,\o$ the computations of a solution of the normal equations and the sums of; squares are based on computing a generalized inverse (X``X)- and the protection matrix $P_X$ = X(X``X)X``. This projection matrix plays a significant role in analyzing the model. The design matrix X can be expressed as a product of two matrices T and $X\o$ namely $X=TX\o$, where $X\o$ is the design matrix of the corresponding balanced model that contains exactly one observation in each cell and T is the matrix indicating the replications of each cell. From this relation the Moore-Pentose inverse $X^+$ can be derived in terms of T and $X^+\o$. Also the projection matrix Px can be obtained in the same manner. Then from these results we can reduce much of the computational efforts to analyze the linear model. We give a simple method for finding the Moore-Penrose inverse of the design matrix for a balanced model. The Moore-Penrose inverse is closely related to the minimum norm least squares solution. The design matrix of the balanced linear model is of a special form. In most cases it can be represented using the Kronecker products of identity matrices and of matrices with all elements equal to unity. Hence the explicit form of the minimum norm least squares solution can be found and from this result the explicit forms of the Moore-Penrose inverse of the design matrix and the projection matrix can be easily obtained. When the model contains the interaction effects we give an iterative procedure for computing the projection matrix. Furthermore using this method we can find the projection matrix of the design matrix for the balanced model whether ...