SummaryWe consider estimating a functional graphical model from multivariate functional observations. In functional data analysis, the classical assumption is that each function has been measured over a densely sampled grid. However, in practice the functions have often been observed, with measurement error, at a relatively small number of points. We propose a class of doubly functional graphical models to capture the evolving conditional dependence relationship among a large number of sparsely or densely sampled functions. Our approach first implements a nonparametric smoother to perform functional principal components analysis for each curve, then estimates a functional covariance matrix and finally computes sparse precision matrices, which in turn provide the doubly functional graphical model. We derive some novel concentration bounds, uniform convergence rates and model selection properties of our estimator for both sparsely and densely sampled functional data in the high-dimensional large-$p$, small-$n$ regime. We demonstrate via simulations that the proposed method significantly outperforms possible competitors. Our proposed method is applied to a brain imaging dataset.

中文翻译：

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高维双功能图形模型

总结我们考虑从多元函数观察中估计一个函数图模型。在功能数据分析中，经典假设是每个功能都在密集采样的网格上进行了测量。然而，在实践中，经常在相对较少的点上观察到这些函数，带有测量误差。我们提出了一类双函数图形模型来捕捉大量稀疏或密集采样函数之间不断发展的条件依赖关系。我们的方法首先实现了一个非参数平滑器来对每条曲线进行函数主成分分析，然后估计函数协方差矩阵，最后计算稀疏精度矩阵，这反过来提供双函数图形模型。我们推导出一些新的浓度界限，我们的估计器在高维大 $p$、小 $n$ 机制中针对稀疏和密集采样的功能数据的统一收敛率和模型选择属性。我们通过模拟证明所提出的方法明显优于可能的竞争对手。我们提出的方法应用于大脑成像数据集。