Bayesian Hierarchical Space-time Model Research Articles

Spatiotemporal heterogeneity and long-term impact of meteorological, environmental, and socio-economic factors on scrub typhus in China from 2006 to 2018.

Large-scale outbreaks of scrub typhus combined with its emergence in new areas as a vector-borne rickettsiosis highlight the ongoing neglect of this disease. This study aims to explore the long-term changes and regional leading factors of scrub typhus in China, with the goal of providing valuable insights for disease prevention and control. This study utilized a Bayesian space-time hierarchical model (BSTHM) to examine the spatiotemporal heterogeneity of scrub typhus and analyze the relationship between environmental factors and scrub typhus in southern and northern China from 2006 to 2018. Additionally, a GeoDetector model was employed to assess the predominant influences of geographical and socioeconomic factors in both regions. Scrub typhus exhibits a seasonal pattern, typically occurring during the summer and autumn months (June to November), with a peak in October. Geographically, the high-risk regions, or hot spots, are concentrated in the south, while the low-risk regions, or cold spots, are located in the north. Moreover, the distribution of scrub typhus is influenced by environment and socio-economic factors. In the north and south, the dominant factors are the monthly normalized vegetation index (NDVI) and temperature. An increase in NDVI per interquartile range (IQR) leads to a 7.580% decrease in scrub typhus risk in northern China, and a 19.180% increase in the southern. Similarly, of 1 IQR increase in temperature reduces the risk of scrub typhus by 10.720% in the north but increases it by 15.800% in the south. In terms of geographical and socio-economic factors, illiteracy rate and altitude are the key determinants in the respective areas, with q-values of 0.844 and 0.882. These results indicated that appropriate climate, environment, and social conditions would increase the risk of scrub typhus. This study provided helpful suggestions and a basis for reasonably allocating resources and controlling the occurrence of scrub typhus.

Progress towards achieving child survival goals in Kenya after devolution: Geospatial analysis with scenario-based projections, 2015-2025.

Subnational projections of under-5 mortality (U5M) have increasingly become an essential planning tool to support Sustainable Development Goals (SDGs) agenda and strategies for improving child survival. To support child health policy, planning, and tracking child development goals in Kenya, we projected U5M at units of health decision making. County-specific annual U5M were estimated using a multivariable Bayesian space-time hierarchical model based on intervention coverage from four alternate intervention scale-up scenarios assuming 1) the highest subnational intervention coverage in 2014, 2) projected coverage based on the fastest county-specific rate of change observed in the period between 2003-2014 for each intervention, 3) the projected national coverage based on 2003-2014 trends and 4) the country-specific targets of intervention coverage relative to business as usual (BAU) scenario. We compared the percentage change in U5M based on the four scale-up scenarios relative to BAU and examined the likelihood of reaching SDG 3.2 target of at least 25 deaths/1,000 livebirths by 2022 and 2025. Projections based on 10 factors assuming BAU, showed marginal reductions in U5M across counties with all the counties except Mandera county not achieving the SDG 3.2 target by 2025. Further, substantial reductions in U5M would be achieved based on the various intervention scale-up scenarios, with 63.8% (30), 74.5% (35), 46.8% (22) and 61.7% (29) counties achieving SDG target for scenarios 1,2,3 and 4 respectively by 2025. Scenario 2 yielded the highest reductions of U5M with individual scale-up of access to improved water, recommended treatment of fever and accelerated HIV prevalence reduction showing considerable impact on U5M reduction (≥ 20%) relative to BAU. Our results indicate that sustaining an ambitious intervention scale-up strategy matching the fastest rate observed between 2003-2014 would substantially reduce U5M in Kenya. However, despite this ambitious scale-up scenario, 25% (12 of 47) of the Kenya's counties would still not achieve SDG 3.2 target by 2025.

