AUC界协方差选择问题的优性
The Goodness of Covariance Selection Problem from AUC Bounds
摘要：
我们进行了一项研究的图形模型，并讨论了模型选择近似的质量制定的问题作为一个检测问题，并检查曲线下面积（AUC）。我们特别关注联合高斯随机向量的模型选择问题。对于高斯随机向量，该问题简化为Dempster [1]在文献中广泛讨论的协方差选择问题。本文给出了包含模型选择问题全部信息的相关近似矩阵的定义，并讨论了具有Toeplitz协方差矩阵的高斯分布的p阶Markov链模型和$p$阶星形网络模型.对于每个模型，我们计算模型协方差矩阵以及高斯分布与其模型之间的KL散度。我们还表明，如果模型阶数p与节点数n成比例，则模型选择是渐近好的，因为节点数n趋于无穷大，因为在这种情况下AUC有界远离1。我们进行了一些模拟，证实了理论分析，也表明，所选择的模型质量增加模型阶数，p，增加。
英文摘要：
We conduct a study of graphical models and discuss the quality of model selection approximation by formulating the problem as a detection problem and examining the area under the curve (AUC). We are specifically looking at the model selection problem for jointly Gaussian random vectors. For Gaussian random vectors, this problem simplifies to the covariance selection problem which is widely discussed in literature by Dempster [1]. In this paper, we give the definition for the correlation approximation matrix (CAM) which contains all information about the model selection problem and discuss the pth order Markov chain model and the $p$th order star network model for the a Gaussian distribution with Toeplitz covariance matrix. For each model, we compute the model covariance matrix as well as the KL divergence between the Gaussian distribution and its model. We also show that if the model order, p, is proportional to the number of nodes, n, then the model selection is asymptotically good as the number of nodes, n, goes to infinity since the AUC in this case is bounded away from one. We conduct some simulations which confirm the theoretical analysis and also show that the selected model quality increases as the model order, p, increases.
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