Variational Bayesian Framework Research Articles

Φ-DVAE: Physics-informed dynamical variational autoencoders for unstructured data assimilation

Incorporating unstructured data into physical models is a challenging problem that is emerging in data assimilation. Traditional approaches focus on well-defined observation operators whose functional forms are typically assumed to be known. This prevents these methods from achieving a consistent model-data synthesis in configurations where the mapping from data-space to model-space is unknown. To address these shortcomings, in this paper we develop a physics-informed dynamical variational autoencoder (Φ-DVAE) to embed diverse data streams into time-evolving physical systems described by differential equations. Our approach combines a standard, possibly nonlinear, filter for the latent state-space model and a VAE, to assimilate the unstructured data into the latent dynamical system. Unstructured data, in our example systems, comes in the form of video data and velocity field measurements, however the methodology is suitably generic to allow for arbitrary unknown observation operators. A variational Bayesian framework is used for the joint estimation of the encoding, latent states, and unknown system parameters. To demonstrate the method, we provide case studies with the Lorenz-63 ordinary differential equation, and the advection and Korteweg-de Vries partial differential equations. Our results, with synthetic data, show that Φ-DVAE provides a data efficient dynamics encoding methodology which is competitive with standard approaches. Unknown parameters are recovered with uncertainty quantification, and unseen data are accurately predicted.

Variational Bayesian Variable Selection for High-Dimensional Hidden Markov Models

The Hidden Markov Model (HMM) is a crucial probabilistic modeling technique for sequence data processing and statistical learning that has been extensively utilized in various engineering applications. Traditionally, the EM algorithm is employed to fit HMMs, but currently, academics and professionals exhibit augmenting enthusiasm in Bayesian inference. In the Bayesian context, Markov Chain Monte Carlo (MCMC) methods are commonly used for inferring HMMs, but they can be computationally demanding for high-dimensional covariate data. As a rapid substitute, variational approximation has become a noteworthy and effective approximate inference approach, particularly in recent years, for representation learning in deep generative models. However, there has been limited exploration of variational inference for HMMs with high-dimensional covariates. In this article, we develop a mean-field Variational Bayesian method with the double-exponential shrinkage prior to fit high-dimensional HMMs whose hidden states are of discrete types. The proposed method offers the advantage of fitting the model and investigating specific factors that impact the response variable changes simultaneously. In addition, since the proposed method is based on the Variational Bayesian framework, the proposed method can avoid huge memory and intensive computational cost typical of traditional Bayesian methods. In the simulation studies, we demonstrate that the proposed method can quickly and accurately estimate the posterior distributions of the parameters with good performance. We analyzed the Beijing Multi-Site Air-Quality data and predicted the PM2.5 values via the fitted HMMs.

A Variational Bayesian Inference-Based En-Decoder Framework for Traffic Flow Prediction

Accurate traffic flow prediction, a hotspot for intelligent transportation research, is the prerequisite for traffic flow prediction for making travel plans. The speed of traffic flow can be affected by roads condition, weather, holidays, etc. Moreover, sensors to catch the information about traffic flow will be interfered with by environmental factors such as illumination, collection time, occlusion, etc. Therefore, the traffic flow in the practical transportation system is complicated, uncertain, and challenging to predict accurately. Motivated from the aforementioned issues and challenges, in this paper, we propose a deep encoder-decoder prediction framework based on variational Bayesian inference. A Bayesian neural network is designed by combining variational inference with Gated Recurrent Units (GRU) which is used as the deep neural network unit of the encoder-decoder framework to mine the intrinsic dynamics of traffic flow. Then, the variational inference is introduced into the multi-head attention mechanism to avoid noise-induced deterioration of prediction accuracy. The proposed model achieves superior prediction performance on the Guangzhou urban traffic flow dataset over the benchmarks, particularly when the long-term prediction.

