Bayesian analysis of semiparametric linear-circular models

In many environmental and agricultural studies, data are collected on both linear and circular random variables, with possible dependence between the variables. Classically, the analysis of such data has been carried out in a classical regression framework. We propose a Bayesian hierarchical framework to handle all forms of uncertainty arising in a linear-circular data set. One novelty of our multivariate linear-circular model is that, marginally, the circular component is assumed to be a mixture model with an unknown number of von Mises (or circular normal) distributions. We use the Dirichlet process to introduce variability in the model dimensionality, and develop a simple Gibbs sampling algorithm for simulating the mixture components. Although we illustrate our methodology on von Mises mixtures, it is widely applicable. We thus avoid complicated reversible-jump Markov chain Monte Carlo methods, which are considered ideal for analyzing mixtures of unknown number of distributions. We illustrate our methodologies with simulated and real data sets. Using pseudo-Bayes factors, we also compare different models associated with both fixed and variable numbers of von Mises distributions. Our findings suggest that models associated with varying numbers of mixture components perform at least as well as those with known numbers of mixture components. We tentatively argue that model averaging associated with variable number of mixture components improves the model’s predictive power, which compensates for the lack of knowledge of the actual number of mixture components.     

A Study of Transport and Mixing in Natural Waters Using ICP-MS: Water-Particle Interactions

Water-particle interactions often may result in non-conservative chemical behavior when waters from different sources mix with one another. The results presented in this paper address the role of these interactions in freshwater and estuarine mixing and support a larger study to develop a method to help resolve flow distribution and water quality questions in surface waters using a source water “fingerprinting” technique. Inductively coupled plasma-mass spectrometry (ICP-MS) is used to “fingerprint” each water source based upon the concentrations and relative proportions of elements in that source. Estimates can then be made of the fractions of various “fingerprinted” waters in water samples that contain a mixture of source waters. Such estimates depend upon the selection of tracers that behave conservatively during mixing; in this paper, results to establish the maximum particle exchange capacity and conservative mixing behavior are presented for samples collected from the Sacramento River-San Francisco Bay-Delta estuary. Elements likely to behave conservatively include boron, sodium, magnesium, potassium, calcium, strontium, and molybdenum.     

Spectral fingerprinting: sediment source discrimination and contribution modelling of artificial mixtures based on VNIR-SWIR spectral properties

Purpose

Knowledge of the origin of suspended sediment is important for improving our understanding of sediment dynamics and thereupon support of sustainable watershed management. An direct approach to trace the origin of sediments is the fingerprinting technique. It is based on the assumption that potential sediment sources can be discriminated and that the contribution of these sources to the sediment can be determined on the basis of distinctive characteristics (fingerprints). Recent studies indicate that visible–near-infrared (VNIR) and shortwave-infrared (SWIR) reflectance characteristics of soil may be a rapid, inexpensive alternative to traditional fingerprint properties (e.g. geochemistry or mineral magnetism).

Materials and methods

To further explore the applicability of VNIR-SWIR spectral data for sediment tracing purposes, source samples were collected in the Isábena watershed, a 445 km² dryland catchment in the central Spanish Pyrenees. Grab samples of the upper soil layer were collected from the main potential sediment source types along with in situ reflectance spectra. Samples were dried and sieved, and artificial mixtures of known proportions were produced for algorithm validation. Then, spectral readings of potential source and artificial mixture samples were taken in the laboratory. Colour coefficients and physically based parameters were calculated from in situ and laboratory-measured spectra. All parameters passing a number of prerequisite tests were subsequently applied in discriminant function analysis for source discrimination and mixing model analyses for source contribution assessment.

Results and discussion

The three source types (i.e. badlands, forest/grassland and an aggregation of other sources, including agricultural land, shrubland, unpaved roads and open slopes) could be reliably identified based on spectral parameters. Laboratory-measured spectral fingerprints permitted the quantification of source contribution to artificial mixtures, and introduction of source heterogeneity into the mixing model decreased accuracies for some source types. Aggregation of source types that could not be discriminated did not improve mixing model results. Despite providing similar discrimination accuracies as laboratory source parameters, in situ derived source information was found to be insufficient for contribution modelling.

