We develop a Bayesian graphical modeling framework for functional data for correlated multivariate random variables observed over a continuous domain. Our method leads to graphical Markov models for functional data which allows the graphs to vary over the functional domain. The model involves estimation of graphical models that evolve functionally in a nonparametric fashion while accounting for within-functional correlations and borrowing strength across functional positions so contiguous locations are encouraged but not forced to have similar graph structure and edge strength. We utilize a strategy that combines nonparametric basis function modeling with modified Bayesian graphical regularization techniques, which induces a new class of hypoexponential normal scale mixture distributions that not only leads to adaptively shrunken estimators of the conditional cross-covariance but also facilitates a thorough theoretical investigation of the shrinkage properties. Our approach scales up to large functional datasets collected on a fine grid. We show through simulations and real data analysis that the Bayesian functional graphical model can efficiently reconstruct the functionally-evolving graphical models by accounting for within-function correlations.
翻译:我们为连续域观测的相关多变量随机变量的功能数据开发了贝叶西亚图形建模框架。 我们的方法为功能数据绘制了图形 Markov 模型,使图形在功能域上变化。 模型涉及对功能性模型进行估算,这些图形模型以非参数方式演化,同时考虑功能性相关关系和在功能位置之间借入强度,因此鼓励但不会被迫具有类似的图形结构和边缘强度。 我们使用一种战略,将非参数性基功能模型与修改的贝叶西亚图形规范化技术相结合,从而产生一种新的低度正常规模混合物分布类别,不仅导致条件跨变量的适应性闪烁估计器,而且还有助于彻底的理论研究收缩特性。 我们的方法将规模提升到在精细网格上收集的大型功能数据集。 我们通过模拟和真实的数据分析显示, Bayesian 功能性图形模型可以通过计算功能内关联来有效地重建功能变化的图形模型。