An Improved Adaptive Iterative Extended Kalman Filter Based on Variational Bayesian

The presence of unknown heavy-tailed noise can lead to inaccuracies in measurements and processes, resulting in instability in nonlinear systems. Various estimation methods for heavy-tailed noise exist. However, these methods often trade estimation accuracy for algorithm complexity and parameter sensitivity. To tackle this challenge, we introduced an improved variational Bayesian (VB)-based adaptive iterative extended Kalman filter. In this VB framework, the inverse Wishart distributionis used as the prior for the state prediction covariance matrix. The system state and noise parameter posterior distributions are then iteratively updated for adaptive estimation. Furthermore, we make adaptive adjustments to the IEKF filter parameters to enhance sensitivity and filtering accuracy, thus ensuring robust prediction estimation. A two-dimensional target tracking and nonlinear numerical UNGM simulation validated our algorithm. Compared to existing algorithms RKF-ML and GA-VB, our method showed significant improvements in RMSEpos and RMSEvel, with increases of 21.81% and 22.11% respectively, and a 49.04% faster convergence speed. These results highlight the method’s reliability and adaptability.

Reward Learning as a Potential Mechanism for Improvement in Schizophrenia Spectrum Disorders Following Cognitive Remediation: Protocol for a Clinical, Nonrandomized, Pre-Post Pilot Study.

Cognitive impairment is common with schizophrenia spectrum disorders. Cognitive remediation (CR) is effective in improving global cognition, but not all individuals benefit from this type of intervention. A better understanding of the potential mechanism of action of CR is needed. One proposed mechanism is reward learning (RL), the cognitive processes responsible for adapting behavior following positive or negative feedback. It is proposed that the structure of CR enhances RL and motivation to engage in increasingly challenging tasks, and this is a potential mechanism by which CR improves cognitive functioning in schizophrenia. Our primary objective is to examine reward processing in individuals with schizophrenia before and after completing CR and to compare this with a group of matched clinical controls. We will assess whether RL mediates the relationship between CR and improved cognitive function and reduced negative symptoms. Potential differences in social RL and nonsocial RL in individuals with schizophrenia will also be investigated and compared with a healthy matched control group. We propose a clinical, nonrandomized, pre-post pilot study comparing the impact of CR on RL and neurocognitive outcomes. The study will use a combination of objective and subjective measures to assess neurocognitive, psychiatric symptoms, and neurophysiological domains. A total of 40 individuals with schizophrenia spectrum disorders (aged 18-35 years) will receive 12 weeks of CR therapy (n=20) or treatment as usual (n=20). Reward processing will be evaluated using a reinforcement learning task with 2 conditions (social reward vs nonsocial reward) at baseline and the 12-week follow-up. Functional magnetic resonance imaging responses will be measured during this task. To validate the reinforcement learning task, RL will also be assessed in 20 healthy controls, matched for age, sex, and premorbid functioning. Mixed-factorial ANOVAs will be conducted to evaluate treatment group differences. For the functional magnetic resonance imaging analysis, computational modeling will allow the estimation of learning parameters at each point in time, during each task condition, for each participant. We will use a variational Bayesian framework to measure how learning occurred during the experimental task and the subprocesses that underlie this learning. Second-level group analyses will examine how learning in patients differs from that observed in control participants and how CR alters learning efficiency and the underlying neural activity. As of September 2023, this study has enrolled 15 participants in the CR group, 1 participant in the treatment-as-usual group, and 11 participants in the healthy control group. Recruitment is expected to be completed by September 2024. Data analysis is expected to be completed and published in early 2025. The results of this study will contribute to the knowledge of CR and RL processes in severe mental illness and the understanding of the systems that impact negative symptoms and cognitive impairments within this population. DERR1-10.2196/52505.

Distributed Sequential State Estimation Over Binary Sensor Networks with Inaccurate Process Noise Covariance: A Variational Bayesian Framework

Distributed Sequential State Estimation Over Binary Sensor Networks with Inaccurate Process Noise Covariance: A Variational Bayesian Framework

Tuning-free filtering for stochastic systems with unmodeled measurement dynamics

This paper applies variational Bayesian to the stochastic system filtering problem for arbitrary measurement disturbances. Considering the system’s robustness when confronted with disturbances, we model its dynamics with a random walk model. Afterward, by assuming an inverse-Gamma distribution, the newly introduced covariance is estimated under a variational Bayesian (VB) framework. The proposed filter improves robustness against additive disturbance and degrades automatically into a Kalman filter when without disturbance. Moreover, this filter avoids the labor-intensive process of tuning parameters. A numerical simulation and an on-site vehicle navigation experiment are given to illustrate the effectiveness of the proposed filter.

