Estimating abundance: a non parametric mark recapture approach for open and closed systems

We present a novel, non-parametric, frequentist approach for capture-recapture data based on a ratio estimator, which offers several advantages. First, as a non-parametric model, it does not require a known underlying distribution for parameters nor the associated assumptions, eliminating the need for post-hoc corrections or additional modeling to account for heterogeneity and other violated assumptions. Second, the model explicitly deals with dependence of trials by considering trials to be dependent; therefore, cluster sampling is handled naturally and additional adjustments are not necessary. Third, it accounts for ordering, utilizing the fact that a system with a small population will have a greater frequency of recaptures “early” in the survey work compared to an identical system with a larger population. We provide mathematical proof that our estimator attains asymptotic minimum variance under open systems. We apply the model to a data set of bottlenose dolphins (Tursiops truncatus) and compare results to those from classic closed models. We show that the model has an impressive rate of convergence and demonstrate that there’s an inverse relationship between population size and the proportion of the population that need to be sampled, while achieving the same degree of accuracy for abundance estimates. The model is flexible and can apply to ecological situations as well as other situations that lend themselves to capture recapture sampling.     

Bayesian analysis of Jolly-Seber type models

We propose the use of finite mixtures of continuous distributions in modelling the process by which new individuals, that arrive in groups, become part of a wildlife population. We demonstrate this approach using a data set of migrating semipalmated sandpipers (Calidris pussila) for which we extend existing stopover models to allow for individuals to have different behaviour in terms of their stopover duration at the site. We demonstrate the use of reversible jump MCMC methods to derive posterior distributions for the model parameters and the models, simultaneously. The algorithm moves between models with different numbers of arrival groups as well as between models with different numbers of behavioural groups. The approach is shown to provide new ecological insights about the stopover behaviour of semipalmated sandpipers but is generally applicable to any population in which animals arrive in groups and potentially exhibit heterogeneity in terms of one or more other processes.     

A generalized estimating equations approach for capture–recapture closed population models

The estimation of population density animal population parameters, such as capture probability, population size, or population density, is an important issue in many ecological applications. Capture–recapture data may be considered as repeated observations that are often correlated over time. If these correlations are not taken into account then parameter estimates may be biased, possibly producing misleading results. We propose a generalized estimating equations (GEE) approach to account for correlation over time instead of assuming independence as in the traditional closed population capture–recapture studies. We also account for heterogeneity among observed individuals and over-dispersion, modelling capture probabilities as a function of covariates. The GEE versions of all closed population capture–recapture models and their corresponding estimating equations are proposed. We evaluate the effect of accounting for correlation structures on capture–recapture model selection based on the quasi-likelihood information criterion (QIC). An example is used for an illustrative application and for comparison to currently used methodology. A Horvitz–Thompson-like estimator is used to obtain estimates of population size based on conditional arguments. A simulation study is conducted to evaluate the performance of the GEE approach in capture-recapture studies. The GEE approach performs well for estimating population parameters, particularly when capture probabilities are high. The simulation results also reveal that estimated population size varies on the nature of the existing correlation among capture occasions.     

On the latent state estimation of nonlinear population dynamics using Bayesian and non-Bayesian state-space models

Nonlinear state-space models have been increasingly applied to study population dynamics and data assimilation in environmental sciences. State-space models can account for process error and measurement error simultaneously to correct for the bias in the estimates of system state and model parameters. However, few studies have compared the performance of different nonlinear state-space models for reconstructing the state of population dynamics from noisy time series. This study compared the performance of the extended Kalman filter (EKF), unscented Kalman filter (UKF) and Bayesian nonlinear state-space models (BNSSM) through simulations. Synthetic population time series were generated using the theta logistic model with known parameters, and normally distributed process and measurement errors were introduced using the Monte Carlo simulations. At higher levels of nonlinearity, the UKF and BNSSM had lower root mean square error (RMSE) than the EKF. The BNSSM performed reliably across all levels of nonlinearity, whereas increased levels of nonlinearity resulted in higher RMSE of the EKF. The Metropolis–Hastings algorithm within the Gibbs algorithm was used to fit the theta logistic model to synthetic time series to estimate model parameters. The estimated posterior distribution of the parameter θ indicated that the 95% credible intervals included the true values of θ (=0.5 and 1.5), but did not include 1.0 and 0.0. Future studies need to incorporate the adaptive Metropolis algorithm to estimate unknown model parameters for broad applications of Bayesian nonlinear state-space models in ecological studies.     

