低秩图模型的 ARMAX 辨识
ARMAX identification of low rank graphical models
AI总结：
针对测量噪声掩盖低秩随机过程秩亏缺结构的问题，提出稀疏加低秩 ARMAX 图模型的最大似然辨识框架，并证明其可辨识性与一致性。
摘要：
在大规模系统中，往往存在复杂的内部关系。这类相互连接的系统可以通过低秩随机过程得到有效描述。当从采样数据中辨识低秩过程的预测模型时，谱密度的秩亏缺特性在实践中常被不可避免的测量噪声所掩盖。然而，现有低秩辨识方法通常没有显式考虑噪声，即使在弱噪声条件下也会导致不可忽略的不准确性。本文研究测量噪声下低秩过程的辨识问题。我们发现，含噪测量模型在潜变量图模型中具有稀疏加低秩结构。具体而言，我们首先将该问题分解为最大熵协方差延拓问题，以及基于带外生输入的自回归滑动平均（ARMAX）模型的低秩图估计问题。为辨识 ARMAX 低秩图模型，我们提出一种基于最大似然的估计方法。在一定条件下证明了该方法的可辨识性和一致性。仿真结果证实，整个算法在参数估计和含噪数据滤波两方面均具有可靠性能。
英文摘要：
In large-scale systems, complex internal relationships are often present. Such interconnected systems can be effectively described by low rank stochastic processes. When identifying a predictive model of low rank processes from sampling data, the rank-deficient property of spectral densities is often obscured by the inevitable measurement noise in practice. However, existing low rank identification approaches often did not take noise into explicit consideration, leading to non-negligible inaccuracies even under weak noise. In this paper, we address the identification issue of low rank processes under measurement noise. We find that the noisy measurement model admits a sparse plus low rank structure in latent-variable graphical models. Specifically, we first decompose the problem into a maximum entropy covariance extension problem, and a low rank graphical estimation problem based on an autoregressive moving-average with exogenous input (ARMAX) model. To identify the ARMAX low rank graphical models, we propose an estimation approach based on maximum likelihood. The identifiability and consistency of this approach are proven under certain conditions. Simulation results confirm the reliable performance of the entire algorithm in both the parameter estimation and noisy data filtering.
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