Multi-scale Modeling of Animal Movement and General Behavior Data Using Hidden Markov Models with Hierarchical Structures

Hidden Markov models (HMMs) are commonly used to model animal movement data and infer aspects of animal behavior. An HMM assumes that each data point from a time series of observations stems from one of N possible states. The states are loosely connected to behavioral modes that manifest themselves at the temporal resolution at which observations are made. Due to advances in tag technology and tracking with digital video recordings, data can be collected at increasingly fine temporal resolutions. Yet, inferences at time scales cruder than those at which data are collected and, which correspond to larger-scale behavioral processes, are not yet answered via HMMs. We include additional hierarchical structures to the basic HMM framework, incorporating multiple Markov chains at various time scales. The hierarchically structured HMMs allow for behavioral inferences at multiple time scales and can also serve as a means to avoid coarsening data. Our proposed framework is one of the first that models animal behavior simultaneously at multiple time scales, opening new possibilities in the area of animal movement and behavior modeling. We illustrate the application of hierarchically structured HMMs in two real-data examples: (i) vertical movements of harbor porpoises observed in the field, and (ii) garter snake movement data collected as part of an experimental design. Supplementary materials accompanying this paper appear online.     

Analysis of cone resistance data with missing observations below stones

When stones prevent the measurement of cone resistance, and missing values below the stones are ignored, then averages can be seriously underestimated. Methods are considered for correcting this bias and an algorithm is proposed in which missing observations are replaced by their expected values. A numerical example gives results in close agreement with those obtained using the optimal, but computationally expensive, method of maximum likelihood estimation. It is recommended that data from incomplete penetrations should not be discarded but should be used, preferably with the proposed algorithm, to reduce the bias in estimates of mean values.     

Evaluating the performance of a block updating MCMC sampler in a simple genetic application

Markov chain Monte Carlo (MCMC) methods have provided an enormous break-through in the analysis of large complex problems such as those which frequently arise in genetic applications. The richness of the inference and the flexibility of an MCMC Bayesian approach in terms of design, data structure that can be analyzed, and models that can be posed, is indisputable. However, despite the strengths of the Bayesian approach, it is important to acknowledge that there are other, often easier, ways of tackling a problem. This is so, especially when simpler, qualitative answers are sought, such as presence or absence of a quantitative trait locus. We critically evaluate the behavior of a Bayesian aCMC block sampler for the detection of a quantitative trait locus by linkage with marker data, and compare it with a traditional least squares method. Some practical issues are illustrated by discussing the pros and cons of a Bayesian block updating sampling scheme versus the least squares method in the context of a simple genetic mapping problem. Depending on the focus of analysis, we show that the MCMC sampler does not always outperform the simpler approach from a frequentist perspective, and, more to the point, may not always perform appropriately in any particular replication.     

Bayesian Spatial Point Process Modeling of Line Transect Data

This paper develops a Bayesian approach for spatial inference on animal density from line transect survey data. We model the spatial distribution of animals within a geographical area of interest by an inhomogeneous Poisson process whose intensity function incorporates both covariate effects and spatial smoothing of residual variation. Independently thinning the animal locations according to their estimated detection probabilities results into another spatial Poisson process for the sightings (the observations). Prior distributions are elicited for all unknown model parameters. Due to the sparsity of data in the application we consider, eliciting sensible prior distributions is important in order to get meaningful estimation results. A reversible jump Markov Chain Monte Carlo (MCMC) algorithm for simulation of the posterior distribution is developed. We present results for simulated data and a real data set of minke whale pods from Antarctic waters. The main advantages of our method compared to design-based analyses are that it can use data arising from sources other than specifically designed surveys and its ability to link covariate effects to variation of animal density. The Bayesian paradigm provides a coherent framework for quantifying uncertainty in estimation results.     

A hierarchical model for compositional data analysis

This article introduces a hierarchical model for compositional analysis. Our approach models both source and mixture data simultaneously, and accounts for several different types of variation: these include measurement error on both the mixture and source data; variability in the sample from the source distributions; and variability in the mixing proportions themselves, generally of main interest. The method is an improvement on some existing methods in that estimates of mixing proportions (including their interval estimates) are sure to lie in the range [0, 1]; in addition, it is shown that our model can help in situations where identification of appropriate source data is difficult, especially when we extend our model to include a covariate. We first study the likelihood surface of a base model for a simple example, and then include prior distributions to create a Bayesian model that allows analysis of more complex situations via Markov chain Monte Carlo sampling from the likelihood. Application of the model is illustrated with two examples using real data: one concerning chemical markers in plants, and another on water chemistry.     

