A modified adaptive Lasso for identifying interactions in the Cox model with the heredity constraint

In many biomedical studies, identifying effects of covariate interactions on survival is a major goal. Important examples are treatment–subgroup interactions in clinical trials, and gene–gene or gene–environment interactions in genomic studies. A common problem when implementing a variable selection algorithm in such settings is the requirement that the model must satisfy the strong heredity constraint, wherein an interaction may be included in the model only if the interaction’s component variables are included as main effects. We propose a modified Lasso method for the Cox regression model that adaptively selects important single covariates and pairwise interactions while enforcing the strong heredity constraint. The proposed method is based on a modified log partial likelihood including two adaptively weighted penalties, one for main effects and one for interactions. A two-dimensional tuning parameter for the penalties is determined by generalized cross-validation. Asymptotic properties are established, including consistency and rate of convergence, and it is shown that the proposed selection procedure has oracle properties, given proper choice of regularization parameters. Simulations illustrate that the proposed method performs reliably across a range of different scenarios.     

Computational inversion of electron micrographs using L2‐gradient flows—convergence analysis

A gradient flow‐based explicit finite element method (L2GF) for reconstructing the 3D density function from a set of 2D electron micrographs has been proposed in recently published papers. The experimental results showed that the proposed method was superior to the other classical algorithms, especially for the highly noisy data. However, convergence analysis of the L2GF method has not been conducted. In this paper, we present a complete analysis on the convergence of L2GF method for the case of using a more general form regularization term, which includes the Tikhonov‐type regularizer and modified or smoothed total variation regularizer as two special cases. We further prove that the L²‐gradient flow method is stable and robust. These results demonstrate that the iterative variational reconstruction method derived from the L²‐gradient flow approach is mathematically sound and effective and has desirable properties. Copyright © 2013 John Wiley & Sons, Ltd.     

分布式L_(1/2)正则化

研究数据集被分割并存储于不同处理器时的特征提取和变量选择问题,其中处理器通过某种网络结构相互连接.提出分布式L_(1/2)正则化方法,基于ADMM算法给出分布式L_(1/2)正则化算法,证明了算法的收敛性.算法通过相邻处理器之间完成信息交互,其变量选择结果与数据集不分割时利用L_(1/2)正则化相同.实验表明,所提出的新算法有效、实用,适合于分布式存储数据处理.     

Cavity identification in linear elasticity and thermoelasticity

The aim of this paper is an analysis of geometric inverse problems in linear elasticity and thermoelasticity related to the identification of cavities in two and three spatial dimensions. The overdetermined boundary data used for the reconstruction are the displacement and temperature on a part of the boundary. We derive identifiability results and directional stability estimates, the latter using the concept of shape derivatives, whose form is known in elasticity and newly derived for thermoelasticity. For numerical reconstructions we use a least‐squares formulation and a geometric gradient descent based on the associated shape derivatives. The directional stability estimates guarantee the stability of the gradient descent approach, so that an iterative regularization is obtained. This iterative scheme is then regularized by a level set approach allowing the reconstruction of multiply connected shapes. Copyright © 2007 John Wiley & Sons, Ltd.     

Regularities in data from factorial experiments

This article documents a meta‐analysis of 113 data sets from published factorial experiments. The study quantifies regularities observed among factor effects and multifactor interactions. Such regularities are known to be critical to efficient planning and analysis of experiments and to robust design of engineering systems. Three previously observed properties are analyzed: effect sparsity, hierarchy, and heredity. A new regularity is introduced and shown to be statistically significant. It is shown that a preponderance of active two‐factor interaction effects are synergistic, meaning that when main effects are used to increase the system response, the interaction provides an additional increase and that when main effects are used to decrease the response, the interactions generally counteract the main effects. © 2006 Wiley Periodicals, Inc. Complexity 11: 32–45, 2006     

USING NETWORK ANALYSIS TO CHARACTERIZE FOREST STRUCTURE

Abstract Network analysis quantifies different structural properties of systems of interrelated parts using a single analytical framework. Many ecological phenomena have network‐like properties, such as the trophic relationships of food webs, geographic structure of metapopulations, and species interactions in communities. Therefore, our ability to understand and manage such systems may benefit from the use of network‐analysis techniques. But network analysis has not been applied extensively to ecological problems, and its suitability for ecological studies is uncertain. Here, we investigate the ability of network analysis to detect spatial patterns of species association in a tropical forest. We use three common graph‐theoretic measures of network structure to quantify the effect of understory tree size on the spatial association of understory species with trees in the canopy: the node degree distribution (NDD), characteristic path length (CPL), and clustering coefficient (CC). We compute the NDD, CPL, and CC for each of seven size classes of understory trees. For significance testing, we compare the observed values to frequency distributions of each statistic computed from randomized data. We find that the ability of network analysis to distinguish observed patterns from those representing randomized data strongly depends on which aspects of structure are investigated. Analysis of NDD finds no significant difference between random and observed networks. However, analysis of CPL and CC detected nonrandom patterns in three and one of the seven size classes, respectively. Network analysis is a very flexible approach that holds promise for ecological studies, but more research is needed to better understand its advantages and limitations.     

