Multivariate Autoregressive Model Research Articles

Markov Model-Based Method to Analyse Time-Varying Networks in EEG Task-Related Data.

The dynamic nature of functional brain networks is being increasingly recognized in cognitive neuroscience, and methods to analyse such time-varying networks in EEG/MEG data are required. In this work, we propose a pipeline to characterize time-varying networks in single-subject EEG task-related data and further, evaluate its validity on both simulated and experimental datasets. Pre-processing is done to remove channel-wise and trial-wise differences in activity. Functional networks are estimated from short non-overlapping time windows within each “trial,” using a sparse-MVAR (Multi-Variate Auto-Regressive) model. Functional “states” are then identified by partitioning the entire space of functional networks into a small number of groups/symbols via k-means clustering.The multi-trial sequence of symbols is then described by a Markov Model (MM). We show validity of this pipeline on realistic electrode-level simulated EEG data, by demonstrating its ability to discriminate “trials” from two experimental conditions in a range of scenarios. We then apply it to experimental data from two individuals using a Brain-Computer Interface (BCI) via a P300 oddball task. Using just the Markov Model parameters, we obtain statistically significant discrimination between target and non-target trials. The functional networks characterizing each ‘state’ were also highly similar between the two individuals. This work marks the first application of the Markov Model framework to infer time-varying networks from EEG/MEG data. Due to the pre-processing, results from the pipeline are orthogonal to those from conventional ERP averaging or a typical EEG microstate analysis. The results provide powerful proof-of-concept for a Markov model-based approach to analyzing the data, paving the way for its use to track rapid changes in interaction patterns as a task is being performed. MATLAB code for the entire pipeline has been made available.

Seizure Prediction by Multivariate Autoregressive Model Order Optimization

Abstract For several decades, researchers are aiming for the detection of precursors of epileptic seizures. A system that is able to issue a warning about an impending seizure could dramatically improve the quality of life of affected patients. In this work, we apply multivariate autoregressive (MVAR) modeling to intracranial electroencephalography (iEEG) recordings of patients with therapy resistant epilepsy. As compared to our previous investigations, we studied the optimal model order of the autoregressive process as a feature for seizure prediction. In a statistical evaluation, we obtain significant results for 17 out of 20 patients.

The Effects of Traffic Composition on Freeway Crash Frequency by Injury Severity: A Bayesian Multivariate Spatial Modeling Approach

This study sets out to investigate the effects of traffic composition on freeway crash frequency by injury severity. A crash dataset collected from Kaiyang Freeway, China, is adopted for the empirical analysis, where vehicles are divided into five categories and crashes are classified into no injury and injury levels. In consideration of correlated spatial effects between adjacent segments, a Bayesian multivariate conditional autoregressive model is proposed to link no-injury and injury crash frequencies to the risk factors, including the percentages of different vehicle categories, daily vehicle kilometers traveled (DVKT), and roadway geometry. The model estimation results show that, compared to Category 5 vehicles (e.g., heavy truck), larger percentages of Categories 1 (e.g., passenger car) and 3 (e.g., medium truck) vehicles would lead to less no-injury crashes and more injury crashes. DVKT, horizontal curvature, and vertical grade are also found to be associated with no-injury and/or injury crash frequencies. The significant heterogeneous and spatial effects for no-injury and injury crashes justify the applicability of the proposed model. The findings are helpful to understand the relationship between traffic composition and freeway safety and to provide suggestions for designing strategies of vehicle safety improvement.

Directional brain networks underlying OM chanting.

OM chanting is an ancient technique of Indian meditation. OM chanting is associated with an experience of relaxation, changes in autonomic balance and deactivation of limbic brain regions. While functional localization is important, how brain regions interact with each other has been shown to underlie various brain functions. Therefore, in this study, we tested the hypothesis that there is reduced communication between deactivated regions during OM chanting. In order to do so, we employed multivariate autoregressive model (MVAR) based Granger causality to obtain directional connectivity between deactivated regions. fMRI scans of 12 right handed healthy volunteers (9 Men) from a previously published study was used in which participants performed OM chanting and a control condition in a block design. We found that outputs from insula, anterior cingulate and orbitofrontal cortices were significantly reduced in OM condition. Of interest is the reduction of outputs from these regions to limbic area amygdala. Modulation of brain regions involved in emotion processing and implicated in major depressive disorder (MDD) raises a potential possibility of OM chanting in the treatment of MDD.

