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15 October 2018, Volume 35 Issue 5 Previous Issue    Next Issue

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Structure Learning of Gaussian Graphical Model with Covariates MA Yan, ZHANG Hai 2018, 35 (5):  489-501.  doi: 10.3969/j.issn.1005-3085.2018.05.001 Abstract ( 182 )   PDF (1093KB) ( 230 )   Save Graphical models is an important tool to study the relationship among variables. Besides the node variables, the additional covariates are frequently recorded together with the data and may influence the dependence relationships. However, most of the existing work on graphical models only considers the node variables. In this paper, we study the problem of reconstructing network structure from the data with covariates by applying the tools of graphical models. In the framework of sparse regularization, we propose a novel sparse Gaussian graphical models to incorporate the covariates information, where the conditional independency relationship between variables are assumed to be a linear function of the covariates. The proposed model is interpretable and easy to solve. We employ the coordinate descent algorithm to solve the model. A series of numerical examples shows that the effect of the covariate is better than that of the non covariate, which indicates the effectiveness and efficiency of the proposed model. Related Articles | Metrics

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A New Non-monotone Conjugate Gradient Method Based on the Trust Region Technique GAO Miao-miao, GONG En-long, SUN Qing-ying, WANG Zhen-zhen, DU Xiao-yu 2018, 35 (5):  502-514.  doi: 10.3969/j.issn.1005-3085.2018.05.002 Abstract ( 192 )   PDF (176KB) ( 249 )   Save In order to effectively solve the large-scale unconstrained optimization problem, based on the trust region technique and the modified quasi-Newton equation, a new non-monotone conjugate gradient algorithm is presented by combining Zhang H. C. and Gu N. Z. strategy in this paper. The trust region technique is applied to ensure the robustness and convergence of the algorithm, and the global convergence property of the algorithm is also analyzed. Under some reasonable conditions, it is proved that the proposed algorithm is linear convergent. Numerical examples indicate that the new algorithm can effectively solve ill-conditioned and large-scale problems. Compared with the algorithm that combines one of the non-monotonic strategies, the new algorithm requires fewer iteration numbers and less running time, and the function value obtained by the new algorithm is closer to the optimal value. Related Articles | Metrics

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Integrated Statistics Pipeline to Mine Key Genes Involved in Tuberculosis from Multiple-omics Data ZHANG Xu, CHEN Dong-dong, YE Zhi-qiang, LI Qi-ming, XIE Jian-ping 2018, 35 (5):  515-522.  doi: 10.3969/j.issn.1005-3085.2018.05.003 Abstract ( 78 )   PDF (7713KB) ( 16 )   Save It is important to define the key host genes participate in the interaction and underlying networks for tuberculosis susceptibility. However, only a handful of host genes have been found and confirmed to date. Two sets of omics data about tuberculosis are analyzed in this paper through different statistical methods such as significance test and cluster analysis. 14 hits are found as most probable genes associated with tuberculosis. These hits were all reported to participate in a variety of important biological processes. What's more, five of them were reported to be directly related with tuberculosis. This indicates that the statistical methodology can be helpful to narrow down the shortlist for tuberculosis disease relevant genes. Related Articles | Metrics

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Convergence Rates for Empirical Bayes Test of Parameter for the Continuous One-parameter Exponential Family HUANG Jin-chao 2018, 35 (5):  523-533.  doi: 10.3969/j.issn.1005-3085.2018.05.004 Abstract ( 139 )   PDF (169KB) ( 233 )   Save The empirical Bayes test problem of parameter for a class of continuous one-parameter exponential family is considered in this paper. By using the recursive kernel esti-mation of probability density function and monotonicity of Bayes test function in the case of independent and identically distributed samples, the EB test function in established for a class of continuous one-parameter exponential family. By the construction method of modifying EB test function, the monotone Bayes test function are constructed. In addition, convergence rates of EB test function are obtained under the suitable conditions, and the results of convergence rate order are improved. Finally, a numerical example is presented to verify the conditions of the theorem. Related Articles | Metrics

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The Multigrid Preconditioned Conjugate Gradient Method for Combined Hybrid Quadrilateral Element WANG Hui-ling, NIE Yu-feng, ZHANG Ling 2018, 35 (5):  534-544.  doi: 10.3969/j.issn.1005-3085.2018.05.005 Abstract ( 101 )   PDF (977KB) ( 275 )   Save Combined hybrid finite elements method, applied to linear elasticity problem, is a stabilized finite element method. It is a large sparse symmetric positive definite system arising from combined hybrid quadrilateral element discretization. In order to solve the system quickly, the multigrid preconditioned conjugate gradient method (MGCGM) is introduced in this paper. Firstly, by choosing the appropriate intergrid transfer operators and smoothing strategy, an effective multigrid preconditioner is obtained. Then, numerical results show that MGCGM proposed is efficient, that is the condition number of the stiffness matrix is greatly decreased by the multigrid preconditioner. Moreover, the method still has a good convergence for the combined hybrid element with high performance in the case of the mesh distortion. Related Articles | Metrics

