Switching Linear Dynamical System Research Articles

Approximate sequential Bayesian filtering to estimate 222Rn emanation from 226Ra sources using spectral time series

Abstract. A new approach to assess the emanation of 222Rn from 226Ra sources based on γ-ray spectrometric measurements is presented. While previous methods have resorted to steady-state treatment of the system, the method presented incorporates well-known radioactive decay kinetics into the inference procedure through the formulation of a theoretically motivated system model. The validity of the 222Rn emanation estimate is thereby extended to regimes of changing source behavior, potentially enabling the development of source surveillance systems in the future. The inference algorithms are based on approximate recursive Bayesian estimation in a switching linear dynamical system, allowing regimes of changing emanation to be identified from the spectral time series while providing reasonable filtering and smoothing performance in steady-state regimes. The derived method is applied to an empirical γ-ray spectrometric time series obtained over 85 d and is able to provide a time series of emanation estimates consistent with the physics of the emanation process.

A Bayesian switching linear dynamical system for estimating seizure chronotypes

Epilepsy is a disorder characterized by paroxysmal transitions between multistable states. Dynamical systems have been useful for modeling the paroxysmal nature of seizures. At the same time, intracranial electroencephalography (EEG) recordings have recently discovered that an electrographic measure of epileptogenicity, interictal epileptiform activity, exhibits cycling patterns ranging from ultradian to multidien rhythmicity, with seizures phase-locked to specific phases of these latent cycles. However, many mechanistic questions about seizure cycles remain unanswered. Here, we provide a principled approach to recast the modeling of seizure chronotypes within a statistical dynamical systems framework by developing a Bayesian switching linear dynamical system (SLDS) with variable selection to estimate latent seizure cycles. We propose a Markov chain Monte Carlo algorithm that employs particle Gibbs with ancestral sampling to estimate latent cycles in epilepsy and apply unsupervised learning on spectral features of latent cycles to uncover clusters in cycling tendency. We analyze the largest database of patient-reported seizures in the world to comprehensively characterize multidien cycling patterns among 1,012 people with epilepsy, spanning from infancy to older adulthood. Our work advances knowledge of cycling in epilepsy by investigating how multidien seizure cycles vary in people with epilepsy, while demonstrating an application of an SLDS to frame seizure cycling within a nonlinear dynamical systems framework. It also lays the groundwork for future studies to pursue data-driven hypothesis generation regarding the mechanistic drivers of seizure cycles.

Online optimization of switched LTI systems using continuous-time and hybrid accelerated gradient flows

This paper studies the design of feedback controllers to steer a switching linear dynamical system to the solution trajectory of a time-varying convex optimization problem. We propose two types of controllers: (i) a continuous controller inspired by the online gradient descent method, and (ii) a hybrid controller that can be interpreted as an online version of Nesterov’s accelerated gradient method with restarts of the state variables. By design, the controllers continuously steer the system toward a time-varying optimal equilibrium point without requiring knowledge of exogenous disturbances affecting the system. For cost functions that are smooth and satisfy the Polyak–Łojasiewicz inequality, we demonstrate that the online gradient-flow controller ensures uniform global exponential stability when the time scales of the system and controller are sufficiently separated and the switching signal of the system varies slowly on average. For cost functions that are strongly convex, we show that the hybrid accelerated controller can outperform the continuous gradient descent method. When the cost function is not strongly convex, we show that the hybrid accelerated method guarantees global practical asymptotic stability.

Increasing State Estimation Accuracy in the Inference Algorithm on a Hybrid Factor Graph Model

We consider an intelligent agent that receives a continuous input from a signal-generating system and aims for estimating the discrete latent state of that system. The agent includes a switching linear dynamical system modeled as a hybrid factor graph for performing state estimation by determining the inference algorithm. We investigate the agent’s performance in relation to two Gaussian mixture reduction methods restricting Gaussian mixture growing while message passing, namely, the naive pruning implemented in the past and the realization of the Kullback-Leibler (KL) discrimination based approach defined by Runnalls (2007). For evaluation, we use simulated data provided with various signal-to-noise ratios. Reviewing the metrics, the agent reaches an improvement in state estimation accuracy by using the KL discrimination based method with minimally increasing computational effort. The findings of our work let us take one step towards establishing intelligent systems for modeling real-world systems with switching behavior capable to analyze noisy data sets.

