Learning the latent network structure from large scale multivariate point process data is an important task in a wide range of scientific and business applications. For instance, we might wish to estimate the neuronal functional connectivity network based on spiking times recorded from a collection of neurons. To characterize the complex processes underlying the observed data, we propose a new and flexible class of nonstationary Hawkes processes that allow both excitatory and inhibitory effects. We estimate the latent network structure using an efficient sparse least squares estimation approach. Using a thinning representation, we establish concentration inequalities for the first and second order statistics of the proposed Hawkes process. Such theoretical results enable us to establish the non-asymptotic error bound and the selection consistency of the estimated parameters. Furthermore, we describe a least squares loss based statistic for testing if the background intensity is constant in time. We demonstrate the efficacy of our proposed method through simulation studies and an application to a neuron spike train data set.
翻译:从大型多点进程数据中学习潜在网络结构是一系列广泛的科学和商业应用中的一项重要任务。 例如,我们不妨根据从神经元收集中记录到的弹射时间来估计神经功能连接网络。 为了描述观测到的数据所基于的复杂过程的特点, 我们提议了一个新的灵活的非静止的鹰类进程, 允许产生刺激和抑制效应。 我们使用高效的稀释最小方位估计方法来估计潜在的网络结构。 我们使用稀释的表示法, 确定拟议霍克斯进程第一和第二顺序统计的集中性不平等。 这些理论结果使我们能够确定非被动错误的界限和估计参数的选定一致性。 此外, 我们描述一个基于最小平方损失的统计, 用于测试, 如果背景强度在时间上保持不变。 我们通过模拟研究和对神经峰值列数据集的应用来展示我们拟议方法的功效。