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     

Difficulties with the lognormal model in mean estimation and testing

A frequent assumption in environmental risk assessment is that the underlying distribution of an analyte concentration is lognormal. However, the distribution of a random variable whose log has a t-distribution has infinite mean. Because of the proximity of the standard normal and t-distribution, this suggests that a distribution such as the gamma or truncated normal, with smaller right tail probabilities, might make a better statistical model for mean estimation than the lognormal. In order to assess the effect of departures from lognormality on lognormal-based statistics, we simulated complete lognormal, truncated normal, and gamma data for various sample sizes and coefficients of variation. In these cases, departures from lognormality were not easily detected with the Shapiro-Wilk test. Various lognormal-based estimates and tests were compared with alternate methods based on the ordinary sample mean and standard error. The examples were also considered in the presence of random left censoring with the mean and standard error of the product limit estimate replacing the ordinary sample mean and standard error. The results suggest that in the estimation of or tests about a mean, if the assumption of lognormality is at all suspect, then lognormal-based approaches may not be as good as the alternative methods.     

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     

Modelling asymmetrically distributed circular data using the wrapped skew-normal distribution

We consider problems of inference for the wrapped skew-normal distribution on the circle. A centered parametrization of the distribution is introduced, and simulation used to compare the performance of method of moments and maximum likelihood estimation for its parameters. Maximum likelihood estimation is shown, in general, to be superior. The operating characteristics of two moment based tests, for wrapped normal and wrapped half-normal parent populations, respectively, are also explored. The former test is easy to apply, maintains the nominal significance level well and is generally highly powerful. The latter test does not hold the nominal significance level so well, although it is very powerful against negatively skew alternatives. Likelihood based tests for the two distributions are also discussed. A real data set from the ornithological literature is used to illustrate the application of the developed methodology and its extension to finite mixture modelling. Received: September 2003/ Revised: April 2005     

A new multimodal and asymmetric bivariate circular distribution

Multimodal and asymmetric bivariate circular data arise in several different disciplines and fitting appropriate distribution plays an important role in the analysis of such data. In this paper, we propose a new bivariate circular distribution which can be used to model both asymmetric and multimodal bivariate circular data simultaneously. In fact the proposed density covers unimodality as well as multimodality, symmetry as well as asymmetry of circular bivariate data. A number of properties of the proposed density are presented. A Bayesian approach with MCMC scheme is employed for statistical inference. Three real datasets and a simulation study are provided to illustrate the performance of the proposed model in comparison with alternative models such as finite mixture Cosine model.     

Validation of ecological state space models using the Laplace approximation

Many statistical models in ecology follow the state space paradigm. For such models, the important step of model validation rarely receives as much attention as estimation or hypothesis testing, perhaps due to lack of available algorithms and software. Model validation is often based on a naive adaptation of Pearson residuals, i.e. the difference between observations and posterior means, even if this approach is flawed. Here, we consider validation of state space models through one-step prediction errors, and discuss principles and practicalities arising when the model has been fitted with a tool for estimation in general mixed effects models. Implementing one-step predictions in the R package Template Model Builder, we demonstrate that it is possible to perform model validation with little effort, even if the ecological model is multivariate, has non-linear dynamics, and whether observations are continuous or discrete. With both simulated data, and a real data set related to geolocation of seals, we demonstrate both the potential and the limitations of the techniques. Our results fill a need for convenient methods for validating a state space model, or alternatively, rejecting it while indicating useful directions in which the model could be improved.     

Sine-skewed axial distributions with an application for fallen tree data

An axis is an undirected line where there is no reason to distinguish one end of the line from the other. Phenomena in nature that can be described as axial data are numerous. In this paper a method of trigonometric moments for the axial normal or axial von Mises distribution as an alternative to the method by Arnold and SenGupta (Environ Ecol Stat 13:271–285, 2006) is discussed. Sine-skewed axial Jones–Pewsey, von Mises and wrapped Cauchy distributions are introduced as special cases of a more general construction of skew axial distributions. As an example the methods are applied to a data set which consists of the orientations of logs on the floor of a primeval spruce forest.     

Modelling species distributions with penalised logistic regressions: A comparison with maximum entropy models

An important aspect of species distribution modelling is the choice of the modelling method because a suboptimal method may have poor predictive performance. Previous comparisons have found that novel methods, such as Maxent models, outperform well-established modelling methods, such as the standard logistic regression. These comparisons used training samples with small numbers of occurrences per estimated model parameter, and this limited sample size may have caused poorer predictive performance due to overfitting. Our hypothesis is that Maxent models would outperform a standard logistic regression because Maxent models avoid overfitting by using regularisation techniques and a standard logistic regression does not. Regularisation can be applied to logistic regression models using penalised maximum likelihood estimation. This estimation procedure shrinks the regression coefficients towards zero, causing biased predictions if applied to the training sample but improving the accuracy of new predictions. We used Maxent and logistic regression (standard and penalised) to analyse presence/pseudo-absence data for 13 tree species and evaluated the predictive performance (discrimination) using presence-absence data. The penalised logistic regression outperformed standard logistic regression and equalled the performance of Maxent. The penalised logistic regression may be considered one of the best methods to develop species distribution models trained with presence/pseudo-absence data, as it is comparable to Maxent. Our results encourage further use of the penalised logistic regression for species distribution modelling, especially in those cases in which a complex model must be fitted to a sample with a limited size.     