Mapping and characterizing areas with high levels of malaria in pregnancy in Brazil: A spatiotemporal analysis

Malaria in pregnancy (MiP) is a public health problem in the Brazilian Amazon region that requires special attention due to associated serious adverse consequences, such as low birth weight, increased prematurity and spontaneous abortion rates. In Brazil, there have been no comprehensive epidemiological studies of MiP. In this study, we aimed to explore the spatial and spatiotemporal patterns of MiP in Brazil and epidemiologically characterize this population of pregnant women over a period of 15 years. We performed a national-scale ecological analysis using a Bayesian space-time hierarchical model to estimate the incidence rates of MiP between 1 January 2004 and 31 December 2018. We mapped the high-incidence clusters among pregnant women aged 10-49 years old using a Poisson model, and we characterized the population based on data from the Epidemiological Surveillance Information System for Malaria (SIVEP-Malaria). We consolidated the data of 61,833 women with MiP in Brazil. Our results showed a reduction of 50·1% (95% CI: 47·3 to 52·9) in the number of malaria cases over the period analysed, with Plasmodium vivax malaria having the highest incidence. MiP was widely distributed throughout the Amazon region, and spatial and spatiotemporal analyses revealed statistically significant clusters in some municipalities of Amazonas, Acre, Rondônia and Pará. In addition, we observed that younger pregnant women had a higher risk of infection, and the administration of appropriate treatment requires more attention. This nationwide study provides robust evidence that, despite the reduction in the number of MiP cases in the country, it remains a serious public health problem, especially for young pregnant women. Our analyses highlight focus areas for strengthening interventions to control and eliminate MiP. FAPESP and CNPq - Brazil.

Association between Global Air Pollution and COVID-19 Mortality: A Study of Forty-Six Cities in the World

Ambient air pollution plays a significant role in an increased risk of incidence and mortality of COVID-19 on a global scale. This study aims to understand the multiscale spatial effect of global air pollution on COVID-19 mortality. Based on forty-six cities from six countries worldwide between 1 April 2020 and 31 December 2020, a Bayesian space–time hierarchical model was used based on the lag effects of seven, fourteen, and twenty-one days to quantify the relative risks of NO2 and PM2.5 on the daily death rates of COVID-19, accounting for the effect of meteorological and human mobility variability based on global and city level. Results show that positive correlations between air pollution and COVID-19 mortality are observed, with the relative risks of NO2 and PM2.5 ranging from 1.006 to 1.014 and from 1.002 to 1.004 with the lag effects of seven, fourteen, and twenty-one days. For the individual city analysis, however, both positive and negative associations are found between air pollution and daily mortality, showing that the relative risks of NO2 and PM2.5 are between 0.754 and 1.245 and between 0.888 and 1.032, respectively. The discrepancies in air pollution risks among cities were demonstrated in this study and further allude to the necessity to explore the uncertainty in the multiscale air pollution–mortality relationship.

Mapping disparities in homicide trends across Brazil: 2000\u20132014

BackgroundHomicides are a major problem in Brazil. Drugs and arms trafficking, and land conflicts are three of the many factors driving homicide rates in Brazil. Understanding long-term spatiotemporal trends and social structural factors associated with homicides in Brazil would be useful for designing policies aimed at reducing homicide rates.MethodsWe obtained data from 2000 to 2014 from the Brazil Ministry of Health (MOH) Mortality Information System and sociodemographic data from the Brazil Institute of Geography and Statistics (IBGE). First, we quantified the rate of change in homicides at the municipality and state levels. Second, we used principal component regression and k-medoids clustering to examine differences in temporal trends across municipalities. Lastly, we used Bayesian hierarchical space-time models to describe spatio-temporal patterns and to assess the contribution of structural factors.ResultsThere were significant variations in homicide rates across states and municipalities. We noted the largest decrease in homicide rates in the western and southeastern states of Sao Paulo, Rio de Janeiro and Espirito Santo, which coincided with an increase in homicide rates in the northeastern states of Ceará, Alagoas, Paraiba, Rio Grande Norte, Sergipe and Bahia during the fifteen-year period. The decrease in homicides in municipalities with populations of at least 250,000 coincided with an increase in municipalities with 25,000 people or less. Structural factors that predicted municipality-level homicide rates included crude domestic product, urbanization, border with neighboring countries and proportion of population aged fifteen to twenty-nine.ConclusionsOur findings support both a dissemination hypothesis and an interiorization hypothesis. These findings should be considered when designing interventions to curb homicide rates.

Relationships Between Opioid-Related Hospitalizations and Intimate Partner Violence and Child Maltreatment Hospitalizations in Pennsylvania Across Space and Time.