A bayesian model for the analysis of lactation curves of dairy goats

This article aims to determine the effect of certain covariates, such as season of kidding, parity, and time of kidding on the characteristics of the lactation curve of Saanen dairy goats. Characteristics investigated are peak milk yield, time of peak milk yield, total milk production, persistency, and the relationship between fat and protein in milk composition, as well as between lactation curves of the same animal in successive years. The analysis is carried out using a hierarchical Bayesian approach, together with Wood’s model, to model lactation. Posterior distributions of quantities of interest are obtained by means of the Markov chain Monte Carlo (MCMC) methods. These clearly illustrate the significant effect of especially parity, but also season and time of kidding on the characteristics of the lactation curve. Total and peak milk yield increase with increasing parity up to about the third or fourth parity, while peak yield is later for first than for later parities. The analysis also enables estimation of lactation characteristics of untested animals, prediction of future characteristics and identification of exceptional animals.     

Bayesian Inference for Multistate ‘Step and Turn’ Animal Movement in Continuous Time

Mechanistic modelling of animal movement is often formulated in discrete time despite problems with scale invariance, such as handling irregularly timed observations. A natural solution is to formulate in continuous time, yet uptake of this has been slow. This lack of implementation is often excused by a difficulty in interpretation. Here we aim to bolster usage by developing a continuous-time model with interpretable parameters, similar to those of popular discrete-time models that use turning angles and step lengths. Movement is defined by a joint bearing and speed process, with parameters dependent on a continuous-time behavioural switching process, creating a flexible class of movement models. Methodology is presented for Markov chain Monte Carlo inference given irregular observations, involving augmenting observed locations with a reconstruction of the underlying movement process. This is applied to well-known GPS data from elk (Cervus elaphus), which have previously been modelled in discrete time. We demonstrate the interpretable nature of the continuous-time model, finding clear differences in behaviour over time and insights into short-term behaviour that could not have been obtained in discrete time.     

A Multivariate Hidden Markov Model for the Identification of Sea Regimes from Incomplete Skewed and Circular Time Series

The identification of sea regimes from environmental multivariate times series is complicated by the mixed linear?Ccircular support of the data, by the occurrence of missing values, by the skewness of some variables, and by the temporal autocorrelation of the measurements. We address these issues simultaneously by a hidden Markov approach, and segment the data into pairs of toroidal and skew-elliptical clusters by means of the inferred sequence of latent states. Toroidal clusters are defined by a class of bivariate von Mises densities, while skew-elliptical clusters are defined by mixed linear models with positive random effects. The core of the classification procedure is an EM algorithm accounting for missing measurements, unknown cluster membership, and random effects as different sources of incomplete information. Moreover, standard simulation routines allow for the efficient computation of bootstrap standard errors. The proposed procedure is illustrated for a multivariate marine time series, and identifies a number of wintertime regimes in the Adriatic Sea.     

Bayesian Clustering of Animal Abundance Trends for Inference and Dimension Reduction

We consider a model-based clustering approach to examining abundance trends in a metapopulation. When examining trends for an animal population with management goals in mind one is often interested in those segments of the population that behave similarly to one another with respect to abundance. Our proposed trend analysis incorporates a clustering method that is an extension of the classic Chinese Restaurant Process, and the associated Dirichlet process prior, which allows for inclusion of distance covariates between sites. This approach has two main benefits: (1) nonparametric spatial association of trends and (2) reduced dimension of the spatio-temporal trend process. We present a transdimensional Gibbs sampler for making Bayesian inference that is efficient in the sense that all of the full conditionals can be directly sampled from save one. To demonstrate the proposed method we examine long term trends in northern fur seal pup production at 19 rookeries in the Pribilof Islands, Alaska. There was strong evidence that clustering of similar year-to-year deviation from linear trends was associated with whether rookeries were located on the same island. Clustering of local linear trends did not seem to be strongly associated with any of the distance covariates. In the fur seal trends analysis an overwhelming proportion of the MCMC iterations produced a 73–79 % reduction in the dimension of the spatio-temporal trend process, depending on the number of cluster groups.     

