Study on Euler characteristic heuristics in distribution theory of random field

随机场分布理论中的欧拉特征启发式研究

基本信息

批准号：
15500194
负责人：
KURIKI Satoshi
金额：
$ 1.92万
依托单位：
The Institute of Statistical Mathematics
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

The tube method and the Euler characteristic heuristic are well known as integral-geometric methods for approximating the distribution of the maximum Pr (max X (t) > a) of a random field X (t), where t is an element of an index set I. Some characteristics of these methods and their applications to multiple comparisons (mainly genomic data analysis) were studied.(i)Approximating formulas for the maximum of a Gaussian random field with inhomogeneous mean and variance were derived. In addition, the validity of this approximation was proved.(ii)The expressions for the order of errors in the tube method and the Euler characteristic heuristic were given explicitly in the case where the random field is Gaussian with homogeneous mean and variance, and the critical radius is attained globally.(iii)Let A be an orthogonally invariant random matrix. By applying the Euler characteristic heuristic to a random field X(h) = h' A h, an approximating formulas as well as its error for the distribution of the largest eigenvalue were given. The errors of the Euler characteristic methods for many random matrices including the Wishart matrix, the Beta matrix, and the inverse Wishart matrix, were shown to be small.(iv)Methods for adjusting the multiplicity were studied in the problem of finding the fatal genes or the QTL analysis. The LOD score (test statistic) can be regarded as a chi-squared field with complicated correlation structure. A simple method for adjusting the multiplicity of p-values were studied.(v)In the problem of finding genes mentioned above, we have to treat the chi-squared random field with two indices in order to detect the interaction (epistasis) between two loci. When the spaces between maker loci are small, the chi-squared random field has the Ornstein-Uhlenbeck correlation structure. Under the approximation that the makers are located at even intervals, the distribution of the maximum of the random field (that is, the adjusted p-value) was given.

试管方法和Euler特征启发式词是众所周知的积分几何方法，用于近似随机字段x（t）的最大PR（最大x（t）> a）的分布，其中t是索引集合的元素I。这些方法的某些特征及其在多个比较（主要的基因组数据分析）上的特征及其应用程序的应用程序。得出了不均匀的平均值和方差。此外，证明了这种近似值的有效性。（ii）在管道方法中的错误顺序和欧拉特征启发式的表达式明确给出，如果随机场是具有均匀的均值和差异的随机场，而关键的半径是全球范围的。（iii）。通过将Euler特性启发式词应用于随机场X（H）= H'A H，近似公式以及其在最大特征值分布的误差中得到了。在发现致命基因或QTL分析的问题中研究了许多随机矩阵的欧拉特征方法的误差，包括WishArt矩阵，Beta矩阵和逆Wishart矩阵，包括WishArt矩阵，Beta矩阵和逆Wishart矩阵的误差。（iv）调整多重性的方法。 LOD评分（测试统计量）可以被视为具有复杂相关结构的卡方场。研究了一种调整p值的多样性的简单方法。（v）在查找上述基因的问题中，我们必须使用两个指数处理卡方随机场，以便检测两个基因座之间的相互作用（epitsasis）。当制造商基因座之间的空间很小时，卡方随机场具有Ornstein-uhlenbeck相关结构。在制造商以均匀的间隔位置的近似值下，给出了随机场的最大值（即调整后的p值）的分布。