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Modeling Seasonal Time Series with Periodic Mean-Reverting Stochastic Differential Equations

Abstract Number:

1222

Submission Type:

Contributed Abstract

Contributed Abstract Type:

Poster

Participants:

GM Fahad Bin Mostafa (1), Linda J.S. Allen (2)

Institutions:

(1) Arizona State University, N/A, (2) Texas Tech University, Lubbock, TX

Co-Author:

Linda J.S. Allen
Texas Tech University

First Author:

GM Fahad Bin Mostafa
Arizona State University

Presenting Author:

GM Fahad Bin Mostafa
Arizona State University

Abstract Text:

Seasonal variation is a key feature of many environmental and biological systems, including infectious disease outbreaks and temperature patterns. Periodic mean-reverting stochastic differential equations (SDEs) effectively model such variability. We present periodic mean-reverting SDE models, $dX(t) = r(\beta(t) - X(t))dt + d\beta(t) + \sigma X^p(t)dW(t),$ for $p = 0, 1/2, 2/3, 5/6, 1$, with periodic mean $\beta(t)$, and fit them to seasonally varying influenza and temperature data. The model with $p = 0$ corresponds to the Ornstein-Uhlenbeck process, while $p = 1/2$ and $p = 1$ relate to the Cox-Ingersoll-Ross (CIR) process and geometric Brownian motion (GBM), and other mean-reverting SDEs $p = 2/3, 5/6$, respectively. We show that the higher-order moments of CIR and GBM processes exhibit periodicity. Novel model-fitting methods combine least squares for $\beta(t)$ estimation and maximum likelihood for $r$ and $\sigma$. Confidence regions are constructed via bootstrapping, and missing data are handled using a modified MissForest algorithm. These models provide a robust framework for capturing seasonal dynamics and offer flexibility in mean function specification

Keywords:

Sponsors:

Section on Statistics in Epidemiology

Tracks:

Statistical Issues in Environmental Epidemiology

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