Approximate and Pseudo-Likelihood Analysis for Logistic Regression Using External Validation Data to Model Log Exposure

A common goal in environmental epidemiologic studies is to undertake logistic regression modeling to associate a continuous measure of exposure with binary disease status, adjusting for covariates. A frequent complication is that exposure may only be measurable indirectly, through a collection of subject-specific variables assumed associated with it. Motivated by a specific study to investigate the association between lung function and exposure to metal working fluids, we focus on a multiplicative-lognormal structural measurement error scenario and approaches to address it when external validation data are available. Conceptually, we emphasize the case in which true untransformed exposure is of interest in modeling disease status, but measurement error is additive on the log scale and thus multiplicative on the raw scale. Methodologically, we favor a pseudo-likelihood (PL) approach that exhibits fewer computational problems than direct full maximum likelihood (ML) yet maintains consistency under the assumed models without necessitating small exposure effects and/or small measurement error assumptions. Such assumptions are required by computationally convenient alternative methods like regression calibration (RC) and ML based on probit approximations. We summarize simulations demonstrating considerable potential for bias in the latter two approaches, while supporting the use of PL across a variety of scenarios. We also provide accessible strategies for obtaining adjusted standard errors to accompany RC and PL estimates.     

Bayesian estimation of survival from mark-recapture data

An understanding of survival patterns is a fundamental component of animal population biology. Mark-recapture models are often used in the estimation of animal survival rates. Maximum likelihood estimation, via either analytic solution or numerical approximation, has typically been used for inference in these models throughout the literature. In this article, a Bayesian approach is outlined and an easily applicable implementation via Markov chain Monte Carlo is described. The method is illustrated using 13 years of mark-recapture data for fulmar petrels on an island in Orkney. Point estimates of survival are similar to the maximum likelihood estimates (MLEs), but the posterior variances are smaller than the corresponding asymptotic variances of the MLEs. The Bayesian approach yields point estimates of 0.9328 for the average annual survival probability and 14.37 years for the expected lifetime of the fulmar petrels. A simple modification that accounts for missing data is also described. The approach is easier to apply than augmentation methods in this case, and simulations indicate that the performance of the estimators is not significantly diminished by the missing data.     

利用优选状态数的MCMC模拟农机装备负载

传统马尔科夫链蒙特卡洛（Markov chain Monte Carlo,MCMC）方法中状态数的选取常依赖于主观经验,用于农机装备负载模拟时,状态数取值不当将导致负载模拟精度降低或算法运行时间冗长。针对此问题,该研究提出一种基于伪损伤一致性的状态数优选方法。首先确定MCMC算法中状态数的初选范围,然后分别计算范围内不同状态数所对应的负载模拟结果,最后以生成的模拟负载与原始载荷之间的损伤一致性为评价准则确定优选状态数。利用拖拉机关键零部件的实测载荷数据对该方法进行验证。结果表明,随着状态数的提高,模拟负载与原始载荷之间的损伤一致性变化趋于平稳,算法运算时长增速不断提高,相比于传统方法,基于优选状态数的MCMC算法能够得到伪损伤差异在1%以内的负载模拟结果,与载荷谱编制的目标需求更加匹配,在保证模拟结果精度的同时有效减少运算成本。该研究能够为农机装备关键零部件的动态仿真分析及可靠性试验提供更加可靠的数据支撑。     

Estimation of sensitivity and specificity of three conditionally dependent diagnostic tests in the absence of a gold standard

This article presents a model to evaluate the accuracy of diagnostic tests. Data from three tests for the detection of EF-positive Streptococcus suis serotype 2 strains in sows were analyzed. The data were collected in a field study in the absence of a gold standard, that is, the true disease status (noninfected or infected) of the tested animals was unknown. Two tests were based on a polymerase chain reaction (PCR); one test was applied to a tonsil swab (taken from the live animal), and the other test was applied to the whole tonsil (collected at slaughter). The third test was based on a bacterial examination (BE) of the whole tonsil. To reduce experimental cost BE was performed only for a subset of the animals in the sample. The model allows for dependence between tests, conditional upon the unknown true disease status of the animals. Accuracy was expressed in terms of sensitivity and specificity of the tests. A Bayesian analysis was performed that incorporated prior information about the accuracy of the tests. The model parameters have a simple interpretation and specification of priors is straightforward. Posterior inference was carried out with Markov chain Monte Carlo (MCMC) methods, employing the Gibbs sampler, as implemented in the WinBUGS program. Different parameterizations to allow for selection and missing values, use of different priors, practical problems in the analysis, and some interesting issues in a joint analysis of the binary (positive or negative) results of PCR and BE and two additional continuous enzyme-linked immunosorbent assays (ELISA) are discussed.     

