External drift kriging of NOx concentrations with dispersion model output in a reduced air quality monitoring network

In the mid nineteen eighties the Dutch NO_x air quality monitoring network was reduced from 73 to 32 rural and city background stations, leading to higher spatial uncertainties. In this study, several other sources of information are being used to help reduce uncertainties in parameter estimation and spatial mapping. For parameter estimation, we used Bayesian inference. For mapping, we used kriging with external drift (KED) including secondary information from a dispersion model. The methods were applied to atmospheric NO_x concentrations on rural and urban scales. We compared Bayesian estimation with restricted maximum likelihood estimation and KED with universal kriging. As a reference we also included ordinary least squares (OLS). Comparison of several parameter estimation and spatial interpolation methods was done by cross-validation. Bayesian analysis resulted in an error reduction of 10 to 20% as compared to restricted maximum likelihood, whereas KED resulted in an error reduction of 50% as compared to universal kriging. Where observations were sparse, the predictions were substantially improved by inclusion of the dispersion model output and by using available prior information. No major improvement was observed as compared to OLS, the cause presumably being that much good information is contained in the dispersion model output, so that no additional spatial residual random field is required to explain the data. In all, we conclude that reduction in the monitoring network could be compensated by modern geostatistical methods, and that a traditional simple statistical model is of an almost equal quality.

Estimating growth charts via nonparametric quantile regression: a practical framework with application in ecology

We discuss a practical and effective framework to estimate reference growth charts via regression quantiles. Inequality constraints are used to ensure both monotonicity and non-crossing of the estimated quantile curves and penalized splines are employed to model the nonlinear growth patterns with respect to age. A companion R package is presented and relevant code discussed to favour spreading and application of the proposed methods.     

Estimation of the spatial covariance in Universal Kriging: Application to forest inventory

This paper presents a simple least squares procedure for estimating the spatial covariance and compares it with the numerically more difficult restricted maximum likelihood procedure. Thereafter, it compares design-based and kriging techniques for the estimation of spatial averages in the context of double sampling, as used in forest inventory, where terrestrial sample plots are combined with auxiliary information based on aerial photographs. A case study illustrates the theory.     

Spatial interpolation of air temperature using environmental context: Application to a crop model

The air temperature is one of the main input data in models for water balance monitoring or crop models for yield prediction. The different phenological stages of plant growth are generally defined according to cumulated air temperature from the sowing date. When these crop models are used at the regional scale, the meteorological stations providing input climatic data are not spatially dense enough or in a similar environment to reflect the crop local climate. Hence spatial interpolation methods must be used. Climatic data, particularly air temperature, are influenced by local environment. Measurements show that the air above dry surfaces is warmer than above wet areas. We propose a method taking into account the environment of the meteorological stations in order to improve spatial interpolation of air temperature. The aim of this study is to assess the impact of these corrected climatic data in crop models. The proposed method is an external drift kriging where the Kriging system is modified to correct local environment effects. The environment of the meteorological stations was characterized using a land use map summarized in a small number of classes considered as a factor influencing local temperature. This method was applied to a region in south-east France (150×250 km) where daily temperatures were measured on 150 weather stations for two years. Environment classes were extracted from the CORINE Landcover map obtained from remote sensing data. Categorical external drift kriging was compared to ordinary kriging by a cross validation study. The gain in precision was assessed for different environment classes and for summer days. We then performed a sensitivity study of air temperature with the crop model STICS. The influence of interpolation corrections on the main outputs as yield or harvest date is discussed. We showed that the method works well for air temperature in summer and can lead to significant correction for yield prediction. For example, we observed by cross validation a bias reduction of 0.5 to 1.0°C (exceptionally 2.5°C for some class), which corresponds to differences in yield prediction from 0.6 to 1.5 t/ha.     

A note on a non-stationary point source spatial model

A point source, non-stationary covariance structure model is proposed, having only one additional parameter over a standard, stationary covariance structure, spatial model. Additionally, the proposed model is demonstrated to fit better than the three extra parameter, point source, non-stationary spatial model proposed by Ecker and De Oliveira (Commun Stat Theory Methods 37:2066–2078, 2008). The proposed model is fit from a Bayesian perspective and illustrated using a house sales dataset from Cedar Falls, Iowa.     