Response Hand and Motor Set Differentially Modulate the Connectivity of Brain Pathways During Simple Uni-manual Motor Behavior.

We investigated the flexible modulation of undirected functional connectivity (uFC) of brain pathways during simple uni-manual responding. Two questions were central to our interests: (1) does response hand (dominant vs. non-dominant) differentially modulate connectivity and (2) are these effects related to responding under varying motor sets. fMRI data were acquired in twenty right-handed volunteers who responded with their right (dominant) or left (non-dominant) hand (blocked across acquisitions). Within acquisitions, the task oscillated between periodic responses (promoting the emergence of motor sets) or randomly induced responses (disrupting the emergence of motor sets). Conjunction analyses revealed eight shared nodes across response hand and condition, time series from which were analyzed. For right hand responses connectivity of the M1 ←→ Thalamus and SMA ←→ Parietal pathways was more significantly modulated during periodic responding. By comparison, for left hand responses, connectivity between five network pairs (including M1 and SMA, insula, basal ganglia, premotor cortex, parietal cortex, thalamus) was more significantly modulated during random responding. uFC analyses were complemented by directed FC based on multivariate autoregressive models of times series from the nodes. These results were complementary and highlighted significant modulation of dFC for SMA → Thalamus, SMA → M1, basal ganglia → Insula and basal ganglia → Thalamus. The results demonstrate complex effects of motor organization and task demand and response hand on different connectivity classes of fMRI data. The brain's sub-networks are flexibly modulated by factors related to motor organization and/or task demand, and our results have implications for assessment of medical conditions associated with motor dysfunction.

A procedure to increase the power of Granger-causal analysis through temporal smoothing

BackgroundHow the human brain coordinates network activity to support cognition and behavior remains poorly understood. New high-resolution recording modalities facilitate a more detailed understanding of the human brain network. Several approaches have been proposed to infer functional networks, indicating the transient coordination of activity between brain regions, from neural time series. One category of approach is based on statistical modeling of time series recorded from multiple sensors (e.g., multivariate Granger causality). However, fitting such models remains computationally challenging as the history structure may be long in neural activity, requiring many model parameters to fully capture the dynamics. New MethodWe develop a method based on Granger causality that makes the assumption that the history dependence varies smoothly. We fit multivariate autoregressive models such that the coefficients of the lagged history terms are smooth functions. We do so by modelling the history terms with a lower dimensional spline basis, which requires many fewer parameters than the standard approach and increases the statistical power of the model. ResultsWe show that this procedure allows accurate estimation of brain dynamics and functional networks in simulations and examples of brain voltage activity recorded from a patient with pharmacoresistant epilepsy. Comparison with Existing MethodThe proposed method has more statistical power than the Granger method for networks of signals that exhibit extended and smooth history dependencies. ConclusionsThe proposed tool permits conditional inference of functional networks from many brain regions with extended history dependence, furthering the applicability of Granger causality to brain network science.

A Spatio-Temporal Model and Inference Tools for Longitudinal Count Data on Multicolor Cell Growth.

Multicolor cell spatio-temporal image data have become important to investigate organ development and regeneration, malignant growth or immune responses by tracking different cell types both in vivo and in vitro. Statistical modeling of image data from common longitudinal cell experiments poses significant challenges due to the presence of complex spatio-temporal interactions between different cell types and difficulties related to measurement of single cell trajectories. Current analysis methods focus mainly on univariate cases, often not considering the spatio-temporal effects affecting cell growth between different cell populations. In this paper, we propose a conditional spatial autoregressive model to describe multivariate count cell data on the lattice, and develop inference tools. The proposed methodology is computationally tractable and enables researchers to estimate a complete statistical model of multicolor cell growth. Our methodology is applied on real experimental data where we investigate how interactions between cancer cells and fibroblasts affect their growth, which are normally present in the tumor microenvironment. We also compare the performance of our methodology to the multivariate conditional autoregressive (MCAR) model in both simulations and real data applications.

