• The Korean Journal of Applied Statistics
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• v.28 no.1
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• pp.9-21
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• 2015

The support vector machine has been successfully applied to various classification areas due to its flexibility and a high level of classification accuracy. However, when analyzing imbalanced data with uneven class sizes, the classification accuracy of SVM may drop significantly in predicting minority class because the SVM classifiers are undesirably biased toward the majority class. The weighted $L_2$-norm SVM was developed for the analysis of imbalanced data; however, it cannot identify irrelevant input variables due to the characteristics of the ridge penalty. Therefore, we propose the weighted $L_1$-norm SVM, which uses lasso penalty to select important input variables and weights to differentiate the misclassification of data points between classes. We demonstrate the satisfactory performance of the proposed method through simulation studies and a real data analysis.

• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.867-875
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• 1998

In this paper we prove that any continuous function on a bounded closed interval of can be approximated by the superposition of a bounded sigmoidal function with a fixed weight. In addition we show that any continuous function over $\mathbb{R}$ which vanishes at infinity can be approximated by the superposition f a bounded sigmoidal function with a weighted norm. Our proof is constructive.

• East Asian mathematical journal
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• v.31 no.5
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• pp.635-652
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• 2015

It is shown that for a general Haar shift operator, and a weight in the $A_2$ weight class, we establish the weighted norm estimate which linearly depends on $A_2$-characteristic $[w]_{A_2}$. Although the result is now well known, we introduce the new method, which is called the iterated Bellman function method, to provide the estimate.

• Korean Journal of Mathematics
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• v.31 no.3
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• pp.305-311
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• 2023

We exhibit some properties of the weighted Berezin transform T_αf on L^∞(B_n) and on L¹(B_n). As the main result, we prove that if f ∈ L^∞(B_n) with lim_k→∞ T^k_αf exists, then there exist unique M-harmonic function g and $h{\in}{\bar{(I-T_{\alpha})L^{\infty}(B_n)}}$ such that f = g + h. We also show that of the norm of weighted Berezin operator T_α on L¹(B_n, ν) converges to 1 as α tends to infinity, where ν is an ordinary Lebesgue measure.

• Journal of the Korean Mathematical Society
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• v.54 no.2
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• pp.627-645
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• 2017

In this paper, we characterize the boundedness, compactness and essential norm of products of differentiation and composition operators from the Bloch space and weighted Dirichlet spaces to analytic Morrey type spaces.

• Structural Engineering and Mechanics
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• v.82 no.6
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• pp.771-783
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• 2022

Structural damage identification (SDI) methods have been proposed to monitor the safety of structures. However, the traditional SDI methods using modal parameters, such as natural frequencies and mode shapes, are not sensitive enough to structural damage. To tackle this problem, this paper proposes a new SDI method based on transmissibility assurance criterion (TAC) and weighted Schatten-p norm regularization. Firstly, the transmissibility function (TF) has been proved a useful damage index, which can effectively detect structural damage under unknown excitations. Inspired by the modal assurance criterion (MAC), TF and MAC are combined to construct a new damage index, so called as TAC, which is introduced into the objective function together with modal parameters. In addition, the weighted Schatten-p norm regularization method is adopted to improve the ill-posedness of the SDI inverse problem. To evaluate the effectiveness of the proposed method, some numerical simulations and experimental studies in laboratory are carried out. The results show that the proposed method has a high SDI accuracy, especially for weak damages of structures, it can precisely achieve damage locations and quantifications with a good robustness.

• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.1043-1057
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• 2000

This paper is devoted to the investigation of mixed Fourier-Bessel transformation (※Equations, See Full-text) We apply the method of [6] which provides the estimate for weighted L(sub)$\infty$-norm of the spherical mean of │f│$^2$ via its weighted L$_1$-norm (generally it is wrong without the requirement of the non-negativity of f). We prove that in the case of Fourier-Bessel transformatin the mentioned method provides (in dependence on the relation between the dimension of the space of non-special variables n and the length of multiindex ν) similar estimates for weighted spherical means of │f│$^2$, the allowed powers of weights are also defined by multiindex ν. Further those estimates are applied to partial differential equations with singular Bessel operators with respect to y$_1$, …, y(sub)m and we obtain the corresponding estimates for solutions of the mentioned equations.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1451-1469
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• 2013

We present various new sharp assertions on multipliers in mixed norm, weighted Hardy and new Lizorkin-Triebel spaces of harmonic functions in higher dimension. Some results are new even in onedimensional case.

• The Korean Journal of Applied Statistics
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• v.25 no.5
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• pp.829-835
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• 2012

When the input features are generated by factors in a classification problem, it is more meaningful to identify important factors, rather than individual features. The $F_{\infty}$-norm support vector machine(SVM) has been developed to perform automatic factor selection in classification. However, the $F_{\infty}$-norm SVM may suffer from estimation inefficiency and model selection inconsistency because it applies the same amount of shrinkage to each factor without assessing its relative importance. To overcome such a limitation, we propose the adaptive $F_{\infty}$-norm ($AF_{\infty}$-norm) SVM, which penalizes the empirical hinge loss by the sum of the adaptively weighted factor-wise $L_{\infty}$-norm penalty. The $AF_{\infty}$-norm SVM computes the weights by the 2-norm SVM estimator and can be formulated as a linear programming(LP) problem which is similar to the one of the $F_{\infty}$-norm SVM. The simulation studies show that the proposed $AF_{\infty}$-norm SVM improves upon the $F_{\infty}$-norm SVM in terms of classification accuracy and factor selection performance.

• Investigative Magnetic Resonance Imaging
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• v.21 no.3
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• pp.154-161
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• 2017

Purpose: To evaluate the diagnostic performance of diffusion-weighted steady-state free precession (DW-SSFP) in comparison to diffusion-weighted echo-planar imaging (DW-EPI) for differentiating the neoplastic and benign osteoporotic vertebral compression fractures. Materials and Methods: The subjects were 40 patients with recent vertebral compression fractures but no history of vertebroplasty, spine operation, or chemotherapy. They had received 3-Tesla (T) spine magnetic resonance imaging (MRI), including both DW-SSFP and DW-EPI sequences. The 40 patients included 20 with neoplastic vertebral fracture and 20 with benign osteoporotic vertebral fracture. In each fracture lesion, we obtained the signal intensity normalized by the signal intensity of normal bone marrow (SI norm) on DW-SSFP and the apparent diffusion coefficient (ADC) on DW-EPI. The correlation between the SI norm and the ADC in each lesion was analyzed using linear regression. The optimal cut-off values for the diagnosis of neoplastic fracture were determined in each sequence using Youden's J statistics and receiver operating characteristic curve analyses. Results: In the neoplastic fracture, the median SI norm on DW-SSFP was higher and the median ADC on DW-EPI was lower than the benign osteoporotic fracture (5.24 vs. 1.30, P = 0.032, and 0.86 vs. 1.48, P = 0.041, respectively). Inverse linear correlations were evident between SI norm and ADC in both neoplastic and benign osteoporotic fractures (r = -0.45 and -0.61, respectively). The optimal cut-off values for diagnosis of neoplastic fracture were SI norm of 3.0 in DW-SSFP with the sensitivity and specificity of 90.4% (95% confidence interval [CI]: 81.0-99.0) and 95.3% (95% CI: 90.0-100.0), respectively, and ADC of 1.3 in DW-EPI with the sensitivity and specificity of 90.5% (95% CI: 80.0-100.0) and 70.4% (95% CI: 60.0-80.0), respectively. Conclusion: In 3-T MRI, DW-SSFP has comparable sensitivity and specificity to DW-EPI in differentiating the neoplastic vertebral fracture from the benign osteoporotic vertebral fracture.