Probabilistic State Space Model Research Articles

A hybrid-driven probabilistic state space model for tool wear monitoring

Tool wear monitoring (TWM) is essential to improve product quality and maintain machining safety. Within recent years, numerous techniques have been developed for TWM, including data-driven methods and physics-based models. However, traditional data-driven methods are highly dependent on measured data. The physics-based models are difficult to model complex machining processes. To solve these problems, a hybrid-driven probabilistic state space model is proposed to improve the accuracy and robustness of TWM. In this work, a sensitive feature extraction scheme is first constructed to eliminate interferences and redundancies for improving monitoring efficiency. Subsequently, the Gaussian process is innovatively developed to integrate the data mining and physical model from a probabilistic perspective. On this basis, the particle filter is then utilized to estimate the model parameters and estimate tool wear conditions. Finally, an adaptive wear state recognition method is further established for the predictive maintenance of cutting tools. The practical data obtained from milling experiments are used to validate the effectiveness of the proposed methodology. The results show that the proposed hybrid-driven method effectively improves tool wear prediction accuracy to over 97.7%, while the constructed probabilistic model successfully evaluates the prediction results with over 95% confidence. Therefore, the developed methodology may provide a promising way for tool health management to improve economic efficiency and maintain machining safety.

Reactive Message Passing for Scalable Bayesian Inference

We introduce reactive message passing (RMP) as a framework for executing schedule-free, scalable, and, potentially, more robust message passing-based inference in a factor graph representation of a probabilistic model. RMP is based on the reactive programming style, which only describes how nodes in a factor graph react to changes in connected nodes. We recognize reactive programming as the suitable programming abstraction for message passing-based methods that improve robustness, scalability, and execution time of the inference procedure and are useful for all future implementations of message passing methods. We also present our own implementation ReactiveMP.jl, which is a Julia package for realizing RMP through minimization of a constrained Bethe free energy. By user-defined specification of local form and factorization constraints on the variational posterior distribution, ReactiveMP.jl executes hybrid message passing algorithms including belief propagation, variational message passing, expectation propagation, and expectation maximization update rules. Experimental results demonstrate the great performance of our RMP implementation compared to other Julia packages for Bayesian inference across a range of probabilistic models. In particular, we show that the RMP framework is capable of performing Bayesian inference for large-scale probabilistic state-space models with hundreds of thousands of random variables on a standard laptop computer.

Decentralised fully probabilistic design for stochastic networks with multiplicative noise

In this paper, we present an innovative decentralised control framework, designed to address stochastic dynamic complex systems that are influenced by multiple multiplicative noise factors. Our advanced approach builds upon the foundation of conventional Decentralised Fully Probabilistic Design (DFPD) by refining the Riccati equation to accommodate multiple noise sources effectively. By embracing the inherent stochastic nature of complex systems, our methodology fully characterises their dynamic behaviours using probabilistic state–space models, delivering a comprehensive representation of subsystem components. Importantly, the DFPD approach also incorporates system and input constraints by characterising their corresponding ideal distributions, ensuring optimal functionality and performance while adhering to permissible boundaries. To further enhance system performance, we introduce a probabilistic message passing architecture that enables seamless communication between neighbouring subsystems and promotes harmonised decision-making among local nodes. To demonstrate the efficacy of our proposed framework, we employ a three-inverted pendulum system as a numerical example and compare its performance to that of the conventional DFPD. Through this comparison, we showcase the advantages of our novel decentralised control approach in handling complex systems with multiple noise factors.

State space model of undrained triaxial test data for Bayesian identification of constitutive model parameters

Soil constitutive model parameters can be identified from triaxial test data. The identification is frequently performed by fitting a constitutive model to triaxial test data from a purely statistical or probabilistic perspective based on an assumption that measurements are independent. This ignores the sequential attribute of triaxial test data and is, hence, not realistic. In this paper, a probabilistic state space model (SSM) is proposed for undrained triaxial test data analysis, with which the sequential data attribute is explicitly considered and a constitutive model (physics) is linked to the SSM model (statistics) in a natural way. Then, constitutive model parameters can be rigorously learned under a Bayesian framework based on the SSM without artificially augmenting them into hidden variables of SSM. Without loss of generality, the modified Cam Clay (MCC) model is taken as an example to develop the SSM and to demonstrate the proposed Bayesian framework, which is illustrated using simulated and real-life data. Results from the proposed Bayesian framework based on the SSM include not only the best estimates of MCC model parameters but also their posterior distributions for quantifying the identification uncertainty, based on which the identifiability of MCC model parameters is discussed and highlighted.

