Finite Mixture Distribution Model Research Articles

Pollution threshold assessment and risk area delineation of heavy metals in soils through the finite mixture distribution model and Bayesian maximum entropy theory

Pollution threshold and high-risk area determination for heavy metals is important for effectively developing pollution control strategies. Based on heavy metal contents in 3627 dense samples, an integrated framework combining the finite mixture distribution model and Bayesian maximum entropy (BME) theory was proposed to assess pollution thresholds, contamination levels and risk areas in an uncertain environment for soil heavy metals. The results showed that the average heavy metal contents were in the order Zn > Cr > Pb > Cu > Ni > As > Cd > Hg, with strong/moderate variation, and the corresponding pollution thresholds were 158.39, 84.29, 47.84, 49.75, 28.95, 18.01, 0.49 and 0.16 mg/kg, respectively. The thresholds were consistently greater than the Zhejiang Province backgrounds but lower than the national risk screening values, except for Cd. Approximately 27.9% of the samples were classified as contaminated at various levels, and they were distributed in the northern, northwestern and eastern regions of the study area. Additionally, 3.73%, 5.34% and 8.22% of the total area were classified as at-risk areas under confidence levels of 95%, 90% and 75%, respectively, through BME theory. The findings provide a reasonable classification system and suggestions for heavy metal pollution management and control.

Distribution of some potentially toxic elements in the soils of the Jharia Coalfield: A probabilistic approach for source identification and risk assessment

AbstractPotentially toxic elements (PTEs) in coal mining soil degrade soil quality and adversely affect people's health. The increasing mining activities in the Jharia Coalfield (JCF) may increase the concentration of PTEs in soils. Therefore, it is vital to have an in‐depth analysis of land degradation caused by them, but there is a lack of accurate and efficient methods to estimate land degradation and health risk analysis in JCF. Therefore, finite mixture distribution model (FMDM), positive matrix factorization (PMF), and Monte Carlo simulation were applied to Jharia's PTEs dataset. The FMDM was employed to distinguish the background concentration and the contamination due to anthropogenic activity. In addition, the contribution of the source was determined through PMF. The FMDM suggested two possible sources of PTEs, while the PMF defines the influence of PTEs more broadly and identifies three sources of PTEs in soil. The results showed that PTEs contribution by a natural source, coal mining, and atmospheric deposition accounted for 45.72%, 27.34%, and 26.94%, respectively. Based on health risk results, this region does not pose a noncarcinogenic risk, but it does pose a carcinogenic risk, especially to infants. JCF comprises a relatively low concentration of PTEs as compared with other possible values for India. Probabilistic health risk assessment studies are scarce for Indian soils, especially in mining areas. In combination, these approaches will provide deeper insights and help identify potential solutions for PTEs contamination in these areas.

A combination of finite mixture distribution model with geo-statistical models to study spatial patterns and hazardous areas of heavy metals in cropland soils of the Guanzhong Plain, Northwest China

An extensive cropland soil investigation was conducted to determine the pollution thresholds and hazardous zones of heavy metals (HMs) in the Guanzhong Plain, by using an integrated approach that combines finite mixture distribution model (FMDM) and geo-statistical analysis. FMDM results demonstrated that Pb, Cr, Ni, and Cu were fitted by binary mixture distributions representing the background and moderate pollution distributions, and Zn was fitted by a triple mixture distribution representing the background, moderate and high contamination distributions. The moderate pollution thresholds of Pb, Cr, Ni, Zn and Cu calculated by FMDM were 29.75, 80.15, 38.60, 81.48 and 27.10 mg kg−1, whereas the cutoff value of Zn high contamination was 97.49 mg kg−1. The moderately polluted thresholds of all five HMs were higher than their background values in the study area, and lower than the corresponding national standards. The indicator kriging simulation showed Pb, Cr, Ni, Zn had <0.1%, 2.6%, <0.1%, 2.9% of total areas exceed contamination cutoff values, whereas the hazardous area of Cu was contiguous, and covered 17.3% of the total area. Overall, 17.5% of the total area surpassed the moderate contamination threshold. The pollution hot spots and hazardous zones of soil HMs were located in the southern part of the Guanzhong Plain, where population and industrial activities are centralized, indicating that anthropogenic activities played a critical role in HMs accumulation in high-risk regions. The combination of geo-statistical and FMDM delineate the thresholds and hazardous area for HMs pollution reliably, and facilitate the improvement of soil environmental management.

