Autologistic models with interpretable parameters

Ecologists are interested in characterizing succession processes, in particular monitoring the spread of invasive species and their effect on resident species. In situations for which binary response variables representing presence or absence of plants are observed over a spatial lattice, it may be desirable to use a model that accounts for the statistical dependence in the data, as well as the effect of potential covariates. One such model is the autologistic regression model. We show that the typical parameterization of the autologistic model presents difficulties in interpreting model parameters across varying levels of statistical dependence, and propose an alternative (centered) parameterization that overcomes this difficulty.We use the centered autologistic model to study the dynamics over time of two species, Rumex acetosella and Lonicera japonica, in an abandoned agricultural field in New Jersey, and compare the results to those obtained from using the traditional autologistic parameterization.     

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 Zero-Inflated Spatial Gamma Process Model With Applications to Disease Mapping

In this paper, we introduce a novel discrete Gamma Markov random field (MRF) prior for modeling spatial relations among regions in geo-referenced health data. Our proposition is incorporated into a generalized linear mixed model zero-inflated (ZI) framework that accounts for excess zeroes not explained by usual parametric (Poisson or Negative Binomial) assumptions. The ZI framework categorizes subjects into low-risk and high-risk groups. Zeroes arising from the low-risk group contributes to structural zeroes, while the high-risk members contributes to random zeroes. We aim to identify explanatory covariates that might have significant effect on (i) the probability of subjects in low-risk group, and (ii) intensity of the high risk group, after controlling for spatial association and subject-specific heterogeneity. Model fitting and parameter estimation are carried out under a Bayesian paradigm through relevant Markov chain Monte Carlo (MCMC) schemes. Simulation studies and application to a real data on hypertensive disorder of pregnancy confirms that our model provides superior fit over the widely used conditionally auto-regressive proposition.     

Autologistic regression model for the distribution of vegetation

Modeling the contagious distribution of vegetation and species in ecology and biogeography has been a challenging issue. Previous studies have demonstrated that the autologistic regression model is a useful approach for describing the distribution because patial correlation can readily be accounted for in the model. So far studies have been mainly restrained to the first-orderautologistic model. However, the first-order correlationmodel may sometimes be insufficient as long-range dispersal/migration can play a significant role in species distribution. In this study, we used the second-order autologistic regression model to model the distributions of the subarctic evergreen woodland and the boreal evergreen forest in British Columbia, Canada, in terms of climate covariates. We investigated and compared three estimation methods for the second-ordermodel—the maximum pseudo-likelihood method, the Monte Carlo likeli hood method, and the Markov chain Monte Carlo stochasti capproximation. Detailed procedures for these methods were developed and their performances were evaluated through simulations. The study demonstrates the importance for including the second-order correlation in the autologistic model for modeling vegetation distribution at the large geographical scale; each of the two vegetations studied was strongly autocorrelated not only in the south-north direction but also in the north west-southeast direction. The study further concluded that the assessment of climate change should be performed on the basis of individual vegetation or species because different vegetation or species likely respond differently to different sets of climate variables.     

On modeling repeated binary responses and time-dependent missing covariates

We develop a novel modeling strategy for analyzing data with repeated binary responses over time as well as time-dependent missing covariates. We assume that covariates are missing at random (MAR). We use the generalized linear mixed logistic regression model for the repeated binary responses and then propose a joint model for time-dependent missing covariates using information from different sources. A Monte Carlo EM algorithm is developed for computing the maximum likelihood estimates. We propose an extended version of the AIC criterion to identify the important factors that m a y explain the binary responses. A real plant dataset is used to motivate and illustrate the proposed methodology.     