Bayesian analysis under ordered functions of parameters

Ordered parameter problems arise in a wide variety of real world situations and are dealt with extensively in the literature. Traditional frequentist methods for dealing with these problems are rather complicated theoretically, especially when sample sizes are small. Bayesian methods are not widely used because high dimensional numerical integration is often required. However, Markov chain Monte Carlo methods provide alternatives to such numerical integration and also deal with ordered parameter problems in a straightforward manner. Little is known about the situation where functions of parameters are ordered. Such problems may seem to be of little practical concern initially, but one can readily see their importance in situations where ordering is placed on the means and variances of several normal or Gamma populations. For the Gamma distribution we will present real examples where we will analyze monthly precipitation data from San Francisco, California and Oakland Mills, Iowa. For the San Francisco data we will simultaneously order both monthly precipitation means and variances. For the Iowa data we will place ordering on seasonal average while still estimating monthly means. Our results show that we would obtain sharper, more accurate inference when order restrictions are employed.     

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.     

Sampling for weed spatial distribution mapping need not be adaptive

Weeds are species of interest for ecologists because they are competitors of the crop for resources but they also play an important role in maintaining biodiversity in agroecosystems. To study their spatial distribution at the field scale, only sampled observations are available due to the cost of sampling. Weeds sampling strategies are static. However, in the domain of spatial sampling, adaptive strategies have also been developed with, for some of them, an important on-line or off-line computational cost. In this article we provide answers to the following question: Are the current adaptive sampling methods efficient enough to motivate a wider use in practice when sampling a weed species at a field scale? We provide a comparison of the behaviour of 8 static strategies and 3 adaptive ones on four criteria: density class estimation, map restoration, spatial aggregation estimation, and sampling duration. From two weeds data sets, we estimated six contrasted Markov Random Field (MRF) models of weed density class spatial distribution and a model for sampling duration. The MRF models were then used to compare the strategies on a large set of simulated maps. Our main finding was that there is no clear gain in using adaptive sampling strategies rather than static ones for the three first criteria, and adaptive strategies were associated to longer sampling duration. This conclusion points out that for weed mapping, it is more important to build a good model of spatial distribution, than to propose complex adaptive sampling strategies.     

Site‐Occupancy Distribution Modeling to Correct Population‐Trend Estimates Derived from Opportunistic Observations

Abstract: Species’ assessments must frequently be derived from opportunistic observations made by volunteers (i.e., citizen scientists). Interpretation of the resulting data to estimate population trends is plagued with problems, including teasing apart genuine population trends from variations in observation effort. We devised a way to correct for annual variation in effort when estimating trends in occupancy (species distribution) from faunal or floral databases of opportunistic observations. First, for all surveyed sites, detection histories (i.e., strings of detection–nondetection records) are generated. Within‐season replicate surveys provide information on the detectability of an occupied site. Detectability directly represents observation effort; hence, estimating detectablity means correcting for observation effort. Second, site‐occupancy models are applied directly to the detection‐history data set (i.e., without aggregation by site and year) to estimate detectability and species distribution (occupancy, i.e., the true proportion of sites where a species occurs). Site‐occupancy models also provide unbiased estimators of components of distributional change (i.e., colonization and extinction rates). We illustrate our method with data from a large citizen‐science project in Switzerland in which field ornithologists record opportunistic observations. We analyzed data collected on four species: the widespread Kingfisher (Alcedo atthis) and Sparrowhawk (Accipiter nisus) and the scarce Rock Thrush (Monticola saxatilis) and Wallcreeper (Tichodroma muraria). Our method requires that all observed species are recorded. Detectability was <1 and varied over the years. Simulations suggested some robustness, but we advocate recording complete species lists (checklists), rather than recording individual records of single species. The representation of observation effort with its effect on detectability provides a solution to the problem of differences in effort encountered when extracting trend information from haphazard observations. We expect our method is widely applicable for global biodiversity monitoring and modeling of species distributions.     

