A cloudy forecast for species distribution models: Predictive uncertainties abound for California birds after a century of climate and land-use change.

Correlative species distribution models are widely used to quantify past shifts in ranges or communities, and to predict future outcomes under ongoing global change. Practitioners confront a wide range of potentially plausible models for ecological dynamics, but most specific applications only consider a narrow set. Here, we clarify that certain model structures can embed restrictive assumptions about key sources of forecast uncertainty into an analysis. To evaluate forecast uncertainties and our ability to explain community change, we fit and compared 39 candidate multi- or joint species occupancy models to avian incidence data collected at 320 sites across California during the early 20th century and resurveyed a century later. We found massive (>20,000 LOOIC) differences in within-time information criterion across models. Poorer fitting models omitting multivariate random effects predicted less variation in species richness changes and smaller contemporary communities, with considerable variation in predicted spatial patterns in richness changes across models. The top models suggested avian environmental associations changed across time, contemporary avian occupancy was influenced by previous site-specific occupancy states, and that both latent site variables and species associations with these variables also varied over time. Collectively, our results recapitulate that simplified model assumptions not only impact predictive fit but may mask important sources of forecast uncertainty and mischaracterize the current state of system understanding when seeking to describe or project community responses to global change. We recommend that researchers seeking to make long-term forecasts prioritize characterizing forecast uncertainty over seeking to present a single best guess. To do so reliably, we urge practitioners to employ models capable of characterizing the key sources of forecast uncertainty, where predictors, parameters and random effects may vary over time or further interact with previous occurrence states.

Concordant and opposing effects of climate and land-use change on avian assemblages in California's most transformed landscapes.

Climate and land-use change could exhibit concordant effects that favor or disfavor the same species, which would amplify their impacts, or species may respond to each threat in a divergent manner, causing opposing effects that moderate their impacts in isolation. We used early 20th century surveys of birds conducted by Joseph Grinnell paired with modern resurveys and land-use change reconstructed from historic maps to examine avian change in Los Angeles and California's Central Valley (and their surrounding foothills). Occupancy and species richness declined greatly in Los Angeles from urbanization, strong warming (+1.8°C), and drying (-77.2 millimeters) but remained stable in the Central Valley, despite large-scale agricultural development, average warming (+0.9°C), and increased precipitation (+11.2 millimeters). While climate was the main driver of species distributions a century ago, the combined impacts of land-use and climate change drove temporal changes in occupancy, with similar numbers of species experiencing concordant and opposing effects.

A flexible and efficient Bayesian implementation of point process models for spatial capture-recapture data.

Spatial capture-recapture (SCR) is now routinely used for estimating abundance and density of wildlife populations. A standard SCR model includes sub-models for the distribution of individual activity centers (ACs) and for individual detections conditional on the locations of these ACs. Both sub-models can be expressed as point processes taking place in continuous space, but there is a lack of accessible and efficient tools to fit such models in a Bayesian paradigm. Here, we describe a set of custom functions and distributions to achieve this. Our work allows for more efficient model fitting with spatial covariates on population density, offers the option to fit SCR models using the semi-complete data likelihood (SCDL) approach instead of data augmentation, and better reflects the spatially continuous detection process in SCR studies that use area searches. In addition, the SCDL approach is more efficient than data augmentation for simple SCR models while losing its advantages for more complicated models that account for spatial variation in either population density or detection. We present the model formulation, test it with simulations, quantify computational efficiency gains, and conclude with a real-life example using non-invasive genetic sampling data for an elusive large carnivore, the wolverine (Gulo gulo) in Norway.

Estimating spatially variable and density-dependent survival using open-population spatial capture-recapture models.

Open-population spatial capture-recapture (OPSCR) models use the spatial information contained in individual detections collected over multiple consecutive occasions to estimate not only occasion-specific density, but also demographic parameters. OPSCR models can also estimate spatial variation in vital rates, but such models are neither widely used nor thoroughly tested. We developed a Bayesian OPSCR model that not only accounts for spatial variation in survival using spatial covariates but also estimates local density-dependent effects on survival within a unified framework. Using simulations, we show that OPSCR models provide sound inferences on the effect of spatial covariates on survival, including multiple competing sources of mortality, each with potentially different spatial determinants. Estimation of local density-dependent survival was possible but required more data due to the greater complexity of the model. Not accounting for spatial heterogeneity in survival led to up to 10% positive bias in abundance estimates. We provide an empirical demonstration of the model by estimating the effect of country and density on cause-specific mortality of female wolverines (Gulo gulo) in central Sweden and Norway. The ability to make population-level inferences on spatial variation in survival is an essential step toward a fully spatially explicit OPSCR model capable of disentangling the role of multiple spatial drivers of population dynamics.