Robust, comparable population metrics through collaborative photo-monitoring of whale sharks Rhincodon typus.

The formulation of conservation policy for species that are rare and migratory requires broad cooperation to ensure that adequate levels of standardized data collection are achieved and that the results of local analyses are comparable. Estimates of apparent survival rate, relative change in abundance, and proportions of newly marked and returning individuals can inform local management decisions while highlighting corresponding changes at other linked research stations. We have applied computer-assisted photo-identification and mark-recapture population modeling to whale sharks Rhincodon typus at Ningaloo Marine Park (NMP), Western Australia, to create a baseline trend for comparison with other regional aggregations of the species. We estimate several ecological parameters of interest, including an average apparent survival rate of 0.55 yr(-1) for sharks newly marked (new) and 0.83 yr(-1) for sharks captured in multiple seasons (philopatric). The average proportion of philopatric sharks is found to be 0.65 of the total population, and we derive an average population growth rate of 1.12 yr(-1) for them. Our analysis uncovered significant heterogeneity in capture and survival probabilities in this study population; our chosen model structures and data analysis account for these influences and demonstrate a good overall fit to the time-series data. The results show good correspondence between capture probability and an available measure of recapture effort, suggesting that unmodeled systematic effects contribute insignificantly to the model fits. We find no evidence of a decline in the whale shark population at NMP, and our results provide metrics of value to their future management. Overall, our study suggests an effective approach to analyzing and modeling mark-recapture data for a rare species using computer-assisted photo-identification and opportunistic data collection from ecotourism to ensure the quality and volume of data required for population analysis.     

The unified neutral theory of biodiversity: do the numbers add up?

Hubbell's unified neutral theory is a zero-sum ecological drift model in which population sizes change at random in a process resembling genetic drift, eventually leading to extinction. Diversity is maintained within the community by speciation. Hubbell's model makes predictions about the distribution of species abundances within communities and the turnover of species from place to place (beta diversity). However, ecological drift cannot be tested adequately against these predictions without independent estimates of speciation rates, population sizes, and dispersal distances. A more practical prediction from ecological drift is that time to extinction of a population of size N is approximately 2N generations. I test this prediction here using data for passerine birds (Passeriformes). Waiting times to speciation and extinction were estimated from genetic divergence between sister populations and a lineage-through-time plot for endemic South American suboscine passerines. Population sizes were estimated from local counts of birds in two large forest plots extrapolated to the area of wet tropical forest in South America and from atlas data on European passerines. Waiting times to extinction (ca. 2 Ma) are much less than twice the product of average population size (4.0 and 14.4 x 10(6) individuals in South America and Europe) and generation length (five and three years) for songbirds, that is, 40 and 86 Ma, respectively. Thus, drift is too slow to account for turnover in regional avifaunas. Presumably, other processes, involving external drivers, such as climate and physiographic change, and internal drivers, such as evolutionary change in antagonistic interactions, predominate. Hubbell's model is historical and geographic, and his perspective importantly links local and regional process and pattern. Ecological reality can be added to the mix while retaining Hubbell's concept of continuity of communities in space and time.     

Sampling variability and estimates of density dependence: a composite-likelihood approach

It is well known that sampling variability, if not properly taken into account, affects various ecologically important analyses. Statistical inference for stochastic population dynamics models is difficult when, in addition to the process error, there is also sampling error. The standard maximum-likelihood approach suffers from large computational burden. In this paper, I discuss an application of the composite-likelihood method for estimation of the parameters of the Gompertz model in the presence of sampling variability. The main advantage of the method of composite likelihood is that it reduces the computational burden substantially with little loss of statistical efficiency. Missing observations are a common problem with many ecological time series. The method of composite likelihood can accommodate missing observations in a straightforward fashion. Environmental conditions also affect the parameters of stochastic population dynamics models. This method is shown to handle such nonstationary population dynamics processes as well. Many ecological time series are short, and statistical inferences based on such short time series tend to be less precise. However, spatial replications of short time series provide an opportunity to increase the effective sample size. Application of likelihood-based methods for spatial time-series data for population dynamics models is computationally prohibitive. The method of composite likelihood is shown to have significantly less computational burden, making it possible to analyze large spatial time-series data. After discussing the methodology in general terms, I illustrate its use by analyzing a time series of counts of American Redstart (Setophaga ruticilla) from the Breeding Bird Survey data, San Joaquin kit fox (Vulpes macrotis mutica) population abundance data, and spatial time series of Bull trout (Salvelinus confluentus) redds count data.     