Regularization of ill-posed constrained minimization problems by iteration methods

The ill-posed minimization problems in Hilbert space with quadratic objective function and closed convex constraint set are considered. For the compact set the regularization methods for such problems are well understood [1, 2] The regularizing properties of some Iteration projection methods for noncompact constraint set are the main issues of this paper. We are looking the gradient projection method for the sphere.     

Applications of cubic B‐splines collocation method for solving nonlinear inverse parabolic partial differential equations

In this article, we discuss a numerical method for solving some nonlinear inverse parabolic partial differential equations with Dirichlet's boundary conditions. The approach used, is based on collocation of cubic B‐splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B‐splines for spatial variable and derivatives, which produce an ill‐posed system. We solve this system using the Tikhonov regularization method. The accuracy of the proposed method is demonstrated by applying it on two test problems. The figures and comparisons have been presented for clarity. Also the stability of this method has been discussed. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 88–104, 2017     

Reconstructing unknown nonlinear boundary conditions in a time‐fractional inverse reaction–diffusion–convection problem

This paper has focused on unknown functions identification in nonlinear boundary conditions of an inverse problem of a time‐fractional reaction–diffusion–convection equation. This inverse problem is generally ill‐posed in the sense of stability, that is, the solution of problem does not depend continuously on the input data. Thus, a combination of the mollification regularization method with Gauss kernel and a finite difference marching scheme will be introduced to solve this problem. The generalized cross‐validation choice rule is applied to find a suitable regularization parameter. The stability and convergence of the numerical method are investigated. Finally, two numerical examples are provided to test the effectiveness and validity of the proposed approach.     

Graphical Aids for the Analysis of Two-Level Nonregular Designs

The complex alias pattern between main effects and two-factor interactions for two-level nonregular designs has been considered a problem when analyzing these designs. If only a few two-factor interactions are active, however, the pattern induced into the estimated main effects contrasts from the active interactions may be very structured. This is, in particular, true for the 12-run and the 20-run Plackett–Burman (PB) designs, probably the two most important designs for physical experimentation. This article presents a graphical method for the analysis of nonregular two-level designs. The method consists of two steps. The first step is called contrast plots interpretation and is directed toward revealing the cause for the pattern observed in contrast plots. The second step is called alias reduction and aims at simplifying the interpretation of the contrast plots by reducing the aliasing caused by effects that, with a high degree of certainty, may be considered active. The method is tested on the 12-run PB design both with simulated and real data and on the 20-run PB design for one particular case with real data. Supplementary materials (MINITAB codes for performing calculations and plots) are available online.     

Variational formulations for Vlasov–Poisson–Fokker–Planck systems

Time‐discrete variational schemes are introduced for both the Vlasov–Poisson–Fokker–Planck (VPFP) system and a natural regularization of the VPFP system. The time step in these variational schemes is governed by a certain Kantorovich functional (or scaled Wasserstein metric). The discrete variational schemes may be regarded as discretized versions of a gradient flow, or steepest descent, of the underlying free energy functionals for these systems. For the regularized VPFP system, convergence of the variational scheme is rigorously established. Copyright © 2000 John Wiley & Sons, Ltd.     

MULTISPECIES AND SINGLE‐SPECIES MODELS OF FISH POPULATION DYNAMICS: COMPARING PARAMETER ESTIMATES

Abstract Population features inferred from single‐species, age‐structured models are compared to those inferred from a multispecies, age‐structured model that includes predator‐prey interactions among three commercially harvested fish species—walleye pollock, Atka mackerel, and Pacific cod—on the Aleutian Shelf, Alaska. The multispecies framework treats the single‐species models and data as a special case of the multispecies model and data. The same data from fisheries and surveys are used to estimate model parameters for both single‐species and multispecies configurations of the model. Additionally, data from stomach samples and predator rations are used to estimate the parameters of the multispecies model. One form of the feeding functional response, predator pre‐emption, was selected using AIC from seven alternative models for how the predation rate changes with the densities of prey and possibly other predators. Differences in estimated population dynamics and productivity between the multispecies and single‐species models were observed. The multispecies model estimated lower mackerel population sizes from 1964–2003 than the single‐species model, while the spawning biomass of pollock was estimated to have declined more than three times faster since 1964 by the multispecies model. The variances around the estimates of spawning biomass were smaller for mackerel and larger for pollock in the multispecies model compared to the single‐species model.     

