【作者】 王琦；

【导师】 赵强；

【作者基本信息】 山东师范大学 ， 统计学， 2019， 硕士

【摘要】 概率图模型方法是统计领域中的有效研究工具之一.在图模型中,节点代表随机变量,节点之间的边反映了随机变量的关联性.图模型可以分为有向图和无向图.无向图被称为马尔科夫随机场,边的有无表示的是随机变量之间的条件独立性.具有一定概率分布的有向无环图又被称作贝叶斯网.对于贝叶斯网,它的所有边都有方向并且不能构成一个回路.本文重点的研究对象就是高斯有向无环图–高斯贝叶斯网.有向无环图通常用来表示随机变量之间的因果关系,它在物理研究和生物工程中有大量的应用.我们通过穷举法估计n个节点的网络的复杂度为O(2~n),早期的启发式算法能够在节点量少的情况下有较好的效果.但随着节点的增加,以穷举法为代表的算法杂度成指数增长,这使得通过采样数据估计有向图是一个NP难问题.另外,数据采样的局限性,从各个节点往往能够采样得到少量的数据.信息的匮乏使得传统的算法很难适应信息时代的发展.随后,一系列以高维数据的低维模型为基础的算法相继提出.这一类算法利用高斯分布的极大似然估计构建了优化函数,利用高维数据的低秩性通过回归算法估计节点之间的权重,从而推测节点之间的连接性.不同场景下所采样得到的高维数据在结构上往往具有相关性,因此,利用这些相关性可以降低回归问题由于数据量较少所带来的复杂程度.一系列联合估计高斯有向图的算法也孕育而生.尽管有较成熟的算法估计高斯图模型,但是由于有向图的观测等价性,仅通过观测数据恢复有向图是非常困难的.因此,本文利用节点之间的偏序先验,在该先验条件下,估计高斯有向图等价于估计网络的骨架.当多重高斯有向图具有相似性结构,本文提出了带相似性结构惩罚项的回归模型用于估计网络的邻接矩阵,并用坐标下降法求解该模型.数值实验证明了该算法的有效性.更多还原

【Abstract】 Probability graph model is one of the effective research tools in the field of statistics.In the graph model,nodes represent random variables,and the edge be-tween nodes reflect the relationship between random variables.The graph model can be divided into the directed graph or the undirected graph.Undirected graph is called a markov random field,the edge of the presence represents conditional independence between the random variables.The directed acyclic graph that has a certain probability is called a bayesian network.For the bayesian network,its all edges have directions and cannot constitute any loop,the edges expresses the causal relationship between the random variables.The research object of this pa-per is the directed acyclic graph(DAG)with gaussian probability-the gaussian bayesian network.It has been widely used in physics research and biological engi-neering.By the exhaustive method,estimating the network that has n nodes,the complexity of the network is O(2~n),Heuristic algorithm can behave better when the number of nodes is small.But with the number of nodes increasing,the complexity of the algorithm exemplified by the heuristics method grows up geometrically.So estimating directed graph becomes a NP-hard problem.Series of algorithms based on the model of the low dimension structure of high dimension data.Those kind of algorithms utilizes the maximum likelihood estimation of Gaussian distribution to construct the optimization function besides using the low rank of high-dimensional data to estimate the weights between nodes by regression algorithm,so the connectivity between nodes can be estimated.The resulting high-dimensional data tends to be structurally related when sampling in different scenarios.Thus these correlations can reduce the degree of illness caused by the amount of data in the regression problem.A series of algorithms for joint estimation of Gaussian directed graphs are also born.Although there are more mature algorithms to estimate Gaussian graphs,it is a ill-conditional problem to recover directed graphs only by observing data due to the observational equiva-lence of directed graphs.A regression model based on the maximum likelihood estimation with similar structure penalty term is proposed in this paper to es-timate the adjacency matrices of the networks.Computationally,the model is solved by coordinate descent method with complexity O(nk~2p).Compared with PC algorithm,numerical experiments demonstrates the algorithm proposed in this paper performs well.natural ordering between nodes and use the similarity struc-ture between different graphs.A regression model based on the maximum likeli-hood estimation with similar structure penalty term is proposed in this paper to estimate the adjacency matrices of the networks.Computationally,the model is solved by coordinate descent method with complexity O(nk~2p).Compared with PC algorithm,numerical experiments demonstrates the algorithm proposed in this paper performs well.更多还原