Time series data sets often contain heterogeneous signals, composed of both continuously changing quantities and discretely occurring events. The coupling between these measurements may provide insights into key underlying mechanisms of the systems under study. To better extract this information, we investigate the asymptotic statistical properties of coupling measures between continuous signals and point processes. We first introduce martingale stochastic integration theory as a mathematical model for a family of statistical quantities that include the phase locking value, a classical coupling measure to characterize complex dynamics. Based on the martingale central limit theorem, we can then derive the asymptotic gaussian distribution of estimates of such coupling measure that can be exploited for statistical testing. Second, based on multivariate extensions of this result and random matrix theory, we establish a principled way to analyze the low-rank coupling between a large number of point processes and continuous signals. For a null hypothesis of no coupling, we establish sufficient conditions for the empirical distribution of squared singular values of the matrix to converge, as the number of measured signals increases, to the well-known Marchenko-Pastur (MP) law, and the largest squared singular value converges to the upper end of the MP support. This justifies a simple thresholding approach to assess the significance of multivariate coupling. Finally, we illustrate with simulations the relevance of our univariate and multivariate results in the context of neural time series, addressing how to reliably quantify the interplay between multichannel local field potential signals and the spiking activity of a large population of neurons.

中文翻译：

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从连续信号和点过程之间的单变量到多变量耦合：数学框架。

时间序列数据集通常包含异构信号，由连续变化的数量和离散发生的事件组成。这些测量之间的耦合可以提供对所研究系统的关键潜在机制的见解。为了更好地提取这些信息，我们研究了连续信号和点过程之间耦合度量的渐近统计特性。我们首先引入鞅随机积分理论作为包括锁相值在内的一系列统计量的数学模型，锁相值是表征复杂动力学的经典耦合度量。基于鞅中心极限定理，我们可以推导出这种耦合度量估计的渐近高斯分布，可用于统计测试。第二，基于该结果的多元扩展和随机矩阵理论，我们建立了一种分析大量点过程和连续信号之间低秩耦合的原理方法。对于无耦合的零假设，我们为矩阵的平方奇异值的经验分布建立充分条件，随着测量信号数量的增加，收敛到众所周知的马尔琴科-帕斯图 (MP) 定律，并且最大平方奇异值收敛到 MP 支持的上端。这证明了一个简单的阈值方法来评估多变量耦合的重要性。最后，我们通过模拟说明了我们的单变量和多变量结果在神经时间序列的背景下的相关性，