Assessment of brain oxygen extraction fraction between normal cognition and mild cognitive impairment in an elderly cohort using quantitative BOLD

AbstractBackgroundThis study is to quantify oxygen extraction fraction (OEF), an important physiological biomarker for brain hemodynamic impairment, in the elderly cohort using non‐invasive quantitative BOLD (qBOLD) modeling, which is an MRI method for assessment of hemodynamic and metabolic properties.Methods56 participants (75.4±7.0yo, 37F), including 46 cognitively normal (CN) individuals (75.2±7.3yo, 29F) and 10 mild‐cognitive‐impairment (MCI) patients (76.0±6.1yo, 7F), were recruited from UC Davis Alzheimer’s Disease Research Center cohort. MRI scans were acquired on a 3T scanner (Siemens Tim Trio) with a 32‐channel head coil, including an Asymmetric Spin Echo (ASE) scan (TR/TE, 2800/66ms; 2.3×2.3.x1.3mm3 resolution; FOV, 224mm; 13 values of the spin echo displacement time, from 0 to 52ms) and a T1‐weighted MPRAGE. T1‐weighted images were segmented into regions of interest (ROI), including gray matter (GM), white matter (WM), and cortical lobes (frontal, parietal, occipital, temporal) using FSL FAST. ASE signal curves were preprocessed and fit to two‐compartment qBOLD model with Variational Bayesian framework to estimate deoxygenated blood volume (DBV) and transverse relaxation rate (R2 ’), which are used to quantify OEF with known physiological constants (Cherukara et al., NeuroImage 2019). Average OEF values were calculated and compared for each ROI between CN and MCI groups using pairwise t‐tests with Bonferroni correction. The correlation between OEF and age were assessed using a linear mixed effects model that accounted for subject clustering.ResultsCompared to CN, reduced DBV and increased OEF were observed throughout most of the brain on the group average level in MCI (Figure 1). In regional analysis, we observed a trend of increased OEF in MCI compared to CN in several ROIs, which was significant in the occipital lobe (p = 0.007) (Figure 2). In Figure 3, OEF was inversely correlated with age in the parietal lobe for CN (p = 0.026). In contrast, there was a positive trend of association between OEF and age for MCI across all regions.ConclusionThis study showed that age has an association with brain oxygen extraction that are differently affected by normal aging and MCI. Higher brain OEF in MCI individuals may reflect changes in oxygen metabolism because of underlying pathology, including vascular injury and compensation for hypoperfusion.

Assessment of brain oxygen extraction fraction between normal cognition and mild cognitive impairment in an elderly cohort using quantitative BOLD

AbstractBackgroundThis study is to quantify oxygen extraction fraction (OEF), an important physiological biomarker for brain hemodynamic impairment, in the elderly cohort using non‐invasive quantitative BOLD (qBOLD) modeling, which is an MRI method for assessment of hemodynamic and metabolic properties.Method56 participants (75.4±7.0yo, 37F), including 46 cognitively normal (CN) individuals (75.2±7.3yo, 29F) and 10 mild‐cognitive‐impairment (MCI) patients (76.0±6.1yo, 7F), were recruited from UC Davis Alzheimer’s Disease Research Center cohort. MRI scans were acquired on a 3T scanner (Siemens Tim Trio) with a 32‐channel head coil, including an Asymmetric Spin Echo (ASE) scan (TR/TE, 2800/66ms; 2.3×2.3.x1.3mm3 resolution; FOV, 224mm; 13 values of the spin echo displacement time, from 0 to 52ms) and a T1‐weighted MPRAGE. T1‐weighted images were segmented into regions of interest (ROI), including gray matter (GM), white matter (WM), and cortical lobes (frontal, parietal, occipital, temporal) using FSL FAST. ASE signal curves were preprocessed and fit to two‐compartment qBOLD model with Variational Bayesian framework to estimate deoxygenated blood volume (DBV) and transverse relaxation rate (R2’), which are used to quantify OEF with known physiological constants (Cherukara et al., NeuroImage 2019). Average OEF values were calculated and compared for each ROI between CN and MCI groups using pairwise t‐tests with Bonferroni correction. The correlation between OEF and age were assessed using a linear mixed effects model that accounted for subject clustering.ResultCompared to CN, reduced DBV and increased OEF were observed throughout most of the brain on the group average level in MCI (Figure 1). In regional analysis, we observed a trend of increased OEF in MCI compared to CN in several ROIs, which was significant in the occipital lobe (p = 0.007) (Figure 2). In Figure 3, OEF was inversely correlated with age in the parietal lobe for CN (p = 0.026). In contrast, there was a positive trend of association between OEF and age for MCI across all regions.ConclusionThis study showed that age has an association with brain oxygen extraction that are differently affected by normal aging and MCI. Higher brain OEF in MCI individuals may reflect changes in oxygen metabolism because of underlying pathology, including vascular injury and compensation for hypoperfusion.