Decoding Movement States in Stepping Cycles Based on Subthalamic LFPs in Parkinsonian Patients.

Gait disturbances are a prominent feature of Parkinson's disease (PD), often refractory to medication or continuous deep brain stimulation (DBS) on basal ganglia targets such as the subthalamic nucleus (STN). Here we sought to identify movement states during stepping cycles, such as left leg stance and right leg stance. To this end we analyzed local field potential (LFP) activity in STN using a combination of the multivariate autoregressive (MAR) model and the Hidden Markov model (HMM). Our results confirm that information is present in the STN related to movement states in stepping cycles, and that it is feasible to decode movement states based on STN LFPs recorded from DBS electrodes. This information can be used to implement temporally flexible stimulation strategies in order to facilitate patterns of neural modulation associated with better gait performance.

Imputing Missing Values in EEG with Multivariate Autoregressive Models.

Wearable measurement for electroencephalogram (EEG) is expected to enable brain-computer interfaces, biomedical engineering, and neuroscience studies in real environments. When wearable devices are in practical use, only the user (subject) can take care of measurement, unlike laboratory- oriented experiments, where experimenters are always with the subject. As a result, measurement troubles such as artifact contamination or electrode impairment cannot be easily corrected, and EEG recordings will become incomplete, including many missing values. If the missing values are imputed (interpolated) and complete data without missing entries are available, we can employ existing signal analysis techniques that assume compete data. In this paper, we propose an EEG signal imputation method based on multivariate autoregressive (MAR) modeling and its iterative estimation and simulation, inspired by the multiple imputation procedure. We evaluated the proposed method with real data with artificial missing entries. Experimental results show that the proposed method outperforms popular baseline interpolation methods. Our iterative scheme is simple yet effective, and can be the foundation for many extensions.

Selection of the optimal order for multivariate autoregressive model of electroencephalograms for patients with epilepsy

Introduction. Brain electrical activity signals (or EEG) by their very nature are non-stationary time series. This basically allows applying a set of mathematical-statistical analysis methods to them. One of the most common methods for signal analyzing is the construction of autoregressive mathematical models and analysis of their parameters in order to obtain additional information about the signal itself or causality between signals. In multivariate autoregressive (MVAR) modeling of EEG, the main issue is the optimal choice of model order. In this work, the approach for selecting the optimal order of MVAR models of brain electrical activity signals of subjects diagnosed with epilepsy is proposed. MVAR modeling. The autoregressive model assumes that the current sample of the discrete signal can be linearly predicted as a weighted sum of its previous samples. MVAR model extends this assumption to multiple time series so that the vector of current samples of all signals is modeled as a linear sum of their previous samples. MVAR models of EEG signals essentially are formed by solving systems of linear equations. The Yule-Walker method of linear equations systems solving is used in this paper. The accuracy of EEG modeling depends on the order of model. Each model order influenced by the amount of delay between current samples and last previous samples used to generate the model. To assess the order and quality of models the Schwarz-Bayes information criterion (SBC) is used in this work taking into account the covariance matrix of the residuals. Additionally, the quality is assessed by Pearson's correlation coefficient between the real and simulated data. In this paper, the MVAR modeling and statistical analysis of the models' optimal orders of the input signal periods before, during and after an epileptic seizure is carried out. Experimental results. Two sets of EEG data with generalized and focal epileptic seizures are used. The first group of patients with focal seizures consists of 26 people and more than 100 epileptic seizures. The second group with generalized seizures consists of 11 people and about 50 epileptic seizures. For EEG signals modeling, values of orders in a range from 1 to 22 are used. Consequently, for each investigated period of signal (before, during and after a seizure), 22 different MVAR models are constructed and compared. After modeling, the obtained models for each order value are evaluated using the SBC criterion. Conclusions. According to the results, it is recommended to choose the order of MVAR models of EEG signals in the predefined range of orders from 11 to 13. Since the sampling rate of the signals used in these experiments is 250 Hz, the specified range of order values indicates that MVAR-modelling of one signal includes information that contains all other signals with a delay of 44-52 ms. Therefore, theoretically, it is possible to allocate functional characteristics of brain electrical activity for patients with epilepsy that occur synchronously in different parts of the brain and spread at an average of 50 ms. Moreover, the ways of further research of electrical brain activity and functional connections of brain regions during epileptic activity are indicated.