Probabilistic message passing control for complex stochastic switching systems

In this paper, we propose a general decentralised probabilistic control framework for a class of complex stochastic systems with switching modes. Probabilistic state space models are exploited to characterise the subsystems’ dynamical behaviours constituting a complex dynamical system, thus providing a complete description of the subsystems components. To address the variations in the operational modes of the subsystems, the Mixture Density Network (MDN) is applied here to identify the subsystems modes and provides estimates for the system dynamic distributions. Besides, to harmonise the actions between the subsystems, the probabilistic message passing methodology is utilised to provide communication between neighbouring subsystems. Based on the MDN model and the neighbours subsystems information via message passing, the general solution of the fully probabilistic decentralised randomised controller which minimises the Kullback-Leibler divergence (KLD) between the actual and its ideal distributions is then obtained. Moreover, a numerical example is presented to illustrate the effectiveness and the usefulness of our novel proposed framework.

Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems

Data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a data-based, probabilistic perspective that enables the quantification of predictive uncertainties. One of the outstanding problems has been the introduction of physical constraints in the probabilistic machine learning objectives. The primary utility of such constraints stems from the undisputed physical laws such as conservation of mass, energy etc. that they represent. Furthermore and apart from leading to physically realistic predictions, they can significantly reduce the requisite amount of training data which for high-dimensional, multiscale systems are expensive to obtain (Small Data regime). We formulate the coarse-graining process by employing a probabilistic state-space model and account for the aforementioned equality constraints as virtual observables in the associated densities. We demonstrate how deep neural nets in combination with probabilistic inference tools can be employed to identify the coarse-grained variables and their evolution model without ever needing to define a fine-to-coarse (restriction) projection and without needing time-derivatives of state variables.We advocate a sparse Bayesian learning perspective which avoids overfitting and reveals the most salient features in the CG evolution law. The formulation adopted enables the quantification of a crucial, and often neglected, component in the CG process, i.e. the predictive uncertainty due to information loss. Furthermore, it is capable of reconstructing the evolution of the full, fine-scale system and therefore the observables of interest need not be selected a priori. We demonstrate the efficacy of the proposed framework by applying it to systems of interacting particles and a series of images of a nonlinear pendulum. In both cases we identify the underlying coarse dynamics and can generate extrapolative predictions including the forming and propagation of a shock for the particle systems and a stable trajectory in the phase space for the pendulum.

Simulating Active Inference Processes by Message Passing.

The free energy principle (FEP) offers a variational calculus-based description for how biological agents persevere through interactions with their environment. Active inference (AI) is a corollary of the FEP, which states that biological agents act to fulfill prior beliefs about preferred future observations (target priors). Purposeful behavior then results from variational free energy minimization with respect to a generative model of the environment with included target priors. However, manual derivations for free energy minimizing algorithms on custom dynamic models can become tedious and error-prone. While probabilistic programming (PP) techniques enable automatic derivation of inference algorithms on free-form models, full automation of AI requires specialized tools for inference on dynamic models, together with the description of an experimental protocol that governs the interaction between the agent and its simulated environment. The contributions of the present paper are two-fold. Firstly, we illustrate how AI can be automated with the use of ForneyLab, a recent PP toolbox that specializes in variational inference on flexibly definable dynamic models. More specifically, we describe AI agents in a dynamic environment as probabilistic state space models (SSM) and perform inference for perception and control in these agents by message passing on a factor graph representation of the SSM. Secondly, we propose a formal experimental protocol for simulated AI. We exemplify how this protocol leads to goal-directed behavior for flexibly definable AI agents in two classical RL examples, namely the Bayesian thermostat and the mountain car parking problems.

