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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.

管方法和欧拉特征启发法是众所周知的用于近似随机场X（t）的最大值Pr（max X（t）> a）的分布的积分几何方法，其中t是指标集I的元素。研究了这些方法的特点及其在多重比较（主要是基因组数据分析）中的应用。（1）导出了均值和方差均非齐次的高斯随机场最大值的近似公式。此外，还证明了这种近似的有效性.（ii）在随机场为齐次均值和方差的高斯分布，且临界半径全局可达的情况下，给出了管法和欧拉特征启发式的误差阶数表达式。（iii）设A是正交不变随机矩阵。将欧拉特征启发式应用于随机场X（h）= h'Ah，给出了最大特征值分布的近似公式及其误差.对Wishart矩阵、Beta矩阵和逆Wishart矩阵等随机矩阵的欧拉特征线方法的误差较小。（iv）在致死基因的发现或QTL分析中，研究了多重性的调整方法。LOD得分（检验统计量）可以被视为具有复杂相关结构的卡方域。研究了一种调整p值多重性的简单方法。(v)In在上述寻找基因的问题中，我们必须用两个指标来处理卡方随机场，以检测两个位点之间的相互作用（上位性）。当标记位点之间的空间很小时，卡方随机场具有Ornstein-Uhlenbeck相关结构。在标记点位于偶数区间的近似下，给出了随机场最大值（即调整后的p值）的分布。