• Proceedings of the Korean Operations and Management Science Society Conference
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• 1996.10a
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• pp.289-292
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• 1996

In Cholesky factorization of the interior point method, dense columns of A matrix make dense Cholesky factor L regardless of sparsity of A matrix. We introduce a method to transform a primal problem to a dual problem in order to preserve the sparsity.

• Journal of the military operations research society of Korea
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• v.24 no.1
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• pp.146-156
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• 1998

Cholesky factorization is known to be inefficient to problems with dense column and network problems in interior point methods. We use the conjugate gradient method and preconditioners to improve the convergence rate of the conjugate gradient method. Several preconditioners were applied to LPABO 5.1 and the results were compared with those of CPLEX 3.0. The conjugate gradient method shows to be more efficient than Cholesky factorization to problems with dense columns and network problems. The incomplete Cholesky factorization preconditioner shows to be the most efficient among the preconditioners.

• Communications for Statistical Applications and Methods
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• v.23 no.6
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• pp.587-592
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• 2016

It is a long-standing issue to correctly determine the number of long-run relationships among time series processes. We revisit nonparametric test for cointegration rank and propose bootstrap refinements. Consistent with model-free nature of the tests, we make use of Cholesky factor bootstrap methods, which require weak conditions for data generating processes. Simulation studies show that the original Breitung's test have difficulty in obtaining the correct size due to dependence in cointegrated errors. Our proposed bootstrapped tests considerably mitigate size distortions and represent a complementary approach to other bootstrap refinements, including sieve methods.

• Communications for Statistical Applications and Methods
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• v.24 no.1
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• pp.81-96
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• 2017

Cumulative logit random effects models are typically used to analyze longitudinal ordinal data. The random effects covariance matrix is used in the models to demonstrate both subject-specific and time variations. The covariance matrix may also be homogeneous; however, the structure of the covariance matrix is assumed to be homoscedastic and restricted because the matrix is high-dimensional and should be positive definite. To satisfy these restrictions two Cholesky decomposition methods were proposed in linear (mixed) models for the random effects precision matrix and the random effects covariance matrix, respectively: modified Cholesky and moving average Cholesky decompositions. In this paper, we use these two methods to model the random effects precision matrix and the random effects covariance matrix in cumulative logit random effects models for longitudinal ordinal data. The methods are illustrated by a lung cancer data set.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.27 no.2
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• pp.276-284
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• 2003

The preconditioned conjugate gradient method is an efficient iterative solution scheme for large size finite element problems. As preconditioning method, we choose an incomplete Cholesky factorization which has efficiency and easiness in implementation in this paper. The incomplete Cholesky factorization mettled sometimes leads to breakdown of the computational procedure that means pivots in the matrix become minus during factorization. So, it is inevitable that a reduction process fur stabilizing and this process will guarantee robustness of the algorithm at the cost of a little computation. Recently incomplete factorization that enhances robustness through increasing diagonal dominancy instead of reduction process has been developed. This method has better efficiency for the problem that has rotational degree of freedom but is sensitive to parameters and the breakdown can be occurred occasionally. Therefore, this paper presents new method that guarantees robustness for this method. Numerical experiment shows that the present method guarantees robustness without further efficiency loss.

• The Korean Journal of Applied Statistics
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• v.28 no.2
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• pp.211-219
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• 2015

Generalized linear mixed models are used to analyze longitudinal categorical data. Random effects specify the serial dependence of repeated outcomes in these models; however, the estimation of a random effects covariance matrix is challenging because of many parameters in the matrix and the estimated covariance matrix should satisfy positive definiteness. Several approaches to model the random effects covariance matrix are proposed to overcome these restrictions: modified Cholesky decomposition, moving average Cholesky decomposition, and partial autocorrelation approaches. We review several approaches and present potential future work.

• The Korean Journal of Applied Statistics
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• v.30 no.1
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• pp.103-117
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• 2017

In longitudinal data analysis, the serial correlation of repeated outcomes must be taken into account using covariance matrix. Modeling of the covariance matrix is important to estimate the effect of covariates properly. However, It is challenging because there are many parameters in the matrix and the estimated covariance matrix should be positive definite. To overcome the restrictions, several Cholesky decomposition approaches for the covariance matrix were proposed: modified autoregressive (AR), moving average (MA), ARMA Cholesky decompositions. In this paper we review them and compare the performance of the approaches using simulation studies.

• Communications for Statistical Applications and Methods
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• v.20 no.3
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• pp.235-240
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• 2013

Generalized linear mixed models(GLMMs) are frequently used for the analysis of longitudinal categorical data when the subject-specific effects is of interest. In GLMMs, the structure of the random effects covariance matrix is important for the estimation of fixed effects and to explain subject and time variations. The estimation of the matrix is not simple because of the high dimension and the positive definiteness; subsequently, we practically use the simple structure of the covariance matrix such as AR(1). However, this strong assumption can result in biased estimates of the fixed effects. In this paper, we introduce Bayesian modeling approaches for the random effects covariance matrix using a modified Cholesky decomposition. The modified Cholesky decomposition approach has been used to explain a heterogenous random effects covariance matrix and the subsequent estimated covariance matrix will be positive definite. We analyze metabolic syndrome data from a Korean Genomic Epidemiology Study using these methods.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.59-80
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• 2003

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.

• Journal of Korean Institute of Industrial Engineers
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• v.29 no.1
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• pp.78-89
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• 2003

Ordering is performed to reduce the amount of fill-ins of the Cholesky factor of a symmetric positive definite matrix. This paper proposes a new ordering algorithm that reduces the fill-ins of the Cholesky factor iteratively by elimination tree rotations and clique separators. Elimination tree rotations have been used mainly to reorder the rows of the permuted matrix for the efficiency of storage space management or parallel processing, etc. In the proposed algorithm, however, they are repeatedly performed to reduce the fill-ins of the Cholesky factor. In addition, we presents a simple method for finding a minimal node separator between arbitrary two nodes of a chordal graph. The proposed reordering procedure using clique separators enables us to obtain another order of rows of which the number of till-ins decreases strictly.