Species distribution modelling—Effect of design and sample size of pseudo-absence observations

We explored the effect of varying pseudo-absence data in species distribution modelling using empirical data for four real species and simulated data for two imaginary species. In all analyses we used a fixed study area, a fixed set of environmental predictors and a fixed set of presence observations. Next, we added pseudo-absence data generated by different sampling designs and in different numbers to assess their relative importance for the output from the species distribution model. The sampling design strongly influenced the predictive performance of the models while the number of pseudo-absences had minimal effect on the predictive performance. We attribute much of these results to the relationship between the environmental range of the pseudo-absences (i.e. the extent of the environmental space being considered) and the environmental range of the presence observations (i.e. under which environmental conditions the species occurs). The number of generated pseudo-absences had a direct effect on the predicted probability, which translated to different distribution areas. Pseudo-absence observations that fell within grid cells with presence observations were purposely included in our analyses. We discourage the practice of excluding certain pseudo-absence data because it involves arbitrary assumptions about what are (un)suitable environments for the species being modelled.     

Use of Community‐Composition Data to Predict the Fecundity and Abundance of Species

Abstract: Species distribution models are critical tools for the prediction of invasive species spread and conservation of biodiversity. The majority of species distribution models have been built with environmental data. Community ecology theory suggests that species co‐occurrence data could also be used to predict current and potential distributions of species. Species assemblages are the products of biotic and environmental constraints on the distribution of individual species and as a result may contain valuable information for niche modeling. We compared the predictive ability of distribution models of annual grassland plants derived from either environmental or community‐composition data. Composition‐based models were built with the presence or absence of species at a site as predictors of site quality, whereas environment‐based models were built with soil chemistry, moisture content, above‐ground biomass, and solar radiation as predictors. The reproductive output of experimentally seeded individuals of 4 species and the abundance of 100 species were used to evaluate the resulting models. Community‐composition data were the best predictors of both the site‐specific reproductive output of sown individuals and the site‐specific abundance of existing populations. Successful community‐based models were robust to omission of data on the occurrence of rare species, which suggests that even very basic survey data on the occurrence of common species may be adequate for generating such models. Our results highlight the need for increased public availability of ecological survey data to facilitate community‐based modeling at scales relevant to conservation.     

Space-time modeling of environmental monitoring data

This study describes and applies statistical methods for space-time modeling of data from environmental monitoring programs, e.g., within areas such as climate change, air pollution and aquatic environment. Such data are often characterized by sparse sampling in both the temporal and spatial dimensions. In order to improve the amount of information on the physical system in question we suggest using statistical modeling methods for monitoring data. Model predictions combined with observations could be analyzed directly to assess the environmental state or as forcing functions for time series models and deterministic, hydrodynamic models. To illustrate the approach we applied the proposed modeling methods to data from the Danish and Swedish marine monitoring programs. Time series with a weekly resolution were predicted from observations of dissolved inorganic nitrogen (DIN) from the Kattegat basin (1993–1997). DIN observations were sparse, irregularly distributed and comprised approximately 10% of the generated time series.     

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.     

Nest selection by snow petrels Pagodroma nivea in East Antarctica: Validating predictive habitat selection models at the continental scale

Little is known on the factors controlling distribution and abundance of snow petrels in Antarctica. Studying habitat selection through modeling may provide useful information on the relationships between this species and its environment, especially relevant in a climate change context, where habitat availability may change. Validating the predictive capability of habitat selection models with independent data is a vital step in assessing the performance of such models and their potential for predicting species’ distribution in poorly documented areas.From the results of ground surveys conducted in the Casey region (2002–2003, Wilkes Land, East Antarctica), habitat selection models based on a dataset of 4000 nests were created to predict the nesting distribution of snow petrels as a function of topography and substrate. In this study, the Casey models were tested at Mawson, 3800 km away from Casey. The location and characteristics of approximately 7700 snow petrel nests were collected during ground surveys (Summer 2004–2005). Using GIS, predictive maps of nest distribution were produced for the Mawson region with the models derived from the Casey datasets and predictions were compared to the observed data. Models performance was assessed using classification matrixes and Receiver operating characteristic (ROC) curves. Overall correct classification rates for the Casey models varied from 57% to 90%. However, two geomorphologically different sub-regions (coastal islands and inland mountains) were clearly distinguished in terms of habitat selection by Casey model predictions but also by the specific variations in coefficients of terms in new models, derived from the Mawson data sets. Observed variations in the snow petrel aggregations were found to be related to local habitat availability.We discuss the applicability of various types of models (GLM, CT) and investigate the effect of scale on the prediction of snow petrel habitats. While the Casey models created with data collected at the nest scale did not perform well at Mawson due to regional variations in nest micro-characteristics, the predictive performance of models created with data compiled at a coarser scale (habitat units) was satisfactory. Substrate type was the most robust predictor of nest presence between Casey and Mawson. This study demonstrate that it is possible to predict at the large scale the presence of snow petrel nests based on simple predictors such as topography and substrate, which can be obtained from aerial photography. Such methodologies have valuable applications in the management and conservation of this top predator and associated resources and may be applied to other Antarctic, Sub-Antarctic and lower latitudes species and in a variety of habitats.