Hierarchical State Space Research Articles

SEMIPARAMETRIC BIVARIATE HIERARCHICAL STATE SPACE MODEL WITH APPLICATION TO HORMONE CIRCADIAN RELATIONSHIP.

The adrenocorticotropic hormone and cortisol play critical roles in stress regulation and the sleep-wake cycle. Most research has been focused on how the two hormones regulate each other in terms of short-term pulses. Few studies have been conducted on the circadian relationship between the two hormones and how it differs between normal and abnormal groups. The circadian patterns are difficult to model as parametric functions. Directly extending univariate functional mixed effects models would result in a large dimensional problem and a challenging nonparametric inference. In this article, we propose a semi-parametric bivariate hierarchical state space model, in which each hormone profile is modeled by a hierarchical state space model, with nonparametric population-average and subject-specific components. The bivariate relationship is constructed by concatenating two latent independent subject-specific random functions specified by a design matrix, leading to a parametric inference on the correlation. We propose a computationally efficient state-space EM algorithm for estimation and inference. We apply the proposed method to a study of chronic fatigue syndrome and fibromyalgia and discover an erratic regulation pattern in the patient group in contrast to a circadian regulation pattern conforming to the day-night cycle in the control group.

Dynamic hierarchical state space forecasting.

In this paper, we aim to both borrow information from existing units and incorporate the target unit's history data in time series forecasting. We consider a situation when we have time series data from multiple units that share similar patterns when aligned in terms of an internal time. The internal time is defined as an index according to evolving features of interest. When mapped back to the calendar time, these time series can span different time intervals that can include the future calendar time of the targeted unit, over which we can borrow the information from other units in forecasting the targeted unit. We first build a hierarchical state space model for the multiple time series data in terms of the internal time, where the shared components capture the similarities among different units while allowing for unit-specific deviations. A conditional state space model is then constructed to incorporate the information of existing units as the prior information in forecasting the targeted unit. By running the Kalman filtering based on the conditional state space model on the targeted unit, we incorporate both the information from the other units and the history of the targeted unit. The forecasts are then transformed from internal time back into calendar time for ease of interpretation. A simulation study is conducted to evaluate the finite sample performance. Forecasting state-level new COVID-19 cases in United States is used for illustration.

Robust IMM Filtering Approach with Adaptive Estimation of Measurement Loss Probability for Surface Target Tracking

A suitable jump Markov system (JMS) filtering approach provides an efficient technique for tracking surface targets. In complex surface target tracking situations, due to the joint influences of lost measurements with an unknown probability and heavy-tailed measurement noise (HTMN), the estimation accuracy of conventional interacting multiple model (IMM) methods may be seriously degraded. Aiming to address the filtering issues in JMSs with HTMNs and random measurement losses, this paper presents an IMM filtering approach with the adaptive estimation of unknown measurement loss probability. In this study, we assumed that the measurement noises obey student’s t-distributions and then proposed Bernoulli random variables (BRVs) to characterize the random measurement loss. Notably, by converting the two likelihood functions from the weighted sum form to exponential multiplication, we established hierarchical Gaussian state space models to directly utilize the variational inference method. The system state vectors, unknown distribution parameters, BRVs, and unknown measurement loss probabilities were estimated simultaneously according to the variational Bayesian inference in the IMM framework. The results of maneuvering target tracking simulations verified that the presented filtering approach demonstrated superior estimation accuracy compared to existing IMM filters.

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.

Spatio-temporal categorization for first-person-view videos using a convolutional variational autoencoder and Gaussian processes.

