Logistic回归模型的统计诊断

统计诊断的主要任务就是通过诊断统计量检测已知观测数据在用既定模型拟合时的合理性,主要是找出数据当中的异常点或强影响点。本文主要研究Logostic回归模型的诊断统计量和诊断统计图。用牛顿迭代法给出Logistic回归模型的极大似然估计值,根据扰动模型得到传统的诊断统计量,结合残差、杠杆值和系数变化三者构造新的诊断统计量,绘制新的诊断统计图,通过模拟研究说明新的诊断统计量的有效性,最后用一个实际案例说明新的诊断方法的应用并进一步验证其优越性。     

可行策略对应的图像拓扑下广义博弈Nash平衡的稳定性

以往关于广义博弈Nash平衡的稳定性的研究,均利用可行策略映射之间的一致度量.现考虑在更弱的度量下,利用可行策略映射图像之间的Hausdorff距离定义度量.在此弱图像拓扑下,证明了广义博弈空间的完备性,以及Nash平衡映射的上半连续性和紧性,进而得到广义博弈Nash平衡的通有稳定性.即在Baire分类的意义下,大多数的广义博弈都是本质的.     

应用图论分析与最优化理论来数据挖掘大规模水牛普里昂蛋白结构数据

图论、最优化理论显然在蛋白质结构的研究中大有用场. 首先, 调查/回顾了研究蛋白质结构的所有图论模型. 其后, 建立了一个图论模型: 让蛋白质的侧链来作为图的顶点, 应用图论的诸如团、 $k$-团、 社群、 枢纽、聚类等概念来建立图的边. 然后, 应用数学最优化的现代摩登数据挖掘算法/方法来分析水牛普里昂蛋白结构的大数据. 成功与令人耳目一新的数值结果将展示给朋友们.     

5-正则图的全控制数的一个注记

设G是不含孤立点的图,S是G的一个顶点子集,若G的每一个顶点都与S中的某顶点邻接,则称S是G的全控制集.G的最小全控制集所含顶点的个数称为G的全控制数,记为γt(G).Thomasse和Yeo证明了若G是最小度至少为5的n阶连通图,则γt(G)≤17n/44.在5-正则图上改进了Thomasse和Yeo的结论,证明了若G是n阶5-正则图,则,γt(G)≤106n/275.     

Constructing near spanning trees with few local inspections

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded‐degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph consisting of edges (for a prespecified constant ), where the decision for different edges should be consistent with the same subgraph . Can this task be performed by inspecting only a constant number of edges in G ? Our main results are:

• We show that if every t‐vertex subgraph of G has expansion then one can (deterministically) construct a sparse spanning subgraph of G using few inspections. To this end we analyze a “local” version of a famous minimum‐weight spanning tree algorithm.
• We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3‐regular graphs of high girth, in which every t‐vertex subgraph has expansion . We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.

© 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 183–200, 2017     

Convergent sequences of sparse graphs: A large deviations approach

Models based on sparse graphs are of interest to many communities: they appear as basic models in combinatorics, probability theory, optimization, statistical physics, information theory, and more applied fields of social sciences and economics. Different notions of similarity (and hence convergence) of sparse graphs are of interest in different communities. In probability theory and combinatorics, the notion of Benjamini‐Schramm convergence, also known as left‐convergence, is used quite frequently. Statistical physicists are interested in the the existence of the thermodynamic limit of free energies, which leads naturally to the notion of right‐convergence. Combinatorial optimization problems naturally lead to so‐called partition convergence, which relates to the convergence of optimal values of a variety of constraint satisfaction problems. The relationship between these different notions of similarity and convergence is, however, poorly understood. In this paper we introduce a new notion of convergence of sparse graphs, which we call Large Deviations or LD‐convergence, and which is based on the theory of large deviations. The notion is introduced by “decorating” the nodes of the graph with random uniform i.i.d. weights and constructing corresponding random measures on and . A graph sequence is defined to be converging if the corresponding sequence of random measures satisfies the Large Deviations Principle with respect to the topology of weak convergence on bounded measures on . The corresponding large deviations rate function can be interpreted as the limit object of the sparse graph sequence. In particular, we can express the limiting free energies in terms of this limit object. We then establish that LD‐convergence implies the other three notions of convergence discussed above, and at the same time establish several previously unknown relationships between the other notions of convergence. In particular, we show that partition‐convergence does not imply left‐ or right‐convergence, and that right‐convergence does not imply partition‐convergence. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 52–89, 2017     

Bounds for pairs in judicious partitioning of graphs

In 2002, Bollobás and Scott posed the following problem: for an integer and a graph G of m edges, what is the smallest f (k, m ) such that V (G ) can be partitioned into V ₁,…,V _k in which for all , where denotes the number of edges with both ends in ? In this paper, we solve this problem asymptotically by showing that . We also show that V (G ) can be partitioned into such that for , where Δ denotes the maximum degree of G . This confirms a conjecture of Bollobás and Scott. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 59–70, 2017     

Sequence-dependent clustering properties of nucleotides fragments in an ionic solution

The chemical nature of the DNA bases is an important factor in sequence-mediated association of DNA molecules. Nucleotides are the fundamental DNA elements and the base identity impacts the molecular properties of nucleotide fragments. It is interesting to study the fundamental nature of nucleotides in DNA, on the basis of base-specific interactions, association, and modes of standard atomic or molecular interactions. With all-atom molecular dynamics simulations of model dinucleotide and tetranucleotide systems having single-stranded dinucleotide or tetranucleotide fragments of varying sequences, we show how the base identity and interactions between the different bases as well as water may affect the clustering properties of nucleotides fragments in an ionic solution. Sequence-dependent differential interactions between the nucleotide fragments, ionic concentration, and elevated temperature are found to influence the clustering properties and dynamics of association. Well-known epigenetic modification of DNA, that is, cytosine methylation also promotes dinucleotide clustering in solution. These observations point to one possible chemical nature of the DNA bases, as well as the importance of the base pairing, base stacking, and ionic interactions in DNA structure formation, and DNA sequence-mediated association. Sequence- and the ionic environment-mediated self-association properties of the dinucleotides indicate its great potential to develop biological nanomaterials for desired applications.     

Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions

For finite-dimensional maps and periodic systems, Palmer rigorously proved Smale horseshoe theorem using shadowing lemma in 1988 [20]. For infinite-dimensional maps and periodic systems, such a proof was completed by Steinlein and Walther in 1990 [30], and Henry in 1994 [9]. For finite-dimensional autonomous systems, such a proof was accomplished by Palmer in 1996 [17]. For infinite-dimensional autonomous systems, the current article offers such a proof. First we prove an Inclination Lemma to set up a coordinate system around a pseudo-orbit. Then we utilize graph transform and the concept of persistence of invariant manifold, to prove the existence of a shadowing orbit.