半参数线性混合效应模型的联合变量选择

多数基于线性混合效应模型的变量选择方法分阶段对固定效应和随机效应进行选择,方法繁琐、易产生模型偏差,且大部分非参数和半参数的线性混合效应模型只涉及非参数部分的光滑度或者固定效应的选择,并未涉及非参变量或随机效应的选择。本文用B样条函数逼近非参数函数部分,从而把半参数线性混合效应模型转化为带逼近误差的线性混合效应模型。对随机效应的协方差矩阵采用改进的乔里斯基分解并重新参数化线性混合效应模型,接着对该模型的极大似然函数施加集群ALASSO惩罚和ALASSO惩罚两类惩罚,该法能实现非参数变量、固定效应和随机效应的联合变量选择,基于该法得出的估计量也满足相合性、稀疏性和Oracle性质。文章最后做了个数值模拟,模拟结果表明,本文提出的估计方法在变量选择的准确性、参数估计的精度两个方面均表现较好。     

A full‐pivoting algorithm for the Cholesky decomposition of two‐electron repulsion and spin‐orbit coupling integrals

A significant reduction in the computational effort for the evaluation of the electronic repulsion integrals (ERI) in ab initio quantum chemistry calculations is obtained by using Cholesky decomposition (CD), a numerical procedure that can remove the zero or small eigenvalues of the ERI positive (semi)definite matrix, while avoiding the calculation of the entire matrix. Conversely, due to its antisymmetric character, CD cannot be directly applied to the matrix representation of the spatial part of the two‐electron spin‐orbit coupling (2e‐SOC) integrals. Here, we present a computational strategy to achieve a Cholesky representation of the spatial part of the 2e‐SOC integrals, and propose a new efficient CD algorithm for both ERI and 2e‐SOC integrals. The proposed algorithm differs from previous CD implementations by the extensive use of a full‐pivoting design, which allows a univocal definition of the Cholesky basis, once the CD δ threshold is made explicit. We show that is the upper limit for the errors affecting the reconstructed 2e‐SOC integrals. The proposed strategy was implemented in the ab initio program Computational Emulator of Rare Earth Systems (CERES), and tested for computational performance on both the ERI and 2e‐SOC integrals evaluation. © 2017 Wiley Periodicals, Inc.     

Gaussian basis sets for molecular applications

The choice of basis set in quantum chemical calculations can have a huge impact on the quality of the results, especially for correlated ab initio methods. This article provides an overview of the development of Gaussian basis sets for molecular calculations, with a focus on four popular families of modern atom‐centered, energy‐optimized bases: atomic natural orbital, correlation consistent, polarization consistent, and def2. The terminology used for describing basis sets is briefly covered, along with an overview of the auxiliary basis sets used in a number of integral approximation techniques and an outlook on possible future directions of basis set design. © 2012 Wiley Periodicals, Inc.     

Cholesky decomposition of a positive semidefinite matrix with known kernel

The Cholesky decomposition of a symmetric positive semidefinite matrix A is a useful tool for solving the related consistent system of linear equations or evaluating the action of a generalized inverse, especially when A is relatively large and sparse. To use the Cholesky decomposition effectively, it is necessary to identify reliably the positions of zero rows or columns of the factors and to choose these positions so that the nonsingular submatrix of A of the maximal rank is reasonably conditioned. The point of this note is to show how to exploit information about the kernel of A to accomplish both tasks. The results are illustrated by numerical experiments.     

Fast solving of weighted pairing least-squares systems

This paper presents a generalization of the “weighted least-squares” (WLS), named “weighted pairing least-squares” (WPLS), which uses a rectangular weight matrix and is suitable for data alignment problems. Two fast solving methods, suitable for solving full rank systems as well as rank deficient systems, are studied. Computational experiments clearly show that the best method, in terms of speed, accuracy, and numerical stability, is based on a special {1, 2, 3}-inverse, whose computation reduces to a very simple generalization of the usual “Cholesky factorization-backward substitution” method for solving linear systems.     

Analytical gradients of the second‐order Møller–Plesset energy using Cholesky decompositions

An algorithm for computing analytical gradients of the second‐order Møller–Plesset (MP2) energy using density fitting (DF) is presented. The algorithm assumes that the underlying canonical Hartree–Fock reference is obtained with the same auxiliary basis set, which we obtain by Cholesky decomposition (CD) of atomic electron repulsion integrals. CD is also used for the negative semidefinite MP2 amplitude matrix. Test calculations on the weakly interacting dimers of the S22 test set (Jure?ka et al., Phys. Chem. Chem. Phys. 2006, 8, 1985) show that the geometry errors due to the auxiliary basis set are negligible. With double‐zeta basis sets, the error due to the DF approximation in intermolecular bond lengths is better than 0.1 pm. The computational time is typically reduced by a factor of 6–7. © 2013 Wiley Periodicals, Inc.     

LU和Cholesky分解的向前舍入误差分析

1引言LU分解可用于解可逆线性系统Ax=b．作为数值代数领域中的重要工具,其舍入误差分析一直为众多学者所关注．事实上,长方矩阵的LU分解也有着广泛的应用,如,确定矩阵数值秩的LU分解(RRLU)[5,7],解等式约束最小二乘问题的直接消去法[3]等问题中都涉及到长方矩阵的LU分解．当A∈Rm×n且秩r≤min{m,n},则在考虑A的LU分解时[4],一般需要确定置换阵∏L,∏R使得A(1):=∏L-A∏R的LU分解能持续qr步,这里当A为亏秩矩阵时,qr=r;否贝qr=r-1．在．A(1)的LU分解的第k(k≤qr)步,需执行如下Gauss消去过程:     

大型稀疏无约束最优化的列分划校正Cholesky因子算法

我们考虑求解无约束优化问题1引言(?)f(x),(1)其中f：D(?)R~n→R为R~n上的二次连续可微函数,且f(x)的二阶Hesse阵H(x)稀疏、正定．为了求解问题(1),我们考虑下列Newton型方法x~(k 1)=x~k-(B~k)~(-1)▽f(x~k),k=0,1,…,(2)其中B~k是和Hesse阵H(x~k)具有相同稀疏性的近似．由于Hesse阵对称,我们假定B~k对称．为了具体说明给定矩阵B的稀疏性,我们使用M来定义指标对(i,j)的集合,其     

ON GROWTH FACTORS OF THE LU FACTORIZATION

In this article, we derive upper bounds of different growth factors for the LU factorization, which are dominated by A11(k)-1A12(k),A21(k)A11(k)-1, where A11(k), A12(k), A21(k), A22(k) are sub-matrices of A. We also derive upper bounds of growth factors for the Cholesky factorization. Numerical examples are presented to verify our findings.     

A Heuristic for Moment-Matching Scenario Generation

In stochastic programming models we always face the problem of how to represent the random variables. This is particularly difficult with multidimensional distributions. We present an algorithm that produces a discrete joint distribution consistent with specified values of the first four marginal moments and correlations. The joint distribution is constructed by decomposing the multivariate problem into univariate ones, and using an iterative procedure that combines simulation, Cholesky decomposition and various transformations to achieve the correct correlations without changing the marginal moments.With the algorithm, we can generate 1000 one-period scenarios for 12 random variables in 16 seconds, and for 20 random variables in 48 seconds, on a Pentium III machine.