Intimate partner violence (IPV) and child maltreatment outcomes are markedly associated with substance abuse disorders. However, few studies have explored these serious family violence outcomes in connection to the opioid epidemic or population-level geographic connections between these variables. This study assesses associations of ZIP code-level IPV and child maltreatment hospitalization outcomes with opioid- and alcohol-related diagnoses as well as economic and demographic neighborhood characteristics. We used 11 years (2004-2014) of ZIP code-level Pennsylvania hospital discharge data and U.S. Census neighborhood characteristics data. As nearby ZIP codes are more likely to be similar than those that are distant, we incorporated spatial autocorrelation using conditionally autoregressive Bayesian hierarchical space-time models. There was a positive relationship between ZIP code-level opioid-related diagnoses and both IPV (relative risk 1.061; 95% credible interval [1.015, 1.106]) and child maltreatment (relative risk 1.055; 95% credible interval [1.035, 1.070]) hospitalizations. There was a positive relationship between alcohol-related diagnoses and IPV but not child maltreatment. Higher median household incomes were associated with lower counts of both IPV and child maltreatment hospitalizations. To illustrate geographic heterogeneity of model estimates, posterior distributions were used to compare variability of effects across ZIP codes. Our findings emphasize the secondary implications of the opioid epidemic in the form of family violence within communities.

Understanding the continuous phenological development at daily time step with a Bayesian hierarchical space-time model: impacts of climate change and extreme weather events

The impacts of climate change and extreme weather events (e.g. frost-, heat-, drought-, and heavy rainfall events) on the continuous phenological development over the entire seasonal cycle remained poorly understood. Previous studies mainly focused on modeling key phenological transition dates (e.g. discrete timing of spring bud-break and fall senescence) based on aggregated climate variables (e.g. mean temperature, growing-degree days). We developed and evaluated a Bayesian Hierarchical Space-Time model for Land Surface Phenology (BHST-LSP) to synthesize remotely sensed vegetation greenness with climate covariates at a daily temporal scale from 1981 to 2014 across the entire conterminous United States. The BHST-LSP model incorporated both temporal and spatial information and exhibited high predictive power in simulating daily phenological development with an overall out-of-sample R2 of 0.80 ± 0.17 and 0.72 ± 0.20 for spring and fall phenology, respectively. The overall out-of-sample normalized root mean square errors were 9.3% ± 6.1% and 9.9% ± 5.2% between the observed and predicted vegetation greenness for spring and fall phenology, respectively. We found that a fast increase of temperature can accelerate the speed of spring green-up while a slow decrease of temperature can lead to a decelerated fall brown-down. Increasing accumulated precipitation can benefit daily phenological development over an entire growing season, while extreme rainfall events can have the opposite effects. More frequent frost events could slow spring leaf expansion and accelerate fall leaf senescence. Impacts of extreme heat events were complex and depended on water availability. Cropland in the Midwest as well as evergreen needleleaf forest along the coastal regions showed relatively strong resistance to drought events compared to other land cover types. The BHST-LSP model can be used to forecast vegetation phenology given future climate projection, thus providing valuable information for adopting climate change adaptation and mitigation measures.

Variability and stability in child maltreatment risk across time and space and its association with neighborhood social & housing vulnerability in New Mexico: A bayesian space-time model

