Estimating the spatial covariance structure using the geoadditive model

In geostatistics, both kriging and smoothing splines are commonly used to generate an interpolated map of a quantity of interest. The geoadditive model proposed by Kammann and Wand (J R Stat Soc: Ser C (Appl Stat) 52(1):1–18, 2003) represents a fusion of kriging and penalized spline additive models. Complex data issues, including non-linear covariate trends, multiple measurements at a location and clustered observations are easily handled using the geoadditive model. We propose a likelihood based estimation procedure that enables the estimation of the range (spatial decay) parameter associated with the penalized splines of the spatial component in the geoadditive model. We present how the spatial covariance structure (covariogram) can be derived from the geoadditive model. In a simulation study, we show that the underlying spatial process and prediction of the spatial map are estimated well using the proposed likelihood based estimation procedure. We present several applications of the proposed methods on real-life data examples.     

Modeling carcass removal time for avian mortality assessment in wind farms using survival analysis

In monitoring studies at wind farms, the estimation of bird and bat mortality caused by collision must take into account carcass removal by scavengers or decomposition. In this paper we propose the use of survival analysis techniques to model the time of carcass removal. The proposed method is applied to data collected in ten Portuguese wind farms. We present and compare results obtained from semiparametric and parametric models assuming four main competing lifetime distributions (exponential, Weibull, log-logistic and log-normal). Both homogeneous parametric models and accelerated failure time models were used. The fitted models enabled the estimation of the carcass persistence rates and the calculation of a scavenging correction factor for avian mortality estimation. Additionally, we discuss the impact that the distributional assumption can have on parameter estimation. The proposed methodology integrates the survival probability estimation problem with the analysis of covariate effects. Estimation is based on the most suitable model while simultaneously accounting for censored observations, diminishing scavenging rate estimation bias. Additionally, the method establishes a standardized statistical procedure for the analysis of carcass removal time in subsequent studies.     

Estimating equations for separable spatial-temporal binary data

For binary data with correlation across space and over time, the literature concerning the estimation of fixed effects in marginal models is limited. In this paper, we model the marginal probability of binary responses in terms of parameters of interest by a logistic function. An estimating equation based on the quasi-likelihood concept is developed to estimate parameters. Under separable correlation models, we show that the quasi-likelihood estimate is asymptotically optimal. A series of simulations is conducted to evaluate how the efficiency varies with the regression coefficients. We also compare the relative efficiency with another estimating equation by simulations. The proposed method is applied to an ecological study of forest decline to test independence of two spatial-temporal binary outcomes.     

An application of Bayesian variable selection to spatial concurrent linear models

Spatial concurrent linear models, in which the model coefficients are spatial processes varying at a local level, are flexible and useful tools for analyzing spatial data. One approach places stationary Gaussian process priors on the spatial processes, but in applications the data may display strong nonstationary patterns. In this article, we propose a Bayesian variable selection approach based on wavelet tools to address this problem. The proposed approach does not involve any stationarity assumptions on the priors, and instead we impose a mixture prior directly on each wavelet coefficient. We introduce an option to control the priors such that high resolution coefficients are more likely to be zero. Computationally efficient MCMC procedures are provided to address posterior sampling, and uncertainty in the estimation is assessed through posterior means and standard deviations. Examples based on simulated data demonstrate the estimation accuracy and advantages of the proposed method. We also illustrate the performance of the proposed method for real data obtained through remote sensing.     

Spatiotemporal modeling of irregularly spaced aerosol optical depth data

Many advancements have been introduced to tackle spatial and temporal structures in data. When the spatial and/or temporal domains are relatively large, assumptions must be made to account for the sheer size of the data. The large data size, coupled with realities that come with observational data, make it difficult for all of these assumptions to be met. In particular, air quality data are very sparse across geographic space and time, due to a limited air pollution monitoring network. These “missing” values make it difficult to incorporate most dimension reduction techniques developed for high-dimensional spatiotemporal data. This article examines aerosol optical depth (AOD), an indirect measure of radiative forcing, and air quality. The spatiotemporal distribution of AOD can be influenced by both natural (e.g., meteorological conditions) and anthropogenic factors (e.g., emission from industries and transport). After accounting for natural factors influencing AOD, we examine the spatiotemporal relationship in the remaining human influenced portion of AOD. The presented data cover a portion of India surrounding New Delhi from 2000–2006. The proposed method is demonstrated showing how it can handle the large spatiotemporal structure containing so much missing data for both meteorologic conditions and AOD over time and space.     

Delimiting synchronous populations from monitoring data

We propose to investigate spatial synchrony in population dynamics from monitoring data. We develop a statistical procedure to delineate populations of sites with synchronous dynamics from short time series. The procedure relies on a new norm, the synchronous total variation norm, which promotes synchrony in the estimation of the sites dynamics. The method is tested on some synthetic data sets and is applied on data from the French breeding bird monitoring program.     