In this study, HcVGH, a method that learns spatio-temporal categories by segmenting first-person-view (FPV) videos captured by mobile robots, is proposed. Humans perceive continuous high-dimensional information by dividing and categorizing it into significant segments. This unsupervised segmentation capability is considered important for mobile robots to learn spatial knowledge. The proposed HcVGH combines a convolutional variational autoencoder (cVAE) with HVGH, a past method, which follows the hierarchical Dirichlet process-variational autoencoder-Gaussian process-hidden semi-Markov model comprising deep generative and statistical models. In the experiment, FPV videos of an agent were used in a simulated maze environment. FPV videos contain spatial information, and spatial knowledge can be learned by segmenting them. Using the FPV-video dataset, the segmentation performance of the proposed model was compared with previous models: HVGH and hierarchical recurrent state space model. The average segmentation F-measure achieved by HcVGH was 0.77; therefore, HcVGH outperformed the baseline methods. Furthermore, the experimental results showed that the parameters that represent the movability of the maze environment can be learned.

Rolling bearing diagnosis method based on improved standardized variable distance fusion hierarchical state space correlation entropy

In recent years, the machine learning method has been successfully applied to the intelligent health monitoring of rolling bearings. However, its effectiveness depends heavily on the vibration signal time series collected. For the purpose of abundantly excavate the fault feature information in time series, some new multi-scale entropy methods are used to solve this problem. However, the multi-scale entropy only considers the low frequency component of the signal, neglects the high frequency component of the signal, loses some fault signals with modulation phenomenon, and reduces the accuracy of fault classification. Therefore, in order to overcome the shortcomings of multi-scale entropy. In this paper, proposing a new hierarchical state space correlation entropy (HSSCEn) to extract the fault features contained in the signal. At the same time, in practice, the fault features extracted by HSSCEn are usually high and nonlinear. Directly input to the classifier for training and testing will reduce the accuracy of machine learning method recognition. Therefore, an improved standardized variable distance classifier is proposed in this paper. It effectively reduces the feature redundancy of signal and improves the accuracy of original classifier. Finally, in order to verify the validity of the fault diagnosis method based on HSSCEn fusion standardized variable distance classifier, two sets of fault classification experiments are carried out in this paper. Compared with existing fault identification classifier algorithms, the proposed method has more advantages in accuracy and stability of fault classification.

Joint parameter and state estimation for stochastic uncertain system with multivariate skew t noises

Due to the pulse interference, measurement outliers and artificial modeling errors, the multivariate skew t noise widely exists in the real environment. However, to date, little attention has been paid to the state estimation for systems in which the process noise and the measurement noise are both modeled as the heavy-tailed and skew non-Gaussian noise. In this paper, the multivariate skew t distribution is utilized to model the heavy-tailed and skew non-Gaussian noise. Then a probabilistic graphical form of the multivariate skew t distribution is given and proved. Based on the probabilistic graphical form, a hierarchical Gaussian state space model for stochastic uncertain systems is proposed, which transforms the estimation problem for systems with the heavy-tailed and skew non-Gaussian noises into the one with a hierarchical Gaussian state space model. Next, given the designed Gaussian state space model, the robust Bayesian filter and smoother based on the variational Bayesian inference are proposed to approximately estimate the system state and the unknown noise parameters. Furthermore, the complexity analysis together with the controllability and observability for stochastic uncertain systems with multivariate skew t noises is given. Finally, the simulation results of the target tracking scenario verify the validity of the proposed algorithms.

Diagnosis and Compensation of Control Program, Sensor and Actuator Failures in Nonlinear Systems Using Hierarchical State Space Checks

Autonomous systems with nonlinear dynamics need to be extremely resilient to errors in sensors, actuators and on-board electronics for the purpose of overall vehicle safety. Prior research has focused on control-theoretic methods with significant computational burden with a focus on failures in actuation. In contrast, we propose the use of hierarchical machine learning driven state space checks that detect and diagnose errors in control program execution, sensors and actuators with high sensitivity and low latency. Each check produces a time-varying error signal that facilitates effect-cause diagnosis of the system, while allowing rapid parameter estimation from each check. Since the checks are over small subsets of system parameters, estimation is fast and accurate. The estimated parameters are then used to reconfigure the system controller parameters for rapid system recovery. We use a quadcopter system to demonstrate and validate our approach. Controller, sensor and actuator errors can be detected, diagnosed and compensated using a common checking platform with low computational overhead. The technique is validated on a quadcopter hardware test vehicle.

