• Journal of applied mathematics & informatics
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• v.8 no.1
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• pp.9-26
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• 2001

In some previous papers the author extended two algorithms proposed by Z. Kovarik for approximate orthogonalization of a finite set of linearly independent vectors from a Hibert space, to the case when the vectors are rows (not necessary linearly independent) of an arbitrary rectangular matrix. In this paper we describe combinations between these two methods and the classical Kaczmarz’s iteration. We prove that, in the case of a consistent least-squares problem, the new algorithms so obtained converge ti any of its solutions (depending on the initial approximation). The numerical experiments described in the last section of the paper on a problem obtained after the discretization of a first kind integral equation ilustrate the fast convergence of the new algorithms. AMS Mathematics Subject Classification : 65F10, 65F20.

• Communications for Statistical Applications and Methods
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• v.28 no.5
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• pp.463-475
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• 2021

Desgagné and de Micheaux (2018) proposed an alternative univariate normality test to the Jarque-Bera test. The proposed statistic is based on the sample second power skewness and kurtosis while the Jarque-Bera statistic uses sample Pearson's skewness and kurtosis that are the third and fourth standardized sample moments, respectively. In this paper, we generalize their statistic to a multivariate version based on orthogonalization or an empirical standardization of data. The proposed multivariate statistic follows chi-squared distribution approximately. A simulation study shows that the proposed statistic has good control of type I error even for a very small sample size when critical values from the approximate distribution are used. It has comparable power to the multivariate version of the Jarque-Bera test with exactly the same idea of the orthogonalization. It also shows much better power for some mixed normal alternatives.

• Journal of the Korean Institute of Telematics and Electronics B
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• v.29B no.1
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• pp.26-33
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• 1992

This paper presents lattice structure of bilinear filter and the conversion equations from lattice parameters to direct-form parameters. Billnear models are attractive for adaptive filtering applications because they can approximate a large class of nonlinear systems adequately, and usually with considerable parsimony in the number of coefficients required. The lattice filter formulation transforms the nonlinear filtering problem into an equivalent multichannel linear filtering problem and then uses multichannel lattice filtering algorithms to solve the nonlinear filtering problem. The lattice filters perform a Gram-Schmidt orthogonalization of the input data and have very good easily extended to more general nonlinear output feedback structures.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.687-696
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• 2010

In this paper, an algorithm for computing the sparse approximate inverse factor of matrix $A^{T}\;A$, where A is an $m\;{\times}\;n$ matrix with $m\;{\geq}\;n$ and rank(A) = n, is proposed. The computation of the inverse factor are done without computing the matrix $A^{T}\;A$. The computed sparse approximate inverse factor is applied as a preconditioner for solving normal equations in conjunction with the CGNR algorithm. Some numerical experiments on test matrices are presented to show the efficiency of the method. A comparison with some available methods is also included.

• Journal of the Korean Institute of Telematics and Electronics B
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• v.29B no.1
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• pp.34-42
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• 1992

This paper presents two fast least-squares lattice algorithms for adaptive nonlinear filters equipped with bilinear system models. The lattice filters perform a Gram-Schmidt orthogonalization of the input data and have very good numerical properties. Furthermore, the computational complexity of the algorithms is an order of magnitude snaller than previously algorithm is an order of magnitude smaller than previously available methods. The first of the two approaches is an equation error algorithm that uses the measured desired response signal directly to comprte the adaptive filter outputs. This method is conceptually very simple`however, it will result in biased system models in the presence of measurement noise. The second approach is an approximate least-squares output error solution. In this case, the past samples of the output of the adaptive system itself are used to produce the filter output at the current time. Results of several experiments that demonstrate and compare the properties of the adaptive bilinear filters are also presented in this paper. These results indicate that the output error algorithm is less sensitive to output measurement noise than the squation error method.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.24 no.9B
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• pp.1785-1794
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• 1999

The affine projection algorithm has known t require less computational complexity than RLS but have much faster convergence than NLMS for speech-like input signals. But the affine projection algorithm is still much more computationally demanding than the LMS algorithm because it requires the matrix inversion. In this paper, we show that the affine projection algorithm can be realized with the Gram-Schmidt orthogonalizaion of input vectors. Using the derived relation, we propose an approximate but much more efficient implementation of the affine projection algorithm. Simulation results show that the proposed algorithm has the convergence speed that is comparable to the affine projection algorithm with only a slight extra calculation complexity beyond that of NLMS.

• Journal of Communications and Networks
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• v.12 no.3
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• pp.222-230
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• 2010

In this paper, we present a closed-form approximation of the average sum rate of zero-forcing (ZF) beamforming (BF) with semi-orthogonal user selection (SUS). We first derive the survival probability associated with the SUS that absolute square of the channel correlation between two users is less than the orthogonalization level threshold (OLT).With this result, each distribution for the number of surviving users at each iteration of the SUS and the number of streams for transmission is calculated. Secondly, the received signal power of ZF-BF is represented as a function of the elements of the upper triangular matrix from QR decomposition of the channel matrix. Thirdly, we approximate the received signal power of ZF-BF with the SUS as the maximum of scaled chisquare random variables where the scaling factor is approximated as a function of both OLT and the number of users in the system. Putting all the above derivations and order statistics together, the approximated ergodic sum rate of ZF-BF with the SUS is shown in a closed form. The simulation results verify that the approximation tightly matches with the sample average for any OLT and even for a small number of users.