Modeling range dynamics in heterogeneous landscapes: invasion of the hemlock woolly adelgid in eastern North America

Range expansion by native and exotic species will continue to be a major component of global change. Anticipating the potential effects of changes in species distributions requires models capable of forecasting population spread across realistic, heterogeneous landscapes and subject to spatiotemporal variability in habitat suitability. Several decades of theory and model development, as well as increased computing power and availability of fine-resolution GIS data, now make such models possible. Still unanswered, however, is the question of how well this new generation of dynamic models will anticipate range expansion. Here we develop a spatially explicit stochastic model that combines dynamic dispersal and population processes with fine-resolution maps characterizing spatiotemporal heterogeneity in climate and habitat to model range expansion of the hemlock woolly adelgid (HWA; Adelges tsugae). We parameterize this model using multiyear data sets describing population and dispersal dynamics of HWA and apply it to eastern North America over a 57-year period (1951-2008). To evaluate the model, the observed pattern of spread of HWA during this same period was compared to model predictions. Our model predicts considerable heterogeneity in the risk of HWA invasion across space and through time, and it suggests that spatiotemporal variation in winter temperature, rather than hemlock abundance, exerts a primary control on the spread of HWA. Although the simulations generally matched the observed current extent of the invasion of HWA and patterns of anisotropic spread, it did not correctly predict when HWA was observed to arrive in different geographic regions. We attribute differences between the modeled and observed dynamics to an inability to capture the timing and direction of long-distance dispersal events that substantially affected the ensuing pattern of spread.     

Temporal limits to simulating the future spread pattern of invasive species: Buddleja davidii in Europe and New Zealand

The spread of invasive species is a major ecological and economic problem. Dynamic spread modelling is a potentially valuable tool to assist regional and central government authorities to monitor and control invasive species. To date a lack of suitable data has meant that most broad scale dispersal models have not been validated with independent datasets, and so their predictive ability and reliability has remained unscrutinised. A dynamic, stochastic dispersal model of the widely invasive plant Buddleja davidii was calibrated on European spread data and then used to project the temporal progression of B. davidii's distribution in New Zealand, starting from several different historical distributions. To assess the model's performance, we constructed an occupancy map based on the average number of simulation realisations that have a population present. The application of Receiver Operating Characteristic (ROC) curves to occupancy maps is introduced, but with specificity substituted by the proportion of available area used in a realisation. A derivative measure, the partial area under these curves when assessed through time (pAUC), is introduced and used to assess overall performance of the spread model. The model was able to attain a high level of model sensitivity, encompassing all of the known locations within the occupancy envelope. However, attempting to simulate the spread of this invasive species beyond a decade had very low model specificity. This is due to several factors, including the exponential process of spread (the further a population spreads the more sites exist from which it can spread stochastically), and the Markovian chain property of the stochastic system whereby differences between realisations compound through time. These features are seen in many reports of spread models, without being explicitly acknowledged. Our measure of pAUC through time allows a model's temporal performance and its specificity to be simultaneously assessed. While the rapid deterioration in model performance limits the utility of this type of modelling for forecasting long-term broad-scale strategic management of biological invasions, it does not necessarily limit its attractiveness for informing smaller scale and shorter term invasion management activities such as surveillance, containment and local eradication.     

Joint modelling of breeding and survival in the kittiwake using frailty models

Assessment of population dynamics is central to population dynamics and conservation. In structured populations, matrix population models based on demographic data have been widely used to assess such dynamics. Although highlighted in several studies, the influence of heterogeneity among individuals in demographic parameters and of the possible correlation among these parameters has usually been ignored, mostly because of difficulties in estimating such individual-specific parameters. In the kittiwake (Rissa tridactyla), a long-lived seabird species, differences in survival and breeding probabilities among individual birds are well documented. Several approaches have been used in the animal ecology literature to establish the association between survival and breeding rates. However, most are based on observed heterogeneity between groups of individuals, an approach that seldom accounts for individual heterogeneity. Few attempts have been made to build models permitting estimation of the correlation between vital rates. For example, survival and breeding probability of individual birds were jointly modelled using logistic random effects models by [Cam, E., Link, W.A., Cooch, E.G., Monnat, J., Danchin, E., 2002. Individual covariation in life-history traits: seeing the trees despite the forest. Am. Naturalist, 159, in press]. This is the only example in wildlife animal populations we are aware of. Here we adopt the survival analysis approaches from epidemiology. We model the survival and the breeding probability jointly using a normally distributed random effect (frailty). Conditionally on this random effect, the survival time is modelled assuming a lognormal distribution, and breeding is modelled with a logistic model. Since the deaths are observed in year-intervals, we also take into account that the data are interval censored. The joint model is estimated using classic frequentist methods and also MCMC techniques in Winbugs. The association between survival and breeding attempt is quantified using the standard deviation of the random frailty parameters. We apply our joint model on a large data set of 862 birds, that was followed from 1984 to 1995 in Brittany (France). Survival is positively correlated with breeding indicating that birds with greater inclination to breed also had higher survival.     

