Probability distributions and statistical inference for axial data

Observations on axes which lack information on the direction of propagation are referred to as axial data. Such data are often encountered in enviromental sciences, e.g. observations on propagations of cracks or on faults in mining walls. Even though such observations are recorded as angles, circular probability models are inappropriate for such data since the constraint that observations lie only in [0, π) needs to be enforced. Probability models for such axial data are argued here to have a general structure stemming from that of wrapping a circular distribution on a semi-circle. In particular, we consider the most popular circular model, the von Mises or circular normal distribution, and derive the corresponding axial normal distribution. Certain properties of this distribution are established. Maximum likelihood estimation of its parameters are shown to be surprisingly, in contrast to trigonometric moment estimation, numerically quite appealing. Finally we illustrate our results by several real life axial data sets. Received: September 2004/ Revised: December 2004     

Analyzing longitudinal circular data by projected normal models: a semi-parametric approach based on finite mixture models

The analysis of circular data has been recently the focus of a wide range of literature, with the general objective of providing reliable parameter estimates in the presence of heterogeneity and/or dependence among observations under a longitudinal setting. In this paper, we extend the variance component model approach to the analysis of longitudinal circular data, defining a mixed effects model for radial projections onto the circle and introducing dependence between projections through a set of correlated random coefficients. Estimation is carried out by numerical integration through an expectation-maximization algorithm without parametric assumptions upon the random coefficients distribution. The resulting model is a finite mixture of projected normal distributions. A simulation study has been carried out to investigate the behavior of the proposed model in a series of empirical situations. The proposed model is computationally parsimonious and, when applied to a real dataset on animal orientation, produces novel results.     

Hidden Markov models for circular and linear-circular time series

We introduce a new class of circular time series based on hidden Markov models. These are compared with existing models, their properties are outlined and issues relating to parameter estimation are discussed. The new models conveniently describe multi-modal circular time series as dependent mixtures of circular distributions. Two examples from biology and meteorology are used to illustrate the theory. Finally, we introduce a hidden Markov model for bivariate linear-circular time series and use it to describe larval movement of the fly Drosophila. Received: September 2003 / Revised: March 2004     

Propagation functions for monitoring distributional changes

In environmental assessment and monitoring, a primary objective of the investigator is to describe the changes occurring in the environmentally important variables over time. Propagation functions have been proposed to describe the distributional changes occurring in the variable of interest at two different times. McDonald et al. (1992, 1995) proposed an estimator of propagation function under the assumption of normality. We conduct a detailed sensitivity analysis of inference based on the normal model. It turns out that this model is appropriate only for small departures from normality whereas, for moderate to large departures, both estimation and testing of hypothesis break down. Non-parametric estimation of the propagation function based on kernel density estimation is also considered and the robustness of the choice of bandwidth for kernel density estimation is investigated. Bootstrapping is employed to obtain confidence intervals for the propagation function and also to determine the critical regions for testing the significance of distributional changes between two sampling epochs. Also studied briefly is the mathematical form and graphical shape of the propagation function for some parametric bivariate families of distributions. Finally, the proposed estimation techniques are illustrated on a data set of tree ring widths.     

物种敏感度分布的非参数核密度估计模型

针对目前物种敏感度分布参数方法建模所存在的缺点,首次提出基于非参数核密度估计方法的物种敏感度分布模型,并提出相应的最优窗宽和检验方法。选用无机汞作为案例研究对象,利用非参数核密度估计方法和3种传统参数模型分别推导了保护我国水生生物的无机汞的急性水质基准值。结果表明,非参数核密度估计方法在推导无机汞水质基准中的稳健性和精确度都大大优于传统参数模型,能够更好地构建物种敏感度分布曲线。该方法的提出丰富了水质基准的理论方法学,为更好地保护水生生物提供了有力的支撑。     

Nonparametric analysis of directional data based on data depth

Data depth is a statistical method whose primary aim is to order data of a reference space according to centrality. This is particularly appealing for directional data because no standard ordering is available, not even in the circular case. Moreover, the depth approach deals in a natural way with the peculiar aspects of directional data, i.e., the lack of a zero-direction and the wrap-around effect. The paper reviews the data depth program concentrating on typical applications to directional distributions and data. Two geometrical depth functions, simplicial depth and Tukey’s depth, are considered. Several depth-based summaries, including angular medians, depth regions and dispersion parameters are illustrated, the latter with some novel results. Examples cover symmetric and asymmetric distributions as well as real-data applications.     

