Chao Estimator Research Articles

Probability density estimation of polynomial chaos and its application in structural reliability analysis

Polynomial chaos expansion (PCE) is a widely used approach for establishing the surrogate model of a time-consuming performance function for the convenience of uncertainty quantification of a stochastic structure. However, it remains difficult to calculate the probability density function (PDF) of the PCE accurately for general cases, though the PDF, as a complete representation of a random variable, is often required in some uncertainty problems. To address this problem, this paper proposes a semi-analytical method to compute the PDF of a PCE. This method derives the closed-form solutions of characteristic functions (CFs) of the first- and second-order PCEs, while an equivalent parabolization technique is proposed to provide the approximate solutions of CFs of higher-order PCEs. Then, the PDF of the PCE can be obtained by the Fourier transform of the resulting CF. Three numerical examples are investigated to demonstrate the accuracy, applicability, and efficiency of the proposed method for probability density estimation of PCE in structural reliability analysis.

NRT1.1B mediates rice plant growth and soil microbial diversity under different nitrogen conditions

Interactions between microorganisms and plants can stimulate plant growth and promote nitrogen cycling. Nitrogen fertilizers are routinely used in agriculture to improve crop growth and yield; however, poor use efficiency impairs the optimal utilization of such fertilizers. Differences in the microbial diversity and plant growth of rice soil under different nitrogen application conditions and the expression of nitrogen-use efficiency-related genes have not been previously investigated. Therefore, this study investigates how nitrogen application and nitrogen-use efficiency-related gene NRT1.1B expression affect the soil microbial diversity and growth indices of two rice varieties, Huaidao 5 and Xinhuai 5. In total, 103,463 and 98,427 operational taxonomic units were detected in the soils of the Huaidao 5 and Xinhuai 5 rice varieties, respectively. The Shannon and Simpson indices initially increased and then decreased, whereas the Chao and abundance-based coverage estimator indices decreased after the application of nitrogen fertilizer. Nitrogen fertilization also reduced soil bacterial diversity and richness, as indicated by the reduced abundances of Azotobacter recorded in the soils of both rice varieties. Nitrogen application initially increased and then decreased the grain number per panicle, yield per plant, root, stem, and leaf nitrogen, total nitrogen content, glutamine synthetase, nitrate reductase, urease, and root activities of both varieties. Plant height showed positive linear trends in response to nitrogen application, whereas thousand-grain weights showed a negative trend. Our findings may be used to optimize nitrogen fertilizer use for rice cultivation and develop crop-variety-specific strategies for nitrogen fertilizer application.

Correction: Estimation of Chaos and Coherence in Frontoparietal, Occipital and Temporal Lobes at Pre, Post and During Focal Cerebral Ischemia

Correction: Estimation of Chaos and Coherence in Frontoparietal, Occipital and Temporal Lobes at Pre, Post and During Focal Cerebral Ischemia

The global dynamics of a new fractional-order chaotic system

This paper investigates the global dynamics of a new 3-dimensional fractional-order (FO) system that presents just cross-product nonlinearities. Firstly, the FO forced Lorenz-84 system is introduced and the stability of its equilibrium points, as well as the chaos control for their stabilization, are addressed. Secondly, dynamical behavior is further analyzed and bifurcation diagrams, phase portraits, and largest Lyapunov exponent (LE) are discussed. Then, the global Mittag-Leffler attractive sets (MLASs) and Mittag-Leffler positive invariant sets (MLPISs) of the FO forced Lorenz-84 system are presented. Finally, the Hamilton energy function (HEF) of the Lorenz-84 system is calculated by using the Helmholtz theorem. The calculation of the Hamilton energy has an essential role on the estimation of chaos in dynamical systems, the guidance of orbits, and stability. In fact, any control action on the dynamical system completely changes the HEF. Numerical simulations are presented for illustrating the theoretical findings.

Applications of fractal interpolants in kernel regression estimations

Generating a function from its finite set of samples is widely studied in data fitting problems. The theory of nonparametric modeling has been developed by many researchers, and several types of estimators have been established in the literature. On the other hand, the fractal theory provides new technologies for making complicated curves and fitting experimental data, and fractal interpolation functions are established as generalizations of traditional interpolation techniques. The aim of this paper is to investigate applications of (smooth) fractal interpolation functions in nonparametric kernel regressions. We discuss a key sufficient condition in detail in the construction of (smooth) fractal interpolation functions. Then the Priestley–Chao estimator is considered, and we establish its fractal perturbation. The raw data we consider in our example is the Crude Oil WTI Futures daily highest price from 2021/10/13 to 2022/7/29. We take 11 samples from the raw data to construct a fractal perturbation of the Priestley–Chao regression function. SLSQP is applied to find the optimal values of hyperparameters so that the given empirical mean squared error is minimum. We see that nonparametric kernel regressions for finite sample data are widely used in applications, and we can decrease the empirical mean squared error further by considering fractal perturbations of these regression functions.

Perturbations in gastrointestinal tract microbiota composition and function in individuals with yellow-greasy tongue coating

ObjectiveTo study the composition and function of tongue coating (TC) and gastrointestinal tract (GIT) microbiota in participants with yellow-greasy tongue coating (YGTC), and to explore the representative metabolite markers and pathways in this group. MethodsSubjects with YGTC or thin-white tongue coating (TWTC) were recruited from December 1, 2021 to October 30, 2022, and the TC and fecal samples were collected. Samples were subjected to both whole-genome shotgun (WGS), and 16S rRNA gene sequencing. The α-diversity analysis, principal component analysis (PCA), and Spearman correlation analysis were performed for two groups. Ultra-performance liquid chromatography combined with tandem mass spectrometry (UPLC–MS/MS) analysis was used to analyze metabolomics and enrichment of metabolic pathways. ResultsThe results revealed 20 YGTC participates and 19 TWTC participates. At the genus level, the dominant bacterial species of TC flora and intestinal flora in the two groups were roughly the same, but the relative kurtosis difference was marked, and the abundance of potentially pathogenic bacteria in TC and fecal samples of YGTC subjects was higher. There were 9 down-regulated microorganisms in the TC samples, 26 down-regulated microorganisms, and 6 up-regulated microorganisms in YGTC subjects. The α-diversity analysis indicated that the Chao and abundance-based coverage estimator (ACE) indices of TC bacteria in the YGTC subjects showed a decreasing trend, but the difference was not statistically significant (P > 0.05). The α-diversity of fecal samples and the Chao and ACE indices decreased significantly (P < 0.05). PCA showed that the microflora structure of TC and fecal samples were significantly different between the two groups. Spearman correlation analysis showed that there was no correlation between TC and fecal microorganisms at phyla and genus levels in the same subjects (P > 0.05). The metabolomics results demonstrated that fumarate reductase, V/A ATPase, and phosphatidylethanolamine were increased, and glycerate-3p, UDP-glucose, and quinone oxidoreductase metabolites were decreased in YGTC TC samples. Inosine monophosphate (IMP), uridine monophosphate (UMP), and gamma-aminobutyric acid (GABA) were increased in YGTC fecal samples, while the contents of ribo-5P, histidine, biotin, and cobalamin were decreased. Metabolic pathway analysis indicated that the abundance of the TC and fecal samples of the YGTC subjects was relatively low in various metabolic pathways, including amino acid metabolism, carbohydrate metabolism, nitrogen metabolism, and energy metabolism. ConclusionStructural and functional changes in TC and GIT microbiota or metabolite markers could be potential biological bases of YGTC formation.

Correction to: Estimation of Chaos and Coherence in Frontoparietal, Occipital and Temporal Lobes at Pre, Post and During Focal Cerebral Ischemia

Correction to: Estimation of Chaos and Coherence in Frontoparietal, Occipital and Temporal Lobes at Pre, Post and During Focal Cerebral Ischemia

Estimation of Chaos and Coherence in Frontoparietal, Occipital and Temporal Lobes at Pre, Post and During Focal Cerebral Ischemia

Estimation of Chaos and Coherence in Frontoparietal, Occipital and Temporal Lobes at Pre, Post and During Focal Cerebral Ischemia

Weak quantitative propagation of chaos via differential calculus on the space of measures

Consider the metric space (P2(Rd),W2) of square integrable laws on Rd with the topology induced by the 2-Wasserstein distance W2. Let Φ:P2(Rd)→R and μ∈P2(Rd). In this work, we consider (a) μN being the empirical measure of N-samples from μ, and the other case in which (b) μN is the empirical measure of marginal laws of the particle system of a McKean–Vlasov PDE (μt)t. The main result of this paper is to show that under suitable regularity conditions, we have |Φ(μ)−EΦ(μN)|=∑j=1k−1Cj Nj+O(1 Nk), for some positive constants C1,…,Ck−1 that do not depend on N, where k corresponds to the degree of smoothness. The case where the samples are i.i.d. is studied using functional derivatives on the space of measures. The case of particle systems relies on an Itô-type formula for the flow of probability measures and is intimately connected to PDEs on the space of measures, called the master equation in the literature of mean-field games. We state general regularity conditions required for each case and analyze the regularity in the case of functionals of the laws of McKean–Vlasov PDEs. Ultimately, this work reveals quantitative estimates of propagation of chaos for interacting particle systems. Furthermore, we are able to provide weak propagation of chaos estimates for ensembles of interacting particles and show that these may have some remarkable properties.

Response of Gut Microbiota, Digestive Enzyme Ability, and Immune Function to Starvation in the Oriental River Prawn Macrobrachium nipponense

In order to evaluate the interplay of nutritional status and the digestive capacity, immune function, and gut microbiota in the oriental river prawn Macrobrachium nipponense, a 14-day starvation trial was conducted to detect the effects of starvation on the activities of digestive enzymes and immune enzymes and the structure of prawn gut microbiota. The adult prawns were randomly assigned to a control group (F group) and starvation group (S group) with three replicates. The F group was normally fed with commercial diet, and the S group was starved for 14 days. The result showed that digestive enzyme activities of trypsin and lipase in the hepatopancreas of the S group were significantly lower than those of the F group after 4 d of starvation ( P &lt; 0.05 ), while the activity of amylase significantly increased after 14 d of starvation ( P &lt; 0.05 ). Moreover, the results of the histological analysis of prawn gut showed that both the height of epithelial cells and microvilli of intestine in the S group were obviously decreased than those in the F group ( P &lt; 0.05 ) after 14 d of starvation. The activities of immune-related enzymes including superoxide dismutase, catalase, and acid phosphatase in the hepatopancreas of the S group significantly decreased after 4 d of starvation, and lysozyme activity of the F group was lower than that of the S group after 7 d of starvation ( P &lt; 0.05 ). In addition, the results of Illumina high-throughput sequencing showed that a total of 14,285 OTUs were obtained and classified into 30 phyla, among which Actinobacteria and Proteobacteria were the predominant microflora in the intestinal microbial communities of both groups. However, the relative abundance of opportunistic pathogens was significantly increased in the S group, while the relative abundance of beneficial bacterium was decreased ( P &lt; 0.05 ). The bacterial richness with the Chao estimator was significantly higher in the F group than in the S group ( P &lt; 0.05 ). Both the results of principal coordinate analysis (PCoA) and nonmetric multidimensional scaling (NMDS) demonstrated that the intestinal microbiota of the F group was separate from those of the S group. The result of functional prediction of the metabolic pathways showed that the intestinal microbiota was enriched in the KEGG pathways of amino acid, carbohydrate, fatty acid, and lipid biosynthesis and degradation at level 2. This result implied that the microbiota of shrimp gut decreased nutrition metabolism under the stress of starvation. Meanwhile, comparing to the F group, the immune-related pathway of enterobacterial common antigen biosynthesis was markedly reduced in the S group ( P &lt; 0.05 ). The result of redundancy analysis (RDA) further confirmed that the activities of digestive and immune enzymes were correlated with the microbial community structure. Finally, the structural equation modeling highlighted that changes in the activities of digestive and immune enzymes were directly related to the gut bacterial community and notably affected prawn growth.

Determining macrophyte species richness and dark diversity sources – A novel approach to improve the biodiversity estimation based on species traits

Biodiversity measures deliver valuable ecological information by reflecting a range of ecosystem processes. However, the accuracy of environmental assessment based on species patterns may often be affected by insufficient survey details. The comprehensive evaluation of plant taxa richness in rivers requires an extensive sampling effort. The use of Hill numbers and Chao estimators improves species diversity assessment based on a feasible number of samples. The aim of this work was to identify macrophyte groups, associated with various species traits, which are rich in species, as well as those whose detection is particularly difficult as it requires an exceptional sampling effort (sources of dark diversity). Analyses were performed with the use of Hill numbers and Chao estimators. It was shown that the field identification of all estimated macrophytes is particularly difficult for low trophy indicators and generally submerged plants, as well as for small-leaved species. A field survey encompassing the full (expected) macrophyte diversity encountered within a river is easiest to perform in the case of free-floating plants and large-leaved macrophytes, as well as for species with high trophic tolerance. The study proved that ecological assessment of rivers based on a small number of sampling units may lead to incorrect diversity estimates. Conversely, the estimation of diversity patterns at the level of the Shannon and Simpson indices does not require extensive sampling, and the extrapolation approach is not needed. The effectiveness of diversity assessment in fluvial ecosystems can be increased by extrapolation of gray diversity which can be considered in planning of monitoring programs. Moreover even estimated dark diversity bight be already efficient to identify ecological pattern and when comparing biodiversity across regions and ecosystems.

Uniform Long-Time and Propagation of Chaos Estimates for Mean Field Kinetic Particles in Non-convex Landscapes

Combining the results of [14] and [10], the trend to equilibrium in large time is studied for a large particle system associated to a Vlasov-Fokker-Planck equation. Under some conditions (that allow non-convex confining potentials) the convergence rate is proven to be independent from the number of particles. From this are derived uniform in time propagation of chaos estimates and an exponentially fast convergence for the nonlinear equation itself.

Plant growth-promoting abilities and community structure of culturable endophytic bacteria from the fruit of an invasive plant Xanthium italicum.

Diversity of endophytic bacterial communities of capsular fruit, upper and lower seeds of an invasive plant Xanthium italicum growing in Xinjiang, China, was investigated. All isolates from the seed capsules, the upper seeds, and the lower seeds were identified by 16S rRNA gene sequencing, and sequences were compared to bacterial databases to define operational taxonomic units (OTUs). Finally, we obtained 316 endophytic isolates corresponding to 58 OTUs based on 16S rRNA gene sequences. The most common OTU corresponded most closely to Bacillus zhangzhouensis and comprised 9.49% of all bacterial isolates. The richness and diversity of endophytes in lower seeds were higher than that of the upper seeds; moreover, the Chao estimator and Shannon index of endophytes in the lower seeds were approximate to that in the seed capsules. Bacillus and Staphylococcus were found as the common taxa in three different tissues that were investigated (OTUs belong to these genera constituted > 70% of the total community). The bacterial endophytic communities differed significantly among these three fruit tissues, especially Bacillus strains, which have been reported to contribute to plant growth promotion and stress resilience to their hosts in harsh environment; abundance of Bacillus species was in the following order: capsules (78 OTUs) > lower seeds (55 OTUs) > upper seeds (40 OTUs). The lower seeds harboring more Bacillus species might be responsible for their earlier seed germination compared with the upper seeds.

Monotonicity preservation properties of kernel regression estimators

Three common classes of kernel regression estimators are considered: the Nadaraya–Watson (NW) estimator, the Priestley–Chao (PC) estimator, and the Gasser–Müller (GM) estimator. It is shown that (i) the GM estimator has a certain monotonicity preservation property for any kernel K, (ii) the NW estimator has this property if and only the kernel K is log concave, and (iii) the PC estimator does not have this property for any kernel K. Other related properties of these regression estimators are discussed.

A correction to make Chao estimator conservative when the number of sampling occasions is finite

Chao estimator is not guaranteed to be asymptotically conservative with finite sampling efforts. A simple correction solves the issue. We illustrate with simulations and examples that the corrected Chao estimator is asymptotically conservative, and has lower standard error.

Estimating the number of farms experienced foot and mouth disease outbreaks using capture-recapture methods

Foot and mouth disease (FMD) is recognized as an endemic disease in Thailand and throughout other countries in Southeast Asia. The underreporting of FMD outbreaks has affected the true status of the disease. This study aimed to determine the number of dairy farms in Chiang Mai that had experienced FMD outbreaks (FMD outbreak farm) during 2015-2016 using capture-recapture (CR) methods. Two independent FMD outbreak data sources including data from the livestock authorities and survey questionnaires were analyzed using Chapman estimator and Chao estimator. Results showed that the estimated number of FMD outbreak farms was 264 (95% CI = 250, 277) and 273 (95% CI = 259, 292) farms based on the Chapman estimator and Chao estimator, respectively. The estimated prevalence of FMD corresponding to the Chapman estimator was lower than the Chao estimator. The active approach of the survey method offered a higher degree of sensitivity compared to the passive method used by the livestock authorities. Estimations from the CR method provided an upper bound for the true number of outbreak farms. This study demonstrated the use of the CR method to estimate the true status of FMD outbreaks. Our proposed approach can potentially be used as a tool to enhance the accuracy and sensitivity of established monitoring and surveillance systems.

Characteristics of microbial community of soil subjected to industrial production of antibiotics.

Ecosystems worldwide are exposed to pollutants connected to the industrial production of pharmaceuticals. The objective of this study was to study the composition and characteristics of the soil microbial communities that had been exposed to long-term selection pressure caused by the industrial production of penicillin G. Soil samples from four sites among the penicillin G production plant were analysed using 16S rRNA profiling via Illumina MiSeq platform and were compared with the control samples from four sites outside the plant. Total metagenomic DNA from the impacted soil was also used for the preparation of E. coli T1R-based fosmid library which was consequently qualitatively tested for the presence of penicillin G acylase (PGA)-encoding genes using the method of sequence homology. Analyses of alpha diversity revealed that the long-term antibiotic presence in the soil significantly increased the microbial diversity and richness in terms of Shannon diversity index (p = 0.002) and Chao estimates (p = 0.004). Principal component analysis showed that the two types of communities (on-site and control) could be separated at the phylum, class and genus level. The on-site soil was enriched in Betaproteobacteria, Deltaproteobacteria, Gemmatimonadetes, Acidobacteria and Planctomycetia, while a significant decrease in Actinobacteria was observed. Metagenomic fosmid library revealed high hit rates in identifying PGAs (14 different genes identified) and confirmed the biotechnological potential of soils impacted by anthropogenic activity. This study offers new insights into the changes in microbial communities of soils exposed to anthropogenic activity as well as indicates that those soils may represent a hotspot for biotechnologically interesting targets.

Information content in pollination network reveals missing interactions

Network analysis is an indispensable part of ecological studies. Specifically, networks have played a pivotal role in studying the diversity, dynamics and functionality of pollination systems. Recording plant-pollinator interaction networks is a laborious task, prone to missing or false negative interactions. Several methods enable the assessment of sampling completeness of the network with the use of species accumulation curves or Chao estimators. However, these methods do not provide a way to identify which interactions might be missed in the field. Methods that enable a more directed and focused field sampling are needed. Such methods would greatly benefit plant-pollinator studies and network studies in general.We additively decomposed the information of an interaction matrix into three parts: the difference in importance of the species, the specificity of the interactions, and the generality of the interactions. This information is exploited by a previously proposed linear filtering method to re-score absent interactions, thus pinpointing the likely missing interactions. We evaluated this approach using null models, intensive cross-validation, as well as external validation with the Web of Life database.By means of a case study, we provide insight into the structure of the network using an information-theoretic approach. We show how to use linear filtering to suggest missing interactions. A thorough evaluation shows that these results are both statistically stable and useful to guide the search for missing interactions in real-world networks.The non-uniformity of pollination interactions can be quantified using information theory and extracted using linear filtering. Our work can be valuable as a way to study different interaction networks as well as a tool to help identifying missing interactions.

Estimating the undetected infections in the Covid-19 outbreak by harnessing capture–recapture methods

ObjectivesA major open question, affecting the decisions of policy makers, is the estimation of the true number of Covid-19 infections. Most of them are undetected, because of a large number of asymptomatic cases. We provide an efficient, easy to compute and robust lower bound estimator for the number of undetected cases. MethodsA modified version of the Chao estimator is proposed, based on the cumulative time-series distributions of cases and deaths. Heterogeneity has been addressed by assuming a geometrical distribution underlying the data generation process. An (approximated) analytical variance of the estimator has been derived to compute reliable confidence intervals at 95% level. ResultsA motivating application to the Austrian situation is provided and compared with an independent and representative study on prevalence of Covid-19 infection. Our estimates match well with the results from the independent prevalence study, but the capture–recapture estimate has less uncertainty involved as it is based on a larger sample size. Results from other European countries are mentioned in the discussion. The estimated ratio of the total estimated cases to the observed cases is around the value of 2.3 for all the analyzed countries. ConclusionsThe proposed method answers to a fundamental open question: “How many undetected cases are going around?”. CR methods provide a straightforward solution to shed light on undetected cases, incorporating heterogeneity that may arise in the probability of being detected.

Structural and functional changes of gut microbiota in ovariectomized rats and their correlations with altered bone mass.

As a critical factor involved in the maintenance of physiological homeostasis, the gut microbiota (GM) reportedly plays a key role in bone development. To date, the association between the GM and steroid deficiency-induced osteoporosis remains poorly understood. Forty female Sprague Dawley rats were divided into an ovariectomy (OVX) or control group. We performed 16S rRNA and metagenome sequencing, to compare diversity, taxonomic differences, and functional genes. The GM composition did not change in the control group and the number of operational taxonomic units increased significantly following ovariectomy. Alpha diversity, determined by ACE estimator, CHAO estimator, the Shannon index, and the Simpson index showed an increasing trend after ovariectomy. Samples in the OVX group were well clustered both pre- and post-ovariectomy, as demonstrated by principal coordinate 1 (PC1) and PC2. Functional genes of GM, including those involved in synthesis and metabolism of carbohydrates and nucleotides, microbial structure, and heme, as well as hemin uptake and utilization, increased at the early stage of osteoporosis. We observed that Ruminococcus flavefaciens exhibited the greatest variation in abundance among the GM and this was also associated with osteoclastic indicators and the estrobolome. Specific changes in fecal microbiota are associated with the pathogenesis of steroid deficiency-induced osteoporosis.