Using global sensitivity analysis of demographic models for ecological impact assessment

Population viability analysis (PVA) is widely used to assess population‐level impacts of environmental changes on species. When combined with sensitivity analysis, PVA yields insights into the effects of parameter and model structure uncertainty. This helps researchers prioritize efforts for further data collection so that model improvements are efficient and helps managers prioritize conservation and management actions. Usually, sensitivity is analyzed by varying one input parameter at a time and observing the influence that variation has over model outcomes. This approach does not account for interactions among parameters. Global sensitivity analysis (GSA) overcomes this limitation by varying several model inputs simultaneously. Then, regression techniques allow measuring the importance of input‐parameter uncertainties. In many conservation applications, the goal of demographic modeling is to assess how different scenarios of impact or management cause changes in a population. This is challenging because the uncertainty of input‐parameter values can be confounded with the effect of impacts and management actions. We developed a GSA method that separates model outcome uncertainty resulting from parameter uncertainty from that resulting from projected ecological impacts or simulated management actions, effectively separating the 2 main questions that sensitivity analysis asks. We applied this method to assess the effects of predicted sea‐level rise on Snowy Plover (Charadrius nivosus). A relatively small number of replicate models (approximately 100) resulted in consistent measures of variable importance when not trying to separate the effects of ecological impacts from parameter uncertainty. However, many more replicate models (approximately 500) were required to separate these effects. These differences are important to consider when using demographic models to estimate ecological impacts of management actions.     

Seed terminal velocity, wind turbulence, and demography drive the spread of an invasive tree in an analytical model

Little is known about the relative importance of mechanistic drivers of plant spread, particularly when long-distance dispersal (LDD) events occur. Most methods to date approach LDD phenomenologically, and all mechanistic models, with one exception, have been implemented through simulation. Furthermore, the few recent mechanistically derived spread models have examined the relative role of different dispersal parameters using simulations, and a formal analytical approach has not yet been implemented. Here we incorporate an analytical mechanistic wind dispersal model (WALD) into a demographic matrix model within an analytical integrodifference equation spread model. We carry out analytical perturbation analysis on the combined model to determine the relative effects of dispersal and demographic traits and wind statistics on the spread of an invasive tree. Models are parameterized using data collected in situ and tested using independent data on historical spread. Predicted spread rates and direction match well the two historical phases of observed spread. Seed terminal velocity has the greatest potential influence on spread rate, and three wind properties (turbulence coefficient, mean horizontal wind speed, and standard deviation of vertical wind speed) are also important. Fecundity has marginal importance for spread rate, but juvenile survival and establishment are consistently important. This coupled empirical/theoretical framework enables prediction of plant spread rate and direction using fundamental dispersal and demographic parameters and identifies the traits and environmental conditions that facilitate spread. The development of an analytical perturbation analysis for a mechanistic spread model will enable multispecies comparative studies to be easily implemented in the future.     

Sensitivity of ecological models to their climate drivers: statistical ensembles for forcing.