Comparing N-mixture models and GLMMs for relative abundance estimation in a citizen science dataset.

To analyze species count data when detection is imperfect, ecologists need models to estimate relative abundance in the presence of unknown sources of heterogeneity. Two candidate models are generalized linear mixed models (GLMMs) and hierarchical N-mixture models. GLMMs are computationally robust but do not explicitly separate detection from abundance patterns. N-mixture models separately estimate detection and abundance via a latent state but are sensitive to violations in assumptions and subject to practical estimation issues. When one can assume that detection is not systematically confounded with ecological patterns of interest, these two models can be viewed as sharing a heuristic framework for relative abundance estimation. Model selection can then determine which predicts observed counts best, for example by AIC. We compared four N-mixture model variants and two GLMM variants for predicting bird counts in local subsets of a citizen science dataset, eBird, based on model selection and goodness-of-fit measures. We found that both GLMMs and N-mixture models-especially N-mixtures with beta-binomial detection submodels-were supported in a moderate number of datasets, suggesting that both tools are useful and that relative fit is context-dependent. We provide faster software implementations of N-mixture likelihood calculations and a reparameterization to interpret unstable estimates for N-mixture models.

Identifying engaging bird species and traits with community science observations.

Identifying rates at which birders engage with different species can inform the impact and efficacy of conservation outreach and the scientific use of community-collected biodiversity data. Species that are thought to be "charismatic" are often prioritized in conservation, and previous researchers have used sociological experiments and digital records to estimate charisma indirectly. In this study, we take advantage of community science efforts as another record of human engagement with animals that can reveal observer biases directly, which are in part driven by observer preference. We apply a multistage analysis to ask whether opportunistic birders contributing to iNaturalist engage more with larger, more colorful, and rarer birds relative to a baseline approximated from eBird contributors. We find that body mass, color contrast, and range size all predict overrepresentation in the opportunistic dataset. We also find evidence that, across 472 modeled species, 52 species are significantly overreported and 158 are significantly underreported, indicating a wide variety of species-specific effects. Understanding which birds are highly engaging can aid conservationists in creating impactful outreach materials and engaging new naturalists. The quantified differences between two prominent community science efforts may also be of use for researchers leveraging the data from one or both of them to answer scientific questions of interest.

Building integral projection models with nonindependent vital rates.

Population dynamics are functions of several demographic processes including survival, reproduction, somatic growth, and maturation. The rates or probabilities for these processes can vary by time, by location, and by individual. These processes can co-vary and interact to varying degrees, e.g., an animal can only reproduce when it is in a particular maturation state. Population dynamics models that treat the processes as independent may yield somewhat biased or imprecise parameter estimates, as well as predictions of population abundances or densities. However, commonly used integral projection models (IPMs) typically assume independence across these demographic processes. We examine several approaches for modelling between process dependence in IPMs and include cases where the processes co-vary as a function of time (temporal variation), co-vary within each individual (individual heterogeneity), and combinations of these (temporal variation and individual heterogeneity). We compare our methods to conventional IPMs, which treat vital rates independent, using simulations and a case study of Soay sheep (Ovis aries). In particular, our results indicate that correlation between vital rates can moderately affect variability of some population-level statistics. Therefore, including such dependent structures is generally advisable when fitting IPMs to ascertain whether or not such between vital rate dependencies exist, which in turn can have subsequent impact on population management or life-history evolution.

To rarefy or not to rarefy: robustness and efficiency trade-offs of rarefying microbiome data.