A relaxation-based approach to damage modeling

Common material models that take into account softening effects due to damage have the problem of ill-posed boundary value problems if no regularization is applied. This condition leads to a non-unique solution for the resulting algebraic system and a strong mesh dependence of the numerical results. A possible solution approach to prevent this problem is to apply regularization techniques that take into account the non-local behavior of the damage [1]. For this purpose a field function is often used to couple the local damage parameter to a non-local level, in which differences between the local and non-local parameter as well as the gradient of the non-local parameter can be penalized. In contrast, we present a novel approach [2] to regularization that no longer needs a non-local level but directly provides mesh-independent results. Due to the new variational approach we are also able to improve the calculation times and convergence behavior. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fast proximal algorithms for nonsmooth convex optimization

In the lines of our previous approach to devise proximal algorithms for nonsmooth convex optimization by applying Nesterov fast gradient concept to the Moreau–Yosida regularization of a convex function, we develop three new proximal algorithms for nonsmooth convex optimization. In these algorithms, the errors in computing approximate solutions for the Moreau–Yosida regularization are not fixed beforehand, while preserving the complexity estimates already established. We report some preliminary computational results to give a first estimate of their performance.     

Iteration methods for convexly constrained ill-posed problems in hilbert space

Minimization problems in Hilbert space with quadratic objective function and closed convex constraint set C are considered. In case the minimum is not unique we are looking for the solution of minimal norm. If a problem is ill-posed, i.e. if the solution does not depend continuously on the data, and if the data are subject to errors then it has to be solved by means of regularization methods. The regularizing properties of some gradient projection methods—i.e. convergence for exact data, order of convergence under additional assumptions on the solution and stability for perturbed data—are the main issues of this paper.     

Computationally efficient algorithm for Gaussian Process regression in case of structured samples

Surrogate modeling is widely used in many engineering problems. Data sets often have Cartesian product structure (for instance factorial design of experiments with missing points). In such case the size of the data set can be very large. Therefore, one of the most popular algorithms for approximation–Gaussian Process regression–can be hardly applied due to its computational complexity. In this paper a computationally efficient approach for constructing Gaussian Process regression in case of data sets with Cartesian product structure is presented. Efficiency is achieved by using a special structure of the data set and operations with tensors. Proposed algorithm has low computational as well as memory complexity compared to existing algorithms. In this work we also introduce a regularization procedure allowing to take into account anisotropy of the data set and avoid degeneracy of regression model.     

Learning Interactions via Hierarchical Group-Lasso Regularization

We introduce a method for learning pairwise interactions in a linear regression or logistic regression model in a manner that satisfies strong hierarchy: whenever an interaction is estimated to be nonzero, both its associated main effects are also included in the model. We motivate our approach by modeling pairwise interactions for categorical variables with arbitrary numbers of levels, and then show how we can accommodate continuous variables as well. Our approach allows us to dispense with explicitly applying constraints on the main effects and interactions for identifiability, which results in interpretable interaction models. We compare our method with existing approaches on both simulated and real data, including a genome-wide association study, all using our R package glinternet.     

Global weak solutions to a ferrofluid flow model

We are concerned with the global solvability of the differential system introduced by Shliomis to describe the flow of a colloidal suspension of magnetized nanoparticles in a nonconducting liquid, under the action of an external magnetic field. The system is a combination of the Navier–Stokes equations, the magnetization equation, and the magnetostatic equations. We prove, by using a method of regularization, the existence of global‐in‐time weak solutions with finite energy to an initial boundary‐value problem and establish the long‐time behaviour of such solutions. The main difficulty is due to the singularity of the gradient magnetic force and the torque. Copyright © 2007 John Wiley & Sons, Ltd.     

TV‐like regularization for backward parabolic problems

This paper investigates an inverse problem for parabolic equations backward in time, which is solved by total‐variation‐like (TV‐like, in abbreviation) regularization method with cost function ∥u_x∥². The existence, uniqueness and stability estimate for the regularization problem are deduced in the linear case. For numerical illustration, the variational adjoint method, which presents a simple method to derive the gradient of the optimization functional, is introduced to reconstruct the unknown initial condition for both linear and nonlinear parabolic equations. The conjugate gradient method is used to iteratively search for the optimal approximation. Numerical results validate the feasibility and effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

The instability of some gradient methods for ill-posed problems

Summary For the solution of linear ill-posed problems some gradient methods like conjugate gradients and steepest descent have been examined previously in the literature. It is shown that even though these methods converge in the case of exact data their instability makes it impossible to base a-priori parameter choice regularization methods upon them.