Linear Gaussian State Space Model Research Articles

Application of A Linear Gaussian State Space Model and The Kalman Filter in Estimating Expectations of The Natural Logarithmic Monthly Returns of Shiba Cryptocurrency

Application of A Linear Gaussian State Space Model and The Kalman Filter in Estimating Expectations of The Natural Logarithmic Monthly Returns of Shiba Cryptocurrency

AI‐guided patient stratification for neurodegenerative disorders using unsupervised trajectory modelling

AbstractBackgroundPredicting neurodegenerative disorders early has major implications for timely clinical management and patient outcomes. Despite advances in medical technology, we still lack tools to precisely stratify patients for targeted interventions. Here, we propose an unsupervised modelling approach based on mixtures of state space models that learns from longitudinal multimodal (brain imaging, cognitive) data to stratify patients into clusters with different clinical outcomes. Our approach has the potential to support dementia prediction based on cost‐effective, non‐invasive digital measures alone.MethodWe trained (with cross‐validation, 4 components selected with elbow method) a mixture of linear Gaussian state space models on 571 trajectories (2‐4 assessments) from ADNI comprising a) state variables: neuroimaging biomarkers (temporal lobe gray matter density, β‐amyloid) and b) observations: cognitive tests (ADNI‐Mem, ADNI‐EF, MoCA, & ADAS‐13) collected at regular 2‐year intervals. The model clustered individuals based on longitudinal changes in biomarkers and the relationship between biomarkers and cognitive scores (figure 1). We validated model‐derived clusters against clinical outcomes (unseen by the model) based on each individual’s final assessment.ResultOur unsupervised modelling approach effectively stratifies individuals by clinical outcome. Cluster A comprises 65% healthy controls and 33% stable MCI patients (individuals with MCI who do not convert to AD in a 3‐year period) while cluster D comprises 77% Alzheimer’s patients (table 1). Profiling these clusters using MMSE score validates our approach: individuals in cluster A maintain high memory scores over 6 years whereas individuals in cluster D decline (figure 2). Further, we find that our model maintains good stratification when tested a) with a single rather than multiple assessments and b) with cognitive data only (i.e. without biomarkers). These results enhance the clinical utility of our approach, as collecting longitudinal and biomarker data may prove challenging in clinical settings.ConclusionOur modelling approach robustly stratifies patients by clinical outcome when trained on longitudinal multimodal data and tested using cognitive scores alone. Our approach has strong potential to generalize to other non‐invasive and cost‐effective data (e.g. digital markers from wearable technologies), enhancing the translational impact of our AI‐guided tool for early and precise patient stratification in clinical practice.

Component-wise iterative ensemble Kalman inversion for static Bayesian models with unknown measurement error covariance

The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of the Kalman filter for high dimensional linear Gaussian state space models. EnKF methods have also been developed for parameter inference of static Bayesian models with a Gaussian likelihood, in a way that is analogous to likelihood tempering sequential Monte Carlo (SMC). These methods are commonly referred to as ensemble Kalman inversion (EKI). Unlike SMC, the inference from EKI is asymptotically biased if the likelihood is non-linear and/or non-Gaussian and if the priors are non-Gaussian. However, it is significantly faster to run. Currently, a large limitation of EKI methods is that the covariance of the measurement error is assumed to be fully known. We develop a new method, which we call component-wise iterative EKI (CW-IEKI), that allows elements of the covariance matrix to be inferred alongside the model parameters at negligible extra cost. This novel method is compared to SMC on a linear Gaussian example as well as four examples with non-linear dynamics (i.e. non-linear function of the model parameters). The non-linear examples include a set of population models applied to synthetic data, a model of nitrogen mineralisation in soil that is based on the Agricultural Production Systems Simulator, a model predicting seagrass decline due to stress from water temperature and light, and a model predicting coral calcification rates. On our examples, we find that CW-IEKI has relatively similar predictive performance to SMC, albeit with greater uncertainty, and it has a significantly faster run time.