Selection of a Geostatistical Method to Interpolate Soil Properties of the State Crop Testing Fields using Attributes of a Digital Terrain Model

The three most common techniques to interpolate soil properties at a field scale—ordinary kriging (OK), regression kriging with multiple linear regression drift model (RK + MLR), and regression kriging with principal component regression drift model (RK + PCR)—were examined. The results of the performed study were compiled into an algorithm of choosing the most appropriate soil mapping technique. Relief attributes were used as the auxiliary variables. When spatial dependence of a target variable was strong, the OK method showed more accurate interpolation results, and the inclusion of the auxiliary data resulted in an insignificant improvement in prediction accuracy. According to the algorithm, the RK + PCR method effectively eliminates multicollinearity of explanatory variables. However, if the number of predictors is less than ten, the probability of multicollinearity is reduced, and application of the PCR becomes irrational. In that case, the multiple linear regression should be used instead.     

试论元胞自动机模型与LUCC时空模拟

对土地利用变化过程的模拟和预测,需要地理信息系统(GIS)等相关技术的支撑。但目前GIS不能完整地表达地理实体的时态信息和时空关系,缺乏时空分析和动态模拟的能力。元胞自动机(Cellular Automata,简称CA)是一种自下而上的动态模拟建模框架,具有模拟地理复杂系统时空演化过程的能力。本文从元胞自动机的原理和特征入手,介绍了CA模型的构造方法,对CA模型应用于土地利用变化过程的动态模拟及预测的可操作性进行了探讨,并分析了其应用现状,存在的问题及前景。     

On Estimation and Selection of Autologistic Regression Models via Penalized Pseudolikelihood

Autologistic regression models are suitable for relating spatial binary responses in ecology to covariates such as environmental factors. For big ecological data, pseudolikelihood estimation is appealing due to its ease of computation, but at least two challenges remain. Although an important issue, it is unclear how model selection may be carried out under pseudolikelihood. In addition, for assessing the variation of pseudolikelihood estimates, parametric bootstrap using Monte Carlo simulation is often used but may be infeasible for very large data sizes. Here both these issues are addressed by developing a penalized pseudolikelihood estimation method and an approximation of the variance of the parameter estimates. A simulation study is conducted to evaluate the performance of the proposed method, followed by a data example in a study of land cover in relation to land ownership characteristics. Extension of these models and methods to spatial-temporal binary data is further discussed. This article has supplementary material online.     

Spatial stochastic volatility for lattice data

Spatial heteroscedasticity may arise jointly with spatial autocorrelation in lattice data collected from agricultural trials and environmental studies. This leads to spatial clustering not only in the level but also in the variation of the data, the latter of which may be very important, for example, in constructing prediction intervals. This article introduces a spatial stochastic volatility (SSV) component into the widely used conditional autoregressive (CAR) model to capture the spatial clustering in heteroscedasticity. The SSV component is a mean zero, conditionally independent Gaussian process given a latent spatial process of the variances. The logarithm of the latent variance process is specified by an intrinsic Gaussian Markov random field. The SSV model relaxes the traditional homoscedasticity assumption for spatial heterogeneity and brings greater flexibility to the popular spatial statistical models. The Bayesian method is used for inference. The full conditional distribution of the heteroscedasticity components can be shown to be log-concave, which facilitates an adaptive rejection sampling algorithm. Application to the well-known wheat yield data illustrates that incorporating spatial stochastic volatility may reveal the spatial heteroscedasticity hidden from existing analyses.     

基于GWR模型的江西省山洪灾害区域异同性研究

[目的]探索山洪灾害空间分布的规律,为江西省山洪灾害防治和各流域的监测和管理提供重要决策支持。[方法]根据山洪灾害的形成机理,从触发因子、孕灾环境、承灾体3个方面选取9个解释变量,将山洪灾害调查数据的5项内容作为反应变量,并作为评价山洪灾害度的指标,利用地理加权回归方法(GWR)构建模型,然后利用GIS技术探讨江西省3个不同区域山洪灾害空间分布的异同性。[结果]同一区域不同山洪灾害度指标的模型之间具有异同性,不同区域同一山洪灾害度指标的模型之间也具有异同性,不同灾害度指标的空间分布也表现出明显的异同性。[结论]在构建各项灾害度指标模型时,不仅要考虑到地域上的差异,也要考虑到不同灾害度指标之间的差异,GWR模型能有效地解释局部空间变化情况和重要解释变量的分异性。     

