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.     

Detecting and Modeling Spatial and Temporal Dependence in Conservation Biology

Abstract: Due to the structuring forces and large-scale physical processes that shape our biosphere, we often find that environmental and ecological data are either spatially or temporally—or both spatially and temporally—dependent. When these data are analyzed, statistical techniques and models are frequently applied that were developed for independent data. We describe some of the detrimental consequences, such as inefficient parameter estimators, biased hypothesis test results, and inaccurate predictions, of ignoring spatial and temporal data dependencies, and we cite an example of adverse statistical results occurring when spatial dependencies were disregarded. We also discuss and recommend available techniques used to detect and model spatial and temporal dependence, including variograms, covariograms, autocorrelation and partial autocorrelation plots, geostatistical techniques, Gaussian autoregressive models, K functions, and ARIMA models, in environmental and ecological research to avoid the aforementioned difficulties.     

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.     

Evolution trend analysis of Nearshore seabed in Yangshan Deep-water Port,China

Yangshan Deep-water Port, the largest deep-water port in China, is located in the sea area of the Qiqu Archipelago adjacent to Hangzhou Bay. It goes deep into the ocean and far from the continent, and plays a key role in the economy and shipping of China. The evolution and stability of the seabed in the Yangshan Deep-water Port have potential influences on the security of port engineering. Based on GIS spatial analysis technology and MATLAB numerical analysis software, this study predicted the short-term evolution trend of the Yangshan Port frontier seabed terrain through the establishment of a modified power function model. The research included: (1) a systematic analysis of the characteristics of the terrain evolution before (1960–1997) and after (1998–2008) the construction of Yangshan Port by using terrain data from the study area; (2) based on the historical erosion and deposition characteristics of Yangshan Port, an improved power function model was established and the reliability of the model was validated to simulate the study area’s frontier seabed evolution trend in 2015. The results show that: (1) before the construction of Yangshan Port, the seabed in the study area had a narrower variation in erosion and deposition, with the ratio of erosion and deposition of the stable region, erosion area and deposition area being 53.7 %, 18.3 % and 28.0 % respectively, overall the area showed a relatively stable erosion and deposition character; (2) after the construction of the port, the erosion and deposition variation ranges of the seabed were sharply amplified, obviously due to man-made interference being stronger than natural evolution. The stable region of erosion and deposition was only 22.7 %, erosion area was 53.8 %, and the deposition area was 23.4 %, which showed an erosion intensity that was larger than the deposition intensity; (3) the established improved power function model can be used in the short-term prediction of the Yangshan Port frontier seabed evolution trend with high prediction accuracy. The results can aid in decision making with regard to coastal protection and prospective construction schemes around Yangshan Port.     

Improving seabed classification from Multi-Beam Echo Sounder (MBES) backscatter data with visual data mining

Multi-Beam Echo Sounders are often used for classification of seabed type, as there exists a strong link between sonar backscatter and sediment characteristics of the seabed. Most of the methods for seabed classification from MBES backscatter create a highly-dimensional data set of statistical features and then use a combination of Principal Component Analysis and k-means clustering to derive classes. This procedure can be time consuming for contemporary large MBES data sets with millions of records. This paper examines the complexity of one of most commonly used classification approaches and suggests an alternative where feature data set is optimised in terms of dimensionality using computational and visual data mining. Both the original and the optimised method are tested on an MBES backscatter data set and validated against ground truth. The study found that the optimised method improves accuracy of classification and reduced complexity of processing. This is an encouraging result, which shows that bringing together methods from acoustic classification, visual data mining, spatial analysis and remote sensing can support the unprecedented increases in data volumes collected by contemporary acoustic sensors.     

Smoothing of land use maps for trend and change detection in urbanization

Urban sprawl and its evolution over relatively short periods of time demands that we develop statistical tools to make best use of the routinely produced land use data from satellites. An efficient smoothing framework to estimate spatial patterns in binary raster maps derived from land use datasets is developed and presented in this paper. The framework is motivated by the need to model urbanization, specifically urban sprawl, and also its temporal evolution. We frame the problem as estimation of a probability of urbanization surface and use Bayesian P-splines as the tool of choice. Once such a probability map is produced, with associated uncertainty, we develop exploratory tools to identify regions of significant change across space and time. The proposal is used to study urbanisation and its development around the city of Bologna, Emilia Romagna, Italy, using land use data from the Cartography Archive of Emilia Romagna Region for the period 1976–2008.     