A Comparison Study of Predictive Models for Electricity Demand in a Diverse Urban Environment

The increasing population migration to urban and peri-urban areas increases basic service needs for cities worldwide. Residential electricity demand increases with more customers and varies with novel uses, such as charging electric vehicles, which may add additional dynamics to the residential electricity demand profile. Widespread installation of smart electricity meters provides fine temporal resolution data reflecting current demands and supports predicting future demands. As part of a demand-side management program, understanding the main drivers of current electricity usage based on demand-driven and exogenous predictors represents a valuable tool for utilities facing new demand scenarios. This work presents the application of multiple models to forecast electricity demand based on the input data and the forecasting horizon. Models with exogenous variables as predictors are part of the input–output category, including a Feed Forward Neural Network, Random Forest, and a Linear Gaussian State Space model. The second category is demand-driven models, where predictors include only previous demand values. The demand-driven models in our analysis include a univariate Nonlinear Autoregressive Neural Network and a Linear Gaussian State Space. Using smart electricity meter data from the greater Chicago area, we compare the performance of the models on two different types of accounts: single- and multi-family residential users when forecasting one and multiple steps. Results show that the Linear Gaussian model reports an R2 of 0.99 compared to an average R2 of 0.92 from the Feed Forward Neural Networks and Random Forest when forecasting single- and multi-family electricity demand one step ahead. However, Nonlinear Autoregressive Neural Networks report an average R2 of 0.85 compared to the Linear Gaussian R2 of 0.58 when forecasting 48 steps. We also found that the most important predictors for single-family demand are temporal variables like weekdays, working and non-working days, and day hours. For multi-family demand, electricity demand at the same hour as the previous week replaces weekdays as a significant predictor. Different forecasting models can assist utilities and city planners to manage demand under different and novel residential electricity usage conditions.

Optimal monitoring and attack detection of networks modeled by Bayesian attack graphs

Early attack detection is essential to ensure the security of complex networks, especially those in critical infrastructures. This is particularly crucial in networks with multi-stage attacks, where multiple nodes are connected to external sources, through which attacks could enter and quickly spread to other network elements. Bayesian attack graphs (BAGs) are powerful models for security risk assessment and mitigation in complex networks, which provide the probabilistic model of attackers’ behavior and attack progression in the network. Most attack detection techniques developed for BAGs rely on the assumption that network compromises will be detected through routine monitoring, which is unrealistic given the ever-growing complexity of threats. This paper derives the optimal minimum mean square error (MMSE) attack detection and monitoring policy for the most general form of BAGs. By exploiting the structure of BAGs and their partial and imperfect monitoring capacity, the proposed detection policy achieves the MMSE optimality possible only for linear-Gaussian state space models using Kalman filtering. An adaptive resource monitoring policy is also introduced for monitoring nodes if the expected predictive error exceeds a user-defined value. Exact and efficient matrix-form computations of the proposed policies are provided, and their high performance is demonstrated in terms of the accuracy of attack detection and the most efficient use of available resources using synthetic Bayesian attack graphs with different topologies.

Bayesian multivariate nonlinear state space copula models

A novel flexible class of multivariate nonlinear non-Gaussian state space models, based on copulas, is proposed. Specifically, it is assumed that the observation equation and the state equation are defined by copula families that are not necessarily equal. Inference is performed within the Bayesian framework, using the Hamiltonian Monte Carlo method. Simulation studies show that the proposed copula-based approach is extremely flexible, since it is able to describe a wide range of dependence structures and, at the same time, allows us to deal with missing data. The application to atmospheric pollutant measurement data shows that the approach is suitable for accurate modeling and prediction of data dynamics in the presence of missing values. Comparison to a Gaussian linear state space model and to Bayesian additive regression trees shows the superior performance of the proposed model with respect to predictive accuracy.

Fine-grained Network Traffic Prediction from Coarse Data

ICT systems provide detailed information on computer network traffic. However, due to storage limitations, some of the information on past traffic is often only retained in an aggregated form. In this paper we show that Linear Gaussian State Space Models yield simple yet effective methods to make predictions based on time series at different aggregation levels. The models link coarse-grained and fine-grained time series to a single model that is able to provide fine-grained predictions. Our numerical experiments show up to 3.7 times improvement in expected mean absolute forecast error when forecasts are made using, instead of ignoring, additional coarse-grained observations. The forecasts are obtained in a Bayesian formulation of the model, which allows for provisioning of a traffic prediction service with highly informative priors obtained from coarse-grained historical data.