MOTIVATION: Microbiome datasets provide rich information about microbial communities. However, vast library size variations across samples present great challenges for proper statistical comparisons. To deal with these challenges, rarefaction is often used in practice as a normalization technique, although there has been debate whether rarefaction should ever be used. Conventional wisdom and previous work suggested that rarefaction should never be used in practice, arguing that rarefying microbiome data is statistically inadmissible. These discussions, however, have been confined to particular parametric models and simulation studies. RESULTS: We develop a semiparametric graphical model framework for grouped microbiome data and analyze in the context of differential abundance testing the statistical trade-offs of the rarefaction procedure, accounting for latent variations and measurement errors. Under the framework, it can be shown rarefaction guarantees that subsequent permutation tests properly control the Type I error. In addition, the loss in sensitivity from rarefaction is solely due to increased measurement error; if the underlying variation in microbial composition is large among samples, rarefaction might not hurt subsequent statistical inference much. We develop the rarefaction efficiency index (REI) as an indicator for efficiency loss and illustrate it with a dataset on the effect of storage conditions for microbiome data. Simulation studies based on real data demonstrate that the impact of rarefaction on sensitivity is negligible when overdispersion is prominent, while low REI corresponds to scenarios in which rarefying might substantially lower the statistical power. Whether to rarefy or not ultimately depends on assumptions of the data generating process and characteristics of the data. AVAILABILITY AND IMPLEMENTATION: Source codes are publicly available at https://github.com/jcyhong/rarefaction. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Multi-scale trend analysis of water quality using error propagation of generalized additive models.

Effective stewardship of ecosystems to sustain current ecological status or mitigate impacts requires nuanced understanding of how conditions have changed over time in response to anthropogenic pressures and natural variability. Detecting and appropriately characterizing changes requires accurate and flexible trend assessment methods that can be readily applied to environmental monitoring datasets. A key requirement is complete propagation of uncertainty through the analysis. However, this is difficult when there are mismatches between sampling frequency, period of record, and trends of interest. Here, we propose a novel application of generalized additive models (GAMs) for characterizing multi-decadal changes in water quality indicators and demonstrate its utility by analyzing a 30-year record of biweekly-to-monthly chlorophyll-a concentrations in the San Francisco Estuary. GAMs have shown promise in water quality trend analysis to separate long-term (i.e., annual or decadal) trends from seasonal variation. Our proposed methods estimate seasonal averages in a response variable with GAMs, extract uncertainty measures for the seasonal estimates, and then use the uncertainty measures with mixed-effects meta-analysis regression to quantify inter-annual trends that account for full propagation of error across methods. We first demonstrate that nearly identical descriptions of temporal changes can be obtained using different smoothing spline formulations of the original time series. We then extract seasonal averages and their standard errors for an a priori time period within each year from the GAM results. Finally, we demonstrate how across-year trends in seasonal averages can be modeled with mixed-effects meta-analysis regression that propagates uncertainties from the GAM fits to the across-year analysis. Overall, this approach leverages GAMs to smooth data with missing observations or varying sample effort across years to estimate seasonal averages and meta-analysis to estimate trends across years. Methods are provided in the wqtrends R package.

Estimating and forecasting spatial population dynamics of apex predators using transnational genetic monitoring.

The ongoing recovery of terrestrial large carnivores in North America and Europe is accompanied by intense controversy. On the one hand, reestablishment of large carnivores entails a recovery of their most important ecological role, predation. On the other hand, societies are struggling to relearn how to live with apex predators that kill livestock, compete for game species, and occasionally injure or kill people. Those responsible for managing these species and mitigating conflict often lack fundamental information due to a long-standing challenge in ecology: How do we draw robust population-level inferences for elusive animals spread over immense areas? Here we showcase the application of an effective tool for spatially explicit tracking and forecasting of wildlife population dynamics at scales that are relevant to management and conservation. We analyzed the world's largest dataset on carnivores comprising more than 35,000 noninvasively obtained DNA samples from over 6,000 individual brown bears (Ursus arctos), gray wolves (Canis lupus), and wolverines (Gulo gulo). Our analyses took into account that not all individuals are detected and, even if detected, their fates are not always known. We show unequivocal quantitative evidence of large carnivore recovery in northern Europe, juxtaposed with the finding that humans are the single-most important factor driving the dynamics of these apex predators. We present maps and forecasts of the spatiotemporal dynamics of large carnivore populations, transcending national boundaries and management regimes.

One size does not fit all: Customizing MCMC methods for hierarchical models using NIMBLE.