Time-varying MVAR algorithms for directed connectivity analysis: Critical comparison in simulations and benchmark EEG data.

Human brain function depends on directed interactions between multiple areas that evolve in the subsecond range. Time-varying multivariate autoregressive (tvMVAR) modeling has been proposed as a way to help quantify directed functional connectivity strengths with high temporal resolution. While several tvMVAR approaches are currently available, there is a lack of unbiased systematic comparative analyses of their performance and of their sensitivity to parameter choices. Here, we critically compare four recursive tvMVAR algorithms and assess their performance while systematically varying adaptation coefficients, model order, and signal sampling rate. We also compared two ways of exploiting repeated observations: single-trial modeling followed by averaging, and multi-trial modeling where one tvMVAR model is fitted across all trials. Results from numerical simulations and from benchmark EEG recordings showed that: i) across a broad range of model orders all algorithms correctly reproduced patterns of interactions; ii) signal downsampling degraded connectivity estimation accuracy for most algorithms, although in some cases downsampling was shown to reduce variability in the estimates by lowering the number of parameters in the model; iii) single-trial modeling followed by averaging showed optimal performance with larger adaptation coefficients than previously suggested, and showed slower adaptation speeds than multi-trial modeling. Overall, our findings identify strengths and weaknesses of existing tvMVAR approaches and provide practical recommendations for their application to modeling dynamic directed interactions from electrophysiological signals.

Selecting Sensors in Biological Fractional-Order Systems

In this paper, we focus on sensor selection, that is, determine the minimum number of state variables that need to be measured, to monitor the evolution of the entire biological system, that is, all state variables, when modeled by discrete-time fractional-order systems (DTFOS) that are subject to modeling errors, process, and measurement noise. These systems are particularly relevant while modeling the spatiotemporal dynamics of processes in which the impact of long-range memory cannot be properly modeled by multivariate autoregressive integrative moving-average models. Therefore, the DTFOS enable a unified state-space framework to model the evolution of several biological (e.g., stem cell growth and bacteria evolution) and physiological signals (e.g., electroencephalogram and electromyogram). Therefore, in this paper, we focus on the solution to four different (yet related) problems of sensor selection for DTFOS, which are motivated by constraints on the data acquisition that are enforced by the detrimental impact of the sensing mechanisms to the biological system, the cost of performing the measurements with the current sensing technology, or spatial constraints that limit the number of sensors that can be deployed. Toward determining the solution to these problems that we show to be NP-hard, we leverage the representation of the DTFOS to derive new objectives and conditions that, ultimately, enable us to efficiently approximate a solution to the different problems by exploiting the submodularity structure, which enables us to establish suboptimality guarantees.

Topographical assessment of neurocortical connectivity by using directed transfer function and partial directed coherence during meditation.

Due to the presence of nonlinearity and volume conduction in electroencephalography (EEG), sometimes it's challenging to find out the actual brain network from neurodynamical alteration. In this paper, two well-known time-frequency brain connectivity measures, namely partial directed coherence (PDC) and directed transfer function (DTF), have been applied to evaluate the performance analysis of EEG signals obtained during meditation. These measures are implemented to the multichannel meditation EEG data to get the directed neural information flow. Mostly the assessment of PDC and DTF is entirely subjective and there are probabilities to have erroneous connectivity estimation. To avoid the subjective evaluation, the performance results are compared in terms of absolute energy, signal-to-noise ratio (SNR) and relative SNR (R-SNR) scale. In most of the cases, the PDC result is found to be more efficient than DTF. The limitation of DTF and PDC in terms of the time-varying multivariate autoregressive (MVAR) model is highlighted. The time-varying MVAR model can track the neurodynamical changes better than any other method. In the present study, we would like to show that the PDC-based connectivity gives a better understanding of the non-symmetric relation in EEG obtained during Kriya Yoga meditation in comparison to DTF. However, it needs to be investigated further to warrant this claim.