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Growth of Meromorphic Solutions of Some Kind of Homogeneous and Non-homogeneous Higher Order Linear Differential Equations with Coefficients Relative to Fejér Gap Series ZHOU Yan-ping, ZHENG Xiu-min 2018, 35 (5):  545-558.  doi: 10.3969/j.issn.1005-3085.2018.05.006 Abstract ( 105 )   PDF (181KB) ( 250 )   Save Nevanlinna theory has been widely applied in the field of complex differential equa-tions. It is an important research subject to explore the relationship between the growth of the coefficients and the growth and value distribution of meromorphic solutions of complex linear differential equations by Nevanlinna theory. Meanwhile, the gap series has some special properties which may play important roles when the gap series appear as the coefficients of certain equation. Therefore, the properties of meromorphic solutions of complex linear differential equations can be investigated by combining with the definition and properties of gap series. In this paper, we consider a kind of the homogeneous and non-homogeneous higher order complex linear differential equation based on Nevanlinna theory and the definition and properties of Fejér gap series. When one of the coefficients is relative to Fejér gap series and the others are entire or meromorphic functions, the estimates on the order of meromorphic solutions of the involved equation are obtained, which promotes and improves the previous research results. Related Articles | Metrics

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The Determination of Heat Sources in 2-D Inverse Steady Heat Problems Based on LS-SVM YU Jia-ju, LI Fu-le, LIU Zhen-bin, WU Zi-ku, LI Juan 2018, 35 (5):  559-569.  doi: 10.3969/j.issn.1005-3085.2018.05.007 Abstract ( 143 )   PDF (389KB) ( 277 )   Save In this paper, the inverse problem of the 2-D steady heat conduction is investigated. Firstly, the training points set is allocated. Then, the approximate solution of the problem is expressed as the form of the combination of the Gauss kernel functions which related to the training points. Moreover, the parameters are optimized basing on the least squares support vector machines. The approximate solution consists of two parts. The first part is a known function that satisfies the boundary condition, and the other part is the combination of Gauss Kernel functions which including regression coefficients. Finally, the validity of the proposed method is verified by two examples which have analytical solutions. The result shows that the proposed method is feasible to solve the inverse problem of the 2-D steady heat conduction. Related Articles | Metrics

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Legendre Function Method for Solving Fractional-order Partial Differential Equations ZHU Shuai, XIE Jia-quan, WU Shi-yue 2018, 35 (5):  570-578.  doi: 10.3969/j.issn.1005-3085.2018.05.008 Abstract ( 132 )   PDF (362KB) ( 521 )   Save Fractional partial differential equations are regarded as one kind of common differ-ential equations to describe the engineering problems. Compared with other traditional methods, the proposed method has a great advantage on computational precision and efficiency. Legendre function method expands the dual functions as basis functions, and transforms the original equations into a system of algebra equations combined with the operational matrices. This paper introduces the method which can accurately stimulate the complex mathematical phenomenon of the engineering problems, and is relatively simple in the function construction and theoretical derivation. Finally, the error analysis is presented to verify the effectiveness and robust. Related Articles | Metrics

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The Anti-reflexive Solution of the Inverse Quadratic Eigenvalue Problem and Its Optimal Approximation SHANG Xiao-lin, ZHANG Lan 2018, 35 (5):  579-587.  doi: 10.3969/j.issn.1005-3085.2018.05.009 Abstract ( 123 )   PDF (123KB) ( 286 )   Save The inverse problem of quadratic eigenvalue is an inverse process of quadratic eigenvalue problem, and it is widely used in the field of structural dynamic model correction. Given part of eigenvalues and eigenvectors, based on the singular value decomposition of matrix, block matrix method and generalized inverse of Moore-Penrose, the inverse quadratic eigenvalue problem of constructing anti-reflexive matrices is considered in this paper. Then, a general expression of solution to the problem is presented. Moreover, the existence and uniqueness of the optimal approximation problem associated with solution set is discussed. Finally, the expression and numerical method are proposed, the correctness of the result is verified by a numerical example. Related Articles | Metrics

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Determining the Discriminating Domain for Hybrid Differential Game with Two Targets and Two Players HAN Yan-li, GAO Yan 2018, 35 (5):  588-600.  doi: 10.3969/j.issn.1005-3085.2018.05.010 Abstract ( 100 )   PDF (140KB) ( 334 )   Save Hybrid differential game combines the control engineering, mathematics and computer science. The research has a high value no matter in theory or in practical applications. In this paper, the discriminating domain of hybrid differential game with two targets and two players is discussed based on this theory. Firstly, the discriminating domain of continuous differential game on a region where the functions are piecewise smooth is discussed using nonsmooth analysis. Then, we find that the problem of determining the discriminating domain can be transformed into determining the solution of the inequalities. Finally, the result is generalized to the hybrid differential game. The novelty is that we propose the method of discriminating domain for differential game which contains two control variables, not one control variable as in differential inclusion. Related Articles | Metrics

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A New Inexact Subgradient Algorithm for the Equilibrium Problem over the Fixed Point Set of a Firmly Nonexpansive Mapping DANG Ya-zheng, LIU Wen-wen 2018, 35 (5):  601-610.  doi: 10.3969/j.issn.1005-3085.2018.05.011 Abstract ( 119 )   PDF (113KB) ( 206 )   Save In this paper, we present a new method for solving equilibrium problem over the fixed point set of a firmly nonexpansive mapping, where the underlying bifunction is continuous but not necessarily monotone. Firstly, we construct a closed ball by introducing some parameters. Then, we calculate the intermediate iterate by the projection of the inexact subgradient onto the closed convex set. The next iterate is obtained as the firmly nonexpansive mapping of a convex combination, which consists of the current iterate and the intermediate iterate. Finally, we analyse the convergence properties and the global convergence of the algorithm under some suitable conditions. Related Articles | Metrics