Site-selection bias and apparent population declines in long-term studies

Detecting population declines is a critical task for conservation biology. Logistical difficulties and the spatiotemporal variability of populations make estimation of population declines difficult. For statistical reasons, estimates of population decline may be biased when study sites are chosen based on abundance of the focal species. In this situation, apparent population declines are likely to be detected even if there is no decline. This site-selection bias is mentioned in the literature but is not well known. We used simulations and real population data to examine the effects of site-selection biases on inferences about population trends. We used a left-censoring method to detect population-size patterns consistent with site-selection bias. The site-selection bias is an important consideration for conservation biologists, and we offer suggestions for minimizing or mitigating it in study design and analysis. Article impact statement: Estimates of population declines are biased if studies begin in large populations, and time-series data show a signature of such an effect.     

Methods for estimating peak physiological performance and correlating performance measures

Estimates of animal performance often use the maximum of a small number of laboratory trials, a method which has several statistical disadvantages. Sample maxima always underestimate the true maximum performance, and the degree of the bias depends on sample size. Here, we suggest an alternative approach that involves estimating a specific performance quantile (e.g., the 0.90 quantile). We use the information on within-individual variation in performance to obtain a sampling distribution for the residual performance measures; we use this distribution to estimate a desired performance quantile for each individual. We illustrate our approach using simulations and with data on sprint speed in lizards. The quantile method has several advantages over the sample maximum: it reduces or eliminates bias, it uses all of the data from each individual, and its accuracy is independent of sample size. Additionally, we address the estimation of correlations between two different performance measures, such as sample maxima, quantiles, or means. In particular, because of sampling variability, we propose that the correlation of sample means does a better job estimating the correlation of population maxima than the estimator which is the correlation of sample maxima.     

Multinomial mixture model with heterogeneous classification probabilities

Royle and Link (Ecology 86(9):2505?C2512, 2005) proposed an analytical method that allowed estimation of multinomial distribution parameters and classification probabilities from categorical data measured with error. While useful, we demonstrate algebraically and by simulations that this method yields biased multinomial parameter estimates when the probabilities of correct category classifications vary among sampling units. We address this shortcoming by treating these probabilities as logit-normal random variables within a Bayesian framework. We use Markov chain Monte Carlo to compute Bayes estimates from a simulated sample from the posterior distribution. Based on simulations, this elaborated Royle-Link model yields nearly unbiased estimates of multinomial and correct classification probability estimates when classification probabilities are allowed to vary according to the normal distribution on the logit scale or according to the Beta distribution. The method is illustrated using categorical submersed aquatic vegetation data.     

Estimation method to identify scalar point source in turbulent flow based on Taylor’s diffusion theory

We introduce a new approach to diffusion-source estimation for quick identification of the unknown source, based on Taylor’s diffusion theory for turbulent transport of passive scalar from a fixed point source. In order to evaluate the method, we used planar laser-induced fluorescence to measure the concentration field of fluorescent dye in water flowing in a channel. We considered two kinds of datasets: basis data and observed data. The former is used to determine the basis functions characterizing the streamwise dependence of variances for three statistics: the mean concentration, root-mean-square (RMS) of fluctuations in the concentration, and RMS of the temporal gradient of the fluctuating concentration. Consistent with Taylor’s theory, we found that the lateral distribution of each statistic was basically Gaussian, and their standard deviations increased as a function of the square root of the distance from the emitted point. Based on these facts, a basis function can be formulated and expected to be valid for estimation of unknown sources. Source estimation was performed with the observed data, which corresponded to limited available information about the concentration from an unknown point source. We confirmed a good prediction accuracy of the proposed method with an averaged bias as small as the turbulent integral scale. Better precision was achieved by employing several statistics simultaneously. In this case, the standard deviation of the estimated source position was assessed at 14 % of the mean distance between the source and measurement points, after 100 source-estimate trials with different datasets. The methodology tested in this paper is expected to be applicable more general and complex environmental diffusion issues involving anisotropic turbulent dispersion, and space–time variable mainstream systems; but its versatility in such systems is currently under investigation.     

Semi-parametric statistical approaches for space-time process prediction

The problem of estimation and prediction of a spatial-temporal stochastic process, observed at regular times and irregularly in space, is considered. A mixed formulation involving a non- parametric component, accounting for a deterministic trend and the effect of exogenous variables, and a parametric component representing the purely spatio-temporal random variation is proposed. Correspondingly, a two-step procedure, first addressing the estimation of the non- parametric component, and then the estimation of the parametric component is developed from the residual series obtained, with spatial-temporal prediction being performed in terms of suitable spatial interpolation of the temporal variation structure. The proposed model formula-tion, together with the estimation and prediction procedure, are applied using a Gaussian ARMA structure for temporal modelling to space-time forecasting from real data of air pollution concentration levels in the region surrounding a power station in northwest Spain.     