Exploiting Spike-and-Slab Prior for Variational Estimation of Nonlinear Systems

Identification of nonlinear dynamic systems remains challenging nowadays. Although the nonlinear auto-regressive with exogenous input (NARX) model is flexible to describe complex nonlinear behaviors, it is critical to select appropriate model terms to obtain a parsimonious description of the system. In this study, a variational Bayesian (VB) approach to the estimation of NARX systems is developed. A sparsity inducing prior is introduced for model parameters, and the sparseness can be automatically determined by the weighting factor of such prior. The Bayesian model for the identification problem is constructed, and an iterative model pruning strategy is formulated to remove redundant terms and address the structure selection problem. Instead of the single-point estimation, the model parameters with their uncertainties are jointly estimated under VB framework. Finally, one numerical example and several benchmark datasets are adopted to illustrate that the developed algorithm can work promisingly.

BayeSeg: Bayesian modeling for medical image segmentation with interpretable generalizability.

Due to the cross-domain distribution shift aroused from diverse medical imaging systems, many deep learning segmentation methods fail to perform well on unseen data, which limits their real-world applicability. Recent works have shown the benefits of extracting domain-invariant representations on domain generalization. However, the interpretability of domain-invariant features remains a great challenge. To address this problem, we propose an interpretable Bayesian framework (BayeSeg) through Bayesian modeling of image and label statistics to enhance model generalizability for medical image segmentation. Specifically, we first decompose an image into a spatial-correlated variable and a spatial-variant variable, assigning hierarchical Bayesian priors to explicitly force them to model the domain-stable shape and domain-specific appearance information respectively. Then, we model the segmentation as a locally smooth variable only related to the shape. Finally, we develop a variational Bayesian framework to infer the posterior distributions of these explainable variables. The framework is implemented with neural networks, and thus is referred to as deep Bayesian segmentation. Quantitative and qualitative experimental results on prostate segmentation and cardiac segmentation tasks have shown the effectiveness of our proposed method. Moreover, we investigated the interpretability of BayeSeg by explaining the posteriors and analyzed certain factors that affect the generalization ability through further ablation studies. Our code is released via https://zmiclab.github.io/projects.html.

Robust information filtering for distributed networks using mixed Gaussian distribution

This paper proposes a robust distributed state estimation algorithm for sensor networks. In a distributed sensor network, sensor nodes may be affected by outliers and thus influence the information fusion results. In this respect, the measurement likelihood and one-step state prediction are modeled by a mixture of Gaussian distributions. In this Gaussian mixture, one Gaussian uses nominal covariance, while the other uses adaptive covariance with larger values and lower probabilities. We build a centralized multi-sensor fusion model in the variational Bayesian framework based on the aforementioned mixture of Gaussian distributions. Then, a consensus strategy is employed to convert the centralized method into a distributed one. Each node completes estimation tasks separately based on the information from itself and nearby nodes. In addition, the weight of sensor information will adaptively update according to the Gaussian distribution that mainly affects the current sensor measurement noise. Finally, simulation experiments demonstrate the effectiveness of the proposed method in handling outlier effects.

