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

Music City Center 

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

Parameter estimation

Mean-reverting stochastic differential equations

Confidence region

Seasonal time series 

Section on Statistics in Epidemiology