Bivariate geostatistical modelling: a review and an application to spatial variation in radon concentrations

We present a comprehensive review of multivariate geostatistical models, focusing on the bivariate case. We compare in detail three approaches, the linear model of coregionalisation, the common component model and the kernel convolution approach, and discuss similarities between them. We demonstrate the merits of the common component class of models as a flexible means for modelling bivariate geostatistical data of the type that frequently arises in environmental applications. In particular, we show how kernel convolution can be used to approximate the common component model, and demonstrate the method using a data-set of calcium and magnesium concentrations in soil samples. We then apply the model to a study of domestic radon concentrations in the city of Winnipeg, Canada, in which exposure was measured at two sites (bedroom and basement) in each residential location. Our analysis demonstrates that in this study the correlation between the two sites within each house dominates the short-range spatial correlation typical of the distribution of radon.     

Identifying factors that influence soil heavy metals by using categorical regression analysis: A case study in Beijing,China

A method was proposed to identify the main influence factors of soil heavy metals. The influence degree of different environmental factors was ranked. Parent material, soil type, land use and industrial activity were main factors. Interactions between some factors obviously affected soil heavy metal distribution. Identifying the factors that influence the heavy metal contents of soil could reveal the sources of soil heavy metal pollution. In this study, a categorical regression was used to identify the factors that influence soil heavy metals. First, environmental factors were associated with soil heavy metal data, and then, the degree of influence of different factors on the soil heavy metal contents in Beijing was analyzed using a categorical regression. The results showed that the soil parent material, soil type, land use type, and industrial activity were the main influencing factors, which suggested that these four factors were important sources of soil heavy metals in Beijing. In addition, population density had a certain influence on the soil Pb and Zn contents. The distribution of soil As, Cd, Pb, and Zn was markedly influenced by interactions, such as traffic activity and land use type, industrial activity and population density. The spatial distribution of soil heavy metal hotspots corresponded well with the influencing factors, such as industrial activity, population density, and soil parent material. In this study, the main factors affecting soil heavy metals were identified, and the degree of their influence was ranked. A categorical regression represents a suitable method for identifying the factors that influence soil heavy metal contents and could be used to study the genetic process of regional soil heavy metal pollution.     

Regression analysis of correlated circular data based on the multivariate von Mises distribution

Environmental and Ecological Statistics - A regression model for correlated circular data is proposed by assuming that samples of angular measurements are drawn from a multivariate von Mises...     

Procedures for outlier detection in circular time series models

It is known that the occurrence of outliers in linear or non-linear time series models may have adverse effects on the modelling and statistical inference of the data. Consequently, extensive research has been conducted on developing outlier detection procedures so that outliers may be properly managed. However, no work has been done on the problem of outliers in circular time series data. This problem is the focus of this paper. The main objective is to develop novel numerical and graphical procedures for detecting these outliers in circular time series data.A number of circular time series models have been proposed including the circular autoregressive model. We extend the iterative outlier detection procedure which has been successfully used in linear time series models to the circular autoregressive model. The proposed procedure shows a good performance when investigated via simulation for the circular autoregressive model of order one. At the same time, several statistical techniques have been used to detect the change of preferred trend in time series data using SLIME and CUSUM plots. While the methods fail to indicate directly the outliers in circular time series data, we use the ideas employed to develop three novel graphical procedures for identifying the outliers. For illustration, we apply the procedures to a particular set of wind direction data. An agreement between the results of the graphical and iterative detection procedures is observed. These procedures could be very useful in improving the modelling and inferential processes for circular time series data.     

