• The Pure and Applied Mathematics
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• v.9 no.1
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• pp.81-90
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• 2002

The least informative (favorable) distributions, minimizing Fisher information for a multivariate location parameter, are derived in the parametric class of the exponential-power spherically symmetric distributions under the following characterizing restrictions; (i) a bounded variance, (ii) a bounded value of a density at the center of symmetry, and (iii) the intersection of these restrictions. In the first two cases, (i) and (ii) respectively, the least informative distributions are the Gaussian and Laplace, respectively. In the latter case (iii) the optimal solution has three branches, with relatively small variances it is the Gaussian, them with intermediate variances. The corresponding robust minimax M-estimators of location are given by the $L_2$-norm, the $L_1$-norm and the $L_{p}$ -norm methods. The properties of the proposed estimators and their adaptive versions ar studied in asymptotics and on finite samples by Monte Carlo.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.125-130
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• 2007

We propose a simple and intuitive method to derive the exact convergence rate of global $L_{2}-norm$ error for strong numerical approximation of stochastic differential equations the result of which has been reported by Hofmann and $M{\"u}ller-Gronbach\;(2004)$. We conclude that any strong numerical scheme of order ${\gamma}\;>\;1/2$ has the same optimal convergence rate for this error. The method clearly reveals the structure of global $L_{2}-norm$ error and is similarly applicable for evaluating the convergence rate of global uniform approximations.

• The Journal of Korea Institute of Information, Electronics, and Communication Technology
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• v.12 no.5
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• pp.521-528
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• 2019

Epilepsy is one of the most prevalent neurological diseases. Electroencephalogram (EEG) signals are widely used for monitoring and diagnosis tool for epileptic seizure. Typically, a huge amount of EEG signals is needed, where they are visually examined by experienced clinicians. In this study, we propose a simple automatic seizure detection framework using intracranial EEG signals. We suggest a sparse approximation based classification (SAC) scheme by solving overdetermined system. L1-norm minimization algorithms are utilized for efficient sparse signal recovery. For evaluation of the proposed scheme, the public EEG dataset obtained by five healthy subjects and five epileptic patients is utilized. The results show that the proposed fast L1-norm minimization based SAC methods achieve the 99.5% classification accuracy which is 1% improved result than the conventional L2 norm based method with negligibly increased execution time (42msec).

• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.735-748
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• 1998

We analyze the error in the p version of the of the finite element method when the effect of the quadrature error is taken in the load vector. We briefly study some results on the $H^{1}$ norm error and present some new results for the error in the $L^{2}$ norm. We inves-tigate the quadrature error due to the numerical integration of the right hand side We present theoretical and computational examples showing the sharpness of our results.

• Geophysics and Geophysical Exploration
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• v.9 no.1
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• pp.44-49
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• 2006

In this study, the resolution capabilities of electrical resistivity tomography (ERT) in the monitoring of $CO_2$ injection are investigated. The pole-pole and bipole-bipole electrode configuration types are used between two uncased boreholes straddling the $CO_2$ plume. Forward responses for an initial pre-injection model and three models for subsequent stages of $CO_2$ injection are calculated for the two different electrode configuration types, noise is added and the theoretical data are inverted with both L1- and L2-norm optimisation. The results show that $CO_2$ volumes over a certain threshold can be detected with confidence. The L1-norm proved superior to the L2-norm in most instances. Normalisation of the inverted models with the pre-injection inverse model gives good images of the regions of changing resistivity, and an integrated measure of the total change in resistivity proves to be a valid measure of the total injected volume.

• Economic and Environmental Geology
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• v.28 no.4
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• pp.433-438
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• 1995

The most popular minimization method is based on the least-squares criterion, which uses the $L_2$ norm to quantify the misfit between observed and synthetic data. The solution of the least-squares problem is the maximum likelihood point of a probability density containing data with Gaussian uncertainties. The distribution of errors in the geophysical data is, however, seldom Gaussian. Using the $L_2$ norm, large and sparsely distributed errors adversely affect the solution, and the estimated model parameters may even be completely unphysical. On the other hand, the least-absolute-deviation optimization, which is based on the $L_1$ norm, has much more robust statistical properties in the presence of noise. The solution of the $L_1$ problem is the maximum likelihood point of a probability density containing data with longer-tailed errors than the Gaussian distribution. Thus, the $L_1$ norm gives more reliable estimates when a small number of large errors contaminate the data. The effect of outliers is further reduced by M-fitting method with Cauchy error criterion, which can be performed by iteratively reweighted least-squares method.

• Communications for Statistical Applications and Methods
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• v.3 no.3
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• pp.271-282
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• 1996

this paper deal with the estimation of the power spectral density function of time series. A kernel estimator which is based on local average is defined and the rates of convergence of the pointwise, $$L_2$-norm; and; $L{\infty}$-norm associated with the estimator are investigated by restricting as to kernels with suitable assumptions. Under appropriate regularity conditions, it is shown that the optimal rate of convergence for 0$N^{-r}$ both in the pointwiseand $$L_2$-norm, while; $N^{r-1}(logN)^{-r}$is the optimal rate in the $L{\infty}-norm$. Some examples are given to illustrate the application of main results.

• Journal of the Korean Statistical Society
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• v.16 no.2
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• pp.80-91
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• 1987

Minimum $L_i$ norm estimation is a robust procedure ins the sense that it leads to an estimator which has greater statistical eficiency than the least squares estimator in the presence of outliers. And the $L_1$ norm estimator has some desirable statistical properties. In this paper a new computational procedure for $L_1$ norm estimation is proposed which combines the idea of reweighted least squares method and the linear programming approach. A modification of the projective transformation method is employed to solve the linear programming problem instead of the simplex method. It is proved that the proposed algorithm terminates in a finite number of iterations.

• Proceedings of the Korean Information Science Society Conference
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• 2000.04a
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• pp.692-694
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• 2000

본 논문에서는 이진 트리 형태를 가지는 다관절체의 균형을 잡거나 이진 트리 모양으로 연결된 네트워크 상에서 단말 노드들의 부하를 균형 있게 하는데 이용할 수 있는 무게 있는 리프 이진 트리 균형 문제를 제안한다. 또한 무게 있는 리프 이진 트리 균형 문제를 리프들의 무게 변화량의 쌍의 {{{{ { l}_{ 1} }}}}-norm, {{{{ { l}_{2 } }}}}-norm, {{{{ { l}_{3 } }}}}-norm 각각을 최소로 하면서 해결하는 방법들을 제안한다. 이 방법들은 무게 있는 리프 이진 트리 균형 문제의 특성을 이용하여 n개 변수를 하나의 변수의 양의 상수배로 나타냄으로써 해결할 수 있음을 보인다.

• Water for future
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• v.22 no.3
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• pp.315-322
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• 1989

The effects of open boundary conditions including the radiation condition are compared by utilizing the $L^{2_}$-norm and RMS error in the numerical modeling of ocean problem. In numerical tests of $M_2$ tide, grid size and bed roughness are considered and analyzed. For the $M_2$ tide test in a simplified bay where the analytical solutions are available, it is found that improved radiating boundary condition(IMPSOM) may increase the reliability of computed results by 40% of $L^{2_}$-norm and 96% of RMS error than the open boundary condition without radiation effect. In case of using the half-size grids, better results are obtained. It is also found that the IMPSOM is applicable with satisfaction when the bottom friction is included.