Spatial analyses of groundwater level differences using geostatistical modeling

The purpose of this study was to determine and evaluate the spatial changes in the depletion of groundwater level differences by using geostatistical methods based on data from 58 groundwater wells during the period from April 1999 to April 2008 in the study area. Geostatistical methods have been used widely as a convenient tool to make decision on the management of groundwater levels. To evaluate the spatial changes in the level of the groundwater, geographic information system is used for the application of universal kriging method with cross-validation leading to the estimation of groundwater levels. The resulting prediction mappings identify the locations of groundwater level fluctuations of the study area. The average range of variogram (spherical model) for the spatial analysis is about 9,200 m. Results of universal kriging for groundwater level differences drops were underestimated by 15 %. Cross-validation errors are within an acceptable level. The maps show that this area of high decrease of groundwater level is located at the southwest. Kriging model helps also to detect sensitively risk prone areas for groundwater withdrawing. Such areas must be protected with an effective management procedure for future groundwater exploitations.     

A capture-recapture model with heterogeneity and behavioural response

We develop the non-parametric maximum likelihood estimator (MLE) of the full M_bh capture-recapture model which utilizes both initial capture and recapture data and permits both heterogeneity (h) between animals and behavioural (b) response to capture. Our MLE procedure utilizes non-parametric maximum likelihood estimation of mixture distributions (Lindsay, 1983; Lindsay and Roeder, 1992) and the EM algorithm (Dempsteret al., 1977). Our MLE estimate provides the first non-parametric estimate of the bivariate capture-recapture distribution.Since non-parametric maximum likelihood estimation exists for submodels M_h (allowing heterogeneity only), Mb (allowing behavioural response only) and M₀ (allowing no changes), we develop maximum likelihood-based model selection, specifically the Akaike information criterion (AIC) (Akaike, 1973). The AIC procedure does well in detecting behavioural response but has difficulty in detecting heterogeneity.     

Finite area smoothing with generalized distance splines

Most conventional spatial smoothers smooth with respect to the Euclidean distance between observations, even though this distance may not be a meaningful measure of spatial proximity, especially when boundary features are present. When domains have complicated boundaries leakage (the inappropriate linking of parts of the domain which are separated by physical barriers) can occur. To overcome this problem, we develop a method of smoothing with respect to generalized distances, such as within domain distances. We obtain the generalized distances between our points and then use multidimensional scaling to find a configuration of our observations in a Euclidean space of 2 or more dimensions, such that the Euclidian distances between points in that space closely approximate the generalized distances between the points. Smoothing is performed over this new point configuration, using a conventional smoother. To mitigate the problems associated with smoothing in high dimensions we use a generalization of thin plate spline smoothers proposed by Duchon (Constructive theory of functions of several variables, pp 85–100, 1977). This general method for smoothing with respect to generalized distances improves on the performance of previous within-domain distance spatial smoothers, and often provides a more natural model than the soap film approach of Wood et al. (J R Stat Soc Ser B Stat Methodol 70(5):931–955, 2008). The smoothers are of the linear basis with quadratic penalty type easily incorporated into a range of statistical models.     

Spatial methods for plot-based sampling of wildlife populations

Classical sampling methods can be used to estimate the mean of a finite or infinite population. Block kriging also estimates the mean, but of an infinite population in a continuous spatial domain. In this paper, I consider a finite population version of block kriging (FPBK) for plot-based sampling. The data are assumed to come from a spatial stochastic process. Minimizing mean-squared-prediction errors yields best linear unbiased predictions that are a finite population version of block kriging. FPBK has versions comparable to simple random sampling and stratified sampling, and includes the general linear model. This method has been tested for several years for moose surveys in Alaska, and an example is given where results are compared to stratified random sampling. In general, assuming a spatial model gives three main advantages over classical sampling: (1) FPBK is usually more precise than simple or stratified random sampling, (2) FPBK allows small area estimation, and (3) FPBK allows nonrandom sampling designs.     

The maximum likelihood based intensity estimate for circular point processes

Identifying the geometrical nature of spatial point patterns plays an important role in many areas of scientific research. Common types of spatial point processes involve random, regular, and cluster patterns. However, some point patterns suggest identifiable geometrical shapes such as a circular or other conic patterns. These patterns may be recognized as either a specific clustered shape or an inhomogeneous point pattern. Less noisy conic shapes, including circular patterns, are heavily discussed in the pattern recognition literature, but the goodness-of-fit of conic-fitting algorithms is rarely discussed for very noisy data. This study addresses a parameter estimation technique for noisy circular point patterns using the maximum likelihood principle. Additionally, a spatial statistical tool known as the L-function is used to investigate whether the fitted location pattern is reasonably attributable to a circular shape. A novel quantity named ‘relative log-error’ (\(\gamma \)) is introduced to quantify the goodness-of-fit for circular model fits. An iteratively re-weighted least squares procedure is introduced and robustness is evaluated under several error structures. Computational efficiency of the current and novel circle-fitting methods is also discussed. The findings are applied to two environmental science data sets.     

