Cyclicity analysis of the Ornstein-Uhlenbeck process

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Kaushik, Vivek

Contributors dc:contributor

• Baryshnikov, Yuliy
• Zharnitsky, Vadim
• DeVille, Lee
• Sowers, Richard

• Cyclicity Analysis
• Ornstein-uhlenbeck Process
• Stochastic Process
• Data Science
• Lead-lag Dynamics
• Time Series

Statement dc:rights

• Copyright 2024 Vivek Kaushik

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/127129

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F127129