Conclusions

The laboratory mixture experiment provides valuable insights into the capabilities and limitations of spectral fingerprint properties. From this study, we conclude that combinations of spectral properties can be used for mixing model analyses of a restricted number of source groups, whereas more straightforward in situ measured source parameters do not seem suitable. However, modelling results based on laboratory parameters also need to be interpreted with care and should not rely on the estimates of mean values only but should consider uncertainty intervals as well.     

A Note on the Royle–Nichols Model for Repeated Detection–Nondetection Data

In this note, it is shown that the integrated likelihood for the Royle–Nichols model with a Poisson mixing distribution can be expressed as a finite rather than an infinite sum of terms. The advantages which so accrue are discussed and explored by means of two examples. The finite sum formulation of the likelihood is also shown to hold for negative binomial and zero-inflated mixing distributions. Results based on these two mixing distributions proved disappointing however and their use is not recommended unless extensive data are available.     

Autologistic regression analysis of spatial-temporal binary data via Monte Carlo maximum likelihood

This article considers logistic regression analysis of binary data that are measured on a spatial lattice and repeatedly over discrete time points. We propose a spatial-temporal autologistic regression model and draw statistical inference via maximum likelihood. Due to an unknown normalizing constant in the likelihood function, we use Monte Carlo to obtain maximum likelihood estimates of the model parameters and predictive distributions at future time points. We also use path sampling to estimate the unknown normalizing constant and approximate an information criterion for model assessment. The methodology is illustrated by the analysis of a dataset of mountain pine beetle outbreaks in western Canada.     

A Non-Gaussian Spatial Generalized Linear Latent Variable Model

We consider a spatial generalized linear latent variable model with and without normality distributional assumption on the latent variables. When the latent variables are assumed to be multivariate normal, we apply a Laplace approximation. To relax the assumption of marginal normality in favor of a mixture of normals, we construct a multivariate density with Gaussian spatial dependence and given multivariate margins. We use the pairwise likelihood to estimate the corresponding spatial generalized linear latent variable model. The properties of the resulting estimators are explored by simulations. In the analysis of an air pollution data set the proposed methodology uncovers weather conditions to be a more important source of variability than air pollution in explaining all the causes of non-accidental mortality excluding accidents.     

Assessing the Impacts of Time-to-Detection Distribution Assumptions on Detection Probability Estimation

Abundance estimates from animal point-count surveys require accurate estimates of detection probabilities. The standard model for estimating detection from removal-sampled point-count surveys assumes that organisms at a survey site are detected at a constant rate; however, this assumption can often lead to biased estimates. We consider a class of N-mixture models that allows for detection heterogeneity over time through a flexibly defined time-to-detection distribution (TTDD) and allows for fixed and random effects for both abundance and detection. Our model is thus a combination of survival time-to-event analysis with unknown-N, unknown-p abundance estimation. We specifically explore two-parameter families of TTDDs, e.g., gamma, that can additionally include a mixture component to model increased probability of detection in the initial observation period. Based on simulation analyses, we find that modeling a TTDD by using a two-parameter family is necessary when data have a chance of arising from a distribution of this nature. In addition, models with a mixture component can outperform non-mixture models even when the truth is non-mixture. Finally, we analyze an Ovenbird data set from the Chippewa National Forest using mixed effect models for both abundance and detection. We demonstrate that the effects of explanatory variables on abundance and detection are consistent across mixture TTDDs but that flexible TTDDs result in lower estimated probabilities of detection and therefore higher estimates of abundance.Supplementary materials accompanying this paper appear on-line.     

空气总悬浮颗粒物浓度的遥感信息模型研究

空气总悬浮颗粒物遥感信息模型是使用遥感信息模型的方法来模拟空气总悬浮颗粒物在空间上的分布。通过对空气总悬浮颗粒物来源和分布影响的因子分析,认为地表覆盖情况因子对空气总悬浮颗粒物来源影响最大,降雨强度和风速因子对空气总悬浮颗粒物分布影响最大,因此根据此三个因子建立了空气总悬浮颗粒物遥感信息模型。然后根据对厦门市高分辨遥感的分类数据和空气总悬浮颗粒物的分布数据得到了空气总悬浮颗粒物遥感信息模型的地理参数。通过对公式结果验证认为该模型较好的模拟了空气总悬浮颗粒的分布,为空气总悬浮颗粒物浓度的分布研究提出一种新思路。     

Using multiple composite fingerprints to quantify source contributions and uncertainties in an arid region

Purpose

Developing targeted protection measures at a watershed scale requires spatially distributed information of sediment sources. Therefore, the objectives of this study are to (1) test and evaluate the ability of multiple composite fingerprints (MCF) to quantify sediment provenance using multiple particle size classes in an arid region; (2) quantify uncertainty of the estimated proportional contributions of sediment sources; and (3) provide decision support information for sediment control in the Danghe Reservoir Watershed.