Mapping lightning/human-caused wildfires occurrence under ignition point location uncertainty

Fire managers need to study fire history in terms of occurrence in order to understand and model the spatial distribution of the causes of ignition. Fire atlases are useful open sources of information, recording each single fire event by means of its geographical position. In such cases the fire event is considered as point-based, rather than area-based data, completely losing its surface nature. Thus, an accurate method is needed to estimate continuous density surfaces from ignition points where location is affected by a certain degree of uncertainty. Recently, the fire scientific community has focused its attention on the kernel density interpolation technique in order to convert point-based data into continuous surface or surface-data. The kernel density technique needs a priori setting of smoothing parameters, such as the bandwidth size. Up to now, the bandwidth size was often based on subjective choices still needing expert knowledge, eventually supported by empirical decisions, thus leading to serious uncertainties. Nonetheless, a geostatistical model able to describe the point concentration and consequently the clustering degree is required. This paper tries to solve such issues by implementing the kernel density adaptive mode. Lightning/human-caused fires occurrence was investigated in the region of Aragón's autonomy over 19 years (1983–2001) using 3428 and 4195 ignition points respectively for the two causes of fire origin. An analytical calibration procedure was implemented to select the most reliable density surfaces to reduce under/over-density estimation, overcoming the current drawbacks to define it by visual inspection or personal interpretation. Besides, ignition point location uncertainty was investigated to check the sensitivity of the proposed model. The different concentration degree and the dissimilar spatial pattern of the two datasets, allow testing the proposed calibration methodology under several conditions. After having discovered the slight sensitivity of the model to the exact point position, the obtained density surfaces for the two causes were combined to discover hotspot areas and spatial patterns of the two causes. Evident differences in spatial location of the origin causes were noted and described. The general trend follows the geographical features and the human activity of the study areas. The proposed technique should be promising to support decision-making in wildfire prevention actions, because of the occurrence map can be used as a response variable in fire risk predicting models.     

A hierarchical model for spatial capture-recapture data

Estimating density is a fundamental objective of many animal population studies. Application of methods for estimating population size from ostensibly closed populations is widespread, but ineffective for estimating absolute density because most populations are subject to short-term movements or so-called temporary emigration. This phenomenon invalidates the resulting estimates because the effective sample area is unknown. A number of methods involving the adjustment of estimates based on heuristic considerations are in widespread use. In this paper, a hierarchical model of spatially indexed capture-recapture data is proposed for sampling based on area searches of spatial sample units subject to uniform sampling intensity. The hierarchical model contains explicit models for the distribution of individuals and their movements, in addition to an observation model that is conditional on the location of individuals during sampling. Bayesian analysis of the hierarchical model is achieved by the use of data augmentation, which allows for a straightforward implementation in the freely available software WinBUGS. We present results of a simulation study that was carried out to evaluate the operating characteristics of the Bayesian estimator under variable densities and movement patterns of individuals. An application of the model is presented for survey data on the flat-tailed horned lizard (Phrynosoma mcallii) in Arizona, USA.     

Species abundance distributions result from body size-energetics relationships

Abundance distributions are a central characteristic of ecosystems. Certain distributions have been derived from theoretical models of community organization, and therefore the fit of data to these distributions has been proposed as a test of these theories. However, it is shown here that the geometric sequence distribution can be derived directly from the empirical relationship between population density and body size, with the assumption of random or uniform body size distributions on a log scale (as holds at local scales). The geometric sequence model provides a good to excellent fit to empirical data. The presence of noise in the relationship between population density and body size creates a curve that begins to approximate a lognormal species abundance distribution as the noise term increases. For continental-scale data in which the body size distribution is not flat, the result of sampling tends again toward the lognormal. Repeat sampling over time smooths out species population fluctuations and damps out the noise, giving a more precise geometric sequence abundance distribution. It is argued that the direct derivation of this distribution from empirical relationships gives it priority over distributions derived from complex theoretical community models.     