BackgroundModeling the spatio-temporal characteristics of substantiated child maltreatment risk has significant implications for child welfare policy. ObjectiveThis study quantifies the spatiotemporal risk of child abuse and neglect in New Mexico at the census tract level over 9 years, identifies areas of increased risk, and evaluates the role of multiple measures of social and housing insecurity on substantiated child maltreatment referrals. Participants and SettingChild maltreatment substantiation data across 499 census tracts from 2007 to 2015 were obtained from the New Mexico Department of Public Health. MethodsSubstantiated referral counts were analyzed within census tracts with Bayesian hierarchical space-time models using Laplace approximation. Standardized incidence ratios, spatial risk, and probability exceedances were calculated and mapped. ResultsMultiple neighborhood structural factors were associated with an increased risk of substantiated child maltreatment, including the eviction rate (Incidence Density Ratio [IDR] = 1.09 [95 % CrI = 1.01–1.12]), rent burden (IDR = 1.11 [95 % CrI = 1.01–1.13]), urban tracts (IDR = 1.36 [95 % CrI = 1.05–1.77]), food desert tracts (IDR = 1.21 [95 % CrI = 1.04–1.41]), low income tracts (IDR = 1.27 [95 % CrI = 1.09–1.49]), percent of households with no vehicle access ([IDR] = 1.27 [95 % CrI = .247–6.47]), and percent of persons with a disability (IDR = 1.05 [95 % CrI = 1.03–1.06]). The racial/ethnic diversity ratio, however, was associated with lower incidence of child maltreatment allegation risk (IDR = .988 [95 % CrI = .982–.995]). ConclusionsPopulation-based child abuse and neglect prevention and intervention efforts should be aided by the characteristics of neighborhoods that demonstrate strong spatial patterns of household and housing vulnerability, particularly in low income, racially segregated neighborhoods.

Trends and risk factors of gastric cancer in Iran (2005–2010)

Background:Gastric cancer (GC) is the first and the third prevalent cancer among males and females in Iran, respectively. The aim of this study was mainly to identify high-risk areas of GC by assessing the spatial and temporal pattern of incidence, and second, to explore some risk factors of GC in ecological setting.Methods:In this cross-sectional ecological study we used Bayesian hierarchical space-time model to measure the relative risk and temporal trends of GC in Iran from 2005 to 2010 based on available data. Data analysis was done by the use of integrated nested Laplace approximation Bayesian approach in R software.Results:Overall trend of GC was significantly decreasing during the study period. Ardabil, Khorasan Razavi, West Azarbaijan, Zanjan, and Mazandaran provinces had the highest risk of incidence. Overweight and smoking were directly and significantly associated with GC risk.Conclusions:During the study period, GC has decreased in Iran. Nevertheless, GC risk was generally high in Northern and Northwestern provinces of Iran. Different health policies according to GC risk and trend are required for each province. Improvements in screening and education programs and conducting further epidemiological studies could help to reduce the incidence of GC in high risk provinces.

A Bayesian Space–Time Hierarchical Model for Remotely Sensed Lattice Data Based on Multiscale Homogeneous Statistical Units

The Bayesian hierarchical model has the outstanding capacity to combine prior knowledge with observations, as well as to generate rich spatiotemporal patterns. However, this advanced method has limited use in remotely sensed lattice data applications due to the large computational burden and unreasonable statistical inferences based on statistical units with a fixed scale. This paper presents a multiscale spatial homogeneous subdivision method and develops a Bayesian space–time hierarchical model (BSTHM) for remotely sensed lattice data. This can solve the above-mentioned limitations by constructing multiscale spatial homogeneous units with good statistical properties. The quantitative criteria for subdivision are provided. The outcome of the BSTHM is not only more reasonable in theory but also much easier to interpret; meanwhile, the computational efficiency is also considerably improved. This novel approach and its merits are illustrated by a case study that examines the spatiotemporal variation of PM2.5 pollution in Asia from 2000 to 2014 using remotely sensed data describing PM2.5 annual mean concentrations. Overall spatial pattern, common time trend, and local variation trend were decomposed and quantificationally estimated from an intricate spatiotemporal process.

Spatiotemporal Risk of Bacillary Dysentery and Sensitivity to Meteorological Factors in Hunan Province, China

Bacillary dysentery remains a public health concern in the world. Hunan Province is one of the provinces having the highest risk of bacillary dysentery in China, however, the spatial-temporal distribution, variation of bacillary dysentery and sensitivity to meteorological factors in there are unclear. In this paper, a Bayesian space-time hierarchical model (BSTHM) was used to detect space-time variation, and effects of meteorological factors between 2010 and 2015. The risk of bacillary dysentery showed apparent spatial-temporal heterogeneity. The highest risk occurred in the summer season. Economically undeveloped mountainous areas in the west and south of the province had the highest incidence rates. Twenty three (18.9%) and 20 (16.4%) counties were identified as hot and cold spots, respectively. Among the hotspots, 11 counties (47.8%) exhibited a rapidly decreasing trend, suggesting they may become low-risk areas in the future. Of the cold spot counties, six (30%) showed a slowly decreasing trend, and may have a higher risk in the future. Among meteorological factors, air temperature, relative humidity, and wind speed all played a significant role in the spatial-temporal distribution of bacillary dysentery risk. These findings can contribute to the implementation of an early warning system for controlling and preventing bacillary dysentery.