Comprehensive source identification and apportionment analysis of five heavy metals in soils in Wenzhou City, China.

The source identification and apportionment of heavy metals (HMs) is a vital issue for restoring contaminated soil. In this study, qualitative approaches [a finite mixture distribution model (FMDM) and raster-based principal components analysis (RB-PCA)] and a quantitative approach [positive matrix factorization (PMF)] were composed to identify and apportion the sources of five HMs (Cd, Hg, As, Pb, Cr) in Wenzhou City, China, using several crucial auxiliary variables. An initial ecological risk assessment suggested that the ecological risk level in the study area was generally considered low, with the greatest contamination contributions coming from Cd and Hg. The result of the FMDM showed that Cd and Pb fit a single log-normal distribution, Hg fit a double log-normal mixed distribution, and As and Cr presented a triple log-normal distribution. Each element was identified and separated from its natural or anthropogenic sources. A map of RB-PCA combined with an analysis of corresponding auxiliary variables suggested that the three main contribution sources in the entire study area were parental materials, industrial and agricultural mixed pollution, and mining exploration activities. Each element was discussed, using the PMF model, with regard to its quantitative contributions. Parental materials contributed to all elements (Cd, Hg, As, Pb, Cr) at 89.22%, 7.31%, 35.84%, 84.81% and 27.42%, respectively. Industrial emissions and agricultural inputs mixed pollution contributed 2.94%, 80.77%, 15.93%, 4.79%, and 25.63%, respectively. Mining activities contributed 7.84%,11.92%, 48.23%, 10.40% and 46.95%, respectively, to the five HMs. Such result could be used efficiently to generate scientific decisions and strategies in terms of decision-making on regulating HM pollution in soils.

Extension of REBMIX algorithm to von Mises parametric family for modeling joint distribution of wind speed and direction

A statistical analysis of the wind speed and wind direction serves as a solid foundation for the wind-induced vibration analysis. The probabilistic modeling of wind speed and direction can effectively characterize the stochastic properties of wind field. The joint distribution model of wind speed and direction involves a circular distribution and has a multimodal characteristic. In this paper, the finite mixture distribution model is introduced and used to represent the joint distribution model that is comprised of the mixture Weibull distributions and von Mises distributions. An extended parameters estimation algorithm for multivariate and multimodal circular distributions is proposed to construct the joint distribution model. The proposed algorithm estimates the component parameters, mixture weight of each component and the number of components successively by an iterative process. The major improvement is accomplished by adding a circular distribution model. The effectiveness of the proposed algorithm is verified with numerical simulations and one-year field monitoring data and compared with the expectation maximization algorithm-based angular-linear approach in terms of the Akaike’s information criterion and computing time. The results indicate that the finite mixture model represents the joint distribution model of wind speed and direction well and that the proposed algorithm has a good and time-saving performance in parameter estimation for multivariate and multimodal models.

Source Identification and Apportionment of Trace Elements in Soils in the Yangtze River Delta, China.

Trace elements pollution has attracted a lot of attention worldwide. However, it is difficult to identify and apportion the sources of multiple element pollutants over large areas because of the considerable spatial complexity and variability in the distribution of trace elements in soil. In this study, we collected total of 2051 topsoil (0–20 cm) samples, and analyzed the general pollution status of soils from the Yangtze River Delta, Southeast China. We applied principal component analysis (PCA), a finite mixture distribution model (FMDM), and geostatistical tools to identify and quantitatively apportion the sources of seven kinds of trace elements (chromium (Cr), cadmium (Cd), mercury (Hg), copper (Cu), zinc (Zn), nickel (Ni), and arsenic (As)) in soil. The PCA results indicated that the trace elements in soil in the study area were mainly from natural, multi-pollutant and industrial sources. The FMDM also fitted three sub log-normal distributions. The results from the two models were quite similar: Cr, As, and Ni were mainly from natural sources caused by parent material weathering; Cd, Cu, and Zu were mainly from mixed sources, with a considerable portion from anthropogenic activities such as traffic pollutants, domestic garbage, and agricultural inputs, and Hg was mainly from industrial wastes and pollutants.