Bayesian spatial modeling of data from avian point count surveys

We present a unified framework for modeling bird survey data collected at spatially replicated survey sites in the form of repeated counts or detection history counts, through which we model spatial dependence in bird density and variation in detection probabilities due to changes in covariates across the landscape. The models have a complex hierarchical structure that makes them suited to Bayesian analysis using Markov chain Monte Carlo (MCMC) algorithms. For computational efficiency, we use a form of conditional autogressive model for modeling spatial dependence. We apply the models to survey data for two bird species in the Great Smoky Mountains National Park. The algorithms converge well for the more abundant and easily detected of the two species, but some simplification of the spatial model is required for convergence for the second species. We show how these methods lead to maps of estimated relative density which are an improvement over those that would follow from past approaches that ignored spatial dependence. This work also highlights the importance of good survey design for bird species mapping studies.     

Maximum likelihood estimation of regression parameters with spatially dependent discrete data

Generalized estimating equations (GEEs) have been successfully used to estimate regression parameters from discrete longitudinal data. GEEs have been adapted for spatially correlated count data with less success. It is convenient to model correlated counts as lognormal-Poisson, where a latent lognormal random process carries all correlation. This model limits correlation and can lead to negative bias of standard errors. Moreover, correlation is not the best dependence measure for highly nonnormal data. This article proposes a model which yields maximum likelihood (ML) estimates of regression parameters when the response is discrete and spatially dependent. This model employs a spatial Gaussian copula, bringing the discrete distribution into the Gaussian geostatistical framework, where correlation completely describes dependence. The model yields a log-likelihood for regression parameters that can be maximized using established numerical methods. The proposed procedure is used to estimate the relationship between Japanese beetle grub counts and soil organic matter. These data exhibit residual correlation well above the lognormal-Poisson correlation limit, so that model is not appropriate. The data and MATLAB code are available online. Simulations demonstrate that negative bias in GEE standard errors leads to nominal 95% confidence coverage less than 62% for moderate or strong spatial dependence, whereas ML coverage remains above 82%.     

城乡结合部土地资源城镇化的空间驱动模式分析

城乡结合部是城市化最为敏感、迅速的特殊地域实体,区域内土地快速城镇化导致耕地资源大量流失,尖锐的土地利用矛盾引发生态环境质量大幅下降,已成为社会各届关注的热点。该文以南京市郊江宁区为例,基于1999年、2010年的2期影像及其他数据,利用Logistic回归模型和地理加权逻辑回归（GWLR,geographically weighted logistic regression）模型,将邻近变量、邻域变量和政策变量3类因素在空间上予以整合,从全局、局域2个尺度探索城乡结合部土地资源城镇化的驱动模式。结果表明：Logistic模型与GWLR模型分别可解释研究区历史时期51%、64%的土地城镇化过程,后者在局部参数估计方面存在优势,各项评价参数都较优,更能够适用于研究区土地城镇化的模拟预测与机理探究;邻近变量中的距开发区、城市中心、次要干道的远近,邻域变量中城镇用地密度以及政策变量中土地交易样点密度是研究区城市扩展的重要解释变量,充分表征了城乡结合部的开发区建设热潮、已城市化地区的集聚效应与边缘式扩展模式,以及土地市场、内部交通网络对城市增长的引导与促进作用;借助GIS工具,GWLR模型能够可视化各驱动因素的空间非平稳性特征,可用来更直观、深刻地探究各变量对研究区土地城镇化的空间驱动模式,这也为耕地资源的保护提供新的视角。总体而言,以土地出让行为及开发区建设为代表的政府主导力量,是研究区土地城镇化的核心动力,这一定程度反应了中国城市郊区化的普遍特征,可为类似城乡结合部的土地管理工作提供有益借鉴。     

