Multivariate Conditional Autoregressive Research Articles

Predicting Returns, Volatilities and Correlations of Stock Indices Using Multivariate Conditional Autoregressive Range and Return Models

This paper extends the conditional autoregressive range (CARR) model to the multivariate CARR (MCARR) model and further to the two-stage MCARR-return model to model and forecast volatilities, correlations and returns of multiple financial assets. The first stage model fits the scaled realised Parkinson volatility measures using individual series and their pairwise sums of indices to the MCARR model to obtain the fitted volatilities. Then covariances are calculated to construct the fitted variance–covariance matrix of returns which are imputed into the stage-two return model to capture the heteroskedasticity of assets’ returns. We investigate different choices of mean functions to describe the volatility dynamics. Empirical applications are based on the Standard and Poor 500, Dow Jones Industrial Average and Dow Jones United States Financial Service Indices. Results show that the stage-one MCARR models using asymmetric mean functions give better in-sample model fits than those based on symmetric mean functions. They also provide better out-of-sample volatility forecasts than those using CARR models based on two robust loss functions. We also find that the stage-two return models with constant means and multivariate Student-t errors give better in-sample fits than the Baba–Engle–Kraft–Kroner generalised autoregressive conditional heteroskedasticity models. The estimates and forecasts of value-at-risk (VaR) and conditional VaR based on the best MCARR-return models for each asset are provided and tested using backtests to confirm the accuracy of the VaR forecasts.

Effect of transit-oriented design on pedestrian and cyclist safety using bivariate spatial models

Introduction: Walking and cycling for transportation provide immense benefits (e.g., health, environmental, social). However, pedestrians and bicyclists are the most vulnerable segment of the traveling public due to the lack of protective structure and difference in body mass compared with motorized vehicles. Numerous studies are dedicated to enhancing active transportation modes, but very few studies are devoted to the safety analysis of the transit stops, which serve as the important modal interface for pedestrians and bicyclists. Method: This study bridges the gap by developing joint models based on the multivariate conditional autoregressive (MCAR) priors with distance-oriented neighboring weight matrix. For this purpose, transit-oriented design (TOD) related data in Los Angeles County were used for model development. Feature selection relying on both random forest (RF) and correlation analysis was employed, which leads to different covariates inputs to each of the two joint models, resulting in increased model flexibility. An integrated nested Laplace approximation (INLA) algorithm was adopted due to its fast, yet robust, analysis. For a comprehensive comparison of the predictive accuracy of models, different evaluation criteria were utilized. Results: The results demonstrate that models with correlation effect perform much better than the models without a correlation of pedestrians and bicyclists. The joint models also aid in the identification of the significant covariates contributing to the safety of each of the two active transportation modes. The findings show that population density, employment density, and bus stop density positively influence bicyclist-involved crashes, suggesting that an increase in population, employment, or the number of bus stops leads to more active modes involved collisions. Practical Applications: The findings of this study may prove helpful in the development and implementation of the safety management process to improve the roadway environment for the active modes in the long run.

Risk factors of concurrent malnutrition among children in Ethiopia: a bivariate spatial modeling approach

Malnutrition among children remains a challenge to public health in Africa, and it is responsible for the increased infant mortality rate in Ethiopia. This study aims to identify the joint determinant factors of malnutrition among under-age five children in Ethiopia and quantify the regional prevalence across the country. A Bayesian hierarchical linear mixed model, with a bivariate conditional autoregressive model, was adopted to account for the joint spatial prevalence pattern of malnutrition among under-age five children in Ethiopia. The data were provided by the Demography and Health Surveys program. The study revealed that gender, wealth index, mothers' education, toilet system, fever, birth order, birth interval, contraceptive use, and diarrhea, are significant risk factors of child malnutrition. The three malnutrition indicators are most prevalent in the Afar region, while stunting and underweight are most prevalent in Amhara, Beneshangul Gumuz, and Tigray, and the wasting and underweight are most prevalent in the Gambela and Somali. Findings revealed that a stunted child is more likely to be underweight than being wasted, and a wasted child is more likely to be underweight than being stunted. The findings also revealed significant changes in the prevalence of malnutrition for different mothers and children age groups. For cost-effective malnutrition intervention, programs for parents, such as awareness creation about the modern types of contraceptives, appropriate birth spacing, the benefits of antenatal and postnatal care, for example, are a propitious method to mitigate malnutrition in high prevalent regions than direct intervention to the children. Abbreviations: BICAR, Bivariate Conditional Autoregressive; DHS, Demographic and Health Survey; EDHS, Ethiopia Demographic and Health Survey; EA, Enumeration Area; GMRF, Gaussian Markov Random Field; HAZ, Height for Age Z-sscore; INLA, Integrated Nexted Laplace Approximation; MCAR, Multivariate Conditional Autoregressive; MCMC, Markov Chain Monte Carlo; SNNPR, Southern Nations, Nationalities, and People's Region; WAZ, Weight for Age Z-score; WHZ, Weight for Height Z-score; WHO, World Health Organization