Variational Bayesian deep network for blind Poisson denoising

Deep learning-based approaches have recently achieved considerable results in Poisson denoising under low-light conditions. However, most existing methods mainly focus on the network architecture design, which lacks physical interpretability and thus unsuitable for blind denoising in real environments with unknown levels of noises. To address this issue, we propose a variational Bayesian deep network for blind Poisson denoising (VBDNet). We mainly consider an approximate posterior form for the noise variance in a variational Bayesian framework and utilize a neural network to parameterize the variance of Poisson noise. For network design, VBDNet is divided into two sub-networks. The noise estimation sub-network is responsible for the Bayesian inference. This network improves the blind denoising ability of the subsequent denoising sub-network by learning Poisson noise characteristics under different noise levels in the training process. A network of U-Net structures implements the denoising sub-network for noise removal. By combining the advantage of Bayesian inference (noise estimation sub-network) and deep learning (denoising sub-network), VBDNet outperforms other state-of-the-art methods on both synthetic and natural data. The code and details are available at https://github.com/HLImg/VBDNet.

Variational Identification of Linearly Parameterized Nonlinear State-Space Systems

A variational Bayesian (VB) approach to the identification of linearly parameterized nonlinear state-space models (LP-NSSMs) is developed in this article. Conjugate priors over the unknown parameters are introduced, and the posterior distributions are approximated under the VB framework. In order to estimate the hidden states of the LP-NSSMs with random parameters, the augmented LP-NSSMs are established such that nonlinear smoothing methods can be applied. An extension to comprehensively tackling data anomalies, such as outliers and missing observations, is also considered to further improve the robustness of the presented method. Finally, a numerical example and the practical benchmarks are adopted to validate the efficacy of the proposed algorithm.

Variational Bayesian Approach to Nonstationary and Oscillatory Slow Feature Analysis With Applications in Soft Sensing and Process Monitoring

Extraction of underlying patterns from measured variables is central to various data-driven control applications, such as soft-sensor modeling, statistical process monitoring, and fault detection and diagnosis. More often than not, the observed variables display nonstationary characteristics and oscillations due to the changes in operating conditions and problems in controller tuning. Such variations pose a great challenge to conventional feature extraction methods. Hence, we present a probabilistic drift-type nonstationary oscillating slow feature model that can separate oscillating patterns and nonstationary variations from measured data. Furthermore, the measurement noise of each variable is independently modeled to account for the fact that not all the observed variables have the same level of uncertainty. Finally, the feature extractor parameters are estimated under a variational Bayesian framework to incorporate the prior information and obtain corresponding posterior distributions. The proposed methodology is applied to solve a fouling monitoring problem for an industrial oil production process.

Modeling and Bayesian inference for processes characterized by abrupt variations

Abrupt variations are often observed in the datasets of chemical processes but they have not been well studied in the literature. This paper proposes a method of modeling and estimating systems characterized by abrupt (impulsive) changes. Abrupt changes may be due to multiple reasons such as disturbances, capacity change, etc. All these cases result in signals that appear to have sudden jumps. But mixed with these jumps are the other dynamic variations characterizing the regular dynamics of the process. For effective modeling, it is important to capture both the jumps and the regular dynamic variations. This paper proposes to model such behavior through a dynamic latent variable (LV) model. The resulting model has two types of LVs characterizing the abrupt and the regular variations. These two behaviors are modeled using a Cauchy and a Gaussian dynamic model respectively. The inference of the LVs and model parameters is done in the variational Bayesian framework.

A Novel Robust IMM Filtering Method for Surface-Maneuvering Target Tracking with Random Measurement Delay

A proper filtering method for jump Markov system (JMS) is an effective approach for tracking a maneuvering target. Since the coexisting of heavy-tailed measurement noises (HTMNs) and one-step random measurement delay (OSRMD) in the complex scenarios of the surface maneuvering target tracking, the effectiveness of typical interacting multiple model (IMM) techniques may decline severely. To solve the state estimation problem in JMSs with HTMN and OSRMD simultaneously, this article designs a novel robust IMM filter utilizing the variational Bayesian (VB) inference framework. This algorithm models the HTMNs as student’s t-distribuitons, and presents a random Bernoulli variable to describe the OSRMD in JMSs. By transforming measurement likelihood function form from weighted summation to exponential product, this paper constructs hierarchical Gaussian state space models. Then, the state vectors, random Bernoulli vairable, and model probability are inferred jointly according to VB inference. The surface maneuvering target tracking simulation example result indicates that the presented IMM filter achieves superior target state estimation accuracy among existing IMM filters.