Associations among plasma metabolite levels and short-term exposure to PM2.5 and ozone in a cardiac catheterization cohort

RationaleExposure to ambient particulate matter (PM) and ozone has been associated with cardiovascular disease (CVD). However, the mechanisms linking PM and ozone exposure to CVD remain poorly understood. ObjectiveThis study explored associations between short-term exposures to PM with a diameter <2.5μm (PM2.5) and ozone with plasma metabolite concentrations. Methods and resultsWe used cross-sectional data from a cardiac catheterization cohort at Duke University, North Carolina (NC), USA, accumulated between 2001 and 2007. Amino acids, acylcarnitines, ketones and total non-esterified fatty acid plasma concentrations were determined in fasting samples. Daily concentrations of PM2.5 and ozone were obtained from a Bayesian space-time hierarchical model, matched to each patient's residential address. Ten metabolites were selected for the analysis based on quality criteria and cluster analysis. Associations between metabolites and PM2.5 or ozone were analyzed using linear regression models adjusting for long-term trend and seasonality, calendar effects, meteorological parameters, and participant characteristics.We found delayed associations between PM2.5 or ozone and changes in metabolite levels of the glycine-ornithine-arginine metabolic axis and incomplete fatty acid oxidation associated with mitochondrial dysfunction. The strongest association was seen for an increase of 8.1μg/m3 in PM2.5 with a lag of one day and decreased mean glycine concentrations (−2.5% [95% confidence interval: −3.8%; −1.2%]). ConclusionsShort-term exposures to ambient PM2.5 and ozone is associated with changes in plasma concentrations of metabolites in a cohort of cardiac catheterization patients. Our findings might help to understand the link between air pollution and cardiovascular disease.

Environmental predictors of stunting among children under-five in Somalia: cross-sectional studies from 2007 to 2010.

BackgroundStunting among children under five years old is associated with long-term effects on cognitive development, school achievement, economic productivity in adulthood and maternal reproductive outcomes. Accurate estimation of stunting and tools to forecast risk are key to planning interventions. We estimated the prevalence and distribution of stunting among children under five years in Somalia from 2007 to 2010 and explored the role of environmental covariates in its forecasting.MethodsData from household nutritional surveys in Somalia from 2007 to 2010 with a total of 1,066 clusters covering 73,778 children were included. We developed a Bayesian hierarchical space-time model to forecast stunting by using the relationship between observed stunting and environmental covariates in the preceding years. We then applied the model coefficients to environmental covariates in subsequent years. To determine the accuracy of the forecasting, we compared this model with a model that used data from all the years with the corresponding environmental covariates.ResultsRainfall (OR = 0.994, 95 % Credible interval (CrI): 0.993, 0.995) and vegetation cover (OR = 0.719, 95 % CrI: 0.603, 0.858) were significant in forecasting stunting. The difference in estimates of stunting using the two approaches was less than 3 % in all the regions for all forecast years.ConclusionStunting in Somalia is spatially and temporally heterogeneous. Rainfall and vegetation are major drivers of these variations. The use of environmental covariates for forecasting of stunting is a potentially useful and affordable tool for planning interventions to reduce the high burden of malnutrition in Somalia.Electronic supplementary materialThe online version of this article (doi:10.1186/s12889-016-3320-6) contains supplementary material, which is available to authorized users.

