Using game theory to detect genes involved in Autism Spectrum Disorder

Microarray technology is a current approach for detecting alterations in the expression of thousands of genes simultaneously between two different biological conditions. Genes of interest are selected on the basis of an obtained p-value, and, thus, the list of candidates may vary depending on the data processing steps taken and statistical tests applied. Using standard approaches to the statistical analysis of microarray data from individuals with Autism Spectrum Disorder (ASD), several genes have been proposed as candidates. However, the lists of genes detected as differentially regulated in published mRNA expression analyses of Autism often do not overlap, owed at least in part to (i) the multifactorial nature of ASD, (ii) the high inter-individual variability of the gene expression in ASD cases, and (iii) differences in the statistical analysis approaches applied. Game theory recently has been proposed as a new method to detect the relevance of gene expression in different conditions. In this work, we test the ability of Game theory, specifically the Shapley value, to detect candidate ASD genes using a microarray experiment in which only a few genes can be detected as dysregulated using conventional statistical approaches. Our results showed that coalitional games significantly increased the power to identify candidates. A further functional analysis demonstrated that groups of these genes were associated with biological functions and disorders previously shown to be related to ASD.     

无重复因析试验中多个散度效应检验的一种改进方法

在无重复因析试验的多个散度效应分析中,常常出现错误识别的现象,即两个显著的散度效应可能在它们的交互列上产生一个错误的散度效应,并且现有的许多方法都存在这样的问题.为了解决这种模棱两可性,McGrath和Lin(2001)提出了一种基于残差样本方差几何平均的检验方法(ML方法),但是这个方法不能应用于零残差样本方差的情形.鉴于此,提出了一种基于修改残差的改进方法,适用于零残差样本方差的情形,并且通过实例验证了方法的合理性.最后,通过模拟和ML方法做了比较.     

时空模型的局部众数回归

时空数据经常含有奇异点或来自重尾分布,此时基于最小二乘的估计方法效果欠佳,需要更稳健的估计方法.本文提出时空模型的基于局部众数(local modal, LM)的局部线性估计方法.理论和数据分析结果都显示,若数据含有奇异点或来自重尾分布,基于局部众数的局部线性方法比基于最小二乘的局部线性方法有效;若数据无奇异点且来自正态分布,两种方法效率渐近一致.本文采用众数期望最大化(modal expectation-maximization, MEM)算法,并在数据相依情形下得出估计量的渐近正态性.     

Bayesian model-based tight clustering for time course data

Cluster analysis has been widely used to explore thousands of gene expressions from microarray analysis and identify a small number of similar genes (objects) for further detailed biological investigation. However, most clustering algorithms tend to identify loose clusters with too many genes. In this paper, we propose a Bayesian tight clustering method for time course gene expression data, which selects a small number of closely-related genes and constructs tight clusters only with these closely-related genes.     

Entropy embedding and fluctuation analysis in genomic manifolds

The complexity of typically high-dimensional genomic data requires computational work prone to integrate different biological information sources through efficient model solutions. Usually, one step involves dimensionality reduction (DR), which requires projecting the input data onto a low dimensional subspace, and often leads to an embedding. Thus, DR should be able to filter out the uninformative dimensions and recover the original variables. This step is of crucial relevance for any reverse engineering and statistical inference attempt to reconstruct the dynamics underlying the biological systems under study, i.e. the interactions between its genes or proteins. DR has become almost a standard practice just following the pre-processing steps typically applied to the experimental measurements (mining, normalization, etc.). In this work, the data for the analysis reflect expression values of genes whose dynamics are affected by perturbation experiments. In particular, the aims are to monitor the response of genes involved in a certain pathway, and then to isolate their biological variability from any possible external influence. Last, it is of interest to control the stability of the system; with this regard, we look at dynamical aspects of data through embedding theory and entropy fluctuation analysis. We demonstrate that a redundant biological system can in principle be reduced to a minimal number of almost independent components. In particular, such structures detect the higher-order statistical dependencies in the training data in addition to the correlations. Two popular DR techniques are compared in relation to their ability to extract the most salient features, allow gene selection, and minimize the various interferences due to algorithmic approximation errors and variable noise covers.     

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

We present an “a posteriori” error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro‐to‐micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space, and the missing macroscopic data are recovered on‐the‐fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single‐scale) dual‐weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal‐oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Deep kernel dimensionality reduction for scalable data integration

Dimensionality reduction is used to preserve significant properties of data in a low-dimensional space. In particular, data representation in a lower dimension is needed in applications, where information comes from multiple high dimensional sources. Data integration, however, is a challenge in itself.In this contribution, we consider a general framework to perform dimensionality reduction taking into account that data are heterogeneous. We propose a novel approach, called Deep Kernel Dimensionality Reduction which is designed for learning layers of new compact data representations simultaneously. The method can be also used to learn shared representations between modalities. We show by experiments on standard and on real large-scale biomedical data sets that the proposed method embeds data in a new compact meaningful representation, and leads to a lower classification error compared to the state-of-the-art methods.     