A novel approach to compare pinniped populations across a broad geographic range

We utilized aerial images and employed photogrammetric methodologies to collect standardized lengths of Steller sea lions (Eumetopias jubatus) terrestrially hauled out. We conducted comparisons among all site types and separately for rookery and haulout site-types between the two distinct population segments (DPSs; eastern and western) and two broad regions within the western DPS experiencing contrasting population abundance trends. An observed adult female index was created from measurements of reproductive females — in the presence of a pup or juvenile — and was applied as a model constraint for “adult females”. We fitted a finite mixture distribution model to the length-frequency data to estimate the proportion population for three delineated age–sex classes (juveniles, adult females, and adult males) and mean length for juveniles and adult males. Estimated proportions reflected what we expected; however, the broad region within the western DPS exhibiting substantial population declines had greater proportion of all age–sex classes on rookery sites than increasing broad region. Adult sea lions were significantly shorter in the eastern DPS than the western area, providing further evidence of morphological differences between the DPSs. We also introduce a less resource-demanding method for estimating population demographics, and potentially vital rates, for pinnipeds across a vast geographic range.

Regime-dependent Characteristics of KOSPI Return

Stylized facts on asset return are fat-tail, asymmetry, volatility clustering and structure changes. This paper simultaneously captures these characteristics by introducing a multi-regime models: Finite mixture distribution and regime switching GARCH model. Analyzing the daily KOSPI return from <TEX>$4^{th}$</TEX> January 2000 to <TEX>$30^{th}$</TEX> June 2014, we find that a two-component mixture of t distribution is a good candidate to describe the shape of the KOSPI return from unconditional and conditional perspectives. Empirical results suggest that the equality assumption on the shape parameter of t distribution yields better discrimination of heterogeneity component in return data. We report the strong regime-dependent characteristics in volatility dynamics with high persistence and asymmetry by employing a regime switching GJR-GARCH model with t innovation model. Compared to two sub-samples, Pre-Crisis (January 2003 ~ December 2007) and Post-Crisis (January 2010 ~ June 2014), we find that the degree of persistence in the Pre-Crisis is higher than in the Post-Crisis along with a strong asymmetry in the low-volatility (high-volatility) regime during the Pre-Crisis (Post-Crisis).

Applications of stochastic models and geostatistical analyses to study sources and spatial patterns of soil heavy metals in a metalliferous industrial district of China

An extensive soil survey was conducted to study pollution sources and delineate contamination of heavy metals in one of the metalliferous industrial bases, in the karst areas of southwest China. A total of 597 topsoil samples were collected and the concentrations of five heavy metals, namely Cd, As (metalloid), Pb, Hg and Cr were analyzed. Stochastic models including a conditional inference tree (CIT) and a finite mixture distribution model (FMDM) were applied to identify the sources and partition the contribution from natural and anthropogenic sources for heavy metal in topsoils of the study area. Regression trees for Cd, As, Pb and Hg were proved to depend mostly on indicators of anthropogenic activities such as industrial type and distance from urban area, while the regression tree for Cr was found to be mainly influenced by the geogenic characteristics. The FMDM analysis showed that the geometric means of modeled background values for Cd, As, Pb, Hg and Cr were close to their background values previously reported in the study area, while the contamination of Cd and Hg were widespread in the study area, imposing potentially detrimental effects on organisms through the food chain. Finally, the probabilities of single and multiple heavy metals exceeding the threshold values derived from the FMDM were estimated using indicator kriging (IK) and multivariate indicator kriging (MVIK). The high probabilities exceeding the thresholds of heavy metals were associated with metalliferous production and atmospheric deposition of heavy metals transported from the urban and industrial areas. Geostatistics coupled with stochastic models provide an effective way to delineate multiple heavy metal pollution to facilitate improved environmental management.