Materials and methods

In total, 66 samples were collected from north alluvial fan, south alluvial fan, and high mountains, and all samples were divided into six particle size groups. A multistep test was used to remove the tracers that were non-conservative, unable to differentiate sources, or highly variable within a source. Based on geochemical properties of distributed source samples and a linear mixing model, a MCF method with multiple particle size tracking was used to estimate proportions of three potential source contributions. More importantly, the uncertainty of sediment source contributions was quantified using the Gaussian first-order approximation.

Results and discussion

The results showed that the MCF method with multiple particle size tracking could obtain relatively accurate estimates of the contributions with an overall mean absolute relative error of 3.5% and a relatively narrow 95% confidence interval. The major contributions were consistently coming from the high mountains for all six particle groups. During these runoff events, the overall estimated mean proportions were 49.0%, 26.5%, and 24.5% from the high mountains, south alluvial fan, and north alluvial fan, respectively. Furthermore, the Gaussian first-order approximation revealed that more than 60% of the total uncertainty contribution was a byproduct of the downstream sediment mixture, while each individual sediment source produced less than 15% of the absolute uncertainty.

Conclusions

Acquiring watershed scale sediment source information is challenging and the MCF method proved accurate. A majority of the contribution uncertainties were associated with the downstream sediment mixture, which is because the sediment sink inherited both spatial and temporal variations of all contributing sources. Consequently, a larger sample size is recommended for sediment mixtures, compared to each sediment source, in order to increase the accuracy of the source proportion estimation.

     

An Approximate Method for Population Toxicokinetic Analysis With Aggregated Data

Standard statistical models for analyzing inter-individual variability in clinical pharmacokinetics (nonlinear mixed effects; hierarchical Bayesian) require individual data. However, for environmental or occupational toxicants only aggregated data are usually available, so toxicokinetic analyses typically ignore population variability. We propose a hierarchical Bayesian approach to estimate inter-individual variability from the observed mean and variance at each time point, using a bivariate normal (or lognormal) approximation to their joint likelihood. Through analysis of both simulated data and real toxicokinetic data from 1,3-butadiene exposures, we conclude that given information on the form of the individual-level model, useful information on inter-individual variability may be obtainable from aggregated data, but that additional sensitivity and identifiability checks are recommended.     

Identification of model components for a class of continuous spatiotemporal models

Environmental data routinely are collected at irregularly spaced monitoring stations and at intermittent times, times which may differ by location. This article introduces a class of continuous-time, continuous-space statistical models that can accommodate many of these more complex environmental processes. This class of models in corporates temporal and spatial variability in a cohesive manner and is broad enough to include temporal processes that are assumed to be generated by stochastic differential equations with possibly temporally and spatially correlated errors. A wide range of ARIMA temporal models and geostatistical spatial models are included in the class of models investigated. Techniques for identifying the structure of the temporal and spatial components of this class of models are detailed. Point estimates of model parameters, asymptotic distributions, and Kalman-filter prediction methods are discussed.     

A linear functional relationship model for circular data with an application to the assessment of ocean wave measurements

This article suggests a linear functional relationship model for comparing two sets of circular data subject to unobservable errors. Unlike the corresponding and relatively well-studied model for linear data, maximum likelihood estimation for this model is very complicated and no explicit solutions are possible. Using a numerical approximation, we are able to solve the likelihood equations approximately, and to obtain good approximations to the likelihood estimates of the parameters. The quality of our estimates and the feasibility of the estimation method are illustrated via simulation. By establishing a parallel with the model for linear data, we are able to explain the various problems occurring in the process of estimation and to substantiate our numerical results. The interest in the model arose in connection with the study of ocean wave data; an application to such data is also given.     

