Bayesian Structure Learning for Stationary Time Series
Bayesian Structure Learning for Stationary Time Series
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平稳时间序列的贝叶斯结构学习
DOI：
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发表时间：
2015
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通讯作者：
E. Fox
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文献类型：
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作者：
Alex Tank;N. Foti;E. Fox
虽然很多工作已经探索了独立数据的概率图模型，但对时间序列的关注较少。在这种设置中的目标是确定整个时间序列之间的条件独立关系，对于平稳序列，在逆谱密度矩阵中由零编码。我们采用贝叶斯方法来进行结构学习，将先验知识放在（i）图结构和（ii）给定图的谱矩阵上。我们利用惠特尔似然近似，并定义了共轭先验的超复杂逆Wishart复值和图形约束的谱矩阵。由于共轭性，我们可以解析地将谱矩阵边缘化，并获得给定图的时间序列的封闭形式的边缘似然。重要的是，我们的分析边际似然允许我们避免复杂谱矩阵本身的推断，并将我们放回标准（贝叶斯）结构学习的框架中。特别是，将这种边际似然与我们的图先验相结合，可以有效地推断时间序列图本身，我们基于随机搜索过程，尽管任何标准方法都可以直接修改为我们的时间序列案例。我们展示了我们的方法分析股票数据和神经影像数据的大脑活动在各种听觉任务。
While much work has explored probabilistic graphical models for independent data, less attention has been paid to time series. The goal in this setting is to determine conditional independence relations between entire time series, which for stationary series, are encoded by zeros in the inverse spectral density matrix. We take a Bayesian approach to structure learning, placing priors on (i) the graph structure and (ii) spectral matrices given the graph. We leverage a Whittle likelihood approximation and define a conjugate prior—the hyper complex inverse Wishart—on the complex-valued and graph-constrained spectral matrices. Due to conjugacy, we can analytically marginalize the spectral matrices and obtain a closed-form marginal likelihood of the time series given a graph. Importantly, our analytic marginal likelihood allows us to avoid inference of the complex spectral matrices themselves and places us back into the framework of standard (Bayesian) structure learning. In particular, combining this marginal likelihood with our graph prior leads to efficient inference of the time series graph itself, which we base on a stochastic search procedure, though any standard approach can be straightforwardly modified to our time series case. We demonstrate our methods on analyzing stock data and neuroimaging data of brain activity during various auditory tasks.