An adaptive edge element method and its convergence for a Saddle‐Point problem from magnetostatics

Reliable and efficient a posteriori error estimates are derived for the edge element discretization of a saddle‐point Maxwell's system. By means of the error estimates, an adaptive edge element method is proposed and its convergence is rigorously demonstrated. The algorithm uses a marking strategy based only on the error indicators, without the commonly used information on local oscillations and the refinement to meet the standard interior node property. Some new ingredients in the analysis include a novel quasi‐orthogonality and a new inf‐sup inequality associated with an appropriately chosen norm. It is shown that the algorithm is a contraction for the sum of the energy error plus the error indicators after each refinement step. Numerical experiments are presented to show the robustness and effectiveness of the proposed adaptive algorithm. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

两水平无重复因析试验散度效应的估计

本文对两水平无重复因析试验给出了散度效应的一种新的估计,称为AMH估计,改进了文献中散度效应的较好的MH估计的一个缺陷,给出并证明了AMH估计的无偏条件,证明了AMH估计比MH估计有更小的方差下界.最后通过模拟试验比较了AMH和MH估计的偏度,方差和均方误差.     

Discontinuous Galerkin finite element heterogeneous multiscale method for advection–diffusion problems with multiple scales

A discontinuous Galerkin finite element heterogeneous multiscale method is proposed for advection–diffusion problems with highly oscillatory coefficients. The method is based on a coupling of a discontinuous Galerkin discretization for an effective advection–diffusion problem on a macroscopic mesh, whose a priori unknown data are recovered from micro finite element calculations on sampling domains within each macro element. The computational work involved is independent of the high oscillations in the problem at the smallest scale. The stability of our method (depending on both macro and micro mesh sizes) is established for both diffusion dominated and advection dominated regimes without any assumptions about the type of heterogeneities in the data. Fully discrete a priori error bounds are derived for locally periodic data. Numerical experiments confirm the theoretical error estimates.     

Clustering and classification based on the L1 data depth

Clustering and classification are important tasks for the analysis of microarray gene expression data. Classification of tissue samples can be a valuable diagnostic tool for diseases such as cancer. Clustering samples or experiments may lead to the discovery of subclasses of diseases. Clustering genes can help identify groups of genes that respond similarly to a set of experimental conditions. We also need validation tools for clustering and classification. Here, we focus on the identification of outliers—units that may have been misallocated, or mislabeled, or are not representative of the classes or clusters.We present two new methods: DDclust and DDclass, for clustering and classification. These non-parametric methods are based on the intuitively simple concept of data depth. We apply the methods to several gene expression and simulated data sets. We also discuss a convenient visualization and validation tool—the relative data depth plot.     

无重复因析试验中散度效应的ML估计

在无重复因析试验中,若因子A,B的散度效应显著,则不论其交互效应AB的散度效应是否显著,其散度效应的现有估计常常是有偏的,从而导致其被错误地识别为显著效应.提出了散度效应的一种新的估计方法(称为ML估计),并给出了ML估计的方差的精确表达形式,证明了在一类模型中,交互效应AB的散度效应的ML估计是无偏的.最后,将ML估计与现有的常用估计进行了比较.     

A two‐level variational multiscale method for incompressible flows based on two local Gauss integrations

In this article, a two‐level variational multiscale method for incompressible flows based on two local Gauss integrations is presented. We solve the Navier–Stokes problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations, then seek a fine grid solution by solving a linearized problem on a fine grid. In computation, we use the two local Gauss integrations to replace the projection operator without adding any variables. Stability analysis is performed, and error estimates of the method are derived. Finally, a series of numerical experiments are also given, which confirm the theoretical analysis and demonstrate the efficiency of the new method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A numerical approach for the Poisson equation in a planar domain with a small inclusion

We consider the Poisson equation in a domain with a small inclusion. We present a simple numerical method, based on asymptotic analysis, which allows to approximate robustly the far field of the solution as the size of the inclusion goes to zero without any mesh adaptation procedure. The discretization is based on a fully standard Galerkin approach such as finite elements. We prove stability and consistency of the numerical method and provide error estimates. We end the paper with numerical experiments illustrating the efficiency of the technique.     