Markov modelling of viral load adjusting for CD4 orthogonal variable and multivariate conditional autoregressive mapping of the HIV immunological outcomes among ART patients in Zimbabwe

BackgroundThis study aimed to jointly model HIV disease progression patterns based on viral load (VL) among adult ART patients adjusting for the time-varying “incremental transients states” variable, and the CD4 cell counts orthogonal variable in a single 5-stage time-homogenous multistate Markov model. We further jointly mapped the relative risks of HIV disease progression outcomes (detectable VL (VL ≥ 50copies/uL) and immune deterioration (CD4 < 350cells/uL) at the last observed visit) conditional not to have died or become loss to follow-up (LTFU).MethodsSecondary data analysis of individual-level patients on ART was performed. Adjusted transition intensities, hazard ratios (HR) and regression coefficients were estimated from the joint multistate model of VL and CD4 cell counts. The mortality and LTFU transition rates defined the extent of patients’ retention in care. Joint mapping of HIV disease progression outcomes after ART initiation was done using the Bayesian intrinsic Multivariate Conditional Autoregressive prior model.ResultsThe viral rebound from the undetectable state was 1.78times more likely compared to viral suppression among patients with VL ranging from 50-1000copies/uL. Patients with CD4 cell counts lower than expected had a higher risk of viral increase above 1000copies/uL and death if their VL was above 1000copies/uL (state 2 to 3 (λ23): HR = 1.83 and (λ34): HR = 1.42 respectively). Regarding the time-varying effects of CD4 cell counts on the VL transition rates, as the VL increased, (λ12 and λ23) the transition rates increased with a decrease in the CD4 cell counts over time. Regardless of the individual’s VL, the transition rates to become LTFU decreased with a decrease in CD4 cell counts. We observed a strong shared geographical pattern of 66% spatial correlation between the relative risks of detectable VL and immune deterioration after ART initiation, mainly in Matabeleland North.ConclusionWith high rates of viral rebound, interventions which encourage ART adherence and continual educational support on the barriers to ART uptake are crucial to achieve and sustain viral suppression to undetectable levels. Area-specific interventions which focus on early ART screening through self-testing, behavioural change campaigns and social support strategies should be strengthened in heavily burdened regions to sustain the undetectable VL. Sustaining undetectable VL lowers HIV transmission in the general population and this is a step towards achieving zero HIV incidences by 2030.

Bayesian Modelling for Spatially Misaligned Health Areal Data: A Multiple Membership Approach

AbstractDiabetes prevalence is on the rise in the United Kingdom, and for public health strategy, estimation of relative disease risk and subsequent mapping is important. We consider an application to London data on diabetes prevalence and mortality. In order to improve the estimation of relative risks, we analyse jointly prevalence and mortality data to ensure borrowing strength over the two outcomes. The available data involve two spatial frameworks, areas (Middle Layer Super Output Areas, MSOAs) and general practices (GPs) recruiting patients from several areas. This raises a spatial misalignment issue that we deal with by employing the multiple membership principle. Specifically, we translate areal spatial effects to explain GP practice prevalence according to proportions of GP populations resident in different areas. A sparse implementation in RStan of both the multivariate conditional autoregressive (MCAR) and generalised MCAR (GMCAR) with multiple membership allows the comparison of these bivariate priors as well as exploring the different implications for the mapping patterns for both outcomes. The necessary causal precedence of diabetes prevalence over mortality allows a specific conditionality assumption in the GMCAR, not always present in the context of disease mapping. Additionally, an area-locality comparison is considered to locate high versus low relative risk clusters.