Robust sensor fusion with heavy-tailed noises

In this paper, the problem of multi-sensor centralized state fusion with heavy-tailed process and measurement noises is considered. In order to improve the sensor fusion estimation, a novel identify-fusion strategy is proposed. An indicator, which is modelled by a Bernoulli prior, for each measurement in each sensor to identify whether the measurement is an outlier. A Student-t based hierarchical Gaussian state space model is then constructed, and the fusion is formulated as a state space estimation problem. The state and unknown parameters in the Student-t based hierarchical Gaussian state space model are jointly inferred by the variational Bayesian technique. Computer simulations are provided to demonstrate the effectiveness of the proposed method.

Tree water balance drives temperate forest responses to drought.

Intensifying drought is increasingly linked to global forest diebacks. Improved understanding of drought impacts on individual trees has provided limited insight into drought vulnerability in part because tree moisture access and depletion is difficult to quantify. In forests, moisture reservoir depletion occurs through water use by the trees themselves. Here, we show that drought impacts on tree fitness and demographic performance can be predicted by tracking the moisture reservoir available to trees as a mass balance, estimated in a hierarchical state-space framework. We apply this model to multiple seasonal droughts with tree transpiration measurements to demonstrate how species and size differences modulate moisture availability across landscapes. The depletion of individual moisture reservoirs can be tracked over the course of droughts and linked to biomass growth and reproductive output. This mass balance approach can predict individual moisture deficit, tree demographic performance, and drought vulnerability throughout forest stands based on measurements from a sample of trees.

Variable effects of climate on forest growth in relation to climate extremes, disturbance, and forest dynamics.

Changes in the frequency, duration, and severity of climate extremes are forecast to occur under global climate change. The impacts of climate extremes on forest productivity and health remain difficult to predict due to potential interactions with disturbance events and forest dynamics-changes in forest stand composition, density, size and age structure over time. Such interactions may lead to non-linear forest growth responses to climate involving thresholds and lag effects. Understanding how forest dynamics influence growth responses to climate is particularly important given stand structure and composition can be modified through management to increase forest resistance and resilience to climate change. To inform such adaptive management, we develop a hierarchical Bayesian state space model in which climate effects on tree growth are allowed to vary over time and in relation to past climate extremes, disturbance events, and forest dynamics. The model is an important step toward integrating disturbance and forest dynamics into predictions of forest growth responses to climate extremes. We apply the model to a dendrochronology data set from forest stands of varying composition, structure, and development stage in northeastern Minnesota that have experienced extreme climate years and forest tent caterpillar defoliation events. Mean forest growth was most sensitive to water balance variables representing climatic water deficit. Forest growth responses to water deficit were partitioned into responses driven by climatic threshold exceedances and interactions with insect defoliation. Forest growth was both resistant and resilient to climate extremes with the majority of forest growth responses occurring after multiple climatic threshold exceedances across seasons and years. Interactions between climate and disturbance were observed in a subset of years with insect defoliation increasing forest growth sensitivity to water availability. Forest growth was particularly sensitive to climate extremes during periods of high stem density following major regeneration events when average inter-tree competition was high. Results suggest the resistance and resilience of forest growth to climate extremes can be increased through management steps such as thinning to reduce competition during early stages of stand development and small-group selection harvests to maintain forest structures characteristic of older, mature stands.

Integrating population dynamics models and distance sampling data: a spatial hierarchical state-space approach.