The Influence of Model Structure on Conclusions about the Viability and Harvesting of Serengeti Wildebeest

We investigate how the viability and harvestability predicted by population models are affected by details of model construction. Based on this analysis we discuss some of the pitfalls associated with the use of classical statistical techniques for resolving the uncertainties associated with modeling population dynamics. The management of the Serengeti wildebeest (Connochaetes taurinus) is used as a case study. We fitted a collection of age-structured and unstructured models to a common set of available data and compared model predictions in terms of wildebeest viability and harvest. Models that depicted demographic processes in strikingly different ways fitted the data equally well. However, upon further analysis it became clear that models that fit the data equally well could nonetheless have very different management implications. In general, model structure had a much larger effect on viability analysis (e.g., time to collapse) than on optimal harvest analysis (e.g., harvest rate that maximizes harvest). Some modeling decisions, such as including age-dependent fertility rates, did not affect management predictions, but others had a strong effect (e.g., choice of model structure). Because several suitable models of comparable complexity fitted the data equally well, traditional model selection methods based on the parsimony principle were not practical for judging the value of alternative models. Our results stress the need to implement analytical frameworks for population management that explicitly consider the uncertainty about the behavior of natural systems.     

The Use of Bayesian Model Averaging to Better Represent Uncertainty in Ecological Models

Abstract:   In conservation biology, uncertainty about the choice of a statistical model is rarely considered. Model-selection uncertainty occurs whenever one model is chosen over plausible alternative models to represent understanding about a process and to make predictions about future observations. The standard approach to representing prediction uncertainty involves the calculation of prediction (or confidence) intervals that incorporate uncertainty about parameter estimates contingent on the choice of a "best" model chosen to represent truth. However, this approach to prediction based on statistical models tends to ignore model-selection uncertainty, resulting in overconfident predictions. Bayesian model averaging (BMA) has been promoted in a range of disciplines as a simple means of incorporating model-selection uncertainty into statistical inference and prediction. Bayesian model averaging also provides a formal framework for incorporating prior knowledge about the process being modeled. We provide an example of the application of BMA in modeling and predicting the spatial distribution of an arboreal marsupial in the Eden region of southeastern Australia. Other approaches to estimating prediction uncertainty are discussed.     

Risk-Based Viable Population Monitoring

Abstract:  We describe risk-based viable population monitoring, in which the monitoring indicator is a yearly prediction of the probability that, within a given timeframe, the population abundance will decline below a prespecified level. Common abundance-based monitoring strategies usually have low power to detect declines in threatened and endangered species and are largely reactive to declines. Comparisons of the population's estimated risk of decline over time will help determine status in a more defensible manner than current monitoring methods. Monitoring risk is a more proactive approach; critical changes in the population's status are more likely to be demonstrated before a devastating decline than with abundance-based monitoring methods. In this framework, recovery is defined not as a single evaluation of long-term viability but as maintaining low risk of decline for the next several generations. Effects of errors in risk prediction techniques are mitigated through shorter prediction intervals, setting threshold abundances near current abundance, and explicitly incorporating uncertainty in risk estimates. Viable population monitoring also intrinsically adjusts monitoring effort relative to the population's true status and exhibits considerable robustness to model misspecification. We present simulations showing that risk predictions made with a simple exponential growth model can be effective monitoring indicators for population dynamics ranging from random walk to density dependence with stable, decreasing, or increasing equilibrium. In analyses of time-series data for five species, risk-based monitoring warned of future declines and demonstrated secure status more effectively than statistical tests for trend.     

Females go where the food is: does the socio-ecological model explain variation in social organisation of solitary foragers?