The Matérn variogram model: Implications for uncertainty propagation and sampling in geostatistical surveys

The Matérn variogram model has been advocated because it is flexible and can represent varied behaviour at small lags. We show how the constraints on the spherical and exponential variogram at short lags ignore a possible source of uncertainty in the variogram and so in kriging surveys, that the Matérn model can describe. Matérn, spherical and exponential variogram models were fitted by maximum likelihood to a set of log₁₀(K) observations made on a regular grid at Broom's Barn Farm, Suffolk, England. The likelihood profiles of the Matérn parameter estimates were asymmetric. Thus the uncertainty of these estimates could only be adequately assessed by a Bayesian approach. The uncertainty of estimated parameters of the Matérn variogram was larger than for the exponential variogram. This is an indication that the assumption of an exponential model limits the behaviour that may be described by the variogram. Thus uncertainty analyses where an exponential variogram is assumed may underestimate the uncertainty of kriged estimates. Bayesian analysis of the kriged estimates of log₁₀(K) at Broom's Barn Farm using the Matérn variogram revealed an observable component of uncertainty due to variogram uncertainty. When an exponential variogram model was used, the estimate of this component of uncertainty was negligible. The Matérn variogram should therefore be used rather than the exponential model when assessing the adequacy of a variogram estimate. A method of designing sample schemes which is suitable for both estimating a Matérn variogram and interpolation is suggested.     

Bayesian Methods for Hierarchical Distance Sampling Models

The few distance sampling studies that use Bayesian methods typically consider only line transect sampling with a half-normal detection function. We present a Bayesian approach to analyse distance sampling data applicable to line and point transects, exact and interval distance data and any detection function possibly including covariates affecting detection probabilities. We use an integrated likelihood which combines the detection and density models. For the latter, densities are related to covariates in a log-linear mixed effect Poisson model which accommodates correlated counts. We use a Metropolis-Hastings algorithm for updating parameters and a reversible jump algorithm to include model selection for both the detection function and density models. The approach is applied to a large-scale experimental design study of northern bobwhite coveys where the interest was to assess the effect of establishing herbaceous buffers around agricultural fields in several states in the US on bird densities. Results were compared with those from an existing maximum likelihood approach that analyses the detection and density models in two stages. Both methods revealed an increase of covey densities on buffered fields. Our approach gave estimates with higher precision even though it does not condition on a known detection function for the density model.     

Generalized Linear Latent Variable Models for Multivariate Count and Biomass Data in Ecology

In this paper we consider generalized linear latent variable models that can handle overdispersed counts and continuous but non-negative data. Such data are common in ecological studies when modelling multivariate abundances or biomass. By extending the standard generalized linear modelling framework to include latent variables, we can account for any covariation between species not accounted for by the predictors, notably species interactions and correlations driven by missing covariates. We show how estimation and inference for the considered models can be performed efficiently using the Laplace approximation method and use simulations to study the finite-sample properties of the resulting estimates. In the overdispersed count data case, the Laplace-approximated estimates perform similarly to the estimates based on variational approximation method, which is another method that provides a closed form approximation of the likelihood. In the biomass data case, we show that ignoring the correlation between taxa affects the regression estimates unfavourably. To illustrate how our methods can be used in unconstrained ordination and in making inference on environmental variables, we apply them to two ecological datasets: abundances of bacterial species in three arctic locations in Europe and abundances of coral reef species in Indonesia.Supplementary materials accompanying this paper appear on-line.     

Likelihood-based inference for clustered line transect data

The uncertainty in estimation of spatial animal density from line transect surveys depends on the degree of spatial clustering in the animal population. To quantify the clustering we model line transect data as independent thinnings of spatial shot-noise Cox processes. Likelihood-based inference is implemented using Markov chain Monte Carlo (MCMC) methods to obtain efficient estimates of spatial clustering parameters. Uncertainty is addressed using parametric bootstrap or by consideration of posterior distributions in a Bayesian setting. Maximum likelihood estimation and Bayesian inference are compared in an example concerning minke whales in the northeast Atlantic.     