利用基因芯片技术筛选差异表达基因的方法研究

为了探讨高维基因芯片基因表达谱数据筛选差异表达基因的方法,分析比较t检验法、秩和检验法、BON法、SIDAK法及ALSU法5种算法的差异表达基因筛选效率;采用模拟实验对t检验法、ALSU法等5种算法进行比较,并使用第一类、第二类错误率、总体错误率、筛选差异表达基因数及其均方根误差等5种指标进行评价;t检验法、秩和检验法计算结果过于灵敏,筛选差异表达基因个数较多,会促使筛选差异表达基因中假阴性事件的发生,BON法、SIDAK法筛选结果过于保守,筛选的差异表达基因个数较少,假阳性事件发生率较为显著,ALSU法能较稳定的抑制第一、二类错误率的发生,同时ALSU法筛选结果受系统扰动误差影响较笺LSU方法能够稳定的、高效的筛选差异表达基因,在使用高纬基因表达谱数据筛选差异表达基因时应首选ALSU法.     

Problems in gene clustering based on gene expression data

In this work, we assess the suitability of cluster analysis for the gene grouping problem confronted with microarray data. Gene clustering is the exercise of grouping genes based on attributes, which are generally the expression levels over a number of conditions or subpopulations. The hope is that similarity with respect to expression is often indicative of similarity with respect to much more fundamental and elusive qualities, such as function. By formally defining the true gene-specific attributes as parameters, such as expected expression across the conditions, we obtain a well-defined gene clustering parameter of interest, which greatly facilitates the statistical treatment of gene clustering. We point out that genome-wide collections of expression trajectories often lack natural clustering structure, prior to ad hoc gene filtering. The gene filters in common use induce a certain circularity to most gene cluster analyses: genes are points in the attribute space, a filter is applied to depopulate certain areas of the space, and then clusters are sought (and often found!) in the “cleaned” attribute space. As a result, statistical investigations of cluster number and clustering strength are just as much a study of the stringency and nature of the filter as they are of any biological gene clusters. In the absence of natural clusters, gene clustering may still be a worthwhile exercise in data segmentation. In this context, partitions can be fruitfully encoded in adjacency matrices and the sampling distribution of such matrices can be studied with a variety of bootstrapping techniques.     

Convergence of an adaptive mixed finite element method for convection-diffusion-reaction equations

We prove the convergence of an adaptive mixed finite element method (AMFEM) for (nonsymmetric) convection-diffusion-reaction equations. The convergence result holds for the cases where convection or reaction is not present in convection- or reaction-dominated problems. A novel technique of analysis is developed by using the superconvergence of the scalar displacement variable instead of the quasi-orthogonality for the stress and displacement variables, and without marking the oscillation dependent on discrete solutions and data. We show that AMFEM is a contraction of the error of the stress and displacement variables plus some quantity. Numerical experiments confirm the theoretical results.     

A posteriori error analysis of the heterogeneous multiscale method for homogenization problems

In this Note we derive a posteriori error estimates for a multiscale method, the so-called heterogeneous multiscale method, applied to elliptic homogenization problems. The multiscale method is based on a macro-to-micro formulation. The macroscopic method discretizes the physical problem in a macroscopic finite element space, while the microscopic method recovers the unknown macroscopic data on the fly during the macroscopic stiffness matrix assembly process. We propose a framework for the analysis allowing to take advantage of standard techniques for a posteriori error estimates at the macroscopic level and to derive residual-based indicators in the macroscopic domain for adaptive mesh refinement. To cite this article: A. Abdulle, A. Nonnenmacher, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Comparison of Methods for Image Analysis on cDNA Microarray Data

Microarrays are part of a new class of biotechnologies which allow the monitoring of expression levels for thousands of genes simultaneously. Image analysis is an important aspect of microarray experiments, one that can have a potentially large impact on subsequent analyses such as clustering or the identification of differentially expressed genes. This article reviews a number of existing image analysis approaches for cDNA microarray experiments and proposes new addressing, segmentation, and background correction methods for extracting information from microarray scanned images. The segmentation component uses a seeded region growing algorithm which makes provision for spots of different shapes and sizes. The background estimation approach is based on an image analysis technique known as morphological opening. These new image analysis procedures are implemented in a software package named Spot, built on the R environment for statistical computing. The statistical properties of the different segmentation and background adjustment methods are examined using microarray data from a study of lipid metabolism in mice. It is shown that in some cases background adjustment can substantially reduce the precision—that is, increase the variability—of low-intensity spot values. In contrast, the choice of segmentation procedure has a smaller impact. The comparison further suggests that seeded region growing segmentation with morphological background correction provides precise and accurate estimates of foreground and background intensities.