Application of Stochastic Models in Identification and Apportionment of Heavy Metal Pollution Sources in the Surface Soils of a Large-Scale Region

As heavy metals occur naturally in soils at measurable concentrations and their natural background contents have significant spatial variations, identification and apportionment of heavy metal pollution sources across large-scale regions is a challenging task. Stochastic models, including the recently developed conditional inference tree (CIT) and the finite mixture distribution model (FMDM), were applied to identify the sources of heavy metals found in the surface soils of the Pearl River Delta, China, and to apportion the contributions from natural background and human activities. Regression trees were successfully developed for the concentrations of Cd, Cu, Zn, Pb, Cr, Ni, As, and Hg in 227 soil samples from a region of over 7.2 × 10(4) km(2) based on seven specific predictors relevant to the source and behavior of heavy metals: land use, soil type, soil organic carbon content, population density, gross domestic product per capita, and the lengths and classes of the roads surrounding the sampling sites. The CIT and FMDM results consistently indicate that Cd, Zn, Cu, Pb, and Cr in the surface soils of the PRD were contributed largely by anthropogenic sources, whereas As, Ni, and Hg in the surface soils mostly originated from the soil parent materials.

Identifying spatial mixture distributions of PM2.5 and PM10 in Taiwan during and after a dust storm

Abstract Atmospheric particulate matter (PM) exhibits a high correlation with public health. The mixture distributions of PM in Taiwan derive mainly from anthropogenic (industrial) and natural (rural) air pollutants. This study determines the mixture distribution of monthly PM2.5 and PM10 data of a dust storm occurred in Taiwan in March 2008 using a finite mixture distribution model (FMDM). The probabilities of a contaminated area are mapped using sequential Gaussian simulation (SGS) with the FMDM cut-off values of PM2.5 and PM10. Results show that PM2.5 and PM10 can be individually fitted using the FMDM. Using the SGS with FMDM cut-off values can delineate high and low concentration areas. Spatial patterns of PM2.5 and PM10 concentrations show seasonal variations caused by dust storms and meteorological conditions. The difference of the mixture distribution of high and low PM concentrations during the dust storm is significant. The area with high PM concentration is delineated in the industrial and urban areas of Taiwan (southwestern, west central, and northern Taiwan). This information can define the boundaries of the PM hotspot areas to control pollutants and reduce public exposure.

A Twin Association Study of Nicotine Dependence with Markers in the CHRNA3 and CHRNA5 Genes

Twin and family studies have provided overwhelming evidence for the genetic basis of individual differences in tobacco initiation (TI), regular smoking (RS) and nicotine dependence (ND). However, only a few genes have been reliably associated with ND. We used a finite mixture distribution model to examine the significance and effect size of the association of previously identified and replicated specific variants in the CHRNA5 and CHRNA3 receptor genes with ND, against the background of genetic and environmental risk factors for ND. We hypothesize that additional phenotypic information in relatives who have not been genotyped can be used to increase the power of detecting the genetic variant. The nicotine measures were assessed by personal interview in female, male and opposite sex twin pairs (N=4,153) from the population-based Virginia Twin Registry. Three SNPs in the CHRNA5 and CHRNA3 receptor genes, previously shown to be significantly associated with ND in this sample, were replicated in the augmented analyses; they accounted for less than one percent of the genetic variance in liability to ND, which is estimated to be over 50% of the phenotypic variance. The significance of these effects was increased by adding twins with phenotype but without genotype data, but gains are limited and variable. The SNPs associated with ND did not show a significant association with either TI or RS and appear to be specific to the addictive stage of ND, characterized by current smoking and smoking a large amount of cigarettes per day. Furthermore, these SNPs did not appear to be associated with the remaining items comprising the FTND scale. This study confirmed a significant contribution of the CHRNA receptor on different forms of tobacco dependence. However, the genetic variant only accounted for little of the total genetic variance for liability to ND. Including phenotypic data on ungenotyped relatives can improve the statistical power to detect the effects of genetic variants when they contribute to individual differences in the phenotype.