Erik Vanem: Bayesian hierarchical space-time models with application to significant wave height

1 Forristall Ocean Engineering, Inc., 101 Chestnut St., Camden, ME 04843, USA rithms for estimating the necessary integrals has been crucial in these applications. The core of this book is the development of a space-time model for significant wave heights. The data come from a wave hindcast for the area of the North Atlantic between Iceland and theUnitedKingdom. Spatial dependence of the data is modeled using a Markov random field. The time evolution is modeled as a Markov chain. Because the main motivation of the study is to determine long-term trends in the data, linear and quadratic terms are explicitly included in the temporal model. Most of the priors are specified as simple Gaussian distributions. The integrals necessary for theBayesian update are calculated using Markov Chain Monte Carlo techniques. These techniques are explained in a detailedAppendix.Other Appendices describe extreme value modeling, Markov random fields, and sampling from a multi-normal distribution. These Appendices are the most useful part of the book. But it would have been helpful to novices tomore fully describe the path from the probability integrals to Bayesian simulations. The book beginswith a literature reviewof 215 references. The review is broad but not deep. Most of the references are described in a sentence or two, and the connections between references are not clear. For example, forecasts of significant wave height are mixed up with statistics of individual waves in a section on short-term models. Many variations of the model were run. As is generally true, changing the prior distributions had very little effect on the results. The basic model sums the time-independent, space-time, seasonal, and secular time components. Taking a log transform of the data leads to a multiplicative model. The author was unable to find a convincing argument to prefer one model over the others. But almost all of the models show an increasing trend of significant wave height over the 42years of hindcast data. The modeled increase in monthly maximum wave heights is about 70cm. The author is reluc-

The sun-hurricane connection: Diagnosing the solar impacts on hurricane frequency over the North Atlantic basin using a space–time model

The authors define a spatio-statistical response of hurricane frequency to the solar cycle. Previous research indicates reduced (increased) hurricane intensities and frequency in the western (eastern) tropical Atlantic. However, no formal quantitative relationship has been spatially established between hurricane frequency and solar activity. The authors use a Bayesian hierarchical space–time model, an increasingly popular approach due to its advantage in facilitating regression modeling of space–time phenomena in the context of large data sets. Regional hurricane frequency over the period 1866–2010 is examined in response to September sunspot number (SSN) while controlling for other relevant climate factors. The response features a 13 % reduction in probability of annual hurricane occurrence for southeastern Cuba, the southern Bahama islands, Haiti, and Jamaica when the SSN is 80 sunspots. In contrast, hurricane risk in regions of the southeastern Atlantic is predicted to increase by 73 % when the SSN is 160 sunspots. The model can be ported to explore other relationships over contiguous space.

Associations between short-term exposure to PM2.5 and metabolite levels

Background: Air pollutants such as particulate matter (PM) have been associated with cardiovascular disease (CVD). However, the mechanisms linking PM exposure to CVD remain poorly understood. Aims: This study aims to explore associations between short-term exposures to PM with a diameter &lt; 2.5 µm (PM2.5) and changes in the plasma concentrations of metabolites. Methods: We used cross-sectional data from a large CVD cohort in North Carolina (NC), USA, between 2001 and 2008. Blood samples were taken from individuals undergoing coronary artery catheterization at a medical center. Mass spectrometric profiling of 60 amino acids and acylcarnitines was performed in fasting plasma samples. Daily average concentrations of PM2.5 mass in each NC census tract were obtained from a Bayesian space-time hierarchical model that fuses observed and modeled PM2.5 concentrations. Residential addresses were given the exposure as estimated at the closest census tract centroid, measured by perpendicular distance. Daily meteorological data were obtained from a monitoring network. Associations between metabolites and PM2.5 were analyzed using linear regression models adjusting for long-term trend and seasonality, calendar effects, meteorological parameters, age, gender and smoking. Results: Data for analysis were available from 2707 individuals. Preliminary results point to associations between increased PM2.5 levels and changes in eight metabolites with a delay of one to four days. For example, an increase of 10µg/m3 in PM2.5 with a lag of two days was associated with a 9.3% increase in mean plasma methionine concentrations (95% confidence interval [95% CI]: 3.2%-15.5%), corresponding to an absolute increase of 2.29 µM (95% CI: 0.79-3.78 µM). This association remained significant after correcting for multiple testing. Conclusions: Results suggest that short-term exposure to ambient PM is associated to changes in plasma concentrations of metabolites in a population of patients with CVD. This proposed abstract does not necessarily reflect EPA policy.

