Bayesian Dynamic Linear Models Research Articles

Damage detection for structural health monitoring using reinforcement and imitation learning

Structural damages are responsible for expenses associated with maintaining the safety and serviceability of infrastructures. Detecting damages is difficult because they often develop over years affecting structural responses in orders of magnitudes smaller than external effects, such as temperature. When damage occurs, structural responses depart from a normal condition to an abnormal one, which is referred to as an anomaly. Existing anomaly detection methodologies lack a mechanism to quantify the probability of rightfully detecting anomalies as a function of the anomaly’s characteristics, e.g. duration and magnitude, and associate them with the severity of structural damages. This paper proposes a framework addressing these challenges by relying on Bayesian dynamic linear models as well as reinforcement and imitation learning approaches. The former allows separating the changes in the structural responses from the ones caused by external effects, while the latter two enable incorporating information obtained from the changes in the structural responses for detecting anomalies. The proposed methodologies are validated using measurements collected on three instrumented bridge spans in Canada. The results show a good performance of the methods proposed in detecting structural damages with different severity levels and lay the foundation for further applications for other civil infrastructures.

On‐Line Warning System for Pipe Burst Using Bayesian Dynamic Linear Models

AbstractPipe breaks are a recurrent problem in water distribution networks and detecting them quickly is crucial to minimize the economic and environmental costs for municipalities. This study presents a burst detection methodology applying Bayesian dynamic linear models (DLMs) on water flow time series combined with an outlier monitoring tool. The model is used to characterize the actual flow and, for each time, a one‐step ahead forecast distribution is obtained recursively before moving onto the next observation. The outlier detection method consists of comparing the main model with an alternative one wherein the mean flow is shifted to a higher value (as bursts tend to increase flow) to evaluate which model best fit the observed data. If the alternative model is favored, a burst alarm is issued. To verify the performance of this approach, the DLM and monitoring tool were applied on 2 yr of flow data from two district meter areas (DMAs) in Halifax (Canada), and a historical break data set is used to assess model accuracy. The model was able to detect up to 75% and 71.2% of the pipe breaks, with a false alarm rate of 5.15% and 12% in the first and second DMA, respectively. Finally, the proposed model allows for straightforward interpretation of model parameters, nonlinear relationship between flow and predictors of interest, naturally describes the uncertainty for future predictions, can easily accommodate missing values and can be tuned to maximize break detection or minimize false alarm rates to adapt to specific objectives of water infrastructure managers.

Time-varying effects of local weather on behavior and probability of breeding deferral in two Arctic-nesting goose populations

Arctic-nesting geese face energetic challenges during spring migration, including ecological barriers and weather conditions (e.g., precipitation and temperature), which in long-lived species can lead to a trade-off to defer reproduction in favor of greater survival. We used GPS location and acceleration data collected from 35 greater white-fronted geese of the North American midcontinent and Greenland populations at spring migration stopovers, and novel applications of Bayesian dynamic linear models to test daily effects of minimum temperature and precipitation on energy expenditure (i.e., overall dynamic body acceleration, ODBA) and proportion of time spent feeding (PTF), then examined the daily and additive importance of ODBA and PTF on probability of breeding deferral using stochastic antecedent models. We expected distinct responses in behavior and probability of breeding deferral between and within populations due to differences in stopover area availability. Time-varying coefficients of weather conditions were variable between ODBA and PTF, and often did not show consistent patterns among birds, indicating plasticity in how individuals respond to conditions. An increase in antecedent ODBA was associated with a slightly increased probability of deferral in midcontinent geese but not Greenland geese. Probability of deferral decreased with increased PTF in both populations. We did not detect any differentially important time periods. These results suggest either that movements and behavior throughout spring migration do not explain breeding deferral or that ecological linkages between bird decisions during spring and subsequent breeding deferral were different between populations and across migration but occurred at different time scales than those we examined.

