Using hidden Markov chains and empirical Bayes change-point estimation for transect data

Consider a lattice of locations in one dimension at which data are observed. We model the data as a random hierarchical process. The hidden process is assumed to have a (prior) distribution that is derived from a two-state Markov chain. The states correspond to the mean values (high and low) of the observed data. Conditional on the states, the observations are modelled, for example, as independent Gaussian random variables with identical variances. In this model, there are four free parameters: the Gaussian variance, the high and low mean values, and the transition probability in the Markov chain. A parametric empirical Bayes approach requires estimation of these four parameters from the marginal (unconditional) distribution of the data and we use the EM-algorithm to do this. From the posterior of the hidden process, we use simulated annealing to find the maximum a posteriori (MAP) estimate. Using a Gibbs sampler, we also obtain the maximum marginal posterior probability (MMPP) estimate of the hidden process. We use these methods to determine where change-points occur in spatial transects through grassland vegetation, a problem of considerable interest to plant ecologists.     

Hierarchical spatial modeling for estimation of population size

Estimation of population size has traditionally been viewed from a finite population sampling perspective. Typically, the objective is to obtain an estimate of the total population count of individuals within some region. Often, some stratification scheme is used to estimate counts on subregions, whereby the total count is obtained by aggregation with weights, say, proportional to the areas of the subregions. We offer an alternative to the finite population sampling approach for estimating population size. The method does not require that the subregions on which counts are available form a complete partition of the region of interest. In fact, we envision counts coming from areal units that are small relative to the entire study region and that the total area sampled is a very small proportion of the total study area. In extrapolating to the entire region, we might benefit from assuming that there is spatial structure to the counts. We implement this by modeling the intensity surface as a realization from a spatially correlated random process. In the case of multiple population or species counts, we use the linear model of coregionalization to specify a multivariate process which provides associated intensity surfaces hence association between counts within and across areal units. We illustrate the method of population size estimation with simulated data and with tree counts from a Southwestern pinyon-juniper woodland data set.     

Bayesian spatial prediction

This paper presents a complete Bayesian methodology for analyzing spatial data, one which employs proper priors and features diagnostic methods in the Bayesian spatial setting. The spatial covariance structure is modeled using a rich class of covariance functions for Gaussian random fields. A general class of priors for trend, scale, and structural covariance parameters is considered. In particular, we obtain analytic results that allow easy computation of the predictive distribution for an arbitrary prior on the parameters of the covariance function using importance sampling. The computations, as well as model diagnostics and sensitivity analysis, are illustrated with a set of precipitation data.     

A spatial zero-inflated poisson regression model for oak regeneration

Ecological counts data are often characterized by an excess of zeros and spatial dependence. Excess zeros can occur in regions outside the range of the distribution of a given species. A zero-inflated Poisson regression model is developed, under which the species range is determined by a spatial probit model, including physical variables as covariates. Within that range, species counts are independently drawn from a Poisson distribution whose mean depends on biotic variables. Bayesian inference for this model is illustrated using data on oak seedling counts. Received: May 2004 / Revised: December 2004     

Prediction of non-methane hydrocarbons in Kuwait using regression and Bayesian kriged Kalman model

This article describes the hierarchical Bayesian approach for predicting average hourly concentrations of ambient non-methane hydrocarbons (NMHC) in Kuwait where records of six monitor stations located in different sites are observed at successive time points. Our objective is to predict the concentration level of NMHC in unmonitored areas. Here an attempt is made for the prediction of unmeasured concentration of NMHC at two additional locations in Kuwait. We will implement a kriged Kalman filter (KKF) hierarchical Bayesian approach assuming a Gaussian random field, a technique that allows the pooling of data from different sites in order to predict the exposure of the NMHC in different regions of Kuwait. In order to increase the accuracy of the KKF we will use other statistical models such as imputation, regression, principal components, and time series analysis in our approach. We considered four different types of imputation techniques to address the missing data. At the primary level, the logarithmic field is modeled as a trend plus Gaussian stochastic residual model. The trend model depends on hourly meteorological predictors which are common to all sites. The residuals are then modeled using KKF, and the prediction equation is derived conditioned on adjoining hours. On this basis we developed a spatial predictive distribution for these residuals at unmonitored sites. By transforming the predicted residuals back to the original data scales, we can impute Kuwait’s hourly non-methane hydrocarbons field.     

