Continuous-time Stochastic Differential Equations Research Articles

Are paranoid ideation and hallucination spectrum experiences differently associated with affect dynamics? A continuous-time modeling approach.

Psychotic experiences have been associated with distortions in affective functioning, including aberrancies in affect dynamics. However, it remains unclear whether the two principal symptom dimensions of psychosis, namely paranoid ideation and hallucination spectrum experiences, are differently associated with affect dynamics, and whether associations hold after statistically controlling for depressive symptoms. We investigate this by using a novel statistical approach, the hierarchical Ornstein-Uhlenbeck (OU) process model. This is a continuous-time stochastic differential equations model in a Bayesian framework that describes dynamics in affective valence and arousal via three core parameters: attractor point, variability, and attractor strength. In a community sample with varying levels of psychotic experiences (n = 116), we measured affective valence and arousal 10 times per day for 7 days, using the experience-sampling method. We found-while statistically controlling for depressive symptoms-credible between-subjects associations between paranoid ideation and attractor points of negative valence and high arousal. We also found a credible positive association between hallucination spectrum experiences and arousal variability. Limited evidence emerged for small associations between paranoid ideation and high valence variability as well as between paranoid ideation and high attractor strengths of valence and arousal. Hallucination spectrum experiences showed some evidence for a small association with high arousal attractor points. The detailed picture of affect dynamics provided by the OU model reveals different cross-sectional affective profiles associated with paranoid ideation versus hallucination spectrum experiences that suggest different affective mechanisms of their formation and maintenance. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Continuous Time Limit of the Stochastic Ensemble Kalman Inversion: Strong Convergence Analysis

The Ensemble Kalman inversion (EKI) method is a method for the estimation of unknown parameters in the context of (Bayesian) inverse problems. The method approximates the underlying measure by an ensemble of particles and iteratively applies the ensemble Kalman update to evolve (the approximation of the) prior into the posterior measure. For the convergence analysis of the EKI it is common practice to derive a continuous version, replacing the iteration with a stochastic differential equation. In this paper we validate this approach by showing that the stochastic EKI iteration converges to paths of the continuous-time stochastic differential equation by considering both the nonlinear and linear setting, and we prove convergence in probability for the former, and convergence in moments for the latter. The methods employed can also be applied to the analysis of more general numerical schemes for stochastic differential equations in general.

Linear quadratic Gaussian control with advanced continuous-time disturbance models for building thermal regulation

This paper introduces a linear quadratic control scheme for a continuous-time system with observations taken at discrete times. Particular attention is given to the derivation of the disturbance terms in the model. Control performance may depend critical on accurate disturbance forecasts. This is the case for building climate control, where solar rays pass through e.g. windows and deliver significant amounts of energy and the dynamics can be very fast, fluctuating, and spontaneous. We thus argue that it is critical for control performance to sufficiently describe and include disturbances in the control description to obtain satisfactory control accuracy. We suggest and derive in details a control framework based on continuous-time stochastic differential equations (SDEs) and linear quadratic Gaussian control using an advanced continuous-time disturbance model to supply disturbance forecasts. The numerical simulation results suggest that control with embedded forecasts handles uncertainties well and provides up to 26% performance improvements compared to standard disturbance mitigation techniques. Furthermore, we demonstrate that the quadratic controller has a useful trade-off between variability in the control signal, economic cost, and variability around the reference point.

Asynchronous Sensor Fusion of GPS, IMU and CAN-Based Odometry for Heavy-Duty Vehicles

In heavy-duty vehicles, multiple signals are available to estimate the vehicle's kinematics, such as Inertial Measurement Unit (IMU), Global Positioning System (GPS) and linear and angular speed readings from wheel tachometers on the internal Controller Area Network (CAN). These signals have different noise variance, bandwidth and sampling rate (being the latter, possibly, irregular). In this paper we present a non-linear sensor fusion algorithm allowing asynchronous sampling and non-causal smoothing. It is applied to achieve accuracy improvements when incorporating odometry measurements from CAN bus to standard GPS+IMU kinematic estimation, as well as the robustness against missing data. Our results show that this asynchronous multi-sensor (GPS+IMU+CAN-based odometry) fusion is advantageous in low-speed manoeuvres, improving accuracy and robustness to missing data, thanks to non-causal filtering. The proposed algorithm is based on Extended Kalman Filter and Smoother, with exponential discretization of continuous-time stochastic differential equations, in order to process measurements at arbitrary time instants; it can provide data to subsequent processing steps at arbitrary time instants, not necessarily coincident with the original measurement ones. Given the extra information available in the smoothing case, its estimation performance is less sensitive to the noise-variance parameter setting, compared to causal filtering. Working Matlab code is provided at the end of this work.

Representation of simulation errors in single step methods using state dependent noise

The local error of single step methods is modelled as a function of the state derivative multiplied by bias and zero-mean white noise terms. The deterministic Taylor series expansion of the local error depends on the state derivative meaning that the local error magnitude is zero in steady state and grows with the rate of change of the state vector. The stochastic model of the local error may include a constant, “catch-all” noise term. A continuous time extension of the local error model is developed and this allows the original continuous time state differential equation to be represented by a combination of the simulation method and a stochastic term. This continuous time stochastic differential equation model can be used to study the propagation of the simulation error in Monte Carlo experiments, for step size control, or for propagating the mean and variance. This simulation error model can be embedded into continuous-discrete state estimation algorithms. Two illustrative examples are included to highlight the application of the approach.

Noise in Neuronal and Electronic Circuits: A General Modeling Framework and Non-Monte Carlo Simulation Techniques.

