Despite the large research effort devoted to learning dependencies between time series, the state of the art still faces a major limitation: existing methods learn partial correlations but fail to discriminate across distinct frequency bands. Motivated by many applications in which this differentiation is pivotal, we overcome this limitation by learning a block-sparse, frequency-dependent, partial correlation graph, in which layers correspond to different frequency bands, and partial correlations can occur over just a few layers. To this aim, we formulate and solve two nonconvex learning problems: the first has a closed-form solution and is suitable when there is prior knowledge about the number of partial correlations; the second hinges on an iterative solution based on successive convex approximation, and is effective for the general case where no prior knowledge is available. Numerical results on synthetic data show that the proposed methods outperform the current state of the art. Finally, the analysis of financial time series confirms that partial correlations exist only within a few frequency bands, underscoring how our methods enable the gaining of valuable insights that would be undetected without discriminating along the frequency domain.

翻译：尽管大量研究工作致力于学习时间序列之间的依赖关系，现有技术仍面临一个主要限制：现有方法虽能学习部分相关性，却无法在不同频带之间进行区分。受许多需要这种区分的应用推动，我们通过学习一个块稀疏、频率相关的部分相关图来克服这一限制，其中各层对应不同频带，部分相关性仅出现在少数几层。为此，我们提出并求解两个非凸学习问题：第一个问题具有闭式解，适用于对部分相关数量有先验知识的情况；第二个问题基于逐次凸逼近的迭代求解，适用于无先验知识的一般情况。合成数据的数值结果表明，所提方法优于现有技术。最后，金融时间序列的分析证实部分相关性仅存在于少数频带内，凸显了我们的方法能获得若不按频域区分则无法发现的宝贵洞见。