基于混合地理加权回归与克里格的区域降水量空间插值方法

基于四川省区域范围内144个气象站点的实测降水数据,在综合考虑空间位置、地形等影响因素的基础上,采用改进的回归克里格模型,即混合地理加权回归克里格模型（MGWRK）对四川省年降水量的空间分布进行空间插值,并与普通克里格（OK）、全局回归克里格（GRK）和地理加权回归克里格（GWRK）等模型的插值效果进行对比分析。结果表明：（1）应用逐步回归法筛选确定的用于回归分析的影响因子组合为经度、纬度和坡度,可有效消除解释变量间的多重共线性,为后续的空间插值奠定基础;（2）同一回归变量在地理加权回归（GWR）与全局回归（GR）两种回归模型中的AICc（修正的赤池信息量准则,Corrected Akaike Information Criterion）值之差（ΔAICc）可用于定量判定各回归变量的空间非平稳性类型,据此将变量坡度设为全局变量,经度和纬度设为局部变量进行处理。在此基础上,通过MGWRK模型对四川省年降水量进行空间插值;（3）MGWRK插值模型综合考虑了空间位置、地形等多个影响因素及其与降水相互关系的空间非平稳性特征,相对于传统的OK和GRK法具有更高的插值精度。     

天气发生器MulGETS和k-NN对区域历史气象场特征重现能力的比较

传统单站点天气发生器未考虑不同站点气象变量间的空间相关性,导致其在区域影响评价中的应用受到限制,而多站点天气发生器可以克服单站点天气发生器的缺点,近年来得到迅速发展。评估和验证多站点天气发生器对区域历史气象场特征的重现能力是开展影响评价的前提和基础。为此,本研究选取MulGETS（参数型）和k-NN（非参数型）发生器为代表模型,利用湘江流域12个气象站点1981−2010年日序列降水量、最高气温、最低气温资料,通过均值、标准差、偏度、极值、空间相关系数、空间连接度和自相关系数等指标的对比,评估了MulGETS和k-NN模型的优缺点及适用性。结果表明：MulGETS和k-NN模型均较好地再现了原气象场的均值、标准差和偏度,k-NN表现稍好于MulGETS。同时k-NN相比MulGETS在保持气象要素空间相关性上具有优势,特别是降水量的空间间歇性。由于算法本身的限制,k-NN无法模拟出超出历史数据范围的极值,而MulGETS具备一定的极值模拟能力。此外,MulGETS和k-NN在重现原始日尺度降水量的自相关性上均存在不足。总体来看,两个模型各具优势和不足,MulGETS更适于极端气象事件模拟,而k-NN可以更好地体现原始气象场的空间差异,实际使用时应根据不同的研究目的选择合适的模型。     

Multilevel Latent Gaussian Process Model for Mixed Discrete and Continuous Multivariate Response Data

We propose a Bayesian model for mixed ordinal and continuous multivariate data to evaluate a latent spatial Gaussian process. Our proposed model can be used in many contexts where mixed continuous and discrete multivariate responses are observed in an effort to quantify an unobservable continuous measurement. In our example, the latent, or unobservable measurement is wetland condition. While predicted values of the latent wetland condition variable produced by the model at each location do not hold any intrinsic value, the relative magnitudes of the wetland condition values are of interest. In addition, by including point-referenced covariates in the model, we are able to make predictions at new locations for both the latent random variable and the multivariate response. Lastly, the model produces ranks of the multivariate responses in relation to the unobserved latent random field. This is an important result as it allows us to determine which response variables are most closely correlated with the latent variable. Our approach offers an alternative to traditional indices based on best professional judgment that are frequently used in ecology. We apply our model to assess wetland condition in the North Platte and Rio Grande River Basins in Colorado. The model facilitates a comparison of wetland condition at multiple locations and ranks the importance of in-field measurements.     