Bayesian Testing of Granger Causality in Functional Time Series

We develop a multivariate functional autoregressive model (MFAR), which captures the cross-correlation among multiple functional time series and thus improves forecast accuracy. We estimate the parameters under the Bayesian dynamic linear models (DLM) framework. In order to test for Granger causality from one FAR series to another we employ Bayes Factor. Motivated by the broad application of functional data in finance, we investigate the causality between the yield curves of USA and UK. Bayes factor values shows that no causal relation exists among the interest rates of these two countries. Furthermore, we illustrate a climatology example, suggesting that the meteorological factors Granger cause pollutant daily levels in Delhi. The Github repository https://www.Bayesian-Testing-Of-Granger-Causality-In-Functional-Time-Series contains the detailed study of simulation and real data applications.

The Gaussian multiplicative approximation for state‐space models

Applications such as structural health monitoring (SHM) often rely on the analysis of time-series using methods such as state-space models (SSM). In this paper, we propose an analytical method called the Gaussian multiplicative approximation (GMA) that is applicable to multiplicative state-space models that are often encountered in practical SHM applications. The method enables the analytical inference of the mean vector and the covariance matrix for the product of two hidden states in the transition and/or observation models using linear estimation theory and the online estimation of model parameters as hidden states. The potential of combining the GMA and Bayesian dynamic linear models (BDLM) is illustrated through the development of (1) a generic component called online autoregressive that can estimate both the state variable and the parameter together; (2) a generic component called trend multiplicative for multiplicative seasonality model to identify non-harmonic periodic pattern whose amplitude changes linearly with time; and (3) a generic component called double kernel regression to identify non-harmonic periodic pattern that involves the product of two periodic kernel regression components. The SHM-based case studies presented confirm that the GMA exceeds the performance of the existing nonlinear Kalman filter methods in terms of accuracy along with the computational cost.

Forecasting daily new infections, deaths and recovery cases due to COVID-19 in Pakistan by using Bayesian Dynamic Linear Models.

The COVID-19 has caused the deadliest pandemic around the globe, emerged from the city of Wuhan, China by the end of 2019 and affected all continents of the world, with severe health implications and as well as financial-damage. Pakistan is also amongst the top badly effected countries in terms of casualties and financial loss due to COVID-19. By 20th March, 2021, Pakistan reported 623,135 total confirmed cases and 13,799 deaths. A state space model called 'Bayesian Dynamic Linear Model' (BDLM) was used for the forecast of daily new infections, deaths and recover cases regarding COVID-19. For the estimation of states of the models and forecasting new observations, the recursive Kalman filter was used. Twenty days ahead forecast show that the maximum number of new infections are 4,031 per day with 95% prediction interval (3,319-4,743). Death forecast shows that the maximum number of the deaths with 95% prediction interval are 81 and (67-93), respectively. Maximum daily recoveries are 3,464 with 95% prediction interval (2,887-5,423) in the next 20 days. The average number of new infections, deaths and recover cases are 3,282, 52 and 1,840, respectively, in the upcoming 20 days. As the data generation processes based on the latest data has been identified, therefore it can be updated with the availability of new data to provide latest forecast.

Dynamic Reliability Prediction of Bridges Based on Decoupled SHM Extreme Stress Data and Improved BDLM

Bridge health monitoring system has produced a huge amount of monitored data (extreme stress data, etc.) in the long-term service periods; how to reasonably predict structural dynamic reliability with these data is one key problem in structural health monitoring (SHM) field. In this paper, considering the coupling, randomness, and time variation of SHM data, firstly, the coupled extreme stress data, which are considered as a time series, are decoupled into high-frequency and low-frequency data with the moving average method. Secondly, Bayesian dynamic linear models (BDLM) without priori monitoring error data (e.g., unknown monitored error variance) are built to dynamically predict the decoupled extreme stress; furthermore, the dynamic reliability of bridge members is predicted with the built BDLM and first-order second moment (FOSM) reliability method. Finally, an actual example is provided to illustrate the feasibility and application of the proposed models and methods. The research results of this paper will provide the theoretical foundations for structural reliability prediction.