Some comments on design-based line-intersect sampling with segmented transects

Line-intersect sampling based on segmented transects is adopted in many forest inventories to quantify important ecological indicators such as coarse woody debris attributes. By assuming a design-based approach, Affleck, Gregoire and Valentine (2005, Environ Ecol Stat 12:139–154) have recently proposed a sampling protocol for this line-intersect setting and have suggested an estimation method based on linear homogeneous estimators. However, their proposal does not encompass the estimation procedure currently adopted in some national forest inventories. Hence, the present paper aims to introduce a unifying perspective for both methods. Moreover, it is shown that the two procedures give rise to coincident estimators for almost all the usual field applications. Finally, some strategies for efficient segmented-transect replications are considered.

Spatially stratified sampling using auxiliary information for geostatistical mapping

This paper presents a method of spatial sampling based on stratification by Local Moran’s I _i calculated using auxiliary information. The sampling technique is compared to other design-based approaches including simple random sampling, systematic sampling on a regular grid, conditional Latin Hypercube sampling and stratified sampling based on auxiliary information, and is illustrated using two different spatial data sets. Each of the samples for the two data sets is interpolated using regression kriging to form a geostatistical map for their respective areas. The proposed technique is shown to be competitive in reproducing specific areas of interest with high accuracy.     

Bootstrap confidence intervals for adaptive cluster sampling design based on Horvitz–Thompson type estimators

Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) introduced several bootstrap methods under adaptive cluster sampling using a Horvitz–Thompson type estimator. Using a simulation study, they showed that their proposed methods provide confidence intervals with highly understated coverage rates. In this article, we first show that their bootstrap methods provide biased bootstrap estimates. We then define two bootstrap methods, based on the method of Gross (Proceeding of the survey research methods section. American Statistical Association, Alexandria, VA, pp 181–184, 1980) and Bootstrap With Replacement, that provide unbiased bootstrap estimates of the population mean with bootstrap variances matching the corresponding unbiased variance estimator. Using a simulation study, we show that the bootstrap confidence intervals based on our proposed methods have better performance than those based on available bootstrap methods, in the sense of having coverage proportion closer to the nominal coverage level. We also compare the proposed intervals to empirical likelihood based intervals in small samples.     

An estimating function approach to the inference of catch-effort models

A class of catch-effort models, which allows for heterogeneous removal probabilities, is proposed for closed populations. The model includes three types of removal probabilities: multiplicative, Poisson and logistic. The usual removal and generalized removal models then become special cases. The equivalence of the proposed model and a special type of capture-recapture model is discussed. A unified estimating function approach is used to estimate the initial population size. For the homogeneous model, the resulting population size estimator based on optimal estimating functions is asymptotically equivalent to the maximum likelihood estimator. One advantage for our approach is that it can be extended to handle the heterogeneous populations in which the maximum likelihood estimators do not exist. The bootstrap method is applied to construct variance estimators and confidence intervals. We illustrate the method by two real data examples. Results of a simulation study investigating the performance of the proposed estimation procedure are presented.     

A statistical evaluation of non-ergodic variogram estimators

Geostatistics is a set of statistical techniques that is increasingly used to characterize spatial dependence in spatially referenced ecological data. A common feature of geostatistics is predicting values at unsampled locations from nearby samples using the kriging algorithm. Modeling spatial dependence in sampled data is necessary before kriging and is usually accomplished with the variogram and its traditional estimator. Other types of estimators, known as non-ergodic estimators, have been used in ecological applications. Non-ergodic estimators were originally suggested as a method of choice when sampled data are preferentially located and exhibit a skewed frequency distribution. Preferentially located samples can occur, for example, when areas with high values are sampled more intensely than other areas. In earlier studies the visual appearance of variograms from traditional and non-ergodic estimators were compared. Here we evaluate the estimators' relative performance in prediction. We also show algebraically that a non-ergodic version of the variogram is equivalent to the traditional variogram estimator. Simulations, designed to investigate the effects of data skewness and preferential sampling on variogram estimation and kriging, showed the traditional variogram estimator outperforms the non-ergodic estimators under these conditions. We also analyzed data on carabid beetle abundance, which exhibited large-scale spatial variability (trend) and a skewed frequency distribution. Detrending data followed by robust estimation of the residual variogram is demonstrated to be a successful alternative to the non-ergodic approach.     