Efficient Model Selection in Switching Linear Dynamic Systems by Graph Clustering

The computation required for a switching Kalman Filter (SKF) increases exponentially with the number of system operation modes. In this paper, a computationally tractable graph representation is proposed for a switching linear dynamic system (SLDS) along with the solution of a minimum-sum optimization problem for clustering to reduce the switching mode cardinality offline, before collecting measurements. It is shown that upon perfect mode detection, the induced error caused by mode clustering can be quantified exactly in terms of the dissimilarity measures in the proposed graph structure. Numerical results verify that clustering based on the proposed framework effectively reduces model complexity given uncertain mode detection and that the induced error can be well approximated if the underlying assumptions are satisfied.

Variable Selection in Switching Dynamic Regression Models

Complex dynamic phenomena in which dynamics is related to events (modes) that cause structural changes over time, are well described by the switching linear dynamical system (SLDS). We extend the SLDS by allowing the measurement noise to be mode-specific, a flexible way to model non stationary data. Additionally, for models that are functions of explanatory variables, we adapt a variable selection method to identify which of them are significant in each mode. Our proposed model is a flexible Bayesian nonparametric model that allows to learn about the number of modes and their location, and within each mode, it identifies the significant variables and estimates the regression coefficients. The model performance is evaluated by simulation and two application examples from a dataset of meteorological time series of Barranquilla, Colombia are presented.

Quantification of Mismatch Error in Randomly Switching Linear State-Space Models

Switching Kalman Filters (SKF) are well known for solving switching linear dynamic system (SLDS), i.e., piece-wise linear estimation problems. Practical SKFs are heuristic, approximate filters and require more computational resources than a single-mode Kalman filter (KF). On the other hand, applying a single-mode mismatched KF to an SLDS results in erroneous estimation. This paper quantifies the average error an SKF can eliminate compared to a mismatched, single-mode KF before collecting measurements. Derivations of the first and second moments of the estimators errors are provided and compared. One can use these derivations to quantify the average performance of filters beforehand and decide which filter to run in operation to have the best performance in terms of estimation error and computation complexity. We further provide simulation results that verify our mathematical derivations.

Context-Based Path Prediction for Targets with Switching Dynamics

Anticipating future situations from streaming sensor data is a key perception challenge for mobile robotics and automated vehicles. We address the problem of predicting the path of objects with multiple dynamic modes. The dynamics of such targets can be described by a Switching Linear Dynamical System (SLDS). However, predictions from this probabilistic model cannot anticipate when a change in dynamic mode will occur. We propose to extract various types of cues with computer vision to provide context on the target’s behavior, and incorporate these in a Dynamic Bayesian Network (DBN). The DBN extends the SLDS by conditioning the mode transition probabilities on additional context states. We describe efficient online inference in this DBN for probabilistic path prediction, accounting for uncertainty in both measurements and target behavior. Our approach is illustrated on two scenarios in the Intelligent Vehicles domain concerning pedestrians and cyclists, so-called Vulnerable Road Users (VRUs). Here, context cues include the static environment of the VRU, its dynamic environment, and its observed actions. Experiments using stereo vision data from a moving vehicle demonstrate that the proposed approach results in more accurate path prediction than SLDS at the relevant short time horizon (1 s). It slightly outperforms a computationally more demanding state-of-the-art method.

