The Matrix Ridge Approximation: Algorithms and Applications

We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matrix factorization plus a ridge term. Moreover, we present probabilistic interpretations using a normal latent variable model and a Wishart model for this approximation approach. The idea behind the latent variable model in turn leads us to an efficient EM iterative method for handling the matrix ridge approximation problem. Finally, we illustrate the applications of the approximation approach in multivariate data analysis. Empirical studies in spectral clustering and Gaussian process regression show that the matrix ridge approximation with the EM iteration is potentially useful

Analysis of gas chromatography/mass spectrometry data for catalytic lignin depolymerization using positive matrix factorization

Various catalytic technologies are being developed to efficiently convert lignin into renewable chemicals. However, due to its complexity, catalytic lignin depolymerization often generates a wide and complex distribution of product compounds. Gas chromatography/mass spectrometry (GC-MS) is a common analytical technique to profile the compounds that comprise lignin depolymerization products. GC-MS is applied not only to determine the product composition, but also to develop an understanding of the catalytic reaction pathways and of the relationships among catalyst structure, reaction conditions, and the resulting compounds generated. Although a very useful tool, the analysis of lignin depolymerization products with GC-MS is limited by the quality and scope of the available mass spectral libraries and the ability to correlate changes in GC-MS chromatograms to changes in lignin structure, catalyst structure, and other reaction conditions. In this study, the GC-MS data of the depolymerization products generated from organosolv hybrid poplar lignin using a copper-doped porous metal oxide catalyst and a methanol/dimethyl carbonate co-solvent was analyzed by applying a factor analysis technique, positive matrix factorization (PMF). Several different solutions for the PMF model were explored. A 13-factor solution sufficiently explains the chemical changes occurring to lignin depolymerization products as a function of lignin, reaction time, catalyst, and solvent. Overall, seven factors were found to represent aromatic compounds, while one factor was defined by aliphatic compounds

Censoring, Factorizations, and Spectral Analysis for Transition Matrices with Block-Repeating Entries

In this paper, we use the Markov chain censoring technique to study infinite state Markov chains whose transition matrices possess block-repeating entries. We demonstrate that a number of important probabilistic measures are invariant under censoring. Informally speaking, these measures involve first passage times or expected numbers of visits to certain levels where other levels are taboo;they are closely related to the so-called fundamental matrix of the Markov chain which is also studied here. Factorization theorems for the characteristic equation of the blocks of the transition matrix are obtained. Necessary and sufficient conditions are derived for such a Markov chain to be positive recurrent, null recurrent, or transient based either on spectral analysis, or on a property of the fundamental matrix. Explicit expressions are obtained for key probabilistic measures, including the stationary probability vector and the fundamental matrix, which could be potentially used to develop various recursivealgorithms for computing these measures.block-Toeplitz transition matrices, factorization of characteristic functions, spectral analysis, fundamental matrix, conditions of recurrence and transience.

Conditioning analysis of block incomplete factorization and its application to elliptic equations

The paper deals with eigenvalue estimates for block incomplete fac- torization methods for symmetric matrices. First, some previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue. Second, upper bounds for the maximum eigenvalue of the preconditioned matrix are further estimated, which presents a substantial im- provement of earlier results. Finally, the results are used to estimate bounds for every eigenvalue of the preconditioned matrices, in particular, for the maximum eigenvalue, when a modified block incomplete factorization is used to solve an elliptic equation with variable coefficients in two dimensions. The analysis yields a new upper bound of type γh−1 for the condition number of the preconditioned matrix and shows clearly how the coefficients of the differential equation influ- ence the positive constant γ

On generating correlated random variables with a given valid or invalid Correlation matrix

In simulation we often have to generate correlated random variables by giving a reference intercorrelation matrix, R or Q. The matrix R is positive definite and a valid correlation matrix. The matrix Q may appear to be a correlation matrix but it may be invalid (negative definite). With R(m,m) it is easy to generate X(n,m), but Q(m,m) cannot give real X(n,m). So, Q has to be converted into the near-most R matrix by some procedure. NJ Higham (2002) provides a method to generate R from Q that satisfies the minimum Frobenius norm condition for (Q-R). Ali Al-Subaihi (2004) gives another method, but his method does not produce an optimal R from Q. In this paper we propose an algorithm to produce an optimal R from Q by minimizing the maximum norm of (Q-R). A Computer program (in FORTRAN) also has been provided. Having obtained R from Q, the paper gives an algorithm to obtain X(n,m) from R(m,m). The proposed algorithm is based on factorization of R, yet it is different from the Kaiser Dichman (1962) procedure. A computer program also has been given.Positive semidefinite; negative definite; maximum norm; frobenius norm; correlated random variables; intercorrelation matrix; correlation matrix; Monte Carlo experiment; multicollinearity; cointegration; computer program; multivariate analysis; simulation; generation of collinear sample data