Improved efficiency of Markov chain Monte Carlo facilitates all aspects of statistical analysis with Bayesian hierarchical models. Identifying strategies to improve MCMC performance is becoming increasingly crucial as the complexity of models, and the run times to fit them, increases. We evaluate different strategies for improving MCMC efficiency using the open-source software NIMBLE (R package nimble) using common ecological models of species occurrence and abundance as examples. We ask how MCMC efficiency depends on model formulation, model size, data, and sampling strategy. For multiseason and/or multispecies occupancy models and for N-mixture models, we compare the efficiency of sampling discrete latent states vs. integrating over them, including more vs. fewer hierarchical model components, and univariate vs. block-sampling methods. We include the common MCMC tool JAGS in comparisons. For simple models, there is little practical difference between computational approaches. As model complexity increases, there are strong interactions between model formulation and sampling strategy on MCMC efficiency. There is no one-size-fits-all best strategy, but rather problem-specific best strategies related to model structure and type. In all but the simplest cases, NIMBLE's default or customized performance achieves much higher efficiency than JAGS. In the two most complex examples, NIMBLE was 10-12 times more efficient than JAGS. We find NIMBLE is a valuable tool for many ecologists utilizing Bayesian inference, particularly for complex models where JAGS is prohibitively slow. Our results highlight the need for more guidelines and customizable approaches to fit hierarchical models to ensure practitioners can make the most of occupancy and other hierarchical models. By implementing model-generic MCMC procedures in open-source software, including the NIMBLE extensions for integrating over latent states (implemented in the R package nimbleEcology), we have made progress toward this aim.

A pathway for multivariate analysis of ecological communities using copulas.

We describe a new pathway for multivariate analysis of data consisting of counts of species abundances that includes two key components: copulas, to provide a flexible joint model of individual species, and dissimilarity-based methods, to integrate information across species and provide a holistic view of the community. Individual species are characterized using suitable (marginal) statistical distributions, with the mean, the degree of over-dispersion, and/or zero-inflation being allowed to vary among a priori groups of sampling units. Associations among species are then modeled using copulas, which allow any pair of disparate types of variables to be coupled through their cumulative distribution function, while maintaining entirely the separate individual marginal distributions appropriate for each species. A Gaussian copula smoothly captures changes in an index of association that excludes joint absences in the space of the original species variables. A permutation-based filter with exact family-wise error can optionally be used a priori to reduce the dimensionality of the copula estimation problem. We describe in detail a Monte Carlo expectation maximization algorithm for efficient estimation of the copula correlation matrix with discrete marginal distributions (counts). The resulting fully parameterized copula models can be used to simulate realistic ecological community data under fully specified null or alternative hypotheses. Distributions of community centroids derived from simulated data can then be visualized in ordinations of ecologically meaningful dissimilarity spaces. Multinomial mixtures of data drawn from copula models also yield smooth power curves in dissimilarity-based settings. Our proposed analysis pathway provides new opportunities to combine model-based approaches with dissimilarity-based methods to enhance understanding of ecological systems. We demonstrate implementation of the pathway through an ecological example, where associations among fish species were found to increase after the establishment of a marine reserve.

Proximity of restored hedgerows interacts with local floral diversity and species' traits to shape long-term pollinator metacommunity dynamics.

Disconnected habitat fragments are poor at supporting population and community persistence; restoration ecologists, therefore, advocate for the establishment of habitat networks across landscapes. Few empirical studies, however, have considered how networks of restored habitat patches affect metacommunity dynamics. Here, using a 10-year study on restored hedgerows and unrestored field margins within an intensive agricultural landscape, we integrate occupancy modelling with network theory to examine the interaction between local and landscape characteristics, habitat selection and dispersal in shaping pollinator metacommunity dynamics. We show that surrounding hedgerows and remnant habitat patches interact with the local floral diversity, bee diet breadth and bee body size to influence site occupancy, via colonisation and persistence dynamics. Florally diverse sites and generalist, small-bodied species are most important for maintaining metacommunity connectivity. By providing the first in-depth assessment of how a network of restored habitat influences long-term population dynamics, we confirm the conservation benefit of hedgerows for pollinator populations and demonstrate the importance of restoring and maintaining habitat networks within an inhospitable matrix.

A century of climate and land-use change cause species turnover without loss of beta diversity in California's Central Valley.