Probabilistic State Space Models – A Theoretical Framework with Practical Relevance

Probabilistic State Space Models – A Theoretical Framework with Practical Relevance

Physics-Constrained, Data-Driven Discovery of Coarse-Grained Dynamics

The combination of high-dimensionality and disparity of time scales encountered in many problems in computational physics has motivated the development of coarse-grained (CG) models. In this paper, we advocate the paradigm of data-driven discovery for extract- ing governing equations by employing fine-scale simulation data. In particular, we cast the coarse-graining process under a probabilistic state-space model where the transition law dic- tates the evolution of the CG state variables and the emission law the coarse-to-fine map. The directed probabilistic graphical model implied, suggests that given values for the fine- grained (FG) variables, probabilistic inference tools must be employed to identify the cor- responding values for the CG states and to that end, we employ Stochastic Variational In- ference. We advocate a sparse Bayesian learning perspective which avoids overfitting and reveals the most salient features in the CG evolution law. The formulation adopted enables the quantification of a crucial, and often neglected, component in the CG process, i.e. the pre- dictive uncertainty due to information loss. Furthermore, it is capable of reconstructing the evolution of the full, fine-scale system. We demonstrate the efficacy of the proposed frame- work in high-dimensional systems of random walkers.

Impacts of eutrophication and oil spills on the Gulf of Finland herring stock

The Baltic Sea is one of the world’s most stressed sea areas. Major threats to the ecosystem include eutrophication and oil spills. The progression of anthropogenic nutrient enrichment is lengthy and gradual, while oil spills cause rapid changes in the system, with varying impact time. We quantify the impact of eutrophication and the key ecological covariates on the population dynamics of the major pelagic fish stock, the Baltic herring (Clupea harengus membras), in the Gulf of Finland. The full life cycle of herring is represented with a probabilistic state-space model. Moreover, we analyse the impact of the oil spill from M/T Antonio Gramsci in 1987 on herring survival. The results confirm impact of the spill on the early life-stage survival; the observed high frequency of malformed herring larvae in surveys signaled elevated mortality of the year class. The optimal July–August chlorophyll a concentration for herring reproduction is approximately 5 μg·L−1. This level is currently exceeded, suggesting recruitment impairment due to eutrophication. The herring stock was also recruitment-overfished. Analysis suggests deceleration of herring growth as salinity descends below 6 psu.

Calibration of potential drop measuring and damage extent prediction by Bayesian filtering and smoothing

Fatigue related damage growth without feasibility of optical assessment can be monitored conveniently by means of the direct current potential drop method in laboratory experiments. By estimating the unknown damage extent of a structure indirectly via observed measurements, the need to relate both quantities, i.e. a calibration of damage extent and measurements, arises. In recent years, Bayesian inference has been applied with a special focus to such inverse problem formulations.In the present paper, a novel approach to the calibration issue is proposed by employing Bayesian filtering and smoothing. A probabilistic state space model incorporating prior information about the damage extent and calibration parameters as well as process describing models is defined and subsequently used to infer the damage extent of fatigue-tested specimens from potential drop measurements. First, the obtained results in the form of joint conditional posterior distribution functions are exploited to facilitate an evaluation of a direct model calibration on the one hand and direct damage extent estimation on the other hand given persistent uncertainties. In a further step, the inferred damage extent estimations and associated uncertainties are propagated in time as to allow an assessment of decision-making-feasibility within the extended scope of structural health monitoring and damage prognosis. A thorough performance analysis in the light of actual damage extend data is undertaken, revealing accurate results.

Forecasting seasonal influenza with a state-space SIR model.

