• Transactions of the Korean Society of Mechanical Engineers A
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• v.25 no.10
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• pp.1605-1612
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• 2001

Least-squares meshfree method is presented. Conventional meshfree methods based on the Galerkin formulation suffer from inaccurate numerical integration. Least-squares formulation exhibits rather different integration-related characteristics. It is demonstrated through numerical examples that least-squares formulation is much more robust to integration errors than the Galerkin's. Therefore efficient meshfree methods can be devised by combining very simple integration algorithms and least-squares formulation.

• Communications for Statistical Applications and Methods
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• v.13 no.2
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• pp.233-241
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• 2006

Classical principal component analysis (PCA) can be formulated as finding the linear subspace that best accommodates multidimensional data points in the sense that the sum of squared residual distances is minimized. As alternatives to such LS (least squares) fitting approach, we produce LMS (least median of squares) and LTS (least trimmed squares)-type PCA by minimizing the median of squared residual distances and the trimmed sum of squares, in a similar fashion to Rousseeuw (1984)'s alternative approaches to LS linear regression. Proposed methods adopt the data-driven optimization algorithm of Croux and Ruiz-Gazen (1996, 2005) that is conceptually simple and computationally practical. Numerical examples are given.

• Honam Mathematical Journal
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• v.37 no.3
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• pp.299-315
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• 2015

In this paper, we study the first-order system least-squares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1291-1310
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• 2013

This paper develops least-squares pseudo-spectral collocation methods for elliptic boundary value problems having interface conditions given by discontinuous coefficients and singular source term. From the discontinuities of coefficients and singular source term, we derive the interface conditions and then we impose such interface conditions to solution spaces. We define two types of discrete least-squares functionals summing discontinuous spectral norms of the residual equations over two sub-domains. In this paper, we show that the homogeneous least-squares functionals are equivalent to appropriate product norms and the proposed methods have the spectral convergence. Finally, we present some numerical results to provide evidences for analysis and spectral convergence of the proposed methods.

• Communications for Statistical Applications and Methods
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• v.25 no.6
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• pp.673-685
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• 2018

Procrustes analysis is a useful technique useful to measure, compare shape differences and estimate a mean shape for objects; however it is based on a least squares criterion and is affected by some outliers. Therefore, we propose two generalized Procrustes analysis methods based on M-estimation and least median of squares estimation that are resistant to object outliers. In addition, two algorithms are given for practical implementation. A simulation study and some examples are used to examine and compared the performances of the algorithms with the least square method. Moreover since these resistant GPA methods are available for higher dimensions, we need some methods to visualize the objects and mean shape effectively. Also since we have concentrated on resistant fitting methods without considering shape distributions, we wish to shape analysis not be sensitive to particular model.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.65 no.9
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• pp.1541-1550
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• 2016

In this paper, we introduce a novel learning methodology of fuzzy clustering-based neural network pattern classifier. Fuzzy clustering-based neural network pattern classifier depicts the patterns of given classes using fuzzy rules and categorizes the patterns on unseen data through fuzzy rules. Least squares estimator(LSE) or weighted least squares estimator(WLSE) is typically used in order to estimate the coefficients of polynomial function, but this study proposes a novel coefficient estimate method which includes advantages of the existing methods. The premise part of fuzzy rule depicts input space as "If" clause of fuzzy rule through fuzzy c-means(FCM) clustering, while the consequent part of fuzzy rule denotes output space through polynomial function such as linear, quadratic and their coefficients are estimated by the proposed local least squares estimator(LLSE)-based learning. In order to evaluate the performance of the proposed pattern classifier, the variety of machine learning data sets are exploited in experiments and through the comparative analysis of performance, it provides that the proposed LLSE-based learning method is preferable when compared with the other learning methods conventionally used in previous literature.

• International Journal of Aeronautical and Space Sciences
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• v.14 no.1
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• pp.85-90
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• 2013

This paper presents a study on terrain referenced navigation (TRN). The extended Kalman filter (EKF) is adopted as a filter method. A Jacobian matrix of measurement equations in the EKF consists of terrain slope terms, and accurate slope estimation is essential to keep filter stability. Two slope estimation methods are proposed in this study. Both methods are based on the least-squares approach. One is planar regression searching the best plane, in the least-squares sense, representing the terrain map over the region, determined by position error covariance. It is shown that the method could provide a more accurate solution than the previously developed linear regression approach, which uses lines rather than a plane in the least-squares measure. The other proposed method is weighted planar regression. Additional weights formed by Gaussian pdf are multiplied in the planar regression, to reflect the actual pdf of the position estimate of EKF. Monte Carlo simulations are conducted, to compare the performance between the previous and two proposed methods, by analyzing the filter properties of divergence probability and convergence speed. It is expected that one of the slope estimation methods could be implemented, after determining which of the filter properties is more significant at each mission.

• The Transactions of the Korean Institute of Electrical Engineers
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• v.33 no.9
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• pp.372-376
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• 1984

This paper discusses on the stochastic adaprive control which uses a least squares method in the parameter adaptation law. Especially we study on the reason why existing methods have adopted modified least squares methods. After examining the performances of these methods for time-varying systems, we propose a new method to deal with such a situation, study on the stability problem, and finally show the effectiveness of the method with a computer simulation example.

• Proceedings of the Korean Society of Surveying, Geodesy, Photogrammetry, and Cartography Conference
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• 2003.10a
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• pp.15-19
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• 2003

The Total Least Squares (TLS) method is introduced in comparison with the conventional Least Squares (LS) method. The principles and mathematical models for both methods are summarized and the comparison results from their applications to a simple geometric example, fitting a straight line to a set of 2D points are presented. As conceptually reasoned, the results clearly indicate that LS is more susceptible of producing wrong parameters with worse precision rather than TLS. For many applications in surveying, can adjustment computation and parameter estimation based on TLS provide better results.

• ETRI Journal
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• v.28 no.2
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• pp.155-161
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• 2006

In this paper, we introduce an escalator (ESC) algorithm based on the least squares (LS) criterion. The proposed algorithm is relatively insensitive to the eigenvalue spread ratio (ESR) of an input signal and has a faster convergence speed than the conventional ESC algorithms. This algorithm exploits the fast adaptation ability of least squares methods and the orthogonalization property of the ESC structure. From the simulation results, the proposed algorithm shows superior convergence performance.