In this thesis, we consider an $N$-dimensional Ornstein-Uhlenbeck (OU) process satisfying the linear stochastic differential equation $d\mathbf x(t) = - \mathbf B\mathbf x(t) dt + \boldsymbol \Sigma d \mathbf w(t).$ Here, $\mathbf B$ is a fixed $N \times N$ circulant friction matrix whose eigenvalues have positive real parts, $\boldsymbol \Sigma$ is a fixed $N \times M$ matrix. We consider a signal propagation model governed by this OU process. In this model, an underlying signal propagates throughout a network consisting of $N$ linked sensors located in space. We interpret the $n$-th component of the OU process as the measurement of the propagating effect made by the $n$-th sensor. The matrix $\mathbf B$ represents the sensor network structure: if $\mathbf B$ has first row $(b_1 \ , \ \dots \ , \ b_N),$ where $b_1>0$ and $b_2 \ , \ \dots \ ,\ b_N \le 0,$ then the magnitude of $b_p$ quantifies how receptive the $n$-th sensor is to activity within the $(n+p-1)$-th sensor. Finally, the $(m,n)$-th entry of the matrix $\mathbf D = \frac{\boldsymbol \Sigma \boldsymbol \Sigma^\text T}{2}$ is the covariance of the component noises injected into the $m$-th and $n$-th sensors. For different choices of $\mathbf B$ and $\boldsymbol \Sigma,$ we investigate whether Cyclicity Analysis enables us to recover the structure of network. Roughly speaking, Cyclicity Analysis studies the lead-lag dynamics pertaining to the components of a multivariate signal. We specifically consider an $N \times N$ skew-symmetric matrix $\mathbf Q,$ known as the lead matrix, in which the sign of its $(m,n)$-th entry captures the lead-lag relationship between the $m$-th and $n$-th component OU processes. We investigate whether the structure of the leading eigenvector of $\mathbf Q,$ the eigenvector corresponding to the largest eigenvalue of $\mathbf Q$ in modulus, reflects the network structure induced by $\mathbf B.$

翻译：本文研究满足线性随机微分方程 $d\mathbf x(t) = - \mathbf B\mathbf x(t) dt + \boldsymbol \Sigma d \mathbf w(t)$ 的 $N$ 维奥恩斯坦-乌伦贝克过程。其中 $\mathbf B$ 为固定 $N \times N$ 循环摩擦矩阵，其特征值具有正实部；$\boldsymbol \Sigma$ 为固定 $N \times M$ 矩阵。我们考察由该 OU 过程控制的信号传播模型：基础信号在由 $N$ 个空间分布的互联传感器构成的网络中传播，OU 过程的第 $n$ 个分量被解释为第 $n$ 个传感器对传播效应的测量值。矩阵 $\mathbf B$ 表征传感器网络结构：若其首行为 $(b_1 \ , \ \dots \ , \ b_N)$（其中 $b_1>0$，$b_2 \ , \ \dots \ ,\ b_N \le 0$），则 $b_p$ 的幅值量化了第 $n$ 个传感器对第 $(n+p-1)$ 个传感器活动的响应程度。矩阵 $\mathbf D = \frac{\boldsymbol \Sigma \boldsymbol \Sigma^\text T}{2}$ 的 $(m,n)$ 元素表示注入第 $m$ 与第 $n$ 个传感器的分量噪声协方差。针对不同的 $\mathbf B$ 与 $\boldsymbol \Sigma$ 选择，我们探究循环性分析能否重建网络结构。简言之，循环性分析研究多元信号各分量间的超前-滞后动态特性。我们特别考察 $N \times N$ 斜对称矩阵 $\mathbf Q$（称为领先矩阵），其 $(m,n)$ 元素的符号捕捉第 $m$ 与第 $n$ 个分量 OU 过程间的超前-滞后关系。我们研究 $\mathbf Q$ 的主特征向量（即对应 $\mathbf Q$ 模最大特征值的特征向量）的结构是否反映由 $\mathbf B$ 导出的网络结构。