The socio-ecological model (SEM) links ecological factors with characteristics of social systems and allows predictions about the relationships between resource distribution, type of competition and social organisation. It has been mainly applied to group-living species but ought to explain variation in social organisation of solitary species as well. The aim of this study was to test basic predictions of the SEM in two solitary primates, which differ in two characteristics of female association patterns: (1) spatial ranging and (2) sleeping associations. Beginning in August 2002, we regularly (re-)captured and marked individuals of sympatric populations of Madame Berthe's and grey mouse lemurs (Microcebus berthae, Microcebus murinus) in Kirindy Forest (Madagascar). We recorded data on spatial patterns, feeding and social behaviour by means of direct observation of radio-collared females. The major food sources of M. berthae occurred in small dispersed patches leading to strong within-group scramble competition and over-dispersed females with a low potential for female associations. In contrast, M. murinus additionally used patchily distributed, high-quality (large) resources facilitating within-group contest competition. The combined influence of less strong within-group scramble and contest as well as between-group contest over non-food resources allowed females of this species to cluster in space. Additionally, we experimentally manipulated the spatial distribution of food sources and found that females adjusted their spatial patterns to food resource distribution. Thus, our results support basic predictions of the SEM and demonstrated that it can also explain variation in social organisation of solitary foragers.     

Developing a Monitoring Program for Invertebrates: Guidelines and a Case Study

Abstract:  Invertebrates provide the majority of ecosystem services; thus, it is important that they be inventoried, monitored, and protected. Nevertheless, inventories, monitoring, and management generally focus on vertebrates and flowering plants. Consequently, there are few guidelines or case studies for invertebrates. We present a procedure for developing a monitoring program for species-rich invertebrates that entails (1) characterizing the community; (2) identifying surrogates for biodiversity; and (3) establishing efficient methods to monitor surrogates and any ecologically important or sensitive taxa. We used these procedures, biodiversity-based statistical advances, and a survey of arthropods to develop a monitoring plan for the forests of Shenandoah National Park, Virginia (U.S.A.). Our case study revealed that mixed hardwood and hemlock forests had significantly different compositions of arthropods in their soil and understory strata. Of the 10 orders tested Coleoptera and Hymenoptera were the only two to pass most of the five surrogate tests, and their combination improved predictions of overall arthropod diversity. Because arthropods represent the majority of macroscopic species in most ecosystems, the ability of this assemblage to predict overall arthropod diversity makes it a powerful surrogate. Of the 11 collecting methods used, the beat-sheet method was the most efficient for monitoring this surrogate assemblage. To complement this coarse-filter approach to monitoring at-risk, invasive, or other important taxa (fine filter), we used ordination analyses to match 66 taxa with the methods that most effectively sampled them. Our methods serve as a model for developing an invertebrate monitoring plan and should facilitate linking such monitoring with ecosystem functions and management.     

Spatial statistical models that use flow and stream distance

We develop spatial statistical models for stream networks that can estimate relationships between a response variable and other covariates, make predictions at unsampled locations, and predict an average or total for a stream or a stream segment. There have been very few attempts to develop valid spatial covariance models that incorporate flow, stream distance, or both. The application of typical spatial autocovariance functions based on Euclidean distance, such as the spherical covariance model, are not valid when using stream distance. In this paper we develop a large class of valid models that incorporate flow and stream distance by using spatial moving averages. These methods integrate a moving average function, or kernel, against a white noise process. By running the moving average function upstream from a location, we develop models that use flow, and by construction they are valid models based on stream distance. We show that with proper weighting, many of the usual spatial models based on Euclidean distance have a counterpart for stream networks. Using sulfate concentrations from an example data set, the Maryland Biological Stream Survey (MBSS), we show that models using flow may be more appropriate than models that only use stream distance. For the MBSS data set, we use restricted maximum likelihood to fit a valid covariance matrix that uses flow and stream distance, and then we use this covariance matrix to estimate fixed effects and make kriging and block kriging predictions. Received: July 2005 / Revised: March 2006     

Estimating Population Size of Elusive Animals with DNA from Hunter-Collected Feces: Four Methods for Brown Bears

Abstract:  Noninvasive genetic methods can be used to estimate animal abundances and offer several advantages over conventional methods. Few attempts have been made, however, to evaluate the accuracy and precision of the estimates. We compared four methods of estimating population size based on fecal sampling. Two methods used rarefaction indices and two were based on capture-mark-recapture (CMR) estimators, one combining genetic and field data. Volunteer hunters and others collected 1904 fecal samples over 2 consecutive years in a large area containing a well-studied population of brown bears ( Ursus arctos ). On our 49,000-km² study area in south-central Sweden, population size estimates ranged from 378 to 572 bears in 2001 and 273 to 433 bears in 2002, depending on the method of estimation used. The estimates from the best model in the program MARK appeared to be the most accurate, based on the minimum population size estimate from radio-marked bears in a subsection of our sampling area. In addition, MARK models included heterogeneity and temporal variation in detection probabilities, which appeared to be present in our samples. All methods, though, incorrectly suggested a biased sex ratio, probably because of sex differences in detection probabilities and low overall detection probabilities. The population size of elusive animals can be estimated reliably over large areas with noninvasive genetic methods, but we stress the importance of an adequate and well-distributed sampling effort. In cases of biased sampling, calibration with independent estimates may be necessary. We recommend that this noninvasive genetic approach, using the MARK models, be used in the future in areas where sufficient numbers of volunteers can be mobilized.     