Stochastic versions of Gompertz, Ricker, and various other dynamics models play a fundamental role in quantifying strength of density dependence and studying long-term dynamics of wildlife populations. These models are frequently estimated using time series of abundance estimates that are inevitably subject to observation error and missing data. This issue can be addressed with a state-space modeling framework that jointly estimates the observed data model and the underlying stochastic population dynamics (SPD) model. In cases where abundance data are from multiple locations with a smaller spatial resolution (e.g., from mark-recapture and distance sampling studies), models are conventionally fitted to spatially pooled estimates of yearly abundances. Here, we demonstrate that a spatial version of SPD models can be directly estimated from short time series of spatially referenced distance sampling data in a unified hierarchical state-space modeling framework that also allows for spatial variance (covariance) in population growth. We also show that a full range of likelihood based inference, including estimability diagnostics and model selection,is feasible in this class of models using a data cloning algorithm. We further show through simulation experiments that the hierarchical state-space framework introduced herein efficiently captures the underlying dynamical parameters and spatial abundance distribution. We apply our methodology by analyzing a time series of line-transect distance sampling data for fin whales (Balaenoptera physalus) off the U.S. west coast. Although there were only seven surveys conducted during the study time frame, 1991-2014, our analysis detected presence of strong density regulation and provided reliable estimates of fin whale densities. In summary, we show that the integrative framework developed herein allows ecologists to better infer key population characteristics such as presence of density regulation and spatial variability in a population's intrinsic growth potential.

Bayesian state space models for dynamic genetic network construction across multiple tissues.

Construction of gene-gene interaction networks and potential pathways is a challenging and important problem in genomic research for complex diseases while estimating the dynamic changes of the temporal correlations and non-stationarity are the keys in this process. In this paper, we develop dynamic state space models with hierarchical Bayesian settings to tackle this challenge for inferring the dynamic profiles and genetic networks associated with disease treatments. We treat both the stochastic transition matrix and the observation matrix time-variant and include temporal correlation structures in the covariance matrix estimations in the multivariate Bayesian state space models. The unevenly spaced short time courses with unseen time points are treated as hidden state variables. Hierarchical Bayesian approaches with various prior and hyper-prior models with Monte Carlo Markov Chain and Gibbs sampling algorithms are used to estimate the model parameters and the hidden state variables. We apply the proposed Hierarchical Bayesian state space models to multiple tissues (liver, skeletal muscle, and kidney) Affymetrix time course data sets following corticosteroid (CS) drug administration. Both simulation and real data analysis results show that the genomic changes over time and gene-gene interaction in response to CS treatment can be well captured by the proposed models. The proposed dynamic Hierarchical Bayesian state space modeling approaches could be expanded and applied to other large scale genomic data, such as next generation sequence (NGS) combined with real time and time varying electronic health record (EHR) for more comprehensive and robust systematic and network based analysis in order to transform big biomedical data into predictions and diagnostics for precision medicine and personalized healthcare with better decision making and patient outcomes.

Analysing time-varying trends in stratospheric ozone time series using the state space approach