Climate and land-use changes are thought to be the greatest threats to biodiversity, but few studies have directly measured their simultaneous impacts on species distributions. We used a unique historic resource-early 20th-century bird surveys conducted by Joseph Grinnell and colleagues-paired with contemporary resurveys a century later to examine changes in bird distributions in California's Central Valley, one of the most intensively modified agricultural zones in the world and a region of heterogeneous climate change. We analyzed species- and community-level occupancy using multispecies occupancy models that explicitly accounted for imperfect detection probability, and developed a novel, simulation-based method to compare the relative influences of climate and land-use covariates on site-level species richness and beta diversity (measured by Jaccard similarity). Surprisingly, we show that mean occupancy, species richness and between-site similarity have remained remarkably stable over the past century. Stability in community-level metrics masked substantial changes in species composition; occupancy declines of some species were equally matched by increases in others, predominantly species with generalist or human-associated habitat preferences. Bird occupancy, richness and diversity within each era were driven most strongly by water availability (precipitation and percent water cover), indicating that both climate and land-use are important drivers of species distributions. Water availability had much stronger effects than temperature, urbanization and agricultural cover, which are typically thought to drive biodiversity decline.

Commentary to: a cross-validation-based approach for delimiting reliable home range estimates.

BACKGROUND: Continued exploration of the performance of the recently proposed cross-validation-based approach for delimiting home ranges using the Time Local Convex Hull (T-LoCoH) method has revealed a number of issues with the original formulation. MAIN TEXT: Here we replace the ad hoc cross-validation score with a new formulation based on the total log probability of out-of-sample predictions. To obtain these probabilities, we interpret the normalized LoCoH hulls as a probability density. The application of the approach described here results in optimal parameter sets that differ dramatically from those selected using the original formulation. The derived metrics of home range size, mean revisitation rate, and mean duration of visit are also altered using the corrected formulation. CONCLUSION: Despite these differences, we encourage the use of the cross-validation-based approach, as it provides a unifying framework governed by the statistical properties of the home ranges rather than subjective selections by the user.

Origin and maintenance of chemical diversity in a species-rich tropical tree lineage.

Plant secondary metabolites play important ecological and evolutionary roles, most notably in the deterrence of natural enemies. The classical theory explaining the evolution of plant chemical diversity is that new defences arise through a pairwise co-evolutionary arms race between plants and their specialized natural enemies. However, plant species are bombarded by dozens of different herbivore taxa from disparate phylogenetic lineages that span a wide range of feeding strategies and have distinctive physiological constraints that interact differently with particular plant metabolites. How do plant defence chemicals evolve under such multiple and potentially contrasting selective pressures imposed by diverse herbivore communities? To tackle this question, we exhaustively characterized the chemical diversity and insect herbivore fauna from 31 sympatric species of Amazonian Protieae (Burseraceae) trees. Using a combination of phylogenetic, metabolomic and statistical learning tools, we show that secondary metabolites that were associated with repelling herbivores (1) were more frequent across the Protieae phylogeny and (2) were found in average higher abundance than other compounds. Our findings suggest that generalist herbivores can play an important role in shaping plant chemical diversity and support the hypothesis that chemical diversity can also arise from the cumulative outcome of multiple diffuse interactions.

Phylogenetic occupancy models integrate imperfect detection and phylogenetic signal to analyze community structure.

Biological communities are structured phylogenetically-closely related species are typically more likely to be found at the same sites. This may be, in part, because they respond similarly to environmental gradients. Accurately surveying biological communities is, however, made difficult by the fact that detection of species is not perfect. In recent years, numerous statistical methods have been developed that aim to overcome deficiencies in the species detection process. However, these methods do not allow investigators to assess phylogenetic community structure. Here, we introduce the phylogenetic occupancy model (POM), which accounts for imperfect species detection while assessing phylogenetic patterns in community structure. Using simulated data sets we show that the POM grants less biased estimates of phylogenetic structure than models without imperfect detection, and can correctly ascertain the effects of species traits on community composition while accounting for evolutionary non-independence of taxa. Integrating phylogenetic methods into widely used occupancy models will help clarify how evolutionary history influences modern day communities.

Warming and provenance limit tree recruitment across and beyond the elevation range of subalpine forest.