BDVCM about the Steel Box Girder Reliability Prediction Based on Monitoring Extreme Stresses

To reasonably predict the steel box girder reliability considering the dynamic dependence among the performance functions corresponding to the failure modes of the multiple monitoring points, this paper firstly adopts the dynamic monitoring extreme stresses of the multiple control points to build the Bayesian Dynamic Vine Copula Model (BDVCM) taking into account the dynamic dependence of the multiple monitoring variables through combining the vine copula technique with Bayesian Dynamic Linear Models (BDLM); secondly, with first-order second-moment method and the built BDVCM, the steel box girder reliability, taking into account dynamic dependence among the performance functions corresponding to the failure modes of the multiple monitoring points, is predicted; finally, the monitoring data from the five sections of an existing steel box girder were provided to illustrate the proposed model and approach. The analytical results illustrated that the predicted results, without considering the dynamic nonlinear dependence among the failure modes of the multiple monitoring points, are conservative.

Comparison of Kullback-Leibler, Hellinger and LINEX with Quadratic Loss Function in Bayesian Dynamic Linear Models: Forecasting of Real Price of Oil

In this paper we intend to examine the application of Kullback-Leibler, Hellinger and LINEX loss function in Dynamic Linear Model using the real price of oil for 106 years of data from 1913 to 2018 concerning the asymmetric problem in filtering and forecasting. We use DLM form of the basic Hoteling Model under Quadratic loss function, Kullback-Leibler, Hellinger and LINEX trying to address the results if we treat the ‘over-estimation’ and ‘under-estimation’ differently. So, we drive one-step-ahead forecast for Dynamic Linear Model under quadratic, LINEX and Kullback-Leibler losses in Bayesian context. With Normal posterior distribution, our results suggest that, the LINEX loss function may provide better forecasts than conventional Quadratic loss function, Hellinger and Kullback-Leibler loss function, especially in case of having volatility and time-varying parameters. JEL classification: C11, C22, C53, C61, Q47

Optimal asset allocation with multivariate Bayesian dynamic linear models

We introduce a fast, closed-form, simulation-free method to model and forecast multiple asset returns and employ it to investigate the optimal ensemble of features to include when jointly predicting monthly stock and bond excess returns. Our approach builds on the Bayesian dynamic linear models of West and Harrison (Bayesian Forecasting and Dynamic Models (1997) Springer), and it can objectively determine, through a fully automated procedure, both the optimal set of regressors to include in the predictive system and the degree to which the model coefficients, volatilities and covariances should vary over time. When applied to a portfolio of five stock and bond returns, we find that our method leads to large forecast gains, both in statistical and economic terms. In particular, we find that relative to a standard no-predictability benchmark, the optimal combination of predictors, stochastic volatility and time-varying covariances increases the annualized certainty equivalent returns of a leverage-constrained power utility investor by more than 500 basis points.

Measuring effects of medication adherence on time-varying health outcomes using Bayesian dynamic linear models.

One of the most significant barriers to medication treatment is patients' non-adherence to a prescribed medication regimen. The extent of the impact of poor adherence on resulting health measures is often unknown, and typical analyses ignore the time-varying nature of adherence. This article develops a modeling framework for longitudinally recorded health measures modeled as a function of time-varying medication adherence. Our framework, which relies on normal Bayesian dynamic linear models (DLMs), accounts for time-varying covariates such as adherence and non-dynamic covariates such as baseline health characteristics. Standard inferential procedures for DLMs are inefficient when faced with infrequent and irregularly recorded response data. We develop an approach that relies on factoring the posterior density into a product of two terms: a marginal posterior density for the non-dynamic parameters, and a multivariate normal posterior density of the dynamic parameters conditional on the non-dynamic ones. This factorization leads to a two-stage process for inference in which the non-dynamic parameters can be inferred separately from the time-varying parameters. We demonstrate the application of this model to the time-varying effect of antihypertensive medication on blood pressure levels for a cohort of patients diagnosed with hypertension. Our model results are compared to ones in which adherence is incorporated through non-dynamic summaries.