Tensor Cubic Smoothing Splines in Designed Experiments Requiring Residual Modelling

Modelling response surfaces using tensor cubic smoothing splines is presented for three designed experiments. The aim is to show how the analyses can be carried out using the asreml software in the R environment, and details of the analyses including the code to do so are presented in a tutorial style. The experiments were all run over time and involve an explanatory quantitative treatment variable; one experiment is a field trial which has a spatial component and involves an additional treatment. Thus, in addition to the response surface for the time by explanatory variable, modelling of temporal and, for the third experiment, of temporal and spatial effects at the residual level is required. A linear mixed model is used for analysis, and a mixed model representation of the tensor cubic smoothing spline is described and seamlessly incorporated in the full linear mixed model. The analyses show the flexibility and capacity of asreml for complex modelling.Supplementary materials accompanying this paper appear online.     

Spatial Variability in Slash Linear Modeling with Finite Second Moment

This article studies the dependence of spatial linear models using a slash distribution with a finite second moment. The parameters of the model are estimated with maximum likelihood by using the EM algorithm. To avoid identifiability problems, the cross-validation, the Trace and the maximum log-likelihood value are used to choose the parameter for adjusting the kurtosis of the slash distribution and the selection of the model to explain the spatial dependence. We present diagnostic techniques of global and local influences for exploring the sensibility of estimators and the presence of possible influential observations. A simulation study is developed to determine the performance of the methodology. The results showed the effectiveness of the choice criteria of the parameter for adjusting the kurtosis and for the selection of the spatial dependence model. It has also showed that the slash distribution provides an increased robustness to the presence of influential observations. As an illustration, the proposed model and its diagnostics are used to analyze an aquifer data. The spatial prediction with and without the influential observations were compared. The results show that the contours of the interpolation maps and prediction standard error maps showed low changes when we removed the influential observations. Thus, this model is a robust alternative in the spatial linear modeling for dependent random variables. Supplementary materials accompanying this paper appear online.     

The Bayesian Group Lasso for Confounded Spatial Data

Generalized linear mixed models for spatial processes are widely used in applied statistics. In many applications of the spatial generalized linear mixed model (SGLMM), the goal is to obtain inference about regression coefficients while achieving optimal predictive ability. When implementing the SGLMM, multicollinearity among covariates and the spatial random effects can make computation challenging and influence inference. We present a Bayesian group lasso prior with a single tuning parameter that can be chosen to optimize predictive ability of the SGLMM and jointly regularize the regression coefficients and spatial random effect. We implement the group lasso SGLMM using efficient Markov chain Monte Carlo (MCMC) algorithms and demonstrate how multicollinearity among covariates and the spatial random effect can be monitored as a derived quantity. To test our method, we compared several parameterizations of the SGLMM using simulated data and two examples from plant ecology and disease ecology. In all examples, problematic levels multicollinearity occurred and influenced sampling efficiency and inference. We found that the group lasso prior resulted in roughly twice the effective sample size for MCMC samples of regression coefficients and can have higher and less variable predictive accuracy based on out-of-sample data when compared to the standard SGLMM.Supplementary materials accompanying this paper appear online.     

Comparison of spatial variables over subregions using a block bootstrap

In environmental and agricultural studies, it is often of interestto compare spatial variables across different regions. Traditional statistical tools that assume independent samples are inadequate because of potential spatial correlations. In this article, spatial dependence is accounted for by a random field model, and a non parametric test is developed to compare the overall distributions of variables in two neighboring regions. Sampling distribution of the test statistic is estimated by a spatial block bootstrap. For illustration, the procedure is applied to study root-lesion nematode populations on a production farm in Wisconsin. Choices of the bootstrap block size are investigated via a simulation study and results of the test are compared to traditional approaches.