Seasonal influenza is a serious public health and societal problem due to its consequences resulting from absenteeism, hospitalizations, and deaths. The overall burden of influenza is captured by the Centers for Disease Control and Prevention's influenza-like illness network, which provides invaluable information about the current incidence. This information is used to provide decision support regarding prevention and response efforts. Despite the relatively rich surveillance data and the recurrent nature of seasonal influenza, forecasting the timing and intensity of seasonal influenza in the U.S. remains challenging because the form of the disease transmission process is uncertain, the disease dynamics are only partially observed, and the public health observations are noisy. Fitting a probabilistic state-space model motivated by a deterministic mathematical model [a susceptible-infectious-recovered (SIR) model] is a promising approach for forecasting seasonal influenza while simultaneously accounting for multiple sources of uncertainty. A significant finding of this work is the importance of thoughtfully specifying the prior, as results critically depend on its specification. Our conditionally specified prior allows us to exploit known relationships between latent SIR initial conditions and parameters and functions of surveillance data. We demonstrate advantages of our approach relative to alternatives via a forecasting comparison using several forecast accuracy metrics.

A Bayesian particle filtering method for brain source localisation

In this paper, we explore the multiple source localisation problem in the cerebral cortex using magnetoencephalography (MEG) data. We model neural currents as point-wise dipolar sources which dynamically evolve over time, then model dipole dynamics using a probabilistic state space model in which dipole locations are strictly constrained to lie within the cortex. Based on the proposed models, we develop a Bayesian particle filtering algorithm for localisation of both known and unknown numbers of dipoles. The algorithm consists of a region of interest (ROI) estimation step for initial dipole number estimation, a Gibbs multiple particle filter (GMPF) step for individual dipole state estimation, and a selection criterion step for selecting the final estimates. The estimated results from the ROI estimation are used to adaptively adjust particle filter's sample size to reduce the overall computational cost. The proposed models and the algorithm are tested in numerical experiments. Results are compared with existing particle filtering methods. The numerical results show that the proposed methods can achieve improved performance metrics in terms of dipole number estimation and dipole localisation.

State and parameter estimation for nonlinear biological phenomena modeled by S-systems

Biological pathways can be modeled as a nonlinear system described by a set of nonlinear ordinary differential equations (ODEs). A central challenge in computational modeling of biological systems is the determination of the model parameters. In such cases, estimating these variables or parameters from other easily obtained measurements can be extremely useful. For example, time-series dynamic genomic data can be used to develop models representing dynamic genetic regulatory networks, which can be used to design intervention strategies to cure major diseases and to better understand the behavior of biological systems. Unfortunately, biological measurements are usually highly affected by errors that hide the important characteristics in the data. Therefore, these noisy measurements need to be filtered to enhance their usefulness in practice. This paper addresses the problem of state and parameter estimation of biological phenomena modeled by S-systems using Bayesian approaches, where the nonlinear observed system is assumed to progress according to a probabilistic state space model. The performances of various conventional and state-of-the-art state estimation techniques are compared. These techniques include the extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), and the developed improved particle filter (IPF). Specifically, two comparative studies are performed. In the first comparative study, the state variables (the enzyme CadA, the transport protein CadB, the regulatory protein CadC and lysine Lys for a model of the Cad System in E. coli (CSEC)) are estimated from noisy measurements of these variables, and the various estimation techniques are compared by computing the estimation root mean square error (RMSE) with respect to the noise-free data. In the second comparative study, the state variables as well as the model parameters are simultaneously estimated. In this case, in addition to comparing the performances of the various state estimation techniques, the effect of the number of estimated model parameters on the accuracy and convergence of these techniques is also assessed. The results of both comparative studies show that the UKF provides a higher accuracy than the EKF due to the limited ability of EKF to accurately estimate the mean and covariance matrix of the estimated states through lineralization of the nonlinear process model. The results also show that the IPF provides a significant improvement over PF because, unlike the PF which depends on the choice of sampling distribution used to estimate the posterior distribution, the IPF yields an optimum choice of the sampling distribution, which also accounts for the observed data. The results of the second comparative study show that, for all techniques, estimating more model parameters affects the estimation accuracy as well as the convergence of the estimated states and parameters. However, the IPF can still provide both convergence as well as accuracy related advantages over other estimation methods.