Real‐time anomaly detection with Bayesian dynamic linear models

A key goal in structural health monitoring is to detect abnormal events in a structure's behavior by interpreting its observed responses over time. The goal is to develop an anomaly detection method that (a) is robust towards false alarm and (b) capable of performing real-time analysis. The majority of anomaly detection approaches are currently operating over batches of data for which the model parameters are assumed to be constant over time and to be equal to the values estimated during a fixed-size training period. This assumption is not suited for the real-time anomaly detection where model parameters need to be treated as time-varying quantities. This paper presents how this issue is tackled by combining Rao-Blackwellized particle filter (RBPF) with the Bayesian dynamic linear models (BDLMs). The BDLMs, which is a special case of state-space models, allow decomposing time series into a vector of hidden state variables. The RBPF employs the sequential Monte Carlo method to learn model parameters continuously as the new observations are collected. The potential of the new approach is illustrated on the displacement data collected from a dam in Canada. The approach succeeds in detecting the anomaly caused by the refection work on the dam as well as the artificial anomalies that are introduced on the original dataset. The new method opens the way for monitoring the structure's health and conditions in real time.

Bayesian prediction of bridge extreme stresses based on DLTM and monitoring coupled data

For predicting dynamic coupled extreme stresses of bridges with monitoring coupled data, this article considers monitoring extreme stress data as a time series, and takes into account its coupling generated by the fusion of non-stationarity and randomness. First, the local polynomial theory is introduced, and the local polynomial order of monitoring coupled extreme stress data is estimated with time-series analysis method. Second, based on time-series analysis results, dynamic linear trend models (DLTM) and the corresponding Bayesian probability recursive processes are given to predict dynamic coupled extreme stresses. Finally, through the illustration of monitoring coupled extreme stress data from an actual bridge, the proposed method, which is compared with the traditional Bayesian dynamic linear models, is proved to be more effective for predicting dynamic coupled extreme stresses of bridges.

A Kernel-Based Method for Modeling Non-harmonic Periodic Phenomena in Bayesian Dynamic Linear Models

Modeling periodic phenomena with accuracy is a key aspect to detect abnormal behaviour in time series for the context of Structural Health Monitoring. Modeling complex non-harmonic periodic pattern currently requires sophisticated techniques and significant computational resources. To overcome these limitations, this paper proposes a novel approach that combines the existing Bayesian Dynamic Linear Models with a kernel-based method for handling periodic patterns in time series. The approach is applied to model the traffic load on the Tamar Bridge and the piezometric pressure under a dam. The results show that the proposed method succeeds in modeling the stationary and non-stationary periodic patterns for both case-studies. Also, it is computationally efficient, versatile, self-adaptive to changing conditions, and capable of handling observations collected at irregular time intervals.

Uncertainty quantification for model parameters and hidden state variables in Bayesian dynamic linear models

The quantification of uncertainty associated with the model parameters and the hidden state variables is a key missing aspect for the existing Bayesian dynamic linear models. This paper proposes two methods for carrying out the uncertainty quantification task: (a) the maximum a posteriori with the Laplace approximation procedure (LAP-P) and (b) the Hamiltonian Monte Carlo procedure (HMC-P). A comparative study of LAP-P with HMC-P is conducted on simulated data as well as real data collected on a dam in Canada. The results show that the LAP-P is capable to provide a reasonable estimation without requiring a high computation cost, yet it is prone to be trapped in local maxima. The HMC-P yields a more reliable estimation than LAP-P, but it is computationally demanding. The estimation results obtained from both LAP-P and HMC-P tend to the same values as the size of the training data increases. Therefore, a deployment of both LAP-P and HMC-P is suggested for ensuring an efficient and reliable estimation. LAP-P should first be employed for the model development and HMC-P should then be used to verify the estimation obtained using LAP-P.