Process monitoring using recurrent Kalman variational auto-encoder for general complex dynamic processes

Recently, latent models have continued to find advantages in statistical process monitoring, especially with multifarious extensions coping with the nonlinearity and dynamics confronted by the pattern extraction. However, when later performing the famous T2 statistic on the patterns, the issue of non-Gaussianity has rarely been focused alongside, hence vitiating the control of Type-I and -II error in practice. In this paper, a model, in the name of recurrent Kalman variational auto-encoder (RKVAE), is theoretically derived for handling these properties simultaneously, and proposed for its use on monitoring general complex processes. The main idea is to build a monitoring panel on the basis of further reasoning the dynamics underlying the nonlinearly extracted Gaussian features. The model hierarchically consists of three parts: the vanilla VAE for encoding latent Gaussian features, a linear Gaussian state-space model (LGSSM) for temporally reasoning these features, and an RNN for recurrently updating the online LGSSM by fusing several candidate parameters. For the latter K in the name, it is the Kalman filter (KF) that facilitates the feature noise filtering and LGSSM identification. The model has, structurally, a good capacity to be extensible from both the depth and width when modelling complex processes. Finally, based on the explicit formula of LGSSM, a detection panel is designed, and two case studies are given to demonstrate its effectiveness in monitoring and relevant analysis.

Numerical Gaussian Process Kalman Filtering for Spatiotemporal Systems

We present a novel Kalman filter for spatiotemporal systems called the numerical Gaussian process Kalman filter (NGPKF). Numerical Gaussian processes have recently been introduced as a physics informed machine learning method for simulating time-dependent partial differential equations without the need for spatial discretization while also providing uncertainty quantification of the simulation resulting from noisy initial data. We formulate numerical Gaussian processes as linear Gaussian state space models. This allows us to derive the recursive Kalman filter algorithm under the numerical Gaussian process state space model. Using two case studies, we show that the NGPKF is more accurate and robust, given enough measurements, than a spatial discretization based KF.

Time series clustering with an EM algorithm for mixtures of linear Gaussian state space models

In this paper, we consider the task of clustering a set of individual time series while modeling each cluster, that is, model-based time series clustering. The task requires a parametric model with sufficient flexibility to describe the dynamics in various time series. To address this problem, we propose a novel model-based time series clustering method with mixtures of linear Gaussian state space models, which have high flexibility. The proposed method uses a new expectation-maximization algorithm for the mixture model to estimate the model parameters, and determines the number of clusters using the Bayesian information criterion. Experiments on a simulated dataset demonstrate the effectiveness of the method in clustering, parameter estimation, and model selection. The method is applied to real datasets commonly used to evaluate time series clustering methods. Results showed that the proposed method produces clustering results that are as accurate or more accurate than those obtained using previous methods.

Sparse Bayesian Estimation of Parameters in Linear-Gaussian State-Space Models

State-space models (SSMs) are a powerful statistical tool for modelling time-varying systems via a latent state. In these models, the latent state is never directly observed. Instead, a sequence of data points related to the state are obtained. The linear-Gaussian state-space model is widely used, since it allows for exact inference when all model parameters are known, however this is rarely the case. The estimation of these parameters is a very challenging but essential task to perform inference and prediction. In the linear-Gaussian model, the state dynamics are described via a state transition matrix. This model parameter is known to behard to estimate, since it encodes the relationships between the state elements, which are never observed. In many applications, this transition matrix is sparse since not all state components directly affect all other state components. However, most parameter estimation methods do not exploit this feature. In this work we propose SpaRJ, a fully probabilistic Bayesian approach that obtains sparse samples from the posterior distribution of the transition matrix. Our method explores sparsity by traversing a set of models that exhibit differing sparsity patterns in the transition matrix. Moreover, we also design new effective rules to explore transition matrices within the same level of sparsity. This novel methodology has strong theoretical guarantees, and unveils the latent structure of the data generating process, thereby enhancing interpretability. The performance of SpaRJ is showcased in example with dimension 144 in the parameter space, and in a numerical example with real data.

An intrinsically motivated learning algorithm based on Bayesian surprise for cognitive radar in autonomous vehicles

IntroductionThis paper proposes a Bayesian surprise learning algorithm that internally motivates the cognitive radar to estimate a target's state (i.e., velocity, distance) from noisy measurements and make decisions to reduce the estimation error gradually. The work exhibits how the sensor learns from experiences, anticipates future responses, and adjusts its waveform parameters to achieve informative measurements based on the Bayesian surprise.MethodsFor a simple vehicle-following scenario where the radar measurements are generated from linear Gaussian state-space models, the article adopts the Kalman filter to carry out state estimation. According to the information within the filter's estimate, the sensor intrinsically assigns a surprise-based reward value to the immediate past action and updates the value-to-go function. Through a series of hypothetical steps, the cognitive radar considers the impact of future transmissions for a prescribed set of waveforms–available from the sensor profile library–to improve the estimation process.Results and discussionNumerous experiments investigate the performance of the proposed design for various surprise-based reward expressions. The robustness of the proposed method is compared to the state-of-the-art for practical and risky driving situations. Results show that the reward functions inspired by estimation credibility measures outperform their competitors when one-step planning is considered. Simulation results also indicate that multiple-step planning does not necessarily lead to lower error, particularly when the environment changes abruptly.

Fitting joint models of longitudinal observations and time to event by sequential Bayesian updating.

Joint modelling of longitudinal measurements and time to event, with longitudinal and event submodels coupled by latent state variables, has wide application in biostatistics. Standard methods for fitting these models require numerical integration to marginalize over the trajectories of the latent states, which is computationally prohibitive for high-dimensional data and for the large data sets that are generated from electronic health records. This paper describes an alternative model-fitting approach based on sequential Bayesian updating, which allows the likelihood to be factorized as the product of the likelihoods of a state-space model and a Poisson regression model. Updates for linear Gaussian state-space models can be efficiently generated with a Kalman filter and the approach can be implemented with existing software. An application to a publicly available data set is demonstrated.

A Bayesian Surprise Approach in Designing Cognitive Radar for Autonomous Driving.

This article proposes the Bayesian surprise as the main methodology that drives the cognitive radar to estimate a target’s future state (i.e., velocity, distance) from noisy measurements and execute a decision to minimize the estimation error over time. The research aims to demonstrate whether the cognitive radar as an autonomous system can modify its internal model (i.e., waveform parameters) to gain consecutive informative measurements based on the Bayesian surprise. By assuming that the radar measurements are constructed from linear Gaussian state-space models, the paper applies Kalman filtering to perform state estimation for a simple vehicle-following scenario. According to the filter’s estimate, the sensor measures the contribution of prospective waveforms—which are available from the sensor profile library—to state estimation and selects the one that maximizes the expectation of Bayesian surprise. Numerous experiments examine the estimation performance of the proposed cognitive radar for single-target tracking in practical highway and urban driving environments. The robustness of the proposed method is compared to the state-of-the-art for various error measures. Results indicate that the Bayesian surprise outperforms its competitors with respect to the mean square relative error when one-step and multiple-step planning is considered.

Learning Forecast-Efficient Yield Curve Factor Decompositions with Neural Networks

Most factor-based forecasting models for the term structure of interest rates depend on a fixed number of factor loading functions that have to be specified in advance. In this study, we relax this assumption by building a yield curve forecasting model that learns new factor decompositions directly from data for an arbitrary number of factors, combining a Gaussian linear state-space model with a neural network that generates smooth yield curve factor loadings. In order to control the model complexity, we define prior distributions with a shrinkage effect over the model parameters, and we present how to obtain computationally efficient maximum a posteriori numerical estimates using the Kalman filter and automatic differentiation. An evaluation of the model’s performance on 14 years of historical data of the Brazilian yield curve shows that the proposed technique was able to obtain better overall out-of-sample forecasts than traditional approaches, such as the dynamic Nelson and Siegel model and its extensions.

Oscillator decomposition of infant fNIRS data.

The functional near-infrared spectroscopy (fNIRS) can detect hemodynamic responses in the brain and the data consist of bivariate time series of oxygenated hemoglobin (oxy-Hb) and deoxygenated hemoglobin (deoxy-Hb) on each channel. In this study, we investigate oscillatory changes in infant fNIRS signals by using the oscillator decompisition method (OSC-DECOMP), which is a statistical method for extracting oscillators from time series data based on Gaussian linear state space models. OSC-DECOMP provides a natural decomposition of fNIRS data into oscillation components in a data-driven manner and does not require the arbitrary selection of band-pass filters. We analyzed 18-ch fNIRS data (3 minutes) acquired from 21 sleeping 3-month-old infants. Five to seven oscillators were extracted on most channels, and their frequency distribution had three peaks in the vicinity of 0.01-0.1 Hz, 1.6-2.4 Hz and 3.6-4.4 Hz. The first peak was considered to reflect hemodynamic changes in response to the brain activity, and the phase difference between oxy-Hb and deoxy-Hb for the associated oscillators was at approximately 230 degrees. The second peak was attributed to cardiac pulse waves and mirroring noise. Although these oscillators have close frequencies, OSC-DECOMP can separate them through estimating their different projection patterns on oxy-Hb and deoxy-Hb. The third peak was regarded as the harmonic of the second peak. By comparing the Akaike Information Criterion (AIC) of two state space models, we determined that the time series of oxy-Hb and deoxy-Hb on each channel originate from common oscillatory activity. We also utilized the result of OSC-DECOMP to investigate the frequency-specific functional connectivity. Whereas the brain oscillator exhibited functional connectivity, the pulse waves and mirroring noise oscillators showed spatially homogeneous and independent changes. OSC-DECOMP is a promising tool for data-driven extraction of oscillation components from biological time series data.

A new filtering inference procedure for a GED state-space volatility model

A vast amount of methods have been developed to make inferences for volatility data, taking into account the stylized facts of rate/return data. However, the common problem is that the latent parameters of many volatility models are high-dimensional and analytically intractable. Therefore, their inference procedure requires numerical approximations using intensive computational techniques, as the Markov chain Monte Carlo, Laplace, and particle filter methods. A common strategy to overcome this problem is model linearization. This approach consists of writing the stochastic volatility model as a linear Gaussian state-space model, leading to an approximated marginal likelihood using the Kalman filter, reducing the problem’s dimensionality. This paper proposes a new filtering inference procedure with an integrated likelihood for a Generalized Error Distribution (GED) state-space volatility model without model linearization. Also, we evaluate the quality of our method approximation and introduce an approximated smoothing procedure. We use the Bayesian methods for making the inference of static parameters and perform a simulation exercise to study the estimators’ properties. Our results show that the proposed model can be reasonably estimated, and the approximation of our method is reasonable. Furthermore, we provide a case study of the pound/dollar and real/dollar exchange rate to illustrate our approach’s performance. For the real/dollar time series, the model captures a high volatility pattern due to the COVID-19 pandemic.

Graphical Inference in Linear-Gaussian State-Space Models

State-space models (SSM) are central to describe time-varying complex systems in countless signal processing applications such as remote sensing, networks, biomedicine, and finance to name a few. Inference and prediction in SSMs are possible when the model parameters are known, which is rarely the case. The estimation of these parameters is crucial, not only for performing statistical analysis, but also for uncovering the underlying structure of complex phenomena. In this paper, we focus on the linear-Gaussian model, arguably the most celebrated SSM, and particularly in the challenging task of estimating the transition matrix that encodes the Markovian dependencies in the evolution of the multi-variate state. We introduce a novel perspective by relating this matrix to the adjacency matrix of a directed graph, also interpreted as the causal relationship among state dimensions in the Granger-causality sense. Under this perspective, we propose a new method called GraphEM based on the well sounded expectation-maximization (EM) methodology for inferring the transition matrix jointly with the smoothing/filtering of the observed data. We propose an advanced convex optimization solver relying on a consensus-based implementation of a proximal splitting strategy for solving the M-step. This approach enables an efficient and versatile processing of various sophisticated priors on the graph structure, such as parsimony constraints, while benefiting from convergence guarantees. We demonstrate the good performance and the interpretable results of GraphEM by means of two sets of numerical examples.

KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics

State estimation of dynamical systems in real-time is a fundamental task in signal processing. For systems that are well-represented by a fully known linear Gaussian state space (SS) model, the celebrated Kalman filter (KF) is a low complexity optimal solution. However, both linearity of the underlying SS model and accurate knowledge of it are often not encountered in practice. Here, we present KalmanNet, a real-time state estimator that learns from data to carry out Kalman filtering under non-linear dynamics with partial information. By incorporating the structural SS model with a dedicated recurrent neural network module in the flow of the KF, we retain data efficiency and interpretability of the classic algorithm while implicitly learning complex dynamics from data. We demonstrate numerically that KalmanNet overcomes non-linearities and model mismatch, outperforming classic filtering methods operating with both mismatched and accurate domain knowledge.

Gaussian-Beta Filters With Unknown Probability of Measurement Loss

Data loss is ubiquitous in practical engineering applications due to communication delay or congestion. Data loss rate is a key metric to evaluate the reliability of state estimation. To jointly estimate system state and data loss rate, we propose a class of Gaussian-Beta filters for linear and moderate nonlinear Gaussian state-space models with unknown probability of measurement loss. In the filters, the arrival of the measurement at each time is formulated as a binary random variable, which is determined by the classical threshold technology. In addition, the hidden state and the unknown probability of measurement loss are modeled as a product of Gaussian and Beta distributions, and the form remains unchanged through recursive operations. Simulation results verify the effectiveness of the proposed Gaussian-Beta filters compared with the existing filtering algorithms.