2026-09-22T08:25:35Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/127129 2025-03-29 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-03-28 without embargo terms The student, Vivek Kaushik, accepted the attached license on 2024-07-24 at 20:23. The student, Vivek Kaushik, submitted this Dissertation for approval on 2024-07-24 at 20:31. This Dissertation was approved for publication on 2024-07-29 at 13:13. DSpace SAF Submission Ingestion Package generated from Vireo submission #21162 on 2025-03-28 at 14:24:42 Cyclicity analysis of the Ornstein-Uhlenbeck process Kaushik, Vivek 2024-07-29 Baryshnikov, Yuliy Zharnitsky, Vadim DeVille, Lee Sowers, Richard Cyclicity Analysis Ornstein-uhlenbeck Process Stochastic Process Data Science Lead-lag Dynamics Time Series eng In this thesis, we consider an N-dimensional Ornstein-Uhlenbeck (OU) process {x(t)}t≥0 satisfying the linear stochastic differential equation dx(t) = −B x(t) dt + Σ dw(t). Here, B is a fixed N × N circulant friction matrix whose eigenvalues have positive real parts, Σ is a fixed N × M matrix for some M ∈ N, and {w(t)}t≥0 is the standard M-dimensional Wiener process. 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. For each t ≥ 0, we interpret xn(t), the n-th component of the OU process at time t, as the measurement of the propagating effect made by the n-th sensor. The matrix B represents the sensor network structure: if B has first row (b1 , ... , bN), where b1 > 0 and b2 , . . . , bN ≤ 0, then the magnitude of bp quantifies how receptive the n-th sensor is to activity within the (n + p − 1)-th sensor, where n + p − 1 is indexed mod N. Finally, the (m,n)-th entry of the matrix D = ΣΣT is the 2 covariance of the component noises injected into the m-th and n-th sensors. For different choices of B and Σ, 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 × N skew-symmetric matrix 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 Q, the eigenvector corresponding to the largest eigenvalue of Q in modulus, reflects the network structure induced by B. 2024-12 Thesis https://hdl.handle.net/2142/127129 Copyright 2024 Vivek Kaushik Mathematics Mathematics University of Illinois at Urbana-Champaign Ph.D. Dissertation