Does space influence on the frequency and severity of the distraction-affected vehicle crashes? An empirical evidence from the Central Ohio

This study investigates spatial dependencies between frequency and within severity of vehicle crashes caused by distracted driving, along with the role of the built and socio-demographic environments in the Columbus Metropolitan Area, Ohio. We adopt a full Bayesian hierarchical framework with Multivariate Conditional Autoregressive Priors to account for the complex spatial correlation structure as well as the unobserved heterogeneity. Using aggregated crash count data (Property Damage Only and Bodily Injuries) for the 414 census tracts, the analysis outcomes reveal that census tracts providing more jobs and having a higher proportion of commercial land use would have higher likelihood of relative crash risks in both severity levels. Inclusion of correlation structure between frequency as well as within crash-severity-level has proven a significant increase on the performance of the model, verifying influences of space on the frequency and severity of distraction-affected vehicle crashes. In addition, this research presents areas of higher relative risks (spatial clusters) that have 1.5 times elevated risk of collision than other census tracts. The identification of areas of excessive risks informs us to devise policies to mitigate negative consequences of distraction-affected crashes.

Modeling of time-dependent safety performance using anonymized and aggregated smartphone-based dangerous driving event data

Safety performance functions (SPFs) are generally used to relate exposure to the expected number of crashes aggregated over a long time (e.g. a year) by holding all other risk factors constant, and to identify hotspots that have excessive crashes regardless of different time periods. However, it is highly likely that the relationships of exposure, risk factors and crash occurrence can vary across different times of day. This study aims to establish time-dependent SPFs for urban roads by using large-scale dangerous driving event data captured by smartphones in different times of day. Multivariate conditional autoregressive (MVCAR) models are developed to jointly account for spatial and temporal dependence of crash observations. Results of two-sample Kolmogorov-Smirnov tests affirm the heterogeneity of the safety effects of dangerous driving events in different time periods. Time-dependent hotspots are identified using potential for safety improvement (PSI) metric. The assumption here is that due to the change of traffic conditions and environment across different times of day, safety hotspots for different time periods should be different from each other. According to the results of Wilcoxon signed-rank tests, hotspots identified by times of day are found to be mostly different from each other. The findings of this study provide insights into temporal effects of risk factors and can support the development of time-dependent safety countermeasures. Besides, this study also shows the potential of leveraging anonymized and aggregated dangerous driving data to assess traffic safety issues.

A multivariate spatial approach to model crash counts by injury severity

Conventional safety models rely on the assumption of independence of crash data, which is frequently violated. This study develops a novel multivariate conditional autoregressive (MVCAR) model to account for the spatial autocorrelation of neighboring sites and the inherent correlation across different crash types. Manhattan, which is the most densely populated urban area of New York City, is used as the study area. Census tracts are used as the basic geographic units to capture crash, transportation, land use, and demo-economic data. The specification of the proposed multivariate model allows for jointly modeling counts of various crash types that are classified according to injury severity. Results of Moran’s I tests show the ability of the MVCAR model to capture the multivariate spatial autocorrelation among different crash types. The MVCAR model is found to outperform the others by presenting the lowest deviance information criterion (DIC) value. It is also found that the unobserved heterogeneity was mostly attributed to spatial factors instead of non-spatial ones and there is a strong shared geographical pattern of risk among different crash types.

Efficient likelihood computations for some multivariate Gaussian Markov random fields

Data collected from spatial locations are often multivariate. Gaussian conditional autoregressive (CAR) models, also known as Gaussian Markov random fields, are frequently used to analyze such continuous data, or as models for the parameters of discrete distributions. Two difficulties in Gaussian maximum likelihood estimation are ensuring that the parameter estimates are allowable values, and computing the likelihood efficiently. It is shown here that, for some commonly-used multivariate CAR models, checking for allowable parameter values can be facilitated, and the likelihood can be computed very quickly.

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.

Integrating multiple data sources in species distribution modeling: a framework for data fusion.

The last decade has seen a dramatic increase in the use of species distribution models (SDMs) to characterize patterns of species' occurrence and abundance. Efforts to parameterize SDMs often create a tension between the quality and quantity of data available to fit models. Estimation methods that integrate both standardized and non-standardized data types offer a potential solution to the tradeoff between data quality and quantity. Recently several authors have developed approaches for jointly modeling two sources of data (one of high quality and one of lesser quality). We extend their work by allowing for explicit spatial autocorrelation in occurrence and detection error using a Multivariate Conditional Autoregressive (MVCAR) model and develop three models that share information in a less direct manner resulting in more robust performance when the auxiliary data is of lesser quality. We describe these three new approaches ("Shared," "Correlation," "Covariates") for combining data sources and show their use in a case study of the Brown-headed Nuthatch in the Southeastern U.S. and through simulations. All three of the approaches which used the second data source improved out-of-sample predictions relative to a single data source ("Single"). When information in the second data source is of high quality, the Shared model performs the best, but the Correlation and Covariates model also perform well. When the information quality in the second data source is of lesser quality, the Correlation and Covariates model performed better suggesting they are robust alternatives when little is known about auxiliary data collected opportunistically or through citizen scientists. Methods that allow for both data types to be used will maximize the useful information available for estimating species distributions.

