计算限制最大似然方法中的对数似然及其导数
Computing log-likelihood and its derivatives for restricted maximum likelihood methods
AI总结：
本文提出通过串行矩阵变换和平均信息分割技术，高效计算线性混合模型中限制最大似然方法的对数似然及其导数，以提高大规模基因组关联分析的统计效率。
摘要：
近年来，大规模基因组关联分析涉及大规模线性混合模型。在混合模型中使用限制最大似然方法量化（协）方差参数会得到一个作为对数似然一阶导数的得分函数。为了获得方差参数的统计高效估计，需要通过牛顿方法找到得分函数的根。大多数雅可比矩阵的元素涉及四个参数矩阵-矩阵乘法的迹。对于大规模数据集，这在计算上是不可行的。通过串行矩阵变换和平均信息分割技术，可以得到一个近似的雅可比矩阵，该矩阵通过将雅可比矩阵的平均值与其期望值分开来获得。在近似的雅可比矩阵中，其元素仅涉及四个矩阵向量乘法，可以通过求解稀疏线性系统的方法用多前导因子分解法高效评估。
英文摘要：
Recent large scale genome wide association analysis involves large scale linear mixed models. Quantifying (co)-variance parameters in the mixed models with a restricted maximum likelihood method results in a score function which is the first derivative of a log-likelihood. To obtain a statistically efficient estimate of the variance parameters, one needs to find the root of the score function via the Newton method. Most elements of the Jacobian matrix of the score involve a trace term of four parametric matrix-matrix multiplications. It is computationally prohibitively for large scale data sets. By a serial matrix transforms and an averaged information splitting technique, an approximate Jacobian matrix can be obtained by splitting the average of the Jacobian matrix and its expected value. In the approximated Jacobian, its elements only involve Four matrix vector multiplications which can be efficiently evaluated by solving sparse linear systems with the multi-frontal factorization method.
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