Nile water pollution and technical efficiency of crop production in Egypt: an assessment using spatial and non-parametric modelling

Agriculture is considered one of the vital activities in Egypt; it consumes about 83 % of the Egyptian Nile water quota. This activity is becoming negatively affected by water pollution causing negative repercussion on land productivity and subsequently food security. This paper assesses the water quality for agriculture along the mainstream of the Nile in Egypt through spatial distributions modelling of total dissolved solids (TDS), using spatial statistical analysis. The study’s sample frame consists of 78 sampling points collected in February 2008 and located on the Nile mainstream and its two branches, Rosetta and Damietta. Exploratory spatial data analysis is carried out on the TDS, followed by plotting and modelling the experimental semi-variogram. Then, cross validation is executed in order to determine the best fitting model. Finally, surface mapping is generated by performing spatial interpolation, using kriging technique. The generated surface map shows that the TDS levels increase from Upper to Lower Egypt, exceeds the standard limit in Beni-Suef and Rosetta branch. In fact, high levels of TDS are known to have a negative effect on Egyptian agriculture through harmfully affecting the soil and consequently the crop yields. Therefore, an analysis of the effect of water pollution on technical efficiency of crop production is conducted using a non-parametric mathematical programming approach to frontier estimation. The results of this estimation indicated that the TDS is overutilized in all governorates except Aswan.     

Accounting for rate instability and spatial patterns in the boundary analysis of cancer mortality maps

Boundary analysis of cancer maps may highlight areas where causative exposures change through geographic space, the presence of local populations with distinct cancer incidences, or the impact of different cancer control methods. Too often, such analysis ignores the spatial pattern of incidence or mortality rates and overlooks the fact that rates computed from sparsely populated geographic entities can be very unreliable. This paper proposes a new methodology that accounts for the uncertainty and spatial correlation of rate data in the detection of significant edges between adjacent entities or polygons. Poisson kriging is first used to estimate the risk value and the associated standard error within each polygon, accounting for the population size and the risk semivariogram computed from raw rates. The boundary statistic is then defined as half the absolute difference between kriged risks. Its reference distribution, under the null hypothesis of no boundary, is derived through the generation of multiple realizations of the spatial distribution of cancer risk values. This paper presents three types of neutral models generated using methods of increasing complexity: the common random shuffle of estimated risk values, a spatial re-ordering of these risks, or p-field simulation that accounts for the population size within each polygon. The approach is illustrated using age-adjusted pancreatic cancer mortality rates for white females in 295 US counties of the Northeast (1970–1994). Simulation studies demonstrate that Poisson kriging yields more accurate estimates of the cancer risk and how its value changes between polygons (i.e., boundary statistic), relatively to the use of raw rates or local empirical Bayes smoother. When used in conjunction with spatial neutral models generated by p-field simulation, the boundary analysis based on Poisson kriging estimates minimizes the proportion of type I errors (i.e., edges wrongly declared significant) while the frequency of these errors is predicted well by the p-value of the statistical test.

禹城地区土壤铅含量空间分布的指示克里格估值

以鲁西北禹城地区土壤重金属Pb含量为例,针对其存在特异值与偏态分布的特点,利用指示克里格法研究了土壤Pb含量小于特定阈值的条件概率分布,分析了该地区土壤Pb含量的空间分布特征,并对普通克里格和指示克里格的估值效果进行交叉验证。结果表明：研究区土壤Pb含量总体较低,未超过国家标准的限值,但不同部位土壤Pb含量差异较大;普通克里格法和指示克里格法形成的土壤Pb含量空间分布格局相似,但普通克里格法的平滑效果强于指示克里格法;研究区域土壤环境质量总体良好,Pb含量高值区主要集中在各乡镇驻地附近的农田;指示克里格对土壤Pb含量空间估值的精度优于普通克里格,指示克里格法更适用于原始数据呈非正态分布且特异值对估值结果有较大影响的土壤属性。本研究结果为该地区土壤环境质量评价和重金属污染防控提供科学依据。     

A copula-based inference to piecewise exponential models under dependent censoring,with application to time to metamorphosis of salamander larvae

In ecology and evolutionary biology, controlled animal experiments are often conducted to measure time to metamorphosis which is possibly censored by the competing risk of death and the follow-up end. This paper considers the problem of estimating the survival function of time-to-event when it is subject to dependent censoring. When the censorship is due to competing risks, the traditional assumption of independent censorship may not be satisfied, and hence, the usual application of the Kaplan–Meier estimator yields a biased estimation for the survival function of the event time. This paper follows an assumed copula approach (Zheng and Klein in Biometrika 82(1):127–138, 1995) to adjust for dependence between the event time of interest and the competing event time. While the literature on an assumed copula approach has mostly focused on semiparametric settings, we alternatively consider a parametric approach with piecewise exponential models for fitting the survival function. We develop maximum likelihood estimation under the piecewise exponential models with an assumed copula. A goodness-of-fit procedure is also developed, which touches upon the identifiability issue of the copula. We conduct simulations to examine the performance of the proposed method and compare it with an existing semiparametric method. The method is applied to real data analysis on time to metamorphosis for salamander larvae living in Hokkaido, Japan (Michimae et al. in Evol Ecol Res 16:617–629, 2014).