Abstract. We describe a hierarchical statistical state space model for ozone profile time series. The time series are from satellite measurements by the Stratospheric Aerosol and Gas Experiment (SAGE) II and the Global Ozone Monitoring by Occultation of Stars (GOMOS) instruments spanning the years 1984–2011. Vertical ozone profiles were linearly interpolated on an altitude grid with 1 km resolution covering 20–60 km. Monthly averages were calculated for each altitude level and 10° wide latitude bins between 60° S and 60° N. In the analysis, mean densities are studied separately for the 25–35, 35–45, and 45–55 km layers. Model variables include the ozone mean level, local trend, seasonal oscillations, and proxy variables for solar activity, the Quasi-Biennial Oscillation (QBO), and the El Niño–Southern Oscillation (ENSO). This is a companion paper to Kyrölä et al. (2013), where a piecewise linear model was used together with the same proxies as in this work (excluding ENSO). The piecewise linear trend was allowed to change at the beginning of 1997 in all latitudes and altitudes. In the modelling of the present paper such an assumption is not needed as the linear trend is allowed to change continuously at each time step. This freedom is also allowed for the seasonal oscillations whereas other regression coefficients are taken independent of time. According to our analyses, the slowly varying ozone background shows roughly three general development patterns. A continuous decay for the whole period 1984–2011 is evident in the southernmost latitude belt 50–60° S in all altitude regions and in 50–60° N in the lowest altitude region 25–35 km. A second pattern, where a recovery after an initial decay is followed by a further decay, is found at northern latitudes from the equator to 50° N in the lowest altitude region (25–35 km) and between 40° N and 60° N in the 35–45 km altitude region. Further ozone loss occurred after 2007 in these regions. Everywhere else a decay is followed by a recovery. This pattern is shown at all altitudes and latitudes in the Southern Hemisphere (10–50° S) and in the 45–55 km layer in the Northern Hemisphere (from the equator to 40° N). In the 45–55 km range the trend, measured as an average change in 10 years, has mostly turned from negative to positive before the year 2000. In those regions where the "V" type of change of the trend is appropriate, the turning point is around the years 1997–2001. To compare results for the trend changes with the companion paper, we studied the difference in trends between the years from 1984 to 1997 and from 1997 to 2011. Overall, the two methods produce very similar ozone recovery patterns with the maximum trend change of 10% in 35–45 km. The state space method (used in this paper) shows a somewhat faster recovery than the piecewise linear model. For the percent change of the ozone density per decade the difference between the results is below three percentage units.

Modeling Bivariate Longitudinal Hormone Profiles by Hierarchical State Space Models

The hypothalamic-pituitary-adrenal (HPA) axis is crucial in coping with stress and maintaining homeostasis. Hormones produced by the HPA axis exhibit both complex univariate longitudinal profiles and complex relationships among different hormones. Consequently, modeling these multivariate longitudinal hormone profiles is a challenging task. In this article, we propose a bivariate hierarchical state space model, in which each hormone profile is modeled by a hierarchical state space model, with both population-average and subject-specific components. The bivariate model is constructed by concatenating the univariate models based on the hypothesized relationship. Because of the flexible framework of state space form, the resultant models not only can handle complex individual profiles, but also can incorporate complex relationships between two hormones, including both concurrent and feedback relationship. Estimation and inference are based on marginal likelihood and posterior means and variances. Computationally efficient Kalman filtering and smoothing algorithms are used for implementation. Application of the proposed method to a study of chronic fatigue syndrome and fibromyalgia reveals that the relationships between adrenocorticotropic hormone and cortisol in the patient group are weaker than in healthy controls. Supplementary materials for this article are available online.

Shoot, shovel and shut up: cryptic poaching slows restoration of a large carnivore in Europe.

Poaching is a widespread and well-appreciated problem for the conservation of many threatened species. Because poaching is illegal, there is strong incentive for poachers to conceal their activities, and consequently, little data on the effects of poaching on population dynamics are available. Quantifying poaching mortality should be a required knowledge when developing conservation plans for endangered species but is hampered by methodological challenges. We show that rigorous estimates of the effects of poaching relative to other sources of mortality can be obtained with a hierarchical state–space model combined with multiple sources of data. Using the Scandinavian wolf (Canis lupus) population as an illustrative example, we show that poaching accounted for approximately half of total mortality and more than two-thirds of total poaching remained undetected by conventional methods, a source of mortality we term as ‘cryptic poaching’. Our simulations suggest that without poaching during the past decade, the population would have been almost four times as large in 2009. Such a severe impact of poaching on population recovery may be widespread among large carnivores. We believe that conservation strategies for large carnivores considering only observed data may not be adequate and should be revised by including and quantifying cryptic poaching.