Identifying the spatial mixture distribution of bird diversity across urban and suburban areas in the metropolis: A case study in Taipei Basin of Taiwan

This study utilized a finite mixture distribution model (FMDM) to identify the distribution of bird diversity in urban and suburban areas of the Taipei basin. The spatial patterns of Shannon's diversity index were estimated using Sequential Gaussian simulation (SGS) with bird investigation data. To validate the results of FMDM, bird groups were classified by two-way indicator species analysis (TWINSPAN) based on bird presence-absence records. The probabilities exceeding FMDM's cut-off values for suburban bird communities were obtained from SGS realizations. The results showed that bird diversity in the Taipei basin could be comprised of two-component mixture distributions for urban and suburban bird communities, respectively. The FMDM bird diversity classifications fit the TWINSPAN groups of urban and suburban species. The spatial maps of bird diversity in the study area provide evidence that a metropolis can cause changes in the spatial characteristics of bird communities at the landscape level. Furthermore, the bird diversity and land use maps could be useful for future conservation and urban planning in Taipei Basin, particularly urban green spaces and wetlands. The maps could also be utilized in network planning of green belts or corridors for ecological conservation.

The use of a finite mixture distribution model in bottled water characterisation and authentication with stable hydrogen, oxygen and carbon isotopes — Case study from Slovenia

This study investigates the comparison of the stable isotopic composition of oxygen (δ 18O), hydrogen (δ 2H) and carbon in dissolved inorganic carbon (δ 13C DIC) in domestic and foreign bottled waters randomly collected from the Slovene market. Empirical distributions of δ 2H, δ 18O and δ 13C DIC were examined. Mathematical models of unimodal and polymodal probability density functions were applied for their description. A normal (Gaussian) finite mixture model as a representation of the polymodal empirical distribution of δ 13C DIC was used as a classification tool. Finally, based on the classification, descriptive statistics of δ 2H, δ 18O and δ 13C DIC for different groups are calculated. Components of the normal finite mixture model of δ 13C DIC represent four groups of bottled water; a group of artificial sparkling bottled water, two groups of still and flavoured bottled water and a group of bottled water originating from deeper aquifers. Corresponding values of δ 2H and δ 18O show that only the group of bottled water originating from deeper aquifers has a distinct stable isotopic composition, where the others have similar empirical distributions. The proposed model showed that δ 13C DIC for bottled water can be used as a strong discrimination parameter for the classification of bottled water and it can be applied as an authentication tool.

Combining a finite mixture distribution model with indicator kriging to delineate and map the spatial patterns of soil heavy metal pollution in Chunghua County, central Taiwan

This study identifies the natural background, anthropogenic background and distribution of contamination caused by heavy metal pollutants in soil in Chunghua County of central Taiwan by using a finite mixture distribution model (FMDM). The probabilities of contaminated area distribution are mapped using single-variable indicator kriging and multiple-variable indicator kriging (MVIK) with the FMDM cut-off values and regulation thresholds for heavy metals. FMDM results indicate that Cr, Cu, Ni and Zn can be individually fitted by a mixture model representing the background and contamination distributions of the four metals in soil. The FMDM cut-off values for contamination caused by the metals are close to the regulation thresholds, except for the cut-off value of Zn. The receiver operating characteristic (ROC) curve validates that indicator kriging and MVIK with FMDM cut-off values can reliably delineate heavy metals contamination, particularly for areas lacking background information and high heavy metal concentrations in soil.

Fiber Length Distribution in Cotton Processing: A Finite Mixture Distribution Model

This paper introduces a new approach to modeling and parameterizing cotton fiber length distribution. The approach uses finite mixture models to derive a parametric expression of the fiber length probability density function. The model was applied to a multitude of empirical length distributions and proved to adequately parameterize the complex distribution patterns, as well as express the intrinsic and process-related factors determining their shape.

Precision and Bias of a Normal Finite Mixture Distribution Model to Analyze Twin Data When Zygosity is Unknown: Simulations and Application to IQ Phenotypes on a Large Sample of Twin Pairs

The classification of twin pairs based on zygosity into monozygotic (MZ) or dizygotic (DZ) twins is the basis of most twin analyses. When zygosity information is unavailable, a normal finite mixture distribution (mixture distribution) model can be used to estimate components of variation for continuous traits. The main assumption of this model is that the observed phenotypes on a twin pair are bivariately normally distributed. Any deviation from normality, in particular kurtosis, could produce biased estimates. Using computer simulations and analyses of a wide range of phenotypes from the U.K. Twins' Early Developments Study (TEDS), where zygosity is known, properties of the mixture distribution model were assessed. Simulation results showed that, if normality assumptions were satisfied and the sample size was large (e.g., 2,000 pairs), then the variance component estimates from the mixture distribution model were unