Chao Estimator Research Articles

Optimizing performance of nonparametric species richness estimators under constrained sampling.

Understanding the functional relationship between the sample size and the performance of species richness estimators is necessary to optimize limited sampling resources against estimation error. Nonparametric estimators such as Chao and Jackknife demonstrate strong performances, but consensus is lacking as to which estimator performs better under constrained sampling. We explore a method to improve the estimators under such scenario. The method we propose involves randomly splitting species‐abundance data from a single sample into two equally sized samples, and using an appropriate incidence‐based estimator to estimate richness. To test this method, we assume a lognormal species‐abundance distribution (SAD) with varying coefficients of variation (CV), generate samples using MCMC simulations, and use the expected mean‐squared error as the performance criterion of the estimators. We test this method for Chao, Jackknife, ICE, and ACE estimators. Between abundance‐based estimators with the single sample, and incidence‐based estimators with the split‐in‐two samples, Chao2 performed the best when CV < 0.65, and incidence‐based Jackknife performed the best when CV > 0.65, given that the ratio of sample size to observed species richness is greater than a critical value given by a power function of CV with respect to abundance of the sampled population. The proposed method increases the performance of the estimators substantially and is more effective when more rare species are in an assemblage. We also show that the splitting method works qualitatively similarly well when the SADs are log series, geometric series, and negative binomial. We demonstrate an application of the proposed method by estimating richness of zooplankton communities in samples of ballast water. The proposed splitting method is an alternative to sampling a large number of individuals to increase the accuracy of richness estimations; therefore, it is appropriate for a wide range of resource‐limited sampling scenarios in ecology.

Functional Responses of Bacterioplankton Diversity and Metabolism to Experimental Bottom-Up and Top-Down Forcings.

We conducted an experimental approach using microcosms to simultaneously examine the functional response of natural freshwater bacterial assemblages to the impact of resources (nutrients) and top-down factors (viruses and grazers) on bacterial physiological state and their community structure. Addition of organic and inorganic nutrients led to the proliferation of high nucleic acid content bacterial cells accompanied by high bacterial growth efficiency (considered as proxy of bacterial carbon metabolism) estimates, suggesting that this subgroup represented the most active fraction of bacterial community and had a high capacity to incorporate carbon into its biomass. However, their rapid growth induced the pressure of viral lytic infection which led to their lysis toward the end of the experiment. In microcosms with flagellates plus viruses, and with viruses alone, the selective removal of metabolically active high nucleic acid cells through viral lysis benefitted the less active low nucleic acid content cells, perhaps via the use of lysis products for its growth and survival. Changes in bacterial physiological state in microcosms were reflected in their community structure which was examined using 16S ribosomal RNA (rRNA) gene sequencing by Illumina's Miseq platform. Chao estimator and Shannon diversity index values suggested that bacterial species richness was highest in the presence of both the top-down factors, indicating a tighter control of bacterioplankton dominants within a relatively stable bacterial community. The increase in bacterial metabolism with nutrient addition followed by subsequent lysis of bacterial dominants indicate that both resources and top-down factors work in concert for the sustenance of stable bacterial communities.

Pyrosequencing investigation into the influence of Cu2+, Zn2+, Fe2+ and I- mixtures on fungal diversity and toxigenic fungal growth in a fermented liquid feed.

A L9(34) orthogonal experiment was conducted to evaluate the influence of 9 mixtures which consisted of Cu2+, Zn2+, Fe2+ and I− ions at different ion concentrations on fungal diversity and toxigenic fungal growth in a Bacillus subtilis-fermented liquid feed (FLF) using pyrosequencing. The maximal Chao estimator and Shannon index were achieved in the FLF with a mixture of Cu2+ (200 mg/kg), Zn2+ (160 mg/kg), Fe2+ (150 mg/kg) and I− (2.4 mg/kg). The minimal relative abundance of Aspergillus was achieved when a mixture of Cu2+ (200 mg/kg), Zn2+, Fe2+ and I− was added to the FLF. Compared with Zn2+, Fe2+ and I−, Cu2+ was the most important ion in inhibiting Aspergillus growth. Adding Zn2+ (160 mg/kg), Cu2+, Fe2+ and I− to the FLF minimized the relative abundance of Fusarium. It was Zn2+ instead of Cu2+ played a critical role in suppressing the growth of Fusarium. The proper use of the mixture of Cu2+, Zn2+, Fe2+ and I− in FLF contributes to inhibit the growth of mycotoxin-producing fungi during storage. The new findings of this study help farmers properly use the mixture of Cu2+, Zn2+, Fe2+ and I− to inhibit the growth of mycotoxin-producing fungi in the production of high quality FLF and alleviate mycotoxins damages to animals and humans.

Chaos Behavior and Estimation of the Unknown Parameters of Stochastic Lattice Gas for Prey-Predator Model with Pair-Approximation

In this paper, the problem of chaos, stability and estimation of unknown parameters of the stochastic lattice gas for prey-predator model with pair-approximation is studied. The result shows that this dynamical system exhibits an oscillatory behavior of the population densities of prey and predator. Using Liapunov stability technique, the estimators of the unknown probabilities are derived, and also the updating rules for stability around its steady states are derived. Furthermore the feedback control law has been as non-linear functions of the population densities. Numerical simulation study is presented graphically.

Specialized Brains Performing Specialized Tasks: Beta/Gamma EEG Non-linear Analysis Reveals Discriminative Differences between the Chaos/no-chaos Content of Specialized Brain's Dynamics

We studied and compared different non-linear features of the human EEG in the context of the search and characterization of specific functional fingerprints of the EEG dynamics that allow differentiating behaviorally recognized specialized professional activities. Sets of EEG data obtained from two groups of professional performers, a group of 10 professional modern dancers and a group of 12 professional chopper pilots, were compared while executing a mental-cognitive task related to their professional specialty. Dancers were asked to use mental imagination to plan choreographic movements and pilots were asked to solve a reduced version of the Raven's matrices visual intelligent test. Both tasks were analyzed during the first two minutes while EEG activity was recorded from 14 mastoid bone referenced electrodes using the brain-computer interface Emotiv® Epoc Research Edition, sampling at 128Hz rate. Dancers and pilots were then compared in term of their long- and short-term Hurst (H) exponent estimation for beta and gamma EEG ranges. We found that non-linear estimation of chaos in the EEG time series allows differentiating between the brain functional strategies employed by differently specialized groups of performers. The group of pilots showed higher estimated H values for the beta band than for gamma band, contrary to what we found for dancers in which estimated H for gamma band showed higher values than for beta range.

Horseshoe chaos and topological entropy estimation in a palaeoclimate dynamical model

In this paper, a three-dimensional palaeoclimate dynamical model by Saltzman is revisited. We find that chaotic behavior displays in this three-dimensional dynamical system in numerical studies. To verify the chaoticity, we resort to Poincaré section and Poincaré map technique to present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoe theory, and give an estimation of the topological entropy.

Application of Spatial and Closed Capture-Recapture Models on Known Population of the Western Derby Eland (Taurotragus derbianus derbianus) in Senegal.

Camera trapping with capture-recapture analyses has provided estimates of the abundances of elusive species over the last two decades. Closed capture-recapture models (CR) based on the recognition of individuals and incorporating natural heterogeneity in capture probabilities are considered robust tools; however, closure assumption is often questionable and the use of an Mh jackknife estimator may fail in estimations of real abundance when the heterogeneity is high and data is sparse. A novel, spatially explicit capture-recapture (SECR) approach based on the location-specific capture histories of individuals overcomes the limitations of closed models. We applied both methods on a closed population of 16 critically endangered Western Derby elands in the fenced 1,060-ha Fathala reserve, Senegal. We analyzed the data from 30 cameras operating during a 66-day sampling period deployed in two densities in grid and line arrays. We captured and identified all 16 individuals in 962 trap-days. Abundances were estimated in the programs CAPTURE (models M0, Mh and Mh Chao) and R, package secr (basic Null and Finite mixture models), and compared with the true population size. We specified 66 days as a threshold in which SECR provides an accurate estimate in all trapping designs within the 7-times divergent density from 0.004 to 0.028 camera trap/ha. Both SECR models showed uniform tendency to overestimate abundance when sampling lasted shorter with no major differences between their outputs. Unlike the closed models, SECR performed well in the line patterns, which indicates promising potential for linear sampling of properly defined habitats of non-territorial and identifiable herbivores in dense wooded savanna conditions. The CR models provided reliable estimates in the grid and we confirmed the advantage of Mh Chao estimator over Mh jackknife when data appeared sparse. We also demonstrated the pooling of trapping occasions with an increase in the capture probabilities, avoiding violation of results.

Evaluating species richness: Biased ecological inference results from spatial heterogeneity in detection probabilities.