Climate niche models project that subalpine forest ranges will extend upslope with climate warming. These projections assume that the climate suitable for adult trees will be adequate for forest regeneration, ignoring climate requirements for seedling recruitment, a potential demographic bottleneck. Moreover, local genetic adaptation is expected to facilitate range expansion, with tree populations at the upper forest edge providing the seed best adapted to the alpine. Here, we test these expectations using a novel combination of common gardens, seeded with two widely distributed subalpine conifers, and climate manipulations replicated at three elevations. Infrared heaters raised temperatures in heated plots, but raised temperatures more in the forest than at or above treeline because strong winds at high elevation reduced heating efficiency. Watering increased season-average soil moisture similarly across sites. Contrary to expectations, warming reduced Engelmann spruce recruitment at and above treeline, as well as in the forest. Warming reduced limber pine first-year recruitment in the forest, but had no net effect on fourth-year recruitment at any site. Watering during the snow-free season alleviated some negative effects of warming, indicating that warming exacerbated water limitations. Contrary to expectations of local adaptation, low-elevation seeds of both species initially recruited more strongly than high-elevation seeds across the elevation gradient, although the low-provenance advantage diminished by the fourth year for Engelmann spruce, likely due to small sample sizes. High- and low-elevation provenances responded similarly to warming across sites for Engelmann spruce, but differently for limber pine. In the context of increasing tree mortality, lower recruitment at all elevations with warming, combined with lower quality, high-provenance seed being most available for colonizing the alpine, portends range contraction for Engelmann spruce. The lower sensitivity of limber pine to warming indicates a potential for this species to become more important in subalpine forest communities in the coming centuries.

Climate and edaphic controllers influence rhizosphere community assembly for a wild annual grass.

The interface between roots and soil, known as the rhizosphere, is a dynamic habitat in the soil ecosystem. Unraveling the factors that control rhizosphere community assembly is a key starting point for understanding the diversity of plant-microbial interactions that occur in soil. The goals of this study were to determine how environmental factors shape rhizosphere microbial communities, such as local soil characteristics and the regional climate, and to determine the relative influence of the rhizosphere on microbial community assembly compared to the pressures imposed by the local and regional environment. We identified the bacteria present in the soil immediately adjacent to the roots of wild oat (A vena spp.) in three California grasslands using deep Illumina 16S sequencing. Rhizosphere communities were more similar to each other than to the surrounding soil communities from which they were derived, despite the fact that the grasslands studied were separated by hundreds of kilometers. The rhizosphere was the dominant factor structuring bacterial community composition (38% variance explained), and was comparable in magnitude to the combined local and regional effects (22% and 21%, respectively). Rhizosphere communities were most influenced by factors related to the regional climate (soil moisture and temperature), while background soil communities were more influenced by soil characteristics (pH, CEC, exchangeable cations, clay content). The Avena core microbiome was strongly phylogenetically clustered according to the metrics NRI and NTI, which indicates that selective processes likely shaped these communities. Furthermore, 17% of these taxa were not detectable in the background soil, even with a robust sequencing depth of approximately 70,000 sequences per sample. These results support the hypothesis that roots select less abundant or possibly rare populations in the soil microbial community, which appear to be lineages of bacteria that have made a physiological tradeoff for rhizosphere competence at the expense of their competitiveness in non-rhizosphere soil.

Monte Carlo estimation of stage structured development from cohort data.

Cohort data are frequently collected to study stage-structured development and mortalities of many organisms, particularly arthropods. Such data can provide information on mean stage durations, among-individual variation in stage durations, and on mortality rates. Current statistical methods for cohort data lack flexibility in the specification of stage duration distributions and mortality rates. In this paper, we present a new method for fitting models of stage-duration distributions and mortality to cohort data. The method is based on a Monte Carlo within MCMC algorithm and provides Bayesian estimates of parameters of stage-structured cohort models. The algorithm is computationally demanding but allows for flexible specifications of stage-duration distributions and mortality rates. We illustrate the algorithm with an application to data from a previously published experiment on the development of brine shrimp from Mono Lake, California, through nine successive stages. In the experiment, three different food supply and temperature combination treatments were studied. We compare the mean duration of the stages among the treatments while simultaneously estimating mortality rates and among-individual variance of stage durations. The method promises to enable more detailed studies of development of both natural and experimental cohorts. An R package implementing the method and which allows flexible specification of stage duration distributions is provided.