Naïve Bayes switching linear dynamical system: A model for dynamic system modelling, classification, and information fusion

The Naïve Bayes Switching Linear Dynamical System (NB-SLDS) is proposed as a novel variant of the switching linear dynamical system (SLDS). The variant models multi-variable systems that undergo regime changes in their dynamics. The model may be applied to identify regime changes or classify systems according to their dynamics. The NB-SLDS provides the means to fuse multiple sequential data sources into a single model. A key feature of the model is that it is able to handle missing and unsynchronised data. Filtering and smoothing algorithms for inference and an expectation maximisation algorithm for parameter learning in the NB-SLDS are presented. The model is demonstrated and compared to the SLDS and hidden Markov model (HMM) in a human action recognition problem.

Switching LDS detection for GNSS‐based train integrity monitoring system

Train integrity whilst in service establishes the foundation for railway safety. This study investigates train integrity detection which reliably deduces whether the train consists remain intact. A switching linear dynamic system (SLDS) based train integrity detection method is proposed for Global Navigation Satellite System (GNSS) based train integrity Monitoring System (TIMS) using the relative distance, velocity and acceleration of the locomotive and the last van. There, Expectation Maximisation (EM) algorithm estimates the parameters of SLDS model while the Gaussian Sum Filter infers train integrity state. After that, to cope with false detection and misdetection, a verification procedure and train parting time estimation are designed. The approach is evaluated with both field trials and simulated data. Results show that the false alarm rate and misdetection rate of SLDS-based integrity detection approach are 0 and 0.09% respectively, which proves better than the estimated train length based detection model and Hidden Markov Model (HMM).

Systemic banking crisis early warning systems using dynamic Bayesian networks

For decades, the literature on banking crisis early-warning systems has been dominated by two methods, namely, the signal extraction and the logit model methods. However, these methods, do not model the dynamics of the systemic banking system. In this study, dynamic Bayesian networks are applied as systemic banking crisis early-warning systems. In particular, the hidden Markov model, the switching linear dynamic system and the naïve Bayes switching linear dynamic system models are considered. These dynamic Bayesian networks provide the means to model system dynamics using the Markovian framework. Given the dynamics, the probability of an impending crisis can be calculated. A unique approach to measuring the ability of a model to predict a crisis is utilised. The results indicate that the dynamic Bayesian network models can provide precise early-warnings compared with the signal extraction and the logit methods.

Understanding City Traffic Dynamics Utilizing Sensor and Textual Observations

Understanding speed and travel-time dynamics in response to various city related events is an important and challenging problem. Sensor data (numerical) containing average speed of vehicles passing through a road link can be interpreted in terms of traffic related incident reports from city authorities and social media data (textual), providing a complementary understanding of traffic dynamics. State-of-the-art research is focused on either analyzing sensor observations or citizen observations; we seek to exploit both in a synergistic manner. We demonstrate the role of domain knowledge in capturing the non-linearity of speed and travel-time dynamics by segmenting speed and travel-time observations into simpler components amenable to description using linear models such as Linear Dynamical System (LDS). Specifically, we propose Restricted Switching Linear Dynamical System (RSLDS) to model normal speed and travel time dynamics and thereby characterize anomalous dynamics. We utilize the city traffic events extracted from text to explain anomalous dynamics. We present a large scale evaluation of the proposed approach on a real-world traffic and twitter dataset collected over a year with promising results.

Mixture of Switching Linear Dynamics to Discover Behavior Patterns in Object Tracks.