Accurate estimates of species richness are necessary to test predictions of ecological theory and evaluate biodiversity for conservation purposes. However, species richness is difficult to measure in the field because some species will almost always be overlooked due to their cryptic nature or the observer's failure to perceive their cues. Common measures of species richness that assume consistent observability across species are inviting because they may require only single counts of species at survey sites. Single-visit estimation methods ignore spatial and temporal variation in species detection probabilities related to survey or site conditions that may confound estimates of species richness. We used simulated and empirical data to evaluate the bias and precision of raw species counts, the limiting forms of jackknife and Chao estimators, and multispecies occupancy models when estimating species richness to evaluate whether the choice of estimator can affect inferences about the relationships between environmental conditions and community size under variable detection processes. Four simulated scenarios with realistic and variable detection processes were considered. Results of simulations indicated that (1) raw species counts were always biased low, (2) single-visit jackknife and Chao estimators were significantly biased regardless of detection process, (3) multispecies occupancy models were more precise and generally less biased than the jackknife and Chao estimators, and (4) spatial heterogeneity resulting from the effects of a site covariate on species detection probabilities had significant impacts on the inferred relationships between species richness and a spatially explicit environmental condition. For a real data set of bird observations in northwestern Alaska, USA, the four estimation methods produced different estimates of local species richness, which severely affected inferences about the effects of shrubs on local avian richness. Overall, our results indicate that neglecting the effects of site covariates on species detection probabilities may lead to significant bias in estimation of species richness, as well as the inferred relationships between community size and environmental covariates.

Limited sampling hampers "big data" estimation of species richness in a tropical biodiversity hotspot.

Macro-scale species richness studies often use museum specimens as their main source of information. However, such datasets are often strongly biased due to variation in sampling effort in space and time. These biases may strongly affect diversity estimates and may, thereby, obstruct solid inference on the underlying diversity drivers, as well as mislead conservation prioritization. In recent years, this has resulted in an increased focus on developing methods to correct for sampling bias. In this study, we use sample-size-correcting methods to examine patterns of tropical plant diversity in Ecuador, one of the most species-rich and climatically heterogeneous biodiversity hotspots. Species richness estimates were calculated based on 205,735 georeferenced specimens of 15,788 species using the Margalef diversity index, the Chao estimator, the second-order Jackknife and Bootstrapping resampling methods, and Hill numbers and rarefaction. Species richness was heavily correlated with sampling effort, and only rarefaction was able to remove this effect, and we recommend this method for estimation of species richness with “big data” collections.

Comparing capture–recapture methods for estimation of the size of small and medium-sized populations using empirical data on commercial turkey farms in Canada

The study objectives were (1) to conduct a systematic review of the performance of capture–recapture methods; (2) to use empirical data to estimate population size in a small-sized population (turkey breeder farms) and a medium-sized population (meat turkey farms) by applying two-source capture–recapture methods (the Lincoln–Petersen, the Chapman, and Chao's lower-bound estimators) and multi-source capture–recapture methods (the log-linear modeling and sample coverage approaches); and (3) to compare the performance of these methods in predicting the true population sizes (2007 data). Our set-up was unique in that we knew the population sizes for turkey breeder farms (99) and meat turkey farms (592) in Canada in 2007, which we applied as our true population sizes, and had surveillance data from the Canadian Notifiable Avian Influenza Surveillance System (2008–2012). We defined each calendar year of sampling as a data source. We confirmed that the two-source capture–recapture methods were sensitive to the violation of the local independence assumption. The log-linear modeling and sample coverage approaches yielded estimates that were closer to the true population sizes than were the estimates provided by the two-source methods for both populations. The performance of both multi-source capture–recapture methods depended on the number of data sources analyzed and the size of the population. Simulation studies are recommended to better understand the limits of each multi-source capture–recapture method.

Preliminary study of insects associated to indoor body decay in Colombia

Preliminary study of insects associated to indoor body decay in Colombia. This is the first report studying insects associated to indoor body decay process of a white pig (Sus scrofa) (Artiodactyla, Suidae) in a controlled indoor environment in an urban area of Florencia city, Amazonia Piedmont, Colombia. For a period of 54 days, 9,220 individuals (immature and adults), distributed in 3 orders, 5 families, 10 genera, and 10 species were collected using entomological nets and tweezers. Five decaying stages are described (fresh, bloated, active decay, advanced decay and remains). During the fresh stage we recorded Cochliomyia macellaria (Fabricius, 1775), Chrysomya albiceps (Wiedemann, 1819), Ophyra aenescens (Wiedemann, 1830), Oxysarcodexia sp., Lepidodexia sp. and Lasiophanes sp.; during the bloating stage C. macellaria, C. albiceps, Lucilia eximia (Wiedemann, 1819), Hemilucillia semidiaphana (Rondani, 1850), Musca domestica Linnaeus, 1758, O. aenescens, Oxysarcodexia sp., Lepidodexia sp., Dermestes maculatus De Geer, 1774 and Lasiphanes sp.; during the active decay C. macellaria, C. albiceps, L. eximia, M. domestica, O. aenescens, Lepidodexia sp. D. maculatus and Lasiophanes sp.; during the advanced decay C. macellaria, C. albiceps, M. domestica, Lepidodexia sp. and Lasiophanes sp.; and during the remains stage C. albiceps, D. maculatus and Lasiophanes sp. The insects were sorted out in 3 ecological categories; necrophagous, predators and parasites and sarco-sapropha- gous. According to Chao and Jack estimators, total richness was observed on day 20, with 100% of the expected species.

