Cyclicity analysis of the Ornstein-Uhlenbeck process

Kaushik, Vivek

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

Description

Title

Cyclicity analysis of the Ornstein-Uhlenbeck process

Author(s)

Kaushik, Vivek

Issue Date

2024-07-29

Director of Research (if dissertation) or Advisor (if thesis)

Baryshnikov, Yuliy

Doctoral Committee Chair(s)

Zharnitsky, Vadim

Committee Member(s)

• DeVille, Lee
• Sowers, Richard

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Cyclicity Analysis
• Ornstein-Uhlenbeck process
• Stochastic process
• Data Science
• Lead-Lag Dynamics
• Time Series

Abstract

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.

Graduation Semester

2024-12

Type of Resource

Thesis

Handle URL

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

Copyright and License Information

Copyright 2024 Vivek Kaushik

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