A zero-inflated Bayesian spatial model with near-locating measurements

Zero-inflated data arise in many contexts. In this paper, we develop a zero-inflated Bayesian hierarchical model which deals with spatial effects, correlation among near-locating measurements as well as excess zeros simultaneously. Inference, including the sampling from the posterior distributions, predictions at new locations, and model selection, is carried out by using computationally efficient Markov chain Monte Carlo techniques. The posterior distributions are simulated using a Gibbs sampler with the embedded ratio-of-uniform method and the slice sampling algorithm. The approach is illustrated via an application to herbaceous data collected in the Missouri Ozark Forest Ecosystem Project. The results from the proposed model are compared with those generated from a non-zero inflated model. The proposed model fully incorporates the information from data collection and provides more reliable inference. A predictive $p$ value is computed for model checking and it indicates that the proposed model fits the data well.     

Measures of preferred direction for environmental and ecological circular data

Circular or angular variables indicating direction or cyclical time can be of great interest to scientists studying ecology, biology or environmental issues. A common problem of interest in circular data is estimating a preferred direction and its corresponding distribution. This problem is complicated by the so-called “wrap-around effect” on the circle, which exists because there is no natural minimum or maximum. The usual statistics employed for linear data are inappropriate for directional data, as they do not account for its circular nature. Three choices for summarizing the preferred direction (the sample circular mean, sample circular median and a circular analog of the Hodges–Lehmann estimator) are discussed, with examples from environmental and ecological applications. Similar to the linear data case, the relative emphases of different methods sometimes yield different measures of preferred direction in the presence of outliers or lack of symmetry in the original data. Received: November 2003 / Revised: June 2004     

Evaluation of potential topsoil productivity

A biomass productivity model based on soil properties and climate is developed from literature and used to evaluate and compare potential effects of mining and reclamation on several soils in the continental United States. Soil productivity is assumed to vary as a product of root distribution function modified by five soil properties: available water, aeration porosity, bulk density, electrical conductivity, and pH. Yield limiting property levels are derived from literature and input soil data are obtained from available USDA Soil Conservation Service information on typical profiles. Modelled values of potential productivity after mining, indicate problems and limitations to be expected. The proposed model can be used as a guide to reclamation strategy, to restore the land to premining conditions, or at times to enhance productivity of a reclaimed area.     

Asymmetric circular-linear multivariate regression models with applications to environmental data

We propose asymmetric angular-linear multivariate regression models, which were motivated by the need to predict some environmental characteristics based on some circular and linear predictors. A measure of fit is provided through the residual analysis. Some applications using data from solar energy radiation experiment and wind energy are given. Received: September 2003 / Revised: February 2005     

Rarity in Neotropical Forest Mammals Revisited

The identification of rare species is an important goal in conservation biology. Recent attempts to classify rare species have emphasized dichotomies in such characteristics as local population density, area of distribution, and degree of ecological specialization. In particular, Arita et al. (1990) dichotomized 100 Neotropical forest mammals according to local population density and area of distribution. Among these species of mammals, mean body mass was significantly associated with local population density and area of distribution. We argue that the effects of body mass should be removed before species are classified with respect to rarity. We re-evaluated the data on Neotropical mammal species, using regression analyses to remove the effects of body mass on population density and area of distribution, followed by analysis of residuals. This new method resulted in substantial changes in the dichotomous classification of rare species. We combined the analysis of regression residuals with a ranking procedure that assumed that local population density and area of distribution were equally important in their effects on rarity. The new ranking technique produced another different classification of the rarity of the Neotropical forest mammal species. A graphical analysis showed that ranked species differed substantially in their degree of rarity, and in the importance of local population density, area of distribution, or both, to their degree of rarity. The ranking method allows the species of greatest concern to be singled out, it can be modified to include additional variables such as niche breadth, and it should be helpful for making conservation decisions.