Bivariate spatial process modeling for constructing indicator or intensity weighted spatial CDFs

A spatial cumulative distribution function (SCDF) gives the proportion of a spatial domain D having the value of some response variable less than a particular level w. This article provides a fully hierarchical approach to SCDF modeling, using a Bayesian framework implemented via Markov chain Monte Carlo (MCMC) methods. The approach generalizes the customary SCDF to accommodate density or indicator weighting. Bivariate spatial processes emerge as a natural approach for framing such a generalization. Indicator weighting leads to conditional SCDFs, useful in studying, for example, adjusted exposure to one pollutant given a specified level of exposure to another. Intensity weighted (or population density weighted) SCDFs are particularly natural in assessments of environmental justice, where it is important to determine if a particular sociodemographic group is being excessively exposed to harmful levels of certain pollutants. MCMC methods (combined with a convenient Kronecker structure) enable straightforward estimates or approximate estimates of bivariate, conditional, and weighted SCDFs. We illustrate our methods with two air pollution datasets, one recording both nitric oxide (NO) and nitrogen dioxide (NO₂) ambient levels at 67 monitoring sites in central and southern California, and the other concerning ozone exposure and race in Atlanta, GA.     

Optimal Estimation Versus MCMC for $$\mathrm{{CO}}_{2}$$ Retrievals

The Orbiting Carbon Observatory-2 (OCO-2) collects infrared spectra from which atmospheric properties are retrieved. OCO-2 operational data processing uses optimal estimation (OE), a state-of-the-art approach to inference of atmospheric properties from satellite measurements. One of the main advantages of the OE approach is computational efficiency, but it only characterizes the first two moments of the posterior distribution of interest. Here we obtain samples from the posterior using a Markov Chain Monte Carlo (MCMC) algorithm and compare this empirical estimate of the true posterior to the OE results. We focus on 600 simulated soundings that represent the variability of physical conditions encountered by OCO-2 between November 2014 and January 2016. We treat the two retrieval methods as ensemble and density probabilistic forecasts, where the MCMC yields an ensemble from the posterior and the OE retrieval result provide the first two moments of a normal distribution. To compare these methods, we apply both univariate and multivariate diagnostic tools and proper scoring rules. The general impression from our study is that when compared to MCMC, the OE retrieval performs reasonably well for the main quantity of interest, the column-averaged \(\mathrm{{CO}}_{2}\) concentration \(X_{\mathrm{{CO}}_{2}}\), but not for the full state vector \(\mathbf {X}\) which includes a profile of \(\mathrm{{CO}}_{2}\) concentrations over 20 pressure levels, as well as several other atmospheric properties.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.     

A simulation study on tests of hypotheses and confidence intervals for fixed effects in mixed models for blocked experiments with missing data

This article considers the analysis of experiments with missing data from various experimental designs frequently used in agricultural research (randomized complete blocks, split plots, strip plots). We investigate the small sample properties of REML-based Wald-type F tests using linear mixed models. Several methods for approximating the denominator degrees of freedom are employed, all of which are available with the MIXED procedure of the SAS System (8.02). The simulation results show that the Kenward-Roger method provides the best control of the Type I error rate and is not inferior to other methods in terms of power.     

基于全局敏感性分析和贝叶斯方法的WOFOST作物模型参数优化

作物模型参数的敏感性分析、标定和验证可以提高模型的效率和精准度,进而为模型应用做好准备工作。该研究结合参数全局敏感性分析方法以及贝叶斯后验估计理论的马尔科夫蒙特卡洛(Markov Chain Monte Carlo,MCMC)方法,以华北栾城站三年的冬小麦观测数据(叶面积和地上生物量)为参照,对WOFOST模型的55个品种参数进行了敏感性分析、筛选和优化。发现:1)对叶面积影响较大的参数为:生育期为0、0.5、0.6和0.75时的比叶面积、生育期为1.5时的最大光合速率、叶面积指数最大增长率;对地上干物质影响较大的参数为:生育期为1.5时的最大光合速率、生育期为0时的比叶面积、35℃时叶面积的生命周期、生育期为0时的散射消光系数、生育期为1.8时的最大光合速率、储存器官的同化物转换效率。2)潜在和雨养产量水平下,最大叶面积和地上生物量对参数的敏感性差异不大。3)马尔科夫蒙特卡洛方法(MCMC)可以对WOFOST模型品种参数较好地优化;设计的3种校正-验证方案中,第1种方案(用1998-1999年作为校正年份,1999-2000年,2000-2001年作为验证年份)模拟效果最好。4)优化后的参数,模型对潜在产量水平模拟较好,一致性指数均大于0.9,相对均方根误差小于20%;而对有水分胁迫的雨养情况下比潜在产量水平的模拟结果差,表明模型对水分胁迫的模拟不足。该研究为WOFOST模型区域应用和模型调整优化提供科学理论依据。