Modeling of carbon credit prices using regime switching approach

In this study, we analyze the price dynamics of carbon certificates that are traded under the European Union's Emissions Trading System (EU-ETS). With the aim of investigating the joint relations among carbon, electricity, and fuel prices, we model historical prices using several methods and incorporating structural changes, such as econometric time series, regime switching, and multivariate vector autoregression models. We compare the results of the structural model with the results of traditional Markov switching and autoregressive models with breaks and present performance analysis based on the mean average percentage error, root mean squared error, and coefficient of determination. According to these performance tests, models with regimes outperform the approaches where breaks are defined using ex ante dummy variables. Moreover, we conclude that among regime switching models, univariate models are better than multivariate counterparts for modeling carbon price series for the analysis of both in-sample and out-of-samples.

Modeling and forecasting of climatic parameters: univariate SARIMA versus multivariate vector autoregression approach

Agriculture sector throughout the world including Bangladesh is extremely vulnerable to the negative consequences of climate change as evident in a good number of studies. Accurate climate forecasting may prove a valuable resource in mitigating these consequences in agriculture. The study aims to identify the best performing forecasting method by comparing the forecasting abilities of univariate seasonal autoregressive integrated moving average (SARIMA) and multivariate vector autoregression (VAR) models in forecasting monthly maximum and minimum temperatures, humidity, and cloud coverage in Bangladesh. Though the univariate time series investigate the influence of the past values of a single time series on the future values of that specific series, the VAR approach forecast multivariate time series simultaneously incorporating the interrelationship among the groups of variables. Monthly forecasts of climatic parameters for in-sample over the period 1972-2008 and out-of-sample from 2009-2013 were generated via a univariate SARIMA and a VAR approach. Different forecast accuracy measures reveal that VAR model give better forecast than univariate SARIMA model. The forecast results using VAR(9) model from January 2014 to December 2021 show that maximum and minimum temperature, as well as humidity are increasing while the cloud coverage is decreasing, that is, consistent with global warming. Moreover, the impulse response function results exhibit the fluctuated and significant dynamic relationships in future among the foresaid climatic variables. Thus, findings of the study can potentially allow Bangladeshi farmers and other actors in the agriculture sector to make proper planning to abate unwanted impacts or reap the expected benefits of favourable climate.J. Bangladesh Agril. Univ. 16(1): 131-143, April 2018

Age Differences in Day-To-Day Speed-Accuracy Tradeoffs: Results from the COGITO Study

ABSTRACTWe examined adult age differences in day-to-day adjustments in speed-accuracy tradeoffs (SAT) on a figural comparison task. Data came from the COGITO study, with over 100 younger and 100 older adults, assessed for over 100 days. Participants were given explicit feedback about their completion time and accuracy each day after task completion. We applied a multivariate vector auto-regressive model of order 1 to the daily mean reaction time (RT) and daily accuracy scores together, within each age group. We expected that participants adjusted their SAT if the two cross-regressive parameters from RT (or accuracy) on day t-1 of accuracy (or RT) on day t were sizable and negative. We found that: (a) the temporal dependencies of both accuracy and RT were quite strong in both age groups; (b) younger adults showed an effect of their accuracy on day t-1 on their RT on day t, a pattern that was in accordance with adjustments of their SAT; (c) older adults did not appear to adjust their SAT; (d) these effects were partly associated with reliable individual differences within each age group. We discuss possible explanations for older adults’ reluctance to recalibrate speed and accuracy on a day-to-day basis.

Detrimental effects of a novel flow regime on the functional trajectory of an aquatic invertebrate metacommunity.

Novel flow regimes resulting from dam operations and overallocation of freshwater resources are an emerging consequence of global change. Yet, anticipating how freshwater biodiversity will respond to surging flow regime alteration requires overcoming two challenges in environmental flow science: shifting from local to riverscape-level understanding of biodiversity dynamics, and from static to time-varying characterizations of the flow regime. Here, we used time-series methods (wavelets and multivariate autoregressive models) to quantify flow-regime alteration and to link time-varying flow regimes to the dynamics of multiple local communities potentially connected by dispersal (i.e., a metacommunity). We studied the Chattahoochee River below Buford dam (Georgia, U.S.A.), and asked how flow regime alteration by a large hydropower dam may control the long-term functional trajectory of the downstream invertebrate metacommunity. We found that seasonal variation in hydropeaking synchronized temporal fluctuations in trait abundance among the flow-altered sites. Three biological trait states describing adaptation to fast flows benefitted from flow management for hydropower, but did not compensate for declines in 16 "loser" traits. Accordingly, metacommunity-wide functional diversity responded negatively to hydropeaking intensity, and stochastic simulations showed that the risk of functional diversity collapse within the next 4years would decrease by 17% if hydropeaking was ameliorated, or by 9% if it was applied every other season. Finally, an analysis of 97 reference and 23 dam-affected river sites across the U.S. Southeast suggested that flow variation at extraneous, human-relevant scales (12-hr, 24-hr, 1-week) is relatively common in rivers affected by hydropower dams. This study advances the notion that novel flow regimes are widespread, and simplify the functional structure of riverine communities by filtering out taxa with nonadaptive traits and by spatially synchronizing their dynamics. This is relevant in the light of ongoing and future hydrologic alteration due to climate non-stationarity and the new wave of dams planned globally.

Using state-space models to predict the abundance of juvenile and adult sea lice on Atlantic salmon

Sea lice are marine parasites affecting salmon farms, and are considered one of the most costly pests of the salmon aquaculture industry. Infestations of sea lice on farms significantly increase opportunities for the parasite to spread in the surrounding ecosystem, making control of this pest a challenging issue for salmon producers. The complexity of controlling sea lice on salmon farms requires frequent monitoring of the abundance of different sea lice stages over time. Industry-based data sets of counts of lice are amenable to multivariate time-series data analyses.In this study, two sets of multivariate autoregressive state-space models were applied to Chilean sea lice data from six Atlantic salmon production cycles on five isolated farms (at least 20 km seaway distance away from other known active farms), to evaluate the utility of these models for predicting sea lice abundance over time on farms. The models were constructed with different parameter configurations, and the analysis demonstrated large heterogeneity between production cycles for the autoregressive parameter, the effects of chemotherapeutant bath treatments, and the process-error variance. A model allowing for different parameters across production cycles had the best fit and the smallest overall prediction errors. However, pooling information across cycles for the drift and observation error parameters did not substantially affect model performance, thus reducing the number of necessary parameters in the model. Bath treatments had strong but variable effects for reducing sea lice burdens, and these effects were stronger for adult lice than juvenile lice. Our multivariate state-space models were able to handle different sea lice stages and provide predictions for sea lice abundance with reasonable accuracy up to five weeks out.

Oil prices, exchange rates and stock markets under uncertainty and regime-switching

We contribute to the ongoing literature on the interactions between oil prices, exchange rates and stock markets by considering the effects of economic policy uncertainty (EUP). Based on a VAR and a multivariate Markov switching vector autoregressive (MS-VAR) models, we show (i) significant interrelations between currency, oil and stock markets; (ii) relationships between the variables are rather non-linear; (iii) links between the variables change from one regime to the next, but they are stronger during volatile periods; and (iv) oil plays an active role in the transmission of price shocks to both the exchange rate and stock markets.

BAYESIAN ANALYSIS OF DYNAMIC FACTOR MODELS USING MULTIVARIATE T DISTRIBUTION

The multivariate t models are symmetric and have heavier tail than the normal distribution and produce robust inference procedures for applications. In this paper, the Bayesian estimation of a dynamic factor model is presented, where the factors follow a multivariate autoregressive model, using the multivariate t distribution. Since the multivariate t distribution is complex, it was represented in this work as a mix of the multivariate normal distribution and a square root of a chi-square distribution. This method allowed the complete dene of all the posterior distributions. The inference on the parameters was made taking a sample of the posterior distribution through a Gibbs Sampler. The convergence was veried through graphical analysis and the convergence diagnostics of Geweke (1992) and Raftery and Lewis (1992).