Interpretable machine learning for high-dimensional trajectories of aging health

We have built a computational model for individual aging trajectories of health and survival, which contains physical, functional, and biological variables, and is conditioned on demographic, lifestyle, and medical background information. We combine techniques of modern machine learning with an interpretable interaction network, where health variables are coupled by explicit pair-wise interactions within a stochastic dynamical system. Our dynamic joint interpretable network (DJIN) model is scalable to large longitudinal data sets, is predictive of individual high-dimensional health trajectories and survival from baseline health states, and infers an interpretable network of directed interactions between the health variables. The network identifies plausible physiological connections between health variables as well as clusters of strongly connected health variables. We use English Longitudinal Study of Aging (ELSA) data to train our model and show that it performs better than multiple dedicated linear models for health outcomes and survival. We compare our model with flexible lower-dimensional latent-space models to explore the dimensionality required to accurately model aging health outcomes. Our DJIN model can be used to generate synthetic individuals that age realistically, to impute missing data, and to simulate future aging outcomes given arbitrary initial health states.

Stochastic turbulence for Burgers equation driven by cylindrical Lévy process

This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by Lévy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity of solutions and their statistical quantities in this stochastic dynamical system. The quantities include moment estimate, structure function and energy spectrum of the turbulent velocity field. Furthermore, we provide qualitative and quantitative properties of the stochastic Burgers equation when the kinematic viscosity [Formula: see text] tends towards zero. The inviscid limit describes the strong stochastic turbulence.

Interpretable Machine Learning of High-Dimensional Aging Health Trajectories

We have built a computational model of individual aging trajectories of health and survival, that contains physical, functional, and biological variables, and is conditioned on demographic, lifestyle, and medical background information. We combine techniques of modern machine learning with an interpretable network approach, where health variables are coupled by an explicit interaction network within a stochastic dynamical system. Our model is scalable to large longitudinal data sets, is predictive of individual high-dimensional health trajectories and survival from baseline health states, and infers an interpretable network of directed interactions between the health variables. The network identifies plausible physiological connections between health variables and clusters of strongly connected heath variables. We use English Longitudinal Study of Aging (ELSA) data to train our model and show that it performs better than traditional linear models for health outcomes and survival. Our model can also be used to generate synthetic individuals that age realistically, to impute missing data, and to simulate future aging outcomes given an arbitrary initial health state.

Fixed points and stability of A class of Stochastic dynamical system driven by Brownian motion

In real life, many models and systems are affected by random phenomena. For this reason, experts and scholars propose to describe these stochastic processes with Brownian motion respectively. In this paper we consider a kind of stochastic Vollterra dynamical systems of nonconvolution type and give some new conditions to ensure that the zero solution is asymptotically stable in mean square by means of fixed point method. The theorems of asymptotically stability in mean square with a necessary conditions are proved. Some results of related papers are improved.

Existence of solutions for fixed point theorem

The development of a mathematical model based on diffusion has received a great dealof attention in recent years, many scientist and mathematician have tried to apply basicknowledge about the differential equation and the boundary condition to explain anapproximate the diffusion and reaction model. The subject of fractional calculus attracted much attentions and is rapidly growing area of research because of itsnumerous applications in engineering and scientific disciplines such as signal processing, nonlinear control theory, viscoelasticity, optimization theory [1], controlled thermonuclear fusion, chemistry, nonlinear biological systems, mechanics,electric networks, fluid dynamics, diffusion, oscillation, relaxation, turbulence, stochastic dynamical system, plasmaphysics, polymer physics, chemical physics, astrophysics, and economics. Therefore, it deserves an independent theoryparallel to the theory of ordinary differential equations (DEs).In the development of non-linear analysis, fixed point theory plays an important role. Also, it has been widely used in different branches of engineering and sciences. Banach fixed point theory is a essential part of mathematical analysis because of its applications in various area such as variational and linear inequalities, improvement and approximation theory. The fixed-point theorem in diffusion equations plays a significant role to construct methods to solve the problems in sciences and mathematics. Although Banach fixed point theory is a vast field of study and is capable of solving diffusion equations. The main motive of the research is solving the diffusion equations by Banach fixed point theorems and Adomian decomposition method. To analysis the drawbacks of the other fixed-point theorems and different solving methods, the related works are reviewed in this paper.