Modeling of nonlinear biological phenomena modeled by S-systems

A central challenge in computational modeling of biological systems is the determination of the model parameters. In such cases, estimating these variables or parameters from other easily obtained measurements can be extremely useful. For example, time-series dynamic genomic data can be used to develop models representing dynamic genetic regulatory networks, which can be used to design intervention strategies to cure major diseases and to better understand the behavior of biological systems. Unfortunately, biological measurements are usually highly infected by errors that hide the important characteristics in the data. Therefore, these noisy measurements need to be filtered to enhance their usefulness in practice. This paper addresses the problem of state and parameter estimation of biological phenomena modeled by S-systems using Bayesian approaches, where the nonlinear observed system is assumed to progress according to a probabilistic state space model. The performances of various conventional and state-of-the-art state estimation techniques are compared. These techniques include the extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), and the developed variational Bayesian filter (VBF). Specifically, two comparative studies are performed. In the first comparative study, the state variables (the enzyme CadA, the model cadBA, the cadaverine Cadav and the lysine Lys for a model of the Cad System in Escherichia coli (CSEC)) are estimated from noisy measurements of these variables, and the various estimation techniques are compared by computing the estimation root mean square error (RMSE) with respect to the noise-free data. In the second comparative study, the state variables as well as the model parameters are simultaneously estimated. In this case, in addition to comparing the performances of the various state estimation techniques, the effect of the number of estimated model parameters on the accuracy and convergence of these techniques is also assessed. The results of both comparative studies show that the UKF provides a higher accuracy than the EKF due to the limited ability of EKF to accurately estimate the mean and covariance matrix of the estimated states through lineralization of the nonlinear process model. The results also show that the VBF provides a relative improvement over PF. This is because, unlike the PF which depends on the choice of sampling distribution used to estimate the posterior distribution, the VBF yields an optimum choice of the sampling distribution, which also utilizes the observed data. The results of the second comparative study show that, for all techniques, estimating more model parameters affects the estimation accuracy as well as the convergence of the estimated states and parameters. The VBF, however, still provides advantages over other methods with respect to estimation accuracy as well convergence.

Bayesian methods for predicting LAI and soil water content

LAI of winter wheat (Triticum aestivum L.) and soil water content of the topsoil (200 mm) and of the subsoil (500 mm) were considered as state variables of a dynamic soil-crop system. This system was assumed to progress according to a Bayesian probabilistic state space model, in which real values of LAI and soil water content were daily introduced in order to correct the model trajectory and reach better future evolution. The chosen crop model was mini STICS which can reduce the computing and execution times while ensuring the robustness of data processing and estimation. To predict simultaneously state variables and model parameters in this non-linear environment, three techniques were used: extended Kalman filtering (EKF), particle filtering (PF), and variational filtering (VF). The significantly improved performance of the VF method when compared to EKF and PF is demonstrated. The variational filter has a low computational complexity and the convergence speed of states and parameters estimation can be adjusted independently. Detailed case studies demonstrated that the root mean square error of the three estimated states (LAI and soil water content of two soil layers) was smaller and that the convergence of all considered parameters was ensured when using VF. Assimilating measurements in a crop model allows accurate prediction of LAI and soil water content at a local scale. As these biophysical properties are key parameters in the crop-plant system characterization, the system has the potential to be used in precision farming to aid farmers and decision makers in developing strategies for site-specific management of inputs, such as fertilizers and water irrigation.

Secure and robust clustering for quantized target tracking in wireless sensor networks

We consider the problem of secure and robust clustering for quantized target tracking in wireless sensor networks (WSN) where the observed system is assumed to evolve according to a probabilistic state space model. We propose a new method for jointly activating the best group of candidate sensors that participate in data aggregation, detecting the malicious sensors and estimating the target position. Firstly, we select the appropriate group in order to balance the energy dissipation and to provide the required data of the target in the WSN. This selection is also based on the transmission power between a sensor node and a cluster head. Secondly, we detect the malicious sensor nodes based on the information relevance of their measurements. Then, we estimate the target position using quantized variational filtering (QVF) algorithm. The selection of the candidate sensors group is based on multi-criteria function, which is computed by using the predicted target position provided by the QVF algorithm, while the malicious sensor nodes detection is based on Kullback-Leibler distance between the current target position distribution and the predicted sensor observation. The performance of the proposed method is validated by simulation results in target tracking for WSN.

