arXiv subjects

Daniel B. Forger

Publications and source records attributed to Daniel B. Forger.

5 recordsLinked to original sources

arXiv2026

An adaptive split-combine Gaussian mixture filter for nonlinear and multimodal state estimation

Filtering combines model predictions with measurements to estimate the probability density function (PDF) of a system state over time. The PDF often becomes highly asymmetric and even multimodal in nonlinear systems with oscillatory or chaotic dynamics. Such non-Gaussian features violate the single-Gaussian assumption underlying Kalman-type filters. To address this problem, Gaussian mixture filtering has been proposed. However, accurately propagating mixture components and adaptively adjusting their number and weights over time remain open challenges. Here, we develop an adaptive split-combine Gaussian mixture filter (AMF) that estimates the time evolution of asymmetric and multimodal PDFs by adaptively splitting and combining Gaussian particles without auxiliary online numerical optimization. Notably, the proposed splitting method guarantees a reduction in variance along a target level-set-point direction of a Gaussian particle. This enables accurate and efficient propagation of particles. We show that AMF consistently outperforms various baseline filters across diverse benchmarks, including single and coupled slow-fast Van der Pol oscillators and the Lorenz attractor. We also propose a parallel implementation of AMF, allowing high-fidelity PDF estimation with practical computational cost.

math.NA↗

arXiv2022

A Level Set Kalman Filter Approach to Estimate the Circadian Phase and its Uncertainty from Wearable Data

The circadian clock is an internal timer that coordinates the daily rhythms of behavior and physiology, including sleep and hormone secretion. Accurately tracking the state of the circadian clock, or circadian phase, holds immense potential for precision medicine. Wearable devices present an opportunity to estimate the circadian phase in the real world, as they can non-invasively monitor various physiological outputs influenced by the circadian clock. However, accurately estimating circadian phase from wearable data remains challenging, primarily due to the lack of methods that integrate minute-by-minute wearable data with prior knowledge of the circadian phase. To address this issue, we propose a framework that integrates multi-time scale physiological data to estimate the circadian phase, along with an efficient implementation algorithm based on Bayesian inference and a new state space estimation method called the level set Kalman filter. Our numerical experiments indicate that our approach outperforms previous methods for circadian phase estimation consistently. Furthermore, our method enables us to examine the contribution of noise from different sources to the estimation, which was not feasible with prior methods. We found that internal noise unrelated to external stimuli is a crucial factor in determining estimation results. Lastly, we developed a user-friendly computational package and applied it to real-world data to demonstrate the potential value of our approach. Our results provide a foundation for systematically understanding the real-world dynamics of the circadian clock.

math.DS↗

arXiv2021

The Level Set Kalman Filter for State Estimation of Continuous-discrete Systems

We propose a new extension of Kalman filtering for continuous-discrete systems with nonlinear state-space models that we name as the level set Kalman filter (LSKF). The LSKF assumes the probability distribution can be approximated as a Gaussian, and updates the Gaussian distribution through a time-update step and a measurement-update step. The LSKF improves the time-update step when compared to existing methods, such as the continuous-discrete cubature Kalman filter (CD-CKF) by reformulating the underlying Fokker-Planck equation as an ordinary differential equation for the Gaussian, thereby avoiding expansion in time. Together with a carefully picked measurement-update method, numerical experiments show that the LSKF has a consistent performance improvement over CD-CKF for a range of parameters, while also simplifies the implementation, as no user-defined timestep subdivision between measurements is required, and the spatial derivatives of the drift function are not explicitly needed.

eess.SY↗

arXiv2019

Seasonality and Light Phase-Resetting in the Mammalian Circadian Rhythm

We study the impact of light on the mammalian circadian system using the theory of phase response curves. Using a recently developed ansatz we derive a low-dimensional macroscopic model for the core circadian clock in mammals. Significantly, the variables and parameters in our model have physiological interpretations and may be compared with experimental results. We focus on the effect of four key factors which help shape the mammalian phase response to light: heterogeneity in the population of oscillators, the structure of the typical light phase response curve, the fraction of oscillators which receive direct light input and changes in the coupling strengths associated with seasonal day-lengths. We find these factors can explain several experimental results and provide insight into the processing of light information in the mammalian circadian system. In particular, we find that the sensitivity of the circadian system to light may be modulated by changes in the relative coupling forces between the light sensing and non-sensing populations. Finally, we show how seasonal day-length, after-effects to light entrainment and seasonal variations in light sensitivity in the mammalian circadian clock are interrelated.

q-bio.NC↗

arXiv2017

Macroscopic Models for Networks of Coupled Biological Oscillators

The study of synchronization in populations of coupled biological oscillators is fundamental to many areas of biology to include neuroscience, cardiac dynamics and circadian rhythms. Studying these systems may involve tracking the concentration of hundreds of variables in thousands of individual cells resulting in an extremely high-dimensional description of the system. However, for many of these systems the behaviors of interest occur on a collective or macroscopic scale. We define a new macroscopic reduction for networks of coupled oscillators motivated by an elegant structure we find in experimental measurements of circadian gene expression and several mathematical models for coupled biological oscillators. We characterize the emergence of this structure through a simple argument and demonstrate its applicability to stochastic and heterogeneous systems of coupled oscillators. Finally, we perform the macroscopic reduction for the heterogeneous stochastic Kuramoto equation and compare the low-dimensional macroscopic model with numerical results from the high-dimensional microscopic model.

q-bio.QM↗