自举正则化奇异相关矩阵
Bootstraps Regularize Singular Correlation Matrices
摘要：
我分析表明，平均值的$k$自举相关矩阵迅速成为正定的$k$增加，这提供了一个简单的方法来规范奇异皮尔逊相关矩阵。如果$n$是对象的数目，$t$是特征的数目，则当$k> \frac{e}{e-1}\frac{n}{t}\simeq 1.58\frac{n}{t}$在大$t$和$n$的极限中时，平均相关矩阵几乎肯定是正定的。对于有限的$n$和$t$，给出了利用$k$引导获得正定相关矩阵的概率.最后，我证明了所需的引导程序的数量总是小于$n$。该方法在$n$的数量级大于数据点$t$的大小的领域中特别相关，例如，金融、遗传学、社会科学或图像处理。
英文摘要：
I show analytically that the average of $k$ bootstrapped correlation matrices rapidly becomes positive-definite as $k$ increases, which provides a simple approach to regularize singular Pearson correlation matrices. If $n$ is the number of objects and $t$ the number of features, the averaged correlation matrix is almost surely positive-definite if $k> \frac{e}{e-1}\frac{n}{t}\simeq 1.58\frac{n}{t}$ in the limit of large $t$ and $n$. The probability of obtaining a positive-definite correlation matrix with $k$ bootstraps is also derived for finite $n$ and $t$. Finally, I demonstrate that the number of required bootstraps is always smaller than $n$. This method is particularly relevant in fields where $n$ is orders of magnitude larger than the size of data points $t$, e.g., in finance, genetics, social science, or image processing.
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