The brain is extremely energy efficient and remarkably robust in what it does despite the considerable variability and noise caused by the stochastic mechanisms in neurons and synapses. Computational modeling is a powerful tool that can help us gain insight into this important aspect of brain mechanism. A deep understanding and computational design tools can help develop robust neuromorphic electronic circuits and hybrid neuroelectronic systems. In this paper, we present a general modeling framework for biological neuronal circuits that systematically captures the nonstationary stochastic behavior of ion channels and synaptic processes. In this framework, fine-grained, discrete-state, continuous-time Markov chain models of both ion channels and synaptic processes are treated in a unified manner. Our modeling framework features a mechanism for the automatic generation of the corresponding coarse-grained, continuous-state, continuous-time stochastic differential equation models for neuronal variability and noise. Furthermore, we repurpose non-Monte Carlo noise analysis techniques, which were previously developed for analog electronic circuits, for the stochastic characterization of neuronal circuits both in time and frequency domain. We verify that the fast non-Monte Carlo analysis methods produce results with the same accuracy as computationally expensive Monte Carlo simulations. We have implemented the proposed techniques in a prototype simulator, where both biological neuronal and analog electronic circuits can be simulated together in a coupled manner.

Joint dynamic pricing and inventory control policy for a stochastic inventory system with perishable products

This paper considers the joint dynamic pricing and inventory control policy for a stochastic inventory system with perishable products. The inventory system, with random disturbance, is modelled as a continuous-time stochastic differential equation. Combined dynamic pricing and production control, a stochastic dynamic optimisation problem that maximises the total discounted profit is developed. By applying the stochastic optimal control method, we formulate the problem of finding the optimal joint dynamic pricing and production schedule as the problem of solving a Hamilton–Jacobi–Bellman (HJB) equation. It is shown that the optimal dynamic pricing and production rate take the linear feedback form of the inventory level, which allows the decision-maker to effectively adjust the strategies as time evolves. In addition, to highlight the advantage of the joint dynamic pricing and production strategy, the case of the optimal production with static price is considered. Numerical examples are given to illustrate the validness of the theoretical results.

Modelling the heat dynamics of a monitored Test Reference Environment for Building Integrated Photovoltaic systems using stochastic differential equations

This paper deals with grey-box modelling of the energy transfer of a double skin Building Integrated Photovoltaic (BIPV) system. Grey-box models are based on a combination of prior physical knowledge and statistics, which enable identification of the unknown parameters in the system and accurate prediction of the most influential variables. The experimental data originates from tests carried out with an air-based BIPV system installed in a Test Reference Environment. BIPV systems represent an interesting application for achieving the requirements of the EU EPBD Directive. Indeed, these systems could reduce the ventilation thermal losses of the building by pre-heating the fresh air. Furthermore, by decreasing PV module temperature, the ventilation air heat extraction can simultaneously increase electrical and thermal energy production of the building. A correct prediction of the PV module temperature and heat transfer coefficients is fundamental in order to improve the thermo-electrical production.The considered grey-box models are composed of a set of continuous time stochastic differential equations, holding the physical description of the system, combined with a set of discrete time measurement equations, which represent the data driven part.In the present work, both one-state and two-state non-linear grey-box models are considered. In order to validate the results, the residuals are analysed for white-noise properties.

Novel Stochastic Modeling and Filtering of the Attitude Quaternion

A novel continuous-time stochastic differential equation (SDE) for spacecraft attitude quaternion kinematics with state-multiplicative noise and a novel continuous-time exact optimal quaternion filter are developed in the framework of Ito (mean-square) calculus. The quaternion Ito SDE contains dissipative terms that ensure the mean-square stability of the process. Closed-form deterministic propagation equations are developed for the second-order moment of the quaternion. The quaternion filter produces the linear minimum-variance unbiased estimate of the quaternion from continuous observations with additive white noise. The filter gain computations include coupled Riccati equations of the estimation error matrix and of the quaternion second-order moment. These computations are not estimate-dependent and can therefore be performed off-line. The special case of gyro error white noise with independent identically distributed components is considered. The case of correlated components can be addressed straightforwardly. Additive gyro biases are easily handled via state augmentation. Extensive Monte-Carlo simulations are performed in order to validate the Ito model and to illustrate the proposed filter accuracy. Comparative simulations of the novel filter and of a standard additive extended Kaiman filter are performed both for attitude-only estimation and for attitude-bias estimation. For practical purposes, the novel modeling and filtering approach shows similar results as the standard approach when the process noise level is low. For high signal-to-noise ratio however, the numerical study suggests that the novel filter can increase the accuracy of a conventional Kaiman by orders of magnitudes.

Grey box modeling of first flush and incoming wastewater at a wastewater treatment plant

On-line measurements of turbidity, UV absorption and flow in the inlet to a Danish wastewater treatment plant are used to establish a dynamic model of the deposition of pollutants in the sewer system and the pollutant mass flow to the treatment plant. The modelling is made using the grey box approach, which is a statistical method that uses known physical relations to formulate the model. The dynamics of the sewer are modelled by means of continuous time stochastic differential equations combined with dry weather diurnal pollutant mass flows. Copyright © 2000 John Wiley & Sons, Ltd.

Convergent sampling of continuous time hereditary stochastic systems

An effective control of hereditary stochastic dynamical processes involving time-varying delays requires identification of the corresponding system parameters. A valid model is needed for an appropriate parameter identification and estimation. This paper presents a regorous derivation of a discrete time stochastic delayed dynamical system from the continuous model. The discrete model will be shown to be a valid representation of the continuous time stochastic differential equation with time-varying delays. The model construction is implemented in three steps. First, the proof of elxistence and uniqueness of a solution to the stochastic delayed differential equation and then, discretization of the continuous model followed by the proof of verification.