Cyclicity Analysis of the Ornstein-Uhlenbeck Process

Abstract

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 $\mathbf B.$