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.     

Improved testing of species traits--environment relationships in the fourth-corner problem

The fourth-corner problem entails estimation and statistical testing of the relationship between species traits and environmental variables from the analysis of three data tables. In a 2008 paper, S. Dray and P. Legendre proposed and evaluated five permutation methods for statistical significance testing, including a new two-step testing procedure. However, none of these attained the correct type I error in all cases of interest. We solve this problem by showing that a small modification of their two-step procedure controls the type I error in all cases. The modification consists of adjusting the significance level from mean square root of alpha to alpha or, equivalently, of reporting the maximum of the individual P. values as the final one. The test is also applicable to the three-table ordination method RLQ.     

Effects of sampling error and temporal correlations in population growth on process variance estimators

Estimates of a population’s growth rate and process variance from time-series data are often used to calculate risk metrics such as the probability of quasi-extinction, but temporal correlations in the data from sampling error, intrinsic population factors, or environmental conditions can bias process variance estimators and detrimentally affect risk predictions. It has been claimed (McNamara and Harding, Ecol Lett 7:16–20, 2004) that estimates of the long-term variance that incorporate observed temporal correlations in population growth are unaffected by sampling error; however, no estimation procedures were proposed for time-series data. We develop a suite of such long-term variance estimators, and use simulated data with temporally autocorrelated population growth and sampling error to evaluate their performance. In some cases, we get nearly unbiased long-term variance estimates despite ignoring sampling error, but the utility of these estimators is questionable because of large estimation uncertainty and difficulties in estimating correlation structure in practice. Process variance estimators that ignored temporal correlations generally gave more precise estimates of the variability in population growth and of the probability of quasi-extinction. We also found that the estimation of probability of quasi-extinction was greatly improved when quasi-extinction thresholds were set relatively close to population levels. Because of precision concerns, we recommend using simple models for risk estimates despite potential biases, and limiting inference to quantifying relative risk; e.g., changes in risk over time for a single population or comparative risk among populations.     

A proposal for a two-step sampling design to oversample units responding to prescribed characteristics

We introduce a novel method to extract a sample from a finite population where units with desired characteristics are over-represented. The approach is both sequential and adaptive and allows, via suitable compositions of predictive and objective functions, to target specific subsets of the population. We consider the problem of estimation and conjecture the validity of a modified Horvitz–Thompson estimator capable to account for the imbalance induced by the targeting procedure. After discussing how to apply the method to the sampling of geographically distributed units, we investigate its potential via simulations.     

Inferring evolutionary signals from ecological data in a plant-pathogen metapopulation

We followed the dynamics of local epidemics in three populations of a natural plant-pathogen system for four sequential years. We characterize the overwintering process with spatial statistics and use a stochastic, spatially explicit, modeling approach with Bayesian parameter estimation to study the spread of the infection during the growing season. Our modeling approach allows us to infer coevolutionary signals from spatiotemporal data on pathogen prevalence. Most importantly, we are able to assess the distribution of resistant hosts within the distribution of all host plants. We show that resistant hosts occur in areas with high pathogen encounter rates, and that the occurrence of resistance correlates with overwintering probability of the pathogen. The estimates for essentially all model parameters are characterized by a large amount of variation over the years and the populations. While the variation in the fraction of resistant hosts and in the force of infection is to a large extent explained by the population, the other model parameters (two parameters describing the shape of the dispersal kernel) vary essentially in an unpredictable manner, suggesting that much of the variation may occur at very fine spatial and temporal scales.     

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).     

Modelling the distribution of plant species using the autologistic regression model

For modeling the distribution of plant species in terms of climate covariates, we consider an autologistic regression model for spatial binary data on a regularly spaced lattice. This model belongs to the class of autologistic models introduced by Besag (1974). Three estimation methods, the coding method, maximum pseudolikelihood method and Markov chain Monte Carlo method are studied and comparedvia simulation and real data examples. As examples, we use the proposed methodology to model the distributions of two plant species in the state of Florida.     

Statistical Estimation and Model Selection of Species-Accumulation Functions

Abstract:  Soberón and Llorente (1993) proposed pure-birth stochastic processes as theoretical models for species-accumulation curves, and these processes have frequently been used to describe the progress of biological inventories. We describe, in algorithmic form, an alternative statistical analysis based on a likelihood approach ( Díaz-Francés & Gorostiza 2002 ) that provides mathematical rigor to the ideas in Soberón and Llorente (1993) and improves the estimation of the models by incorporating the facts that the variance of the error is not constant and that the observations are correlated. Additionally, we used the likelihood ratios between candidate models as an objective procedure for model selection, allowing comparison between the goodness of fit of various models. The software for these statistical methods can now be downloaded off the Internet. We used two examples of butterfly data sets to illustrate the use of the methods and the software.