An integrated approach for predicting fates of reintroductions with demographic data from multiple populations

We devised a novel approach to model reintroduced populations whereby demographic data collected from multiple sites are integrated into a Bayesian hierarchical model. Integrating data from multiple reintroductions allows more precise population-growth projections to be made, especially for populations for which data are sparse, and allows projections that account for random site-to-site variation to be made before new reintroductions are attempted. We used data from reintroductions of the North Island Robin (Petroica longipes), an endemic New Zealand passerine, to 10 sites where non-native mammalian predators are controlled. A comparison of candidate models that we based on deviance information criterion showed that rat-tracking rate (an index of rat density) was a useful predictor of robin fecundity and adult female survival, that landscape connectivity and a binary measure of whether sites were on a peninsula were useful predictors of apparent juvenile survival (probably due to differential dispersal away from reintroduction sites), and that there was unexplained random variation among sites in all demographic rates. We used the two best supported models to estimate the finite rate of increase (λ) for populations at each of the 10 sites, and for a proposed reintroduction site, under different levels of rat control. Only three of the reintroduction sites had λ distributions completely >1 for either model. At two sites, λ was expected to be >1 if rat-tracking rates were <5%. At the other five reintroduction sites, λ was predicted to be close to 1, and it was unclear whether growth was expected. Predictions of λ for the proposed reintroduction site were less precise than for other sites because distributions incorporated the full range of site-to-site random variation in vital rates. Our methods can be applied to any species for which postrelease data on demographic rates are available and potentially can be extended to model multiple species simultaneously.     

A Bayesian hierarchical model for monitoring harbor seal changes in Prince William Sound, Alaska

Bayesian hierarchical models were used to assess trends of harbor seals, Phoca vitulina richardsi, in Prince William Sound, Alaska, following the 1989 Exxon Valdez oil spill. Data consisted of 4–10 replicate observations per year at 25 sites over 10 years. We had multiple objectives, including estimating the effects of covariates on seal counts, and estimating trend and abundance, both per site and overall. We considered a Bayesian hierarchical model to meet our objectives. The model consists of a Poisson regression model for each site. For each observation the logarithm of the mean of the Poisson distribution was a linear model with the following factors: (1) intercept for each site and year, (2) time of year, (3) time of day, (4) time relative to low tide, and (5) tide height. The intercept for each site was then given a linear trend model for year. As part of the hierarchical model, parameters for each site were given a prior distribution to summarize overall effects. Results showed that at most sites, (1) trend is down; counts decreased yearly, (2) counts decrease throughout August, (3) counts decrease throughout the day, (4) counts are at a maximum very near to low tide, and (5) counts decrease as the height of the low tide increases; however, there was considerable variation among sites. To get overall trend we used a weighted average of the trend at each site, where the weights depended on the overall abundance of a site. Results indicate a 3.3% decrease per year over the time period.     

Major PM10 source location by a spatial multivariate receptor model

We present a multivariate receptor model for identifying the spatial location of major PM₁₀ pollution sources through the concentrations at multiple monitoring stations. We build on a mixed multiplicative log-normal factor model adjusting the source contributions for meteorological covariates and for temporal correlation and considering source profiles as compositional Gaussian random fields, to account for the variability induced by the spatial distribution of the monitoring sites. Taking a Bayesian approach to estimation, the proposed hierarchical model is implemented and used to analyze average daily PM₁₀ concentration measurements from 13 monitoring sites in Taranto, Italy, for the period April–December 2005. Three major sources of pollution are identified and characterized in terms of their spatial and temporal behavior and in relation to meteorological data.     