Forward modeling and inverse estimation for nonlinear filtering

AbstractThe highly accurate state estimation of nonlinear systems is needed in many different application fields. In many existing nonlinear filters, the uncompensated nonlinear approximation error and widely used numerical sampling methods both seriously affect the estimation accuracy and numerical stability. In this paper, based on the variational Bayesian framework, a novel iterative nonlinear filter consisting of forward modeling and inverse estimation is proposed. In the forward modeling stage, a linear Gaussian regression process with distributed variational parameters (VPs) is designed to fit the whole nonlinear measurement likelihood. The distributed VPs consider both first and second moments information, so as to compensate for approximation error from the whole likelihood perspective, instead of only the nonlinear function approximation, which can further improve the accuracy of Bayesian posterior update. In the linear Gaussian regression process, the original nonlinear measurement function (NMF) is just viewed as a prior and no longer directly participates in the modeling of the measurement likelihood. Hence, in the inverse estimation stage, when maximizing the evidence lower bound to update the posteriors of the state and VPs, we can obtain closed‐form solutions without numerical sampling methods to approximate the integral of NMF required in many existing methods, which can guarantee numerical stability. Moreover, to avoid the random setting of hyperparameters and to block the propagation of the accumulated error, we propose two rules, estimation and lower bound consistencies, to conduct the hyperparameters' initialization at the beginning of iteration at each sampling time and also derive an optimization method to update hyperparameters during iteration. Finally, the performance of our proposed method is demonstrated in the nonlinear orbital estimation problems.

Density-adaptive registration of pointclouds based on Dirichlet Process Gaussian Mixture Models

We propose an algorithm for rigid registration of pre- and intra-operative patient anatomy, represented as pointclouds, during minimally invasive surgery. This capability is essential for development of augmented reality systems for guiding such interventions. Key challenges in this context are differences in the point density in the pre- and intra-operative pointclouds, and potentially low spatial overlap between the two. Solutions, correspondingly, must be robust to both of these phenomena. We formulated a pointclouds registration approach which considers the pointclouds after rigid transformation to be observations of a global non-parametric probabilistic model named Dirichlet Process Gaussian Mixture Model. The registration problem is solved by minimizing the Kullback-Leibler divergence in a variational Bayesian inference framework. By this means, all unknown parameters are recursively inferred, including, importantly, the optimal number of mixture model components, which ensures the model complexity efficiently matches that of the observed data. By presenting the pointclouds as KDTrees, both the data and model are expanded in a coarse-to-fine style. The scanning weight of each point is estimated by its neighborhood, imparting the algorithm with robustness to point density variations. Experiments on several datasets with different levels of noise, outliers and pointcloud overlap show that our method has a comparable accuracy, but higher efficiency than existing Gaussian Mixture Model methods, whose performance is sensitive to the number of model components.

A Variational Bayesian Blind Calibration Approach for Air Quality Sensor Deployments

Air pollution has become a global threat to urban environments and public health. Low-cost air quality sensor systems have been deployed to support fine-grained monitoring, and in-field calibration methods are necessary to assure the accuracy of sensor observation. Existing methods use in-field reference to train some specific calibration models. Nevertheless, collecting sufficient reference data after deployment is challenging. First, the synchronized data pair of reference observation is hard to obtain. Thus, many blind calibration approaches rely on some alternative reference like a historical reference before sensor observation. However, since the relationship model between alternative reference and sensor observation becomes more complicated, the amount of collected data is still insufficient to learn the complex model. To address the above challenge, we propose a Variational Bayesian Blind Calibration Algorithm. Our method introduces a supplement set that only involves historical reference and synchronized reference measurement without sensor observation to alleviate the few-reference pressure. Large amounts of the introduced supplement sets have been collected in different cities by accurate stations, which we adopt to conduct prediction tasks and build a Bayesian model to learn how to combine the formulated prediction task and target calibration task in a theoretically better way. Furthermore, we design a variational Bayesian framework to make good use of the easily obtained supplement set to train a better prediction model for improving calibration performance and relieving the pressure of collecting costly reference measurements after deployment. Evaluations of real-world and synthetic datasets show that the proposed approach has a better performance than previous baselines.