Invariants for a Dynamical System with Strong Random Perturbations

In this chapter we consider the invariant method for stochastic system with strong perturbations, and its application to many different tasks related to dynamical systems with invariants. This theory allows constructing the mathematical model (deterministic and stochastic) of actual process if it has invariant functions. These models have a kind of jump-diffusion equations system (stochastic differential Itô equations with a Wiener and a Poisson paths). We show that an invariant function (with probability 1) for stochastic dynamical system under strong perturbations exists. We consider a programmed control with Prob. 1 for stochastic dynamical systems – PSP1. We study the construction of stochastic models with invariant function based on deterministic model with invariant one and show the results of numerical simulation. The concept of a first integral for stochastic differential equation Itô introduce by V. Doobko, and the generalized Itô – Wentzell formula for jump-diffusion function proved us, play the key role for this research.

A novel approach for discovering stochastic models behind data applied to El Niño–Southern Oscillation

Stochastic differential equations (SDEs) are ubiquitous across disciplines, and uncovering SDEs driving observed time series data is a key scientific challenge. Most previous work on this topic has relied on restrictive assumptions, undermining the generality of these approaches. We present a novel technique to uncover driving probabilistic models that is based on kernel density estimation. The approach relies on few assumptions, does not restrict underlying functional forms, and can be used even on non-Markov systems. When applied to El Niño–Southern Oscillation (ENSO), the fitted empirical model simulations can almost perfectly capture key time series properties of ENSO. This confirms that ENSO could be represented as a two-variable stochastic dynamical system. Our experiments provide insights into ENSO dynamics and suggest that state-dependent noise does not play a major role in ENSO skewness. Our method is general and can be used across disciplines for inverse and forward modeling, to shed light on structure of system dynamics and noise, to evaluate system predictability, and to generate synthetic datasets with realistic properties.

A Dynamic Variance-Based Triggering Scheme for Distributed Cooperative State Estimation over Wireless Sensor Networks

Wireless sensor networks (WSNs) have been spawning many new applications where cooperative state estimation is essential. In this paper, the problem of performing cooperative state estimation for a discrete linear stochastic dynamical system over wireless sensor networks with a limitation on the sampling and communication rate is considered, where distributed sensors cooperatively sense a linear dynamical process and transmit observations each other via a common wireless channel. Firstly, a novel dynamic variance-based triggering scheme (DVTS) is designed to schedule the sampling of each sensor and the transmission of its local measurement. In contrast to the existing static variance-based triggering scheme (SVTS), the newly proposed DVTS can lead to the larger average intertrigger time interval and thus fewer total triggering number with almost approximate estimation accuracy. Second, a new Riccati equation of the prediction variance iteration for each estimator is obtained, which switches dynamically among the modes related to the variance of the previous step and the recently received measurements from other sensors. Furthermore, the stability issue is also mainly investigated. Finally, simulation results show the effectiveness and advantage of the proposed strategy.

Forecasting Individual Aging Trajectories and Survival with an Interpretable Network Model

We have built a computational model of individual aging trajectories of health and survival, containing physical, functional, and biological variables, conditioned on demographic, lifestyle, and medical background information. We combine techniques of modern machine learning with a network approach, where the health variables are coupled by an interaction network within a stochastic dynamical system. The resulting model is scalable to large longitudinal data sets, is predictive of individual high-dimensional health trajectories and survival, and infers an interpretable network of interactions between the health variables. The interaction network gives us the ability to identify which interactions between variables are used by the model, demonstrating that realistic physiological connections are inferred. We use English Longitudinal Study of Aging (ELSA) data to train our model and show that it performs better than standard linear models for health outcomes and survival, while also revealing the relevant interactions. Our model can be used to generate synthetic individuals that age realistically from input data at baseline, as well as the ability to probe future aging outcomes given an arbitrary initial health state.