Application of Rice’s theory to recurrence statistics of concentration fluctuations in dispersing plumes

Rice’s theory for the statistical properties of random noise currents has been employed in the context of concentration fluctuations in dispersing plumes. Within this context, the theory has been extended to calculate the distribution of excursion times above a small threshold for arbitrary spacings between an up-crossing and the successive down-crossing. This approach has then been applied to a second-order stochastic model for the evolution of odour concentrations and their time derivative (simple model), and to the superstatistics extension of this model [Reynolds (2007) Phys. Fluids]. In agreement with the measurements of Yee and coworkers [Yee et al. (1993) Boundary-Layer Meteorol. 65, Yee et al. (1994) J. Appl. Meteorol. 33 ], both formulations predict a distribution of excursion times that can be well approximated by a power-law profile with exponent close to −3/2. For the superstatistical model the power-law profile extends over approximately three or more decades, for the simple model this range is smaller. Compared to the simple model, predictions for the superstatistical model are in a better agreement with the measurements.     

Environmental filtering and land-use history drive patterns in biomass accumulation in a mediterranean-type landscape

Aboveground biomass (AGB) reflects multiple and often undetermined ecological and land-use processes, yet detailed landscape-level studies of AGB are uncommon due to the difficulty in making consistent measurements at ecologically relevant scales. Working in a protected mediterranean-type landscape (Jasper Ridge Biological Preserve, California, USA), we combined field measurements with remotely sensed data from the Carnegie Airborne Observatory's light detection and ranging (lidar) system to create a detailed AGB map. We then developed a predictive model using a maximum of 56 explanatory variables derived from geologic and historic-ownership maps, a digital elevation model, and geographic coordinates to evaluate possible controls over currently observed AGB patterns. We tested both ordinary least-squares regression (OLS) and autoregressive approaches. OLS explained 44% of the variation in AGB, and simultaneous autoregression with a 100-m neighborhood improved the fit to an r2 = 0.72, while reducing the number of significant predictor variables from 27 variables in the OLS model to 11 variables in the autoregressive model. We also compared the results from these approaches to a more typical field-derived data set; we randomly sampled 5% of the data 1000 times and used the same OLS approach each time. Environmental filters including incident solar radiation, substrate type, and topographic position were significant predictors of AGB in all models. Past ownership was a minor but significant predictor, despite the long history of conservation at the site. The weak predictive power of these environmental variables, and the significant improvement when spatial autocorrelation was incorporated, highlight the importance of land-use history, disturbance regime, and population dynamics as controllers of AGB.     

Bayesian entropy for spatial sampling design of environmental data

We develop a spatial statistical methodology to design national air pollution monitoring networks with good predictive capabilities while minimizing the cost of monitoring. The underlying complexity of atmospheric processes and the urgent need to give credible assessments of environmental risk create problems requiring new statistical methodologies to meet these challenges. In this work, we present a new method of ranking various subnetworks taking both the environmental cost and the statistical information into account. A Bayesian algorithm is introduced to obtain an optimal subnetwork using an entropy framework. The final network and accuracy of the spatial predictions is heavily dependent on the underlying model of spatial correlation. Usually the simplifying assumption of stationarity, in the sense that the spatial dependency structure does not change location, is made for spatial prediction. However, it is not uncommon to find spatial data that show strong signs of nonstationary behavior. We build upon an existing approach that creates a nonstationary covariance by a mixture of a family of stationary processes, and we propose a Bayesian method of estimating the associated parameters using the technique of Reversible Jump Markov Chain Monte Carlo. We apply these methods for spatial prediction and network design to ambient ozone data from a monitoring network in the eastern US.     

New approaches to modelling fish-habitat relationships

Ecologists often develop models that describe the relationship between faunal communities and their habitat. Coral reef fishes have been the focus of numerous such studies, which have used a wide range of statistical tools to answer an equally wide range of questions. Here, we apply a series of both conventional statistical techniques (linear and generalized additive regression models) and novel machine-learning techniques (the support vector machine and three ensemble techniques used with regression trees) to predict fish species richness, biomass, and diversity from a range of habitat variables. We compare the techniques in terms of their predictive performance, and we compare a subset of the models in terms of the influence each habitat variable has for the predictions. Prediction errors are estimated by cross-validation, and variable importance is assessed using permutations of individual variable values. For predictions of species richness and diversity the tree-based models generally and the random forest model specifically are superior (produce the lowest errors). These model types are all able to model both nonlinear and interaction effects. The linear model, unable to model either effect type, performs the worst (produces the highest errors). For predictions of biomass, the generalized additive model is superior, and the support vector machine performs the worst. Depth range, the difference between maximum and minimum water depth at a given site, is identified as the most important variable in the majority of models predicting the three fish community variables. However, variable importance is highly dependent upon model type, which leads to questions regarding the interpretation of variable importance and its proper use as an indicator of causality. The representation of ecological relationships by tree-based ensemble learners will improve predictive performance, and provide a new avenue for exploring ecological relationships, both statistical and causal.     

Quantifying parameter uncertainty in a coral reef model using Metropolis-Coupled Markov Chain Monte Carlo

Coral reefs are threatened ecosystems, so it is important to have predictive models of their dynamics. Most current models of coral reefs fall into two categories. The first is simple heuristic models which provide an abstract understanding of the possible behaviour of reefs in general, but do not describe real reefs. The second is complex simulations whose parameters are obtained from a range of sources such as literature estimates. We cannot estimate the parameters of these models from a single data set, and we have little idea of the uncertainty in their predictions.We have developed a compromise between these two extremes, which is complex enough to describe real reef data, but simple enough that we can estimate parameters for a specific reef from a time series. In previous work, we fitted this model to a long-term data set from Heron Island, Australia, using maximum likelihood methods. To evaluate predictions from this model, we need estimates of the uncertainty in our parameters. Here, we obtain such estimates using Bayesian Metropolis-Coupled Markov Chain Monte Carlo. We do this for versions of the model in which corals are aggregated into a single state variable (the three-state model), and in which corals are separated into four state variables (the six-state model), in order to determine the appropriate level of aggregation. We also estimate the posterior distribution of predicted trajectories in each case.In both cases, the fitted trajectories were close to the observed data, but we had doubts about the biological plausibility of some parameter estimates. We suggest that informative prior distributions incorporating expert knowledge may resolve this problem. In the six-state model, the posterior distribution of state frequencies after 40 years contained two divergent community types, one dominated by free space and soft corals, and one dominated by acroporid, pocilloporid, and massive corals. The three-state model predicts only a single community type. We conclude that the three-state model hides too much biological heterogeneity, but we need more data if we are to obtain reliable predictions from the six-state model. It is likely that there will be similarly large, but currently unevaluated, uncertainty in the predictions of other coral reef models, many of which are much more complex and harder to fit to real data.     

The evaluation strip: A new and robust method for plotting predicted responses from species distribution models

Increasing use is being made in conservation management of statistical models that couple extensive collections of species and environmental data to make predictions of the geographic distributions of species. While the relationships fitted between a species and its environment are relatively transparent for many of these modeling techniques, others are more ‘black box’ in character, only producing geographic predictions and providing minimal or untraditional summaries of the fitted relationships on which these predictions are based. This in turn prevents robust evaluation of the ecological sensibility of such models, a necessary process if model predictions are to be treated with confidence. Here we propose a new but simple method for visualizing modeled responses that can be implemented with any modeling method, and demonstrate its application using five common methods applied to the prediction of an Australian tree species. This is achieved by insetting an “evaluation strip” into the spatial data layers, which, after predictions have been made, can be clipped out and used for creating plots of the modelled responses. We present findings of the application strip for algorithms GLMs, GAMs, CLIM, DOMAIN and MARS. Evaluation strips can be constructed to investigate either uni-variate responses, or the simultaneous variation in predicted values in relation to two variables. The latter option is particularly useful for evaluating responses in models that allow the fitting of complex interaction terms.