Combining functional data with hierarchical Gaussian process models

Gaussian process models have been used in applications ranging from machine learning to fisheries management. In the Bayesian framework, the Gaussian process is used as a prior for unknown functions, allowing the data to drive the relationship between inputs and outputs. In our research, we consider a scenario in which response and input data are available from several similar, but not necessarily identical, sources. When little information is known about one or more of the populations it may be advantageous to model all populations together. We present a hierarchical Gaussian process model with a structure that allows distinct features for each source as well as shared underlying characteristics. Key features and properties of the model are discussed and demonstrated in a number of simulation examples. The model is then applied to a data set consisting of three populations of Rotifer Brachionus calyciflorus Pallas. Specifically, we model the log growth rate of the populations using a combination of lagged population sizes. The various lag combinations are formally compared to obtain the best model inputs. We then formally compare the leading hierarchical Gaussian process model with the inferential results obtained under the independent Gaussian process model.     

A multivariate prediction of spatial process with non-stationary covariance for Kuwait non-methane hydrocarbons levels

Predicting unmeasured realizations of multivariate spatial process responses is a fundamental problem in environmetrics. The study of levels of air pollutants is important for understanding and improving air quality in major urban areas. This research aims to handle the prediction in a Bayesian framework for non-methane hydrocarbons NMHC pollutant for the State of Kuwait where records of six monitor stations located in different sites are observed at successive time points. Our objective is to study the distribution level of NMHC with respect to time and metreological parameters and space and use this distribution to predict the concentration of NMHC in other sites of Kuwait using the minimum amount of data (reducing the cost). We will implement a hierarchical Bayesian approach assuming Gaussian random field technique that allows us to pool the data from different sites in predicting the exposure of the non-methane hydrocarbons in different regions of Kuwait.     

Lifting a veil on diversity: a Bayesian approach to fitting relative-abundance models.

Bayesian methods incorporate prior knowledge into a statistical analysis. This prior knowledge is usually restricted to assumptions regarding the form of probability distributions of the parameters of interest, leaving their values to be determined mainly through the data. Here we show how a Bayesian approach can be applied to the problem of drawing inference regarding species abundance distributions and comparing diversity indices between sites. The classic log series and the lognormal models of relative- abundance distribution are apparently quite different in form. The first is a sampling distribution while the other is a model of abundance of the underlying population. Bayesian methods help unite these two models in a common framework. Markov chain Monte Carlo simulation can be used to fit both distributions as small hierarchical models with shared common assumptions. Sampling error can be assumed to follow a Poisson distribution. Species not found in a sample, but suspected to be present in the region or community of interest, can be given zero abundance. This not only simplifies the process of model fitting, but also provides a convenient way of calculating confidence intervals for diversity indices. The method is especially useful when a comparison of species diversity between sites with different sample sizes is the key motivation behind the research. We illustrate the potential of the approach using data on fruit-feeding butterflies in southern Mexico. We conclude that, once all assumptions have been made transparent, a single data set may provide support for the belief that diversity is negatively affected by anthropogenic forest disturbance. Bayesian methods help to apply theory regarding the distribution of abundance in ecological communities to applied conservation.     

Efficient statistical mapping of avian count data

We develop a spatial modeling framework for count data that is efficient to implement in high-dimensional prediction problems. We consider spectral parameterizations for the spatially varying mean of a Poisson model. The spectral parameterization of the spatial process is very computationally efficient, enabling effective estimation and prediction in large problems using Markov chain Monte Carlo techniques. We apply this model to creating avian relative abundance maps from North American Breeding Bird Survey (BBS) data. Variation in the ability of observers to count birds is modeled as spatially independent noise, resulting in over-dispersion relative to the Poisson assumption. This approach represents an improvement over existing approaches used for spatial modeling of BBS data which are either inefficient for continental scale modeling and prediction or fail to accommodate important distributional features of count data thus leading to inaccurate accounting of prediction uncertainty.     