A hierarchical state space approach to affective dynamics

Linear dynamical system theory is a broad theoretical framework that has been applied in various research areas such as engineering, econometrics and recently in psychology. It quantifies the relations between observed inputs and outputs that are connected through a set of latent state variables. State space models are used to investigate the dynamical properties of these latent quantities. These models are especially of interest in the study of emotion dynamics, with the system representing the evolving emotion components of an individual. However, for simultaneous modeling of individual and population differences, a hierarchical extension of the basic state space model is necessary. Therefore, we introduce a Bayesian hierarchical model with random effects for the system parameters. Further, we apply our model to data that were collected using the Oregon adolescent interaction task: 66 normal and 67 depressed adolescents engaged in a conflict-oriented interaction with their parents and second-to-second physiological and behavioral measures were obtained. System parameters in normal and depressed adolescents were compared, which led to interesting discussions in the light of findings in recent literature on the links between cardiovascular processes, emotion dynamics and depression. We illustrate that our approach is flexible and general: The model can be applied to any time series for multiple systems (where a system can represent any entity) and moreover, one is free to focus on various components of this versatile model.

Tree growth inference and prediction when the point of measurement changes: modelling around buttresses in tropical forests

Abstract:Estimation of tree growth is generally based on repeated diameter measurements. A buttress at the height of measurement will lead to overestimates of tree diameter. Because buttresses grow up the trunk through time, it has become common practice to increase the height of measurement, to ensure that measurements remain above the buttress. However, tapering of the trunk means that increasing measurement height will bias estimates of diameter downward by up to 10% per m of height. This bias could affect inference concerning species differences and climate effects on tree demography and on biomass accumulation. Here we introduce a hierarchical state space method that allows formal integration of data on diameter taken at different heights and can include individual variation, temporal effects or other covariates. We illustrate our approach using species from Barro Colorado Island, Panama, and La Selva, Costa Rica. Results include trends that are consistent with some of those previously reported for climate responses and changes over time, but differ in relative magnitude. By including the full data-set and accounting for bias and variation among individuals and over time, our approach allows for quantification of climate responses and the uncertainty associated with measurements and the underlying growth process.

Geodetic inversion for space-time distribution of fault slip with time-varying smoothing regularization

SUMMARY We have developed a new geodetic inversion method for space–time distribution of fault slip velocity with time-varying smoothing regularization in order to reconstruct accurate time histories of aseismic fault slip transients. We introduce a temporal smoothing regularization on slip and slip velocity through a Bayesian state space approach in which the strength of regularization (temporal smoothness of slip velocity) is controlled by a hyperparameter. The time-varying smoothing regularization is realized by treating the hyperparameter as a time-dependent stochastic variable and adopting a hierarchical Bayesian state space model, in which a prior distribution on the hyperparameter is introduced in addition to a conventional Bayesian state space model. We have tested this inversion method on two synthetic data sets generated by simulated aseismic slip transients. Results show that our method reproduces well both rapid changes of slip velocity and steady-state velocity without significant oversmoothing and undersmoothing, which has been hard to overcome by the conventional Bayesian approach with time-independent smoothing regularization. Application of this method to transient deformation in 2002 caused by a silent earthquake off the Boso peninsula, Japan, also shows similar advantages of this method over the conventional approach.

A statistical analysis of memory CD8 T cell differentiation: An application of a hierarchical state space model to a short time course microarray experiment

CD8 T cells are specialized immune cells that play an important role in the regulation of antiviral immune response and the generation of protective immunity. In this paper we investigate the differentiation of memory CD8 T cells in the immune response using a short time course microarray experiment. Structurally, this experiment is similar to many in that it involves measurements taken on independent samples, in one biological group, at a small number of irregularly spaced time points, and exhibiting patterns of temporal nonstationarity. To analyze this CD8 T-cell experiment, we develop a hierarchical state space model so that we can: (1) detect temporally differentially expressed genes, (2) identify the direction of successive changes over time, and (3) assess the magnitude of successive changes over time. We incorporate hidden Markov models into our model to utilize the information embedded in the time series and set up the proposed hierarchical state space model in an empirical Bayes framework to utilize the population information from the large-scale data. Analysis of the CD8 T-cell experiment using the proposed model results in biologically meaningful findings. Temporal patterns involved in the differentiation of memory CD8 T cells are summarized separately and performance of the proposed model is illustrated in a simulation study.