We present a novel non-parametric Bayesian model to jointly discover the dynamics of low-level actions and high-level behaviors of tracked objects. In our approach, actions capture both linear, low-level object dynamics, and an additional spatial distribution on where the dynamic occurs. Furthermore, behavior classes capture high-level temporal motion dependencies in Markov chains of actions, thus each learned behavior is a switching linear dynamical system. The number of actions and behaviors is discovered from the data itself using Dirichlet Processes. We are especially interested in cases where tracks can exhibit large kinematic and spatial variations, e.g. person tracks in open environments, as found in the visual surveillance and intelligent vehicle domains. The model handles real-valued features directly, so no information is lost by quantizing measurements into 'visual words', and variations in standing, walking and running can be discovered without discrete thresholds. We describe inference using Markov Chain Monte Carlo sampling and validate our approach on several artificial and real-world pedestrian track datasets from the surveillance and intelligent vehicle domain. We show that our model can distinguish between relevant behavior patterns that an existing state-of-the-art hierarchical model for clustering and simpler model variants cannot. The software and the artificial and surveillance datasets are made publicly available for benchmarking purposes.

Switching LDS-based approach for process fault detection and classification

Dynamic and uncertainty are two main features of the industrial processes data which should be paid attentions when carrying out process monitoring and fault diagnosis. As a typical dynamic Bayesian network model, linear dynamic system (LDS) can efficiently deal with both dynamic and uncertain features of the process data. This paper proposes a switching form of the LDS model for fault detection and classification. A novel and convenient learning algorithm is developed for parameter estimation of the switching LDS model, and the Gaussian Sum Filtering method is introduced for online fault classification. Besides, a switching LDS based threshold statistic is defined for unknown fault detection. Detailed comparative studies are carried out on the Tennessee Eastman (TE) benchmark process among Fisher Discriminant Analysis (FDA), Support Vector Machines (SVM), Hidden Markov Model (HMM), and the proposed method. Simulation results show the superiority of switching LDS over other three methods in terms of fault classification. Furthermore, switching LDS model provides an additional online fault classification mechanism. Simulation results in the same process demonstrate that switching LDS approach can achieve accurate detection rate of the unknown fault with minimal false alarms.

Maritime piracy situation modelling with dynamic Bayesian networks

A generative model for modelling maritime vessel behaviour is proposed. The model is a novel variant of the dynamic Bayesian network (DBN). The proposed DBN is in the form of a switching linear dynamic system (SLDS) that has been extended into a larger DBN. The application of synthetic data fabrication of maritime vessel behaviour is considered. Behaviour of various vessels in a maritime piracy situation is simulated. A means to integrate information from context based external factors that influence behaviour is provided. Simulated observations of the vessels kinematic states are generated. The generated data may be used for the purpose of developing and evaluating counter-piracy methods and algorithms. A novel methodology for evaluating and optimising behavioural models such as the proposed model is presented. The log-likelihood, cross entropy, Bayes factor and the Bhattacharyya distance measures are applied for evaluation. The results demonstrate that the generative model is able to model both spatial and temporal datasets.

Augmented switching linear dynamical system model for gas concentration estimation with MOX sensors in an open sampling system.

In this paper, we introduce a Bayesian time series model approach for gas concentration estimation using Metal Oxide (MOX) sensors in Open Sampling System (OSS). Our approach focuses on the compensation of the slow response of MOX sensors, while concurrently solving the problem of estimating the gas concentration in OSS. The proposed Augmented Switching Linear System model allows to include all the sources of uncertainty arising at each step of the problem in a single coherent probabilistic formulation. In particular, the problem of detecting on-line the current sensor dynamical regime and estimating the underlying gas concentration under environmental disturbances and noisy measurements is formulated and solved as a statistical inference problem. Our model improves, with respect to the state of the art, where system modeling approaches have been already introduced, but only provided an indirect relative measures proportional to the gas concentration and the problem of modeling uncertainty was ignored. Our approach is validated experimentally and the performances in terms of speed of and quality of the gas concentration estimation are compared with the ones obtained using a photo-ionization detector.

Identifying effective connectivity parameters in simulated fMRI: a direct comparison of switching linear dynamic system, stochastic dynamic causal, and multivariate autoregressive models.