Bayesian Dynamic Linear Models for Estimation of Phenological Events from Remote Sensing Data

Estimating the timing of the occurrence of events that characterize growth cycles in vegetation from time series of remote sensing data is desirable for a wide area of applications. For example, the timings of plant life cycle events are very sensitive to weather conditions and are often used to assess the impacts of changes in weather and climate. Likewise, understanding crop phenology can have a large impact on agricultural strategies. To study phenology using remote sensing data, the timings of annual phenological events must be estimated from noisy time series that may have many missing values. Many current state-of-the-art methods consist of smoothing time series and estimating events as features of smoothed curves. A shortcoming of many of these methods is that they do not easily handle missing values and require imputation as a preprocessing step. In addition, while some currently used methods may be extendable to allow for temporal uncertainty quantification, uncertainty intervals are not usually provided with phenological event estimates. We propose methodology utilizing Bayesian dynamic linear models to estimate the timing of key phenological events from remote sensing data with uncertainty intervals. We illustrate the methodology on weekly vegetation index data from 2003 to 2007 over a region of southern India, focusing on estimating the timing of start of season and peak of greenness. Additionally, we present methods utilizing the Bayesian formulation and MCMC simulation of the model to estimate the probability that more than one growing season occurred in a given year. Supplementary materials accompanying this paper appear online.

Optimal Asset Allocation with Multivariate Bayesian Dynamic Linear Models

We introduce a simulation-free method to model and forecast multiple asset returns and employ it to investigate the optimal ensemble of features to include when jointly predicting monthly stock and bond excess returns. Our approach builds on the Bayesian Dynamic Linear Models of West and Harrison (1997), and it can objectively determine, through a fully automated procedure, both the optimal set of regressors to include in the predictive system and the degree to which the model coefficients, volatilities, and covariances should vary over time. When applied to a portfolio of five stock and bond returns, we find that our method leads to large forecast gains, both in statistical and economic terms. In particular, we find that relative to a standard no-predictability benchmark, the optimal combination of predictors, stochastic volatility, and time-varying covariances increases the annualized certainty equivalent returns of a leverage-constrained power utility investor by more than 500 basis points.

Time-variant reliability prediction of bridge system based on BDGCM and SHM data

Structural health monitoring systems produce a large number of monitored data in the long-term service periods. To properly predict the dynamic reliability of bridge system based on these structural health monitoring data, the paper is to propose (a) Bayesian Dynamic Gaussian Copula Model obtained with Bayesian dynamic linear models and Gaussian Copula function for modeling the time-dependent nonlinear correlation coefficients between monitored variables; (b) time-variant reliability prediction approach of bridge system based on Bayesian Dynamic Gaussian Copula Model; and (c) an actual bridge is provided to illustrate the feasibility and application of the proposed models and methods.

Quantifying Temporal Trends in Fisheries Abundance Using Bayesian Dynamic Linear Models: A Case Study of Riverine Smallmouth Bass Populations

AbstractDetecting temporal changes in fish abundance is an essential component of fisheries management. Because of the need to understand short‐term and nonlinear changes in fish abundance, traditional linear models may not provide adequate information for management decisions. This study highlights the utility of Bayesian dynamic linear models (DLMs) as a tool for quantifying temporal dynamics in fish abundance. To achieve this goal, we quantified temporal trends of Smallmouth Bass Micropterus dolomieu catch per effort (CPE) from rivers in the mid‐Atlantic states, and we calculated annual probabilities of decline from the posterior distributions of annual rates of change in CPE. We were interested in annual declines because of recent concerns about fish health in portions of the study area. In general, periods of decline were greatest within the Susquehanna River basin, Pennsylvania. The declines in CPE began in the late 1990s—prior to observations of fish health problems—and began to stabilize toward the end of the time series (2011). In contrast, many of the other rivers investigated did not have the same magnitude or duration of decline in CPE. Bayesian DLMs provide information about annual changes in abundance that can inform management and are easily communicated with managers and stakeholders.

Structural Health Monitoring with dependence on non-harmonic periodic hidden covariates

In Structural Health Monitoring, non-harmonic periodic hidden covariate typically arises when an observed structural response depends on unobserved external effects such as temperature or loading. This paper addresses this challenge by proposing a new extension to Bayesian Dynamic Linear Models (BDLMs) for handling situations where non-harmonic periodic hidden covariates may influence the observed responses of structures. The potential of the new approach is illustrated on the data recorded on a dam in Canada. A model employing the proposed approach is compared to another that only uses a superposition of harmonic hidden components available from the existing BDLMs. The comparative study shows that the proposed approach succeeds in estimating hidden covariates and has a better predictive performance than the existing method using a superposition of harmonic hidden components.