Kullback-Leibler距离作为多元数据过滤信息的度量
Kullback-Leibler distance as a measure of the information filtered from multivariate data
摘要：
我们表明，Kullback-Leibler距离是一个很好的措施，通过使用有限的数据集估计的相关矩阵的统计不确定性。对于多元高斯变量的相关矩阵，我们分析确定的Kullback-Leibler距离的样本相关矩阵从参考模型的预期值，我们表明，预期值也是已知的，当特定的模型是未知的。我们建议利用Kullback-Leibler距离来估计相关滤波过程从相关矩阵中提取的信息。我们还展示了如何使用这个距离来衡量的稳定性过滤程序的统计不确定性。我们解释了我们的方法的有效性，通过比较四个过滤程序，其中两个是基于谱分析和其他两个层次聚类。我们比较这些技术都适用于模拟的因素模型和经验数据。我们调查的能力，这些过滤程序在恢复模型的相关矩阵从模拟。我们讨论了这种能力的异质性的模型参数和数据序列的长度。我们还表明，这两种谱技术通常比基于层次聚类的技术更能提供有关样本相关矩阵的信息，而后者在统计不确定性方面更稳定。
英文摘要：
We show that the Kullback-Leibler distance is a good measure of the statistical uncertainty of correlation matrices estimated by using a finite set of data. For correlation matrices of multivariate Gaussian variables we analytically determine the expected values of the Kullback-Leibler distance of a sample correlation matrix from a reference model and we show that the expected values are known also when the specific model is unknown. We propose to make use of the Kullback-Leibler distance to estimate the information extracted from a correlation matrix by correlation filtering procedures. We also show how to use this distance to measure the stability of filtering procedures with respect to statistical uncertainty. We explain the effectiveness of our method by comparing four filtering procedures, two of them being based on spectral analysis and the other two on hierarchical clustering. We compare these techniques as applied both to simulations of factor models and empirical data. We investigate the ability of these filtering procedures in recovering the correlation matrix of models from simulations. We discuss such an ability in terms of both the heterogeneity of model parameters and the length of data series. We also show that the two spectral techniques are typically more informative about the sample correlation matrix than techniques based on hierarchical clustering, whereas the latter are more stable with respect to statistical uncertainty.
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