Bayesian estimation of age-specific bird nest survival rates with categorical covariates

The populations of many North American landbirds are showing signs of declining. Gathering information on breeding productivity allows critical assessment of population performance and helps identify good habitat-management practices. He (Biometrics (2003) 59 962–973) proposed a Bayesian model to estimate the age-specific nest survival rates. The model allows irregular visiting schedule under the assumption that the observed nests have homogeneous nest survival. Because nest survival studies are often conducted in different sites and time periods, it is not realistic to assume homogeneous nest survival. In this paper, we extend He’s model by incorporating these factors as categorical covariates. The simulation results show that the Bayesian hierarchical model can produce satisfactory estimates on nest survival and capture different factor effects. Finally the model is applied to a Missouri red-winged blackbird data set.     

Hierarchical Bayesian space-time models

Space-time data are ubiquitous in the environmental sciences. Often, as is the case with atmo- spheric and oceanographic processes, these data contain many different scales of spatial and temporal variability. Such data are often non-stationary in space and time and may involve many observation/prediction locations. These factors can limit the effectiveness of traditional space- time statistical models and methods. In this article, we propose the use of hierarchical space-time models to achieve more flexible models and methods for the analysis of environmental data distributed in space and time. The first stage of the hierarchical model specifies a measurement- error process for the observational data in terms of some 'state' process. The second stage allows for site-specific time series models for this state variable. This stage includes large-scale (e.g. seasonal) variability plus a space-time dynamic process for the anomalies'. Much of our interest is with this anomaly proc ess. In the third stage, the parameters of these time series models, which are distributed in space, are themselves given a joint distribution with spatial dependence (Markov random fields). The Bayesian formulation is completed in the last two stages by speci- fying priors on parameters. We implement the model in a Markov chain Monte Carlo framework and apply it to an atmospheric data set of monthly maximum temperature.     

Bewertung heterogener Altlasten durch Probenahme, statistische und geostatistische Modellierung

A slack heap in Saxony serves as an example for the model procedure of evaluation heterogeneous waste sites via sampling as well as statistical and geostatistical data analysis. This paper demonstrates the usefulness of the known regulations for sampling and of the tried and proven geostatistical methods developed for assessing homogeneous wastes and the concornitantly contaminated soil. Emphasized here are the contents of iron and sulfur. Considered in addition are the concentrations of arsenic and the heat loss. To establish a conclusion, the variograms of the original data are used to evaluate the waste site in different directions and at determined depths. With the aid of a spot interpolation, further data can be calculated for additional sites which have not been sampled and can be used consequently for displaying the spatial variability. Finally, the statistical data analysis is applied using the standard methods of random sampling as an alternative to the geostatistical modeling. In this manner, the consequences for the sampling strategy can be demonstrated.     

Analyzing spatial count data,with an application to weed counts

Count data on a lattice may arise in observational studies of ecological phenomena. In this paper a hierarchical spatial model is used to analyze weed counts. Anisotropy is introduced, and a bivariate extension of the model is presented.     

Residential radon concentration in the Abruzzo region (Italy): a different perspective for identifying radon prone areas

Indoor radon is an important risk factor for human health. Indeed radon inhalation is considered the second cause of lung cancer after smoking. During the last decades, in many countries huge efforts have been made in order to measuring, mapping and predicting radon levels in dwellings. Various researches have been devoted to identify those areas within the country where high radon concentrations are more likely to be found. Data collected through indoor radon surveys have been analysed adopting various statistical approaches, among which hierarchical Bayesian models and geostatistical tools are worth noting. The essential goal of this paper regards the identification of high radon concentration areas (the so-called radon prone areas) in the Abruzzo Region (Italy). In order to accurately pinpoint zones deserving attention for mitigation purpose, we adopt spatial cluster detection techniques, traditionally employed in epidemiology. As a first step, we assume that indoor radon measurements do not arise from a continuous spatial process; thus the geographic locations of dwellings where the radon measurements have been taken can be viewed as a realization of a spatial point process. Following this perspective, we adopt and compare recent cluster detection techniques: the simulated annealing scan statistic, the case event approach based on distance regression on the selection order and the elliptic spatial scan statistic. The analysis includes data collected during surveys carried out by the Regional Agency for the Environment Protection of Abruzzo (ARTA) in 1,861 random sampled dwellings across 277 municipalities of the Abruzzo region. The radon prone areas detected by the selected approaches are provided along with the summary statistics of the methods. Finally, the methodologies considered in this paper are tested on simulated data in order to evaluate their power and the precision of cluster location detection.     

