Study on Euler characteristic heuristics in distribution theory of random field

管法和欧拉特征启发式被称为近似随机场X (t)的最大Pr （max X (t) > a）分布的积分几何方法，其中t是索引集i的一个元素。研究了这些方法的一些特征及其在多重比较（主要是基因组数据分析）中的应用。(i)推导了均值和方差均不均匀的高斯随机场最大值的近似公式。此外，还证明了该近似的有效性。（ii）明确地给出了随机场为均匀高斯，均方差齐次，且全局得到临界半径的情况下，管法误差阶数和欧拉特征启发式的表达式。（iii）设A为正交不变随机矩阵。将欧拉特征启发式应用于随机场X(h) = h' a h，给出了最大特征值分布的近似公式及其误差。结果表明，对于包括Wishart矩阵、Beta矩阵和逆Wishart矩阵在内的许多随机矩阵，欧拉特征方法的误差都很小。(4)研究了致死性基因寻找或QTL分析中多重性的调整方法。LOD分数（检验统计量）可以看作是一个具有复杂相关结构的卡方场。研究了一种调整p值复数的简便方法。(5)在上述寻找基因的问题中，为了检测两个基因座之间的相互作用（上位性），我们必须用两个指标来处理卡方随机场。当maker基因座间距较小时，卡方随机场具有Ornstein-Uhlenbeck相关结构。在制造者均匀分布的近似条件下，给出了随机场最大值（即调整后的p值）的分布。

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.