On the Chao and Zelterman estimators in a binomial mixture model

Data from a surveillance system can be used to estimate the size of a disease population. For certain surveillance systems, a binomial mixture model arises as a natural choice. The Chao estimator estimates a lower bound of the population size. The Zelterman estimator estimates a parameter that is neither a lower bound nor an upper bound. By comparing the Chao estimator and the Zelterman estimator both theoretically and numerically, we conclude that the Chao estimator is better.

Poincaré type inequalities for group measure spaces and related transportation cost inequalities

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H. We prove Lp Poincaré inequalities for the group measure space L∞(ΩH,γ)⋊G, where both the group action and the Gaussian measure space (ΩH,γ) are associated with the representation α. The idea of proof comes from Pisierʼs method on the boundedness of Riesz transform and Lust-Piquardʼs work on spin systems. Then we deduce a transportation type inequality from the Lp Poincaré inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffelʼs compact quantum metric spaces.

Capture–Recapture Method for Estimating Annual Incidence of Imported Dengue, France, 2007–2010

Imported dengue cases pose the public health risk for local circulation in European areas, especially southeast France, where the Aedes mosquito is established. Using a capture–recapture method with Chao’s estimator, we estimated the annual incidence of dengue fever and the completeness of existing mandatory notification and laboratory network surveillance systems. During 2007–2010, >8,300 cases with laboratory evidence of recent dengue infection were diagnosed. Of these cases, 4,500 occurred in 2010, coinciding with intense epidemics in the French West Indies. Over this 4-year period, 327 cases occurred in southeast France during the vector activity period. Of these, 234 cases occurred in 2010, most of them potentially viremic. Completeness of the mandatory notification and laboratory network systems were ≈10% and 40%, respectively, but higher in southeast areas during May–November (32% and 69%, respectively). Dengue surveillance systems in France provide complementary information that is essential to the implementation of control measures.

A Generalization of Chao's Estimator for Covariate Information

This note generalizes Chao's estimator of population size for closed capture-recapture studies if covariates are available. Chao's estimator was developed under unobserved heterogeneity in which case it represents a lower bound of the population size. If observed heterogeneity is available in form of covariates we show how this information can be used to reduce the bias of Chao's estimator. The key element in this development is the understanding and placement of Chao's estimator in a truncated Poisson likelihood. It is shown that a truncated Poisson likelihood (with log-link) with all counts truncated besides ones and twos is equivalent to a binomial likelihood (with logit-link). This enables the development of a generalized Chao estimator as the estimated, expected value of the frequency of zero counts under a truncated (all counts truncated except ones and twos) Poisson regression model. If the regression model accounts for the heterogeneity entirely, the generalized Chao estimator is asymptotically unbiased. A simulation study illustrates the potential in gain of bias reduction. Comparisons of the generalized Chao estimator with the homogeneous zero-truncated Poisson regression approach are supplied as well. The method is applied to a surveillance study on the completeness of farm submissions in Great Britain.

On Population Size Estimators in the Poisson Mixture Model

Estimating population sizes via capture-recapture experiments has enormous applications. The Poisson mixture model can be adopted for those applications with a single list in which individuals appear one or more times. We compare several nonparametric estimators, including the Chao estimator, the Zelterman estimator, two jackknife estimators and the bootstrap estimator. The target parameter of the Chao estimator is a lower bound of the population size. Those of the other four estimators are not lower bounds, and they may produce lower confidence limits for the population size with poor coverage probabilities. A simulation study is reported and two examples are investigated.