Bayesian areal wombling via adjacency modeling

Recently, public health professionals and other geostatistical researchers have shown increasing interest in boundary analysis, the detection or testing of zones or boundaries that reveal sharp changes in the values of spatially oriented variables. For areal data (i.e., data which consist only of sums or averages over geopolitical regions), Lu and Carlin (Geogr Anal 37: 265–285, 2005) suggested a fully model-based framework for areal wombling using Bayesian hierarchical models with posterior summaries computed using Markov chain Monte Carlo (MCMC) methods, and showed the approach to have advantages over existing non-stochastic alternatives. In this paper, we develop Bayesian areal boundary analysis methods that estimate the spatial neighborhood structure using the value of the process in each region and other variables that indicate how similar two regions are. Boundaries may then be determined by the posterior distribution of either this estimated neighborhood structure or the regional mean response differences themselves. Our methods do require several assumptions (including an appropriate prior distribution, a normal spatial random effect distribution, and a Bernoulli distribution for a set of spatial weights), but also deliver more in terms of full posterior inference for the boundary segments (e.g., direct probability statements regarding the probability that a particular border segment is part of the boundary). We illustrate three different remedies for the computing difficulties encountered in implementing our method. We use simulation to compare among existing purely algorithmic approaches, the Lu and Carlin (2005) method, and our new adjacency modeling methods. We also illustrate more practical modeling issues (e.g., covariate selection) in the context of a breast cancer late detection data set collected at the county level in the state of Minnesota.     

Modeling and spatial prediction of pre-settlement patterns of forest distribution using witness tree data

At the time of European settlement, land surveys were conducted progressively westward throughout the United States. Outside of the original 13 colonies, surveys generally followed the Public Land Survey system in which trees, called witness trees, were regularly recorded at 1 mi by 1 mi grid intersections. This unintentional sampling provides insight into the composition and structure of pre-European settlement forests, which is used as baseline data to assess forest change following settlement. In this paper, a model for the Public Land Surveys of east central Alabama is developed. Assuming that the locations of trees of each species are realized from independent Poisson processes whose respective log intensities are linear functions of environmental covariates (i.e., elevation, landform, and physiographic province), the species observed at the survey grid intersections are independently sampled from a generalized logistic regression model. If all 68 species found in the survey were included, the model would be highly over-parameterized, so only the distribution of the most common taxon, pines, will be considered at this time. To assess the impact of environmental factors not included in the model, a hidden Gaussian random field shall be added as a random effect. A Markov Chain Monte Carlo algorithm is developed for Bayesian inference on model parameters, and for Bayes posterior prediction of the spatial distribution of pines in east central Alabama. Received: June 2004 / Revised: November 2004     

Modeling and risk assessment for soil temperatures beneath prescribed forest fires

Prescribed fire is a management tool used by wildland resource management organizations in many ecosystems to reduce hazardous fuels and to achieve a host of other objectives. To study the effects of fire in naturally accumulating fuel conditions, the ambient soil temperature is monitored beneath prescribed burns. In this study we developed a stochastic model for temperature profiles (values at 15 minute intervals) recorded at four depths beneath the soil during a large prescribed burn study. The model was used to assess the temporal fit of the data to particular solutions of the heat equation. We used a random effects model to assess the effects of observed site characteristics on maximum temperatures and to estimate risks of temperatures exceeding critical levels in future similar prescribed fires. Contour plots of estimated risks of temperatures exceeding 60°C for a range of fuel levels and soil depths indicated high risks of occurrence, especially when the moisture levels are low. However, the natural variability among sites seems to be large, even after controlling fuel and moisture levels, resulting in large standard errors of predicted risks.