Global and regional numerical models for terrestrial ecosystem dynamics require fine spatial resolution and temporally complete historical climate fields as input variables. However, because climate observations are unevenly spaced and have incomplete records, such fields need to be estimated. In addition, uncertainty in these fields associated with their estimation are rarely assessed. Ecological models are usually driven with a geostatistical model's mean estimate (kriging) of these fields without accounting for this uncertainty, much less evaluating such errors in terms of their propagation in ecological simulations. We introduce a Bayesian statistical framework to model climate observations to create spatially uniform and temporally complete fields, taking into account correlation in time and space, spatial heterogeneity, lack of normality, and uncertainty about all these factors. A key benefit of the Bayesian model is that it generates uncertainty measures for the generated fields. To demonstrate this method, we reconstruct historical monthly precipitation fields (a driver for ecological models) on a fine resolution grid for a climatically heterogeneous region in the western United States. The main goal of this work is to evaluate the sensitivity of ecological models to the uncertainty associated with prediction of their climate drivers. To assess their numerical sensitivity to predicted input variables, we generate a set of ecological model simulations run using an ensemble of different versions of the reconstructed fields. We construct such an ensemble by sampling from the posterior predictive distribution of the climate field. We demonstrate that the estimated prediction error of the climate field can be very high. We evaluate the importance of such errors in ecological model experiments using an ensemble of historical precipitation time series in simulations of grassland biogeochemical dynamics with an ecological numerical model, Century. We show how uncertainty in predicted precipitation fields is propagated into ecological model results and that this propagation had different modes. Depending on output variable, the response of model dynamics to uncertainty in inputs ranged from uncertainty in outputs that matched that of inputs to those that were muted or that were biased, as well as uncertainty that was persistent in time after input errors dropped.     

Using a social–ecological framework to inform the implementation of conservation plans

One of the key determinants of success in biodiversity conservation is how well conservation planning decisions account for the social system in which actions are to be implemented. Understanding elements of how the social and ecological systems interact can help identify opportunities for implementation. Utilizing data from a large‐scale conservation initiative in southwestern of Australia, we explored how a social–ecological system framework can be applied to identify how social and ecological factors interact to influence the opportunities for conservation. Using data from semistructured interviews, an online survey, and publicly available data, we developed a conceptual model of the social–ecological system associated with the conservation of the Fitz‐Stirling region. We used this model to identify the relevant variables (remnants of vegetation, stakeholder presence, collaboration between stakeholders, and their scale of management) that affect the implementation of conservation actions in the region. We combined measures for these variables to ascertain how areas associated with different levels of ecological importance coincided with areas associated with different levels of stakeholder presence, stakeholder collaboration, and scales of management. We identified areas that could benefit from different implementation strategies, from those suitable for immediate conservation action to areas requiring implementation over the long term to increase on‐the‐ground capacity and identify mechanisms to incentivize implementation. The application of a social–ecological framework can help conservation planners and practitioners facilitate the integration of ecological and social data to inform the translation of priorities for action into implementation strategies that account for the complexities of conservation problems in a focused way.     

Population Viability Estimates: Theory and Practice for a Wild Gorilla Population

The logic of demographic modeling, the apparent simplicity of its quantifiably substantiated answers, and the ready availability of software correlate with increasing use of demographic modeling as the means of applying biology to the conservation of potentially endangered populations. I investigated that use by considering a small population (about 300 individuals) of a large, forest-dwelling mammal of the tropics, the Virunga gorilla ( Gorilla gorilla ) of Zaire, Uganda, and Rwanda. Because censuses of forest populations are so inaccurate and data on variance of some parameters takes so long to collect, models might not be broadly applicable. Therefore, simple demographic indices of potential extinction should replace sophisticated models. The current best index could be problematic, however, because it is based on detecting adult mortality, perhaps the most difficult demographic parameter to measure. Models of the Virunga gorilla population that incorporate aspects of demographic heterogeneity valuably indicate genetic and demographic persistence for several hundred years. Deterministic change in habitat is a greater threat than stochastic demographic variation, and yet our ecological ignorance is such that we could not begin to model the consequences of removal of even the main food plant. We must add to our ability to model outcomes of demographic perturbation a far greater understanding of the processes by which the perturbations occur. Demography allows us to model demographic response to demographic change, but we usually need ecology to tell us how the threat produced the demographic change in the first place. In a time of change, accurate prediction requires ecological understanding of process as well as demographic understanding of outcome.     

Fitting complex population models by combining particle filters with Markov chain Monte Carlo

We show how a recent framework combining Markov chain Monte Carlo (MCMC) with particle filters (PFMCMC) may be used to estimate population state-space models. With the purpose of utilizing the strengths of each method, PFMCMC explores hidden states by particle filters, while process and observation parameters are estimated using an MCMC algorithm. PFMCMC is exemplified by analyzing time series data on a red kangaroo (Macropus rufus) population in New South Wales, Australia, using MCMC over model parameters based on an adaptive Metropolis-Hastings algorithm. We fit three population models to these data; a density-dependent logistic diffusion model with environmental variance, an unregulated stochastic exponential growth model, and a random-walk model. Bayes factors and posterior model probabilities show that there is little support for density dependence and that the random-walk model is the most parsimonious model. The particle filter Metropolis-Hastings algorithm is a brute-force method that may be used to fit a range of complex population models. Implementation is straightforward and less involved than standard MCMC for many models, and marginal densities for model selection can be obtained with little additional effort. The cost is mainly computational, resulting in long running times that may be improved by parallelizing the algorithm.     