Robust estimation of microbial diversity in theory and in practice

Quantifying diversity is of central importance for the study of structure, function and evolution of microbial communities. The estimation of microbial diversity has received renewed attention with the advent of large-scale metagenomic studies. Here, we consider what the diversity observed in a sample tells us about the diversity of the community being sampled. First, we argue that one cannot reliably estimate the absolute and relative number of microbial species present in a community without making unsupported assumptions about species abundance distributions. The reason for this is that sample data do not contain information about the number of rare species in the tail of species abundance distributions. We illustrate the difficulty in comparing species richness estimates by applying Chao's estimator of species richness to a set of in silico communities: they are ranked incorrectly in the presence of large numbers of rare species. Next, we extend our analysis to a general family of diversity metrics ('Hill diversities'), and construct lower and upper estimates of diversity values consistent with the sample data. The theory generalizes Chao's estimator, which we retrieve as the lower estimate of species richness. We show that Shannon and Simpson diversity can be robustly estimated for the in silico communities. We analyze nine metagenomic data sets from a wide range of environments, and show that our findings are relevant for empirically-sampled communities. Hence, we recommend the use of Shannon and Simpson diversity rather than species richness in efforts to quantify and compare microbial diversity.

Mixture Models for Estimating the Number of Drug Users in Thailand 2005-2007

It is difficult to measure the sizes of illegal drug user populations directly by using the survey method because of many “hidden drug addicts” and the difficulty of receiving a true response. Systematic and routine information on treatment episodes of drug users is adopted to estimate the population size in this study. Mixture models of zero-truncated Poisson distributions using the nonparametric maximum likelihood estimators (NPMLE) by means of capture-recapture repeated count data were used to project the number of drug users. The method was applied to surveillance data of drug users identified by treatment episodes in over 1140 health treatment centers in Thailand from the Bureau of Health Service System Development, Ministry of Public Health. We presented how this mixture model could be utilized to construct the unobserved frequency of drug users with no treatment episode and further estimated the total population size of drug users in the country from 2005 to 2007. The result of simulation was confirmed that mixture model is suitable when population is large. By means of mixture models, the estimations for the number of drug users were fitted with excellent goodness-of-fit values and we were also compared to the conventional Chao estimates. The NPMLE for the total number of drug users in Thailand 2005, 2006, and 2007 were 184,045 (95% CI: 181,297-86,793), 230,665 (95% CI: 226,611-234,719), 299,670 (95% CI: 294,217-305,123), respectively, also 125,265 (95% CI: 123,092-127,142), 166,287 (95% CI: 163,222-169,352), 228,898 (95% CI: 224,766 - 233,030) for the number of methamphetamine (Yaba) users, and 11,559 (95% CI: 10,234-12,884), 11,333 (95% CI: 9276-13,390), 8953 (95% CI: 7878-10,028) for the number of heroin users, respectively. The numbers of marijuana, kratom-plant, opium, and inhalant users were underestimated because their symptoms were mild and not severe enough to remedy in health treatment centers which led to the smaller size of the total number of drug users. The well-estimated sizes of heroin and methamphetamine addicts are high reliable because they are based on clearly evident count with a severe addiction problem to health treatment centers. The estimation by means of mixture models can be recommended to monitor drug demand trend and drug health service routinely; it is easy to calculate via the available programs MIXTP based on request.

Estimating species richness: still a long way off!

AbstractXu et al., in this issue of the Journal of Vegetation Science, compare several species richness estimators. All the non‐parametric estimators, such as Chao and jackknife estimators, underestimated the true number, whereas all the area‐based models, based on species–area curves, overestimated it. No reliable method yet exists to predict the number of species in an area that is appreciably larger than the one(s) sampled.

Diversidad de enterobacterias asociadas a frutos de tomate (Lycopersicum sculentum Mill) y suelos de invernadero

The aim of this study was to evaluate the diversity of Enterobacteriaceae present in soil and tomato fruits from three greenhouses with fertirigation system. These crop systems are an important alternative for production in protected agriculture; however, there is little information about the microbiological quality of the fruit and its relationship with chemical soil characteristics. Soil evaluations consisted of analyzing organic matter content and pH. In the microbiological analysis were isolated and identified enterobacterias organisms from composite samples of soil and fruits at different stages of maturity (0, 50 and 100%). Culture media used was selective, differential and confirmatory testing with VITEK system. Enteropathogenic E. coli (EPEC) and enterotoxigenic E. coli (ETEC) were characterized genotypically, amplifying the lngA and bfpA genes by the technique of polymerase chain reaction (PCR). Diversity index (Simpson (D), Shannon-Wiener (H') and Chao estimator (SChao1) were calculated with the identified species. The species Enterobacter cloacae, Citrobacter freundii and C. brakii had a higher frequency of isolation, EPEC and ETEC were identified in soil samples and in fruits with 100% maturity. In soil, H' indices were positively correlated with the highest organic matter percentages. In fruit, although H 'and D showed less diverse bacterial communities, the isolation of ETEC and Shigella boydii on the fruit surface compromise their safety because they are usually consumed raw.