{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/127129"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/127129","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Cyclicity analysis of the Ornstein-Uhlenbeck process","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-03-28 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-03-28 without embargo terms","abstract_has_math":false,"creators":["Kaushik, Vivek"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Baryshnikov, Yuliy","Zharnitsky, Vadim","DeVille, Lee","Sowers, Richard"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-07-29","date_published":"2024-07-29","updated_at":"2026-07-22T22:25:03Z","subjects":["Cyclicity Analysis","Ornstein-uhlenbeck Process","Stochastic Process","Data Science","Lead-lag Dynamics","Time Series"],"languages":["en","eng"],"rights":["Copyright 2024 Vivek Kaushik"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/127129","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Baryshnikov, Yuliy","Zharnitsky, Vadim","DeVille, Lee","Sowers, Richard"]},{"key":"dc:creator","label":"Author","values":["Kaushik, Vivek"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2024-07-29","2024-12"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Cyclicity Analysis","Ornstein-uhlenbeck Process","Stochastic Process","Data Science","Lead-lag Dynamics","Time Series"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2024 Vivek Kaushik"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/127129"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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","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."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Cyclicity analysis of the Ornstein-Uhlenbeck process"]}]}],"canonical_facts":{"dc:contributor":["Baryshnikov, Yuliy","Zharnitsky, Vadim","DeVille, Lee","Sowers, Richard"],"dc:creator":["Kaushik, Vivek"],"dc:date":["2024-07-29","2024-12"],"dc:description":["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","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."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/127129"],"dc:language":["en","eng"],"dc:rights":["Copyright 2024 Vivek Kaushik"],"dc:subject":["Cyclicity Analysis","Ornstein-uhlenbeck Process","Stochastic Process","Data Science","Lead-lag Dynamics","Time Series"],"dc:title":["Cyclicity analysis of the Ornstein-Uhlenbeck process"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:03Z"}