Optimal sensor and path selection for target tracking in wireless sensor networks

ABSTRACTThis paper addresses target tracking in wireless sensor networks where the nonlinear observed system is assumed to progress according to a probabilistic state space model. Thus, we propose to improve the use of the quantized variational filtering by jointly selecting the optimal candidate sensor that participates in target localization and its best communication path to the cluster head. In the current work, firstly, we select the optimal sensor in order to provide the required data of the target and to balance the energy dissipation in the wireless sensor networks. This selection is also based on the local cluster node density and their transmission power. Secondly, we select the best communication path that achieves the highest signal‐to‐noise ratio at the cluster head; then, we estimate the target position using quantized variational filtering algorithm. The best communication path is designed to reduce the communication cost, which leads to a significant reduction of energy consumption and an accurate target tracking. The optimal sensor selection is based on mutual information maximization under energy constraints, which is computed by using the target position predictive distribution provided by the quantized variational filtering algorithm. The simulation results show that the proposed method outperforms the quantized variational filtering under sensing range constraint, binary variational filtering, and the centralized quantized particle filtering. Copyright © 2012 John Wiley & Sons, Ltd.

Reprogramming of Escherichia coli K-12 Metabolism during the Initial Phase of Transition from an Anaerobic to a Micro-Aerobic Environment

BackgroundMany bacteria undergo transitions between environments with differing O2 availabilities as part of their natural lifestyles and during biotechnological processes. However, the dynamics of adaptation when bacteria experience changes in O2 availability are understudied. The model bacterium and facultative anaerobe Escherichia coli K-12 provides an ideal system for exploring this process.Methods and FindingsTime-resolved transcript profiles of E. coli K-12 during the initial phase of transition from anaerobic to micro-aerobic conditions revealed a reprogramming of gene expression consistent with a switch from fermentative to respiratory metabolism. The changes in transcript abundance were matched by changes in the abundances of selected central metabolic proteins. A probabilistic state space model was used to infer the activities of two key regulators, FNR (O2 sensing) and PdhR (pyruvate sensing). The model implied that both regulators were rapidly inactivated during the transition from an anaerobic to a micro-aerobic environment. Analysis of the external metabolome and protein levels suggested that the cultures transit through different physiological states during the process of adaptation, characterized by the rapid inactivation of pyruvate formate-lyase (PFL), a slower induction of pyruvate dehydrogenase complex (PDHC) activity and transient excretion of pyruvate, consistent with the predicted inactivation of PdhR and FNR.ConclusionPerturbation of anaerobic steady-state cultures by introduction of a limited supply of O2 combined with time-resolved transcript, protein and metabolite profiling, and probabilistic modeling has revealed that pyruvate (sensed by PdhR) is a key metabolic signal in coordinating the reprogramming of E. coli K-12 gene expression by working alongside the O2 sensor FNR during transition from anaerobic to micro-aerobic conditions.

Adaptive quantized target tracking in wireless sensor networks

This paper addresses target tracking in wireless sensor networks (WSN) where the observed system is assumed to evolve according to a probabilistic state space model. We propose to improve the use of the variational filtering (VF) by optimally quantizing the data collected by the sensors. Recently, VF has been proved to be suitable to the communication constraints of WSN. Its efficiency relies on the fact that the online update of the filtering distribution and its compression are executed simultaneously. However, this problem has been used only for binary sensor networks neglecting the transmission energy consumption in a WSN and the information relevance of sensor measurements. Our proposed method is intended to jointly estimate the target position and optimize the quantization level under fixed and variable transmitting power. At each sampling instant, the adaptive method provides not only the estimate of the target position by using the VF but gives also the optimal number of quantization bits per observation. The adaptive quantization is achieved by minimizing the predicted Cramer---Rao bound if the transmitting power is constant for all sensors, and optimizing the power scheduling under distortion constraint if this power is variable. The computation of the predicted Cramer---Rao bound is based on the target position predictive distribution provided by the VF algorithm. The proposed adaptive quantization scheme suggests that the sensors with bad channels or poor observation qualities should decrease their quantization resolutions or simply become inactive in order to save energy.