(The) projection matrix of the linear models using eigenvectors

Abstract

When a linear model $y = X\beta + e$ is given, the orthogonal projection matrix $P_x = X(X``X)^-X``$ plays and impotrant role in analyzing the linear model. The matrix $P_x$ can be obtained by computing the Moore-Penrose inverse of $X``X$. It is well known that the design matrix $X$ can be expressed as a product $X = TX_o$, where $T$ is a replication matrix and $X_o$ is the design matrix of the corresponding balanced model that contains only one observation in each cell. From this relation the Moore-Penrose inverse of $X``X$ can be drived using the nonzero eigenvalues of $X``_oX_o$ and the corresponding eigenvectors since $X``_oX_o$ is symmetric and positive semi-definite. The purpose of this thesis is to develop an efficient procedure for computing the orthogonal projection matrix of an unbalanced model without empty cell based on the corresponding balanced design matrix $X_o$ whether there are interactions or not. We describe a explicit form of the orthogonal projection matrix of design matrix for a balanced model. This form is based on the spectral decomposition of $X``_oX_o$ for the corresponding balanced model. Also from the corresponding balanced design matrix $X_o$ we can easily find a concrete form of the orthogonal projection matrix for an unbalanced model. Let $Q_r\Lambda_rQ``_r = X``_oX_o$ be a spectral decomposition of $X``_oX_o$ where $\Lambda_r$ is a diagonal matrix of the nonzero eigenval ues of $X``_oX_o$ and $Q_r$ is a matrix whose columns are standardized eigenvectors corresponding to the nonzero eigenvalues. Also letting $Z=X_oQ_r\Lambda_r^{-1/2}$ gives $P_{X_o} = ZZ``$ and $P_X = TZ(Z``DZ)^{-1}Z``T``$, where $D$ is a diagonal matrix of cell frequencies and $D = T``T$. It is sufficient to compute the matrix $Z$ in order to obtain the orthogonal projection matrix of a given unbalanced model. As well, the matrix $Z$ can be obtained by the regular rule that depends on only the number of levels of main factors whether there are interactions or no...