Bayesian hierarchical spatio-temporal modelling of trends and future projections in the ocean wave climate with a $$\text{ CO }_2$$ CO 2 regression component

Bad weather and rough seas continue to be a major cause for ship losses and is thus a significant contributor to the risk to maritime transportation. This stresses the importance of taking severe sea state conditions adequately into account, with due treatment of the uncertainties involved, in ship design and operation in order to enhance safety. Hence, there is a need for appropriate stochastic models describing the variability of sea states. These should also incorporate realistic projections of future return levels of extreme sea states, taking into account long-term trends related to climate change and inherent uncertainties. The stochastic ocean wave model presented in this paper exploits the flexible framework of Bayesian hierarchical space-time models. It allows modelling of complex dependence structures in space and time and incorporation of physical features and prior knowledge, yet at the same time remains intuitive and easily interpreted. Furthermore, by taking a Bayesian approach, the uncertainties of the model parameters are also taken into account. A regression component with $$\text{ CO }_2$$ as an explanatory variable has been introduced in order to extract long-term trends in the data. The model has been fitted by monthly maximum significant wave height data for an area in the North Atlantic ocean. The different components of the model will be outlined in the paper, and the results will be discussed. Furthermore, a discussion of possible extensions to the model will be given.

Bayesian hierarchical modelling of North Atlantic windiness

Abstract. Extreme weather conditions represent serious natural hazards to ship operations and may be the direct cause or contributing factor to maritime accidents. Such severe environmental conditions can be taken into account in ship design and operational windows can be defined that limits hazardous operations to less extreme conditions. Nevertheless, possible changes in the statistics of extreme weather conditions, possibly due to anthropogenic climate change, represent an additional hazard to ship operations that is less straightforward to account for in a consistent way. Obviously, there are large uncertainties as to how future climate change will affect the extreme weather conditions at sea and there is a need for stochastic models that can describe the variability in both space and time at various scales of the environmental conditions. Previously, Bayesian hierarchical space-time models have been developed to describe the variability and complex dependence structures of significant wave height in space and time. These models were found to perform reasonably well and provided some interesting results, in particular, pertaining to long-term trends in the wave climate. In this paper, a similar framework is applied to oceanic windiness and the spatial and temporal variability of the 10-m wind speed over an area in the North Atlantic ocean is investigated. When the results from the model for North Atlantic windiness is compared to the results for significant wave height over the same area, it is interesting to observe that whereas an increasing trend in significant wave height was identified, no statistically significant long-term trend was estimated in windiness. This may indicate that the increase in significant wave height is not due to an increase in locally generated wind waves, but rather to increased swell. This observation is also consistent with studies that have suggested a poleward shift of the main storm tracks.

Stochastic modelling of long-term trends in the wave climate and its potential impact on ship structural loads

This paper presents a stochastic model in space and time for significant wave height, a Bayesian hierarchical space-time model. The model consists of different components in a hierarchical manner including a component to model the contribution from long-term trends in the wave climate. As far as the authors are aware, no such model of significant wave height to date exploits the flexible framework of Bayesian hierarchical space-time models, which allow modelling of complex dependence structures in space and time and incorporation of physical features and prior knowledge, yet at the same time remains intuitive and easily interpreted. Furthermore, including a trend component in the model is a novel feature.The model presented in this paper has been fitted to significant wave height data for monthly maxima over an area in the North Atlantic ocean, and aims at describing the temporal and spatial variability of the data over a period of more than 44 years for the chosen area. In particular, the model identifies long-term trends present in the data. Subsequently, it will be explored how the results from the model can be linked to structural loads and response calculations. The proposed approach is illustrated by an example showing the potential impact of the estimated long-term trends of significant wave height on the wave-induced structural loads of an oil tanker.

Modelling the effect of climate change on the wave climate of the world’s oceans

This paper analyses the trends and the future projections of significant wave height in several ocean areas at different parts of the world. It uses a stochastic Bayesian hierarchical space-time model, with a regression component with atmospheric levels of CO2 as covariates in order to estimate the expected long-term trends and make future projections towards the year 2100. The model was initially developed for an area in the North Atlantic ocean, and has been found to perform reasonably well there, and it is now investigated how the model performs for other ocean areas. 11 new ocean areas have been analysed with the model, and this paper presents the results pertaining to the estimated long-term trends and future projections of monthly maximum significant wave height for each of the 12 ocean areas.