Linear models of coregionalization for multivariate lattice data: Order-dependent and order-free cMCARs.

This paper concerns with multivariate conditional autoregressive models defined by linear combination of independent or correlated underlying spatial processes. Known as linear models of coregionalization, the method offers a systematic and unified approach for formulating multivariate extensions to a broad range of univariate conditional autoregressive models. The resulting multivariate spatial models represent classes of coregionalized multivariate conditional autoregressive models that enable flexible modelling of multivariate spatial interactions, yielding coregionalization models with symmetric or asymmetric cross-covariances of different spatial variation and smoothness. In the context of multivariate disease mapping, for example, they facilitate borrowing strength both over space and cross variables, allowing for more flexible multivariate spatial smoothing. Specifically, we present a broadened coregionalization framework to include order-dependent, order-free, and order-robust multivariate models; a new class of order-free coregionalized multivariate conditional autoregressives is introduced. We tackle computational challenges and present solutions that are integral for Bayesian analysis of these models. We also discuss two ways of computing deviance information criterion for comparison among competing hierarchical models with or without unidentifiable prior parameters. The models and related methodology are developed in the broad context of modelling multivariate data on spatial lattice and illustrated in the context of multivariate disease mapping. The coregionalization framework and related methods also present a general approach for building spatially structured cross-covariance functions for multivariate geostatistics.

Spatio-temporal model for crop yield forecasting

ABSTRACTThis paper proposes a linear mixed model (LMM) with spatial effects, trend, seasonality and outliers for spatio-temporal time series data. A linear trend, dummy variables for seasonality, a binary method for outliers and a multivariate conditional autoregressive (MCAR) model for spatial effects are adopted. A Bayesian method using Gibbs sampling in Markov Chain Monte Carlo is used for parameter estimation. The proposed model is applied to forecast rice and cassava yields, a spatio-temporal data type, in Thailand. The data have been extracted from the Office of Agricultural Economics, Ministry of Agriculture and Cooperatives of Thailand. The proposed model is compared with our previous model, an LMM with MCAR, and a log transformed LMM with MCAR. We found that the proposed model is the most appropriate, using the mean absolute error criterion. It fits the data very well in both the fitting part and the validation part for both rice and cassava. Therefore, it is recommended to be a primary model for forecasting these types of spatio-temporal time series data.

Linear models of coregionalization for multivariate lattice data: a general framework for coregionalized multivariate CAR models.

We present a general coregionalization framework for developing coregionalized multivariate Gaussian conditional autoregressive (cMCAR) models for Bayesian analysis of multivariate lattice data in general and multivariate disease mapping data in particular. This framework is inclusive of cMCARs that facilitate flexible modelling of spatially structured symmetric or asymmetric cross-variable local interactions, allowing a wide range of separable or non-separable covariance structures, and symmetric or asymmetric cross-covariances, to be modelled. We present a brief overview of established univariate Gaussian conditional autoregressive (CAR) models for univariate lattice data and develop coregionalized multivariate extensions. Classes of cMCARs are presented by formulating precision structures. The resulting conditional properties of the multivariate spatial models are established, which cast new light on cMCARs with richly structured covariances and cross-covariances of different spatial ranges. The related methods are illustrated via an in-depth Bayesian analysis of a Minnesota county-level cancer data set. We also bring a new dimension to the traditional enterprize of Bayesian disease mapping: estimating and mapping covariances and cross-covariances of the underlying disease risks. Maps of covariances and cross-covariances bring to light spatial characterizations of the cMCARs and inform on spatial risk associations between areas and diseases. Copyright © 2016 John Wiley & Sons, Ltd.