The Odd Log-Logistic Student <Emphasis Type="Italic">t</Emphasis> Distribution: Theory and Applications

The normal distribution is most used in analysis of experiments. However, it is not suitable to apply in situations where the data have evidence of bimodality or heavier tails than the normal distribution. So, we propose a new four-parameter model called the odd log-logistic Student t distribution as an alternative to the normal and Student t distributions. The new distribution can be symmetric, platykurtic, mesokurtic or leptokurtic and may be unimodal or bimodal. Its various structural properties can be determined from the linear representation of its density function. The estimation of the model parameters is performed by maximum likelihood. The proposed distribution can be used as an alternative for randomized complete block design, thus providing analysis of real data more realistic than other special regression models. We perform a sensitivity analysis to detect influential or outlying observations, and construct generated envelopes from the residuals to select appropriate models. We illustrate the importance of the proposed model by means of three real data sets in analysis of experiments carried out in different regions of Brazil.     

似然函数形式对水稻物候期模型品种参数校正的影响

针对传统高斯正态似然函数(Gaussian likelihood function,GLF)在观测数据存在测量误差和模型算法结构复杂时无法描述模型残差异方差特点,造成马尔科夫链蒙特卡洛(Markov chain Monte Carlo,MCMC)算法进行模型参数校正时结果存在偏差的问题,通过引入变异系数(coefficient of variation,CV)变换的高斯似然函数(GLF with CV transformation,GLF-CV)和BC(Box-Cox)变换的高斯似然函数(GLF with BC transformation,GLF-BC)对观测数据和模型结构造成的异方差进行特征描述,并比较了参数校正效果及模型不确定度(uncertainty ratio,UR)。以2004—2009年高要雪花粘(早熟)、2001—2004年兴化武育粳3号(中熟)、1991—2004年六安汕优63号(晚熟)3个生态点的田间栽培试验数据为基础,RiceGrow和Oryza2000物候期模型为对象,利用仿射不变马尔科夫链蒙特卡洛集成采样(ensemble sampling for affin...     

Empirical Bayes analysis of variance component models for microarray data

A gene-by-gene mixed model analysis is a useful statistical method for assessing significance for microarray gene differential expression. While a large amount of data on thousands of genes are collected in a microarray experiment, the sample size for each gene is usually small, which could limit the statistical power of this analysis. In this report, we introduce an empirical Bayes (EB) approach for general variance component models applied to microarray data. Within a linear mixed model framework, the restricted maximum likelihood (REML) estimates of variance components of each gene are adjusted by integrating information on variance components estimated from all genes. The approach starts with a series of single-gene analyses. The estimated variance components from each gene are transformed to the “ANOVA components”. This transformation makes it possible to independently estimate the marginal distribution of each “ANOVA component.” The modes of the posterior distributions are estimated and inversely transformed to compute the posterior estimates of the variance components. The EB statistic is constructed by replacing the REML variance estimates with the EB variance estimates in the usual t statistic. The EB approach is illustrated with a real data example which compares the effects of five different genotypes of male flies on post-mating gene expression in female flies. In a simulation study, the ROC curves are applied to compare the EB statistic and two other statistics. The EB statistic was found to be the most powerful of the three. Though the null distribution of the EB statistic is unknown, a t distribution may be used to provide conservative control of the false positive rate.     

Non-additive effects vary with the number of component residues and their mixing proportions during residue mixture decomposition: A microcosm study

Poplar leaf litter and crop residues (leaves and stems) of two main crops (soybean and maize) collected from semiarid agroforestry systems of Northeast China were used in our microcosm study. The aims were to examine whether non-additive effects (synergistic or antagonistic) between poplar leaf litter and crop residues exist during decomposition and to identify the influence of residue mixing proportion on the incidence of non-additive effects of residue mixture for the same plant residues. We determined residue decomposition rate by measuring mass loss and N release. Synergistic effects between poplar leaf litter and crop residues were more common than additive effects in terms of mass loss and N release. Moreover, the interactive effects between tree leaf litter and crop residues on decomposition varied with the number of component residues and their mixing proportion. Three-residue mixtures produced synergistic effects on mass loss and N release, although two-residue mixtures showed an additive effect in some cases. In addition, as compared with equal proportion, mixing residues with unequal proportion increased the incidence of non-additive effects during decomposition of residue mixture. These findings highlight that residue decomposition dynamics in ecosystems should be assessed on the basis of plant residue mixtures and their mixing proportions, which may help us better understand nutrient dynamics and guide our decisions on nutrient management.