The number and variety of connectivity estimation methods is likely to continue to grow over the coming decade. Comparisons between methods are necessary to prune this growth to only the most accurate and robust methods. However, the nature of connectivity is elusive with different methods potentially attempting to identify different aspects of connectivity. Commonalities of connectivity definitions across methods upon which base direct comparisons can be difficult to derive. Here, we explicitly define “effective connectivity” using a common set of observation and state equations that are appropriate for three connectivity methods: dynamic causal modeling (DCM), multivariate autoregressive modeling (MAR), and switching linear dynamic systems for fMRI (sLDSf). In addition while deriving this set, we show how many other popular functional and effective connectivity methods are actually simplifications of these equations. We discuss implications of these connections for the practice of using one method to simulate data for another method. After mathematically connecting the three effective connectivity methods, simulated fMRI data with varying numbers of regions and task conditions is generated from the common equation. This simulated data explicitly contains the type of the connectivity that the three models were intended to identify. Each method is applied to the simulated data sets and the accuracy of parameter identification is analyzed. All methods perform above chance levels at identifying correct connectivity parameters. The sLDSf method was superior in parameter estimation accuracy to both DCM and MAR for all types of comparisons.

Bayesian Nonparametric Inference of Switching Dynamic Linear Models

Many complex dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes. We consider two such models: the switching linear dynamical system (SLDS) and the switching vector autoregressive (VAR) process. Our Bayesian nonparametric approach utilizes a hierarchical Dirichlet process prior to learn an unknown number of persistent, smooth dynamical modes. We additionally employ automatic relevance determination to infer a sparse set of dynamic dependencies allowing us to learn SLDS with varying state dimension or switching VAR processes with varying autoregressive order. We develop a sampling algorithm that combines a truncated approximation to the Dirichlet process with efficient joint sampling of the mode and state sequences. The utility and flexibility of our model are demonstrated on synthetic data, sequences of dancing honey bees, the IBOVESPA stock index, and a maneuvering target tracking application.

Speech Feature Mapping based on Switching Linear Dynamic System

Signals originated from the same speech source usually appear differently depending on a variety of acoustic effects such as the background noises, linear or nonlinear distortions incurred by the recording devices or reverberations. These acoustical effects result in mismatches between the trained speech recognition models and the input speech. One of the well-known approaches to reduce this mismatch is to map the distorted speech feature to its clean counterpart. The mapping function is usually trained based on a set of stereo data which consists of the simultaneous recordings obtained in both the reference and target conditions. In this paper, we propose the switching linear dynamic system (SLDS) as a useful model for speech feature sequence mapping. In contrast to the conventional vector-to-vector mapping algorithms, SLDS can describe sequence-to-sequence mapping in a systematic way. The proposed approach is applied to robust speech recognition in various environmental conditions and shows a dramatic improvement in recognition performance.

Identification and validation of effective connectivity networks in functional magnetic resonance imaging using switching linear dynamic systems

Dynamic connectivity networks identify directed interregional interactions between modeled brain regions in neuroimaging. However, problems arise when the regions involved in a task and their interconnections are not fully known a priori. Objective measures of model adequacy are necessary to validate such models. We present a connectivity formalism, the Switching Linear Dynamic System (SLDS), that is capable of identifying both Granger-Geweke and instantaneous connectivity that vary according to experimental conditions. SLDS explicitly models the task condition as a Markov random variable. The series of task conditions can be estimated from new data given an identified model providing a means to validate connectivity patterns. We use SLDS to model functional magnetic resonance imaging data from five regions during a finger alternation task. Using interregional connectivity alone, the identified model predicted the task condition vector from a different subject with a different task ordering with high accuracy. In addition, important regions excluded from a model can be identified by augmenting the model state space. A motor task model excluding primary motor cortices was augmented with a new neural state constrained by its connectivity with the included regions. The augmented variable time series, convolved with a hemodynamic kernel, was compared to all brain voxels. The right primary motor cortex was identified as the best region to add to the model. Our results suggest that the SLDS model framework is an effective means to address several problems with modeling connectivity including measuring overall model adequacy and identifying important regions missing from models.