Spatial Temporal Conditional Auto-Regressive Model: A New Autoregressive Matrix

In the study of geographical patterns of disease, multivariate areal data models proposed so far in the literature (Ma and Carlin, 2007; Carlin and Banerjee, 2003; Knorr-Held and Best, 2001) have allowed to handle several features of a phenomenon at the same time. In this paper, we propose a new model for areal data, the Spatial Temporal Conditional Auto-Regressive (STCAR) model, that allows to handle the spatial dependence between sites as well as the temporal dependence among the realizations, in the presence ofmeasurements recorded at each spatial location in a time interval. Inspired by the Generalized Multivariate Conditional Auto-Regressive (GMCAR) model published by Jin, Carlin, and Banerjee (2005), the STCAR model reduces the unknown parameters to the single parameter of spatial association estimated at every period considered. Unlike the Vector Auto-Regressive (VAR) model proposed by Sims (1980), in addition, its space-time autoregressive matrix takes into account the spatial localization of the realizations sampled. Moreover, we already know that the main areas of application of these modelsrelate to disease mapping, disease clustering, ecological analysis (Lawson, Browne, and Vidal Rodeiro, 2003). In this work, however, the STCAR model is applied in business, exploiting the analogy between the danger of contracting a particular disease and the risk of falling into bankruptcy, in order to “reconstruct” the spatial temporal distribution of expected bankruptcies of small and medium enterprises of the province of Lecce (Italy).

A Poisson-lognormal conditional-autoregressive model for multivariate spatial analysis of pedestrian crash counts across neighborhoods

This work examines the relationship between 3-year pedestrian crash counts across Census tracts in Austin, Texas, and various land use, network, and demographic attributes, such as land use balance, residents’ access to commercial land uses, sidewalk density, lane-mile densities (by roadway class), and population and employment densities (by type). The model specification allows for region-specific heterogeneity, correlation across response types, and spatial autocorrelation via a Poisson-based multivariate conditional auto-regressive (CAR) framework and is estimated using Bayesian Markov chain Monte Carlo methods. Least-squares regression estimates of walk-miles traveled per zone serve as the exposure measure. Here, the Poisson-lognormal multivariate CAR model outperforms an aspatial Poisson-lognormal multivariate model and a spatial model (without cross-severity correlation), both in terms of fit and inference.Positive spatial autocorrelation emerges across neighborhoods, as expected (due to latent heterogeneity or missing variables that trend in space, resulting in spatial clustering of crash counts). In comparison, the positive aspatial, bivariate cross correlation of severe (fatal or incapacitating) and non-severe crash rates reflects latent covariates that have impacts across severity levels but are more local in nature (such as lighting conditions and local sight obstructions), along with spatially lagged cross correlation. Results also suggest greater mixing of residences and commercial land uses is associated with higher pedestrian crash risk across different severity levels, ceteris paribus, presumably since such access produces more potential conflicts between pedestrian and vehicle movements. Interestingly, network densities show variable effects, and sidewalk provision is associated with lower severe-crash rates.

Geographic risk modeling of childhood cancer relative to county-level crops, hazardous air pollutants and population density characteristics in Texas

BackgroundChildhood cancer has been linked to a variety of environmental factors, including agricultural activities, industrial pollutants and population mixing, but etiologic studies have often been inconclusive or inconsistent when considering specific cancer types. More specific exposure assessments are needed. It would be helpful to optimize future studies to incorporate knowledge of high-risk locations or geographic risk patterns. The objective of this study was to evaluate potential geographic risk patterns in Texas accounting for the possibility that multiple cancers may have similar geographic risks patterns.MethodsA spatio-temporal risk modeling approach was used, whereby 19 childhood cancer types were modeled as potentially correlated within county-years. The standard morbidity ratios were modeled as functions of intensive crop production, intensive release of hazardous air pollutants, population density, and rapid population growth.ResultsThere was supportive evidence for elevated risks for germ cell tumors and "other" gliomas in areas of intense cropping and for hepatic tumors in areas of intense release of hazardous air pollutants. The risk for Hodgkin lymphoma appeared to be reduced in areas of rapidly growing population. Elevated spatial risks included four cancer histotypes, "other" leukemias, Central Nervous System (CNS) embryonal tumors, CNS other gliomas and hepatic tumors with greater than 95% likelihood of elevated risks in at least one county.ConclusionThe Bayesian implementation of the Multivariate Conditional Autoregressive model provided a flexible approach to the spatial modeling of multiple childhood cancer histotypes. The current study identified geographic factors supporting more focused studies of germ cell tumors and "other" gliomas in areas of intense cropping, hepatic cancer near Hazardous Air Pollutant (HAP) release facilities and specific locations with increased risks for CNS embryonal tumors and for "other" leukemias. Further study should be performed to evaluate potentially lower risk for Hodgkin lymphoma and malignant bone tumors in counties with rapidly growing population.