Hidden process models for animal population dynamics.

Hidden process models are a conceptually useful and practical way to simultaneously account for process variation in animal population dynamics and measurement errors in observations and estimates made on the population. Process variation, which can be both demographic and environmental, is modeled by linking a series of stochastic and deterministic subprocesses that characterize processes such as birth, survival, maturation, and movement. Observations of the population can be modeled as functions of true abundance with realistic probability distributions to describe observation or estimation error. Computer-intensive procedures, such as sequential Monte Carlo methods or Markov chain Monte Carlo, condition on the observed data to yield estimates of both the underlying true population abundances and the unknown population dynamics parameters. Formulation and fitting of a hidden process model are demonstrated for Sacramento River winter-run chinook salmon (Oncorhynchus tshawytsha).     

Managing plant population spread: prediction and analysis using a simple model

Models can be used to direct the management of population spread for the control of invasives or to encourage species of conservation value. Analytical models are attractive because of their theoretical basis and limited data requirements, but there is concern that their simplicity may limit their practical utility. We address the applied use of simple models in a study of a declining annual herb, Rhinanthus minor. We parameterized a population-spread model using field data on demography and dispersal for four management systems: grazed only (GR), hay-cut once (H1), hay-cut twice (H2), and hay-cut with autumn grazing (HG). Within a replicated experiment we measured spread rates of introduced R. minor populations over eight years. The modeled and measured spread rates were very similar in terms of both patterns of management effects and absolute values, so that in both cases HG > H2, H1 > GR. The treatments affected both dispersal and demography (establishment and survival) and so we used decomposition approaches to analyze the major causes of differences in population spread. Increased dispersal under hay-cutting was more important than demographic changes and accounted for approximately 70% of the differences in spread rate between the hay-cut and grazed-only treatments. Furthermore, management effects on the tail of the dispersal curve were by far the most critical in governing spread. This study suggests that simple models can be used to inform practical conservation management, and we demonstrate straightforward uses of our model to predict the impacts of different management strategies. While simple models can give accurate projections, we emphasize that they must be parameterized with high-quality data gathered at the appropriate spatial scale.     

Ecological boundary detection using Bayesian areal wombling

The study of ecological boundaries and their dynamics is of fundamental importance to much of ecology, biogeography, and evolution. Over the past two decades, boundary analysis (of which wombling is a subfield) has received considerable research attention, resulting in multiple approaches for the quantification of ecological boundaries. Nonetheless, few methods have been developed that can simultaneously (1) analyze spatially homogenized data sets (i.e., areal data in the form of polygons rather than point-reference data); (2) account for spatial structure in these data and uncertainty associated with them; and (3) objectively assign probabilities to boundaries once detected. Here we describe the application of a Bayesian hierarchical framework for boundary detection developed in public health, which addresses these issues but which has seen limited application in ecology. As examples, we analyze simulated spread data and the historic pattern of spread of an invasive species, the hemlock woolly adelgid (Adelges tsugae), using county-level summaries of the year of first reported infestation and several covariates potentially important to influencing the observed spread dynamics. Bayesian areal wombling is a promising approach for analyzing ecological boundaries and dynamics related to changes in the distributions of native and invasive species.     

Combining functional data with hierarchical Gaussian process models

Gaussian process models have been used in applications ranging from machine learning to fisheries management. In the Bayesian framework, the Gaussian process is used as a prior for unknown functions, allowing the data to drive the relationship between inputs and outputs. In our research, we consider a scenario in which response and input data are available from several similar, but not necessarily identical, sources. When little information is known about one or more of the populations it may be advantageous to model all populations together. We present a hierarchical Gaussian process model with a structure that allows distinct features for each source as well as shared underlying characteristics. Key features and properties of the model are discussed and demonstrated in a number of simulation examples. The model is then applied to a data set consisting of three populations of Rotifer Brachionus calyciflorus Pallas. Specifically, we model the log growth rate of the populations using a combination of lagged population sizes. The various lag combinations are formally compared to obtain the best model inputs. We then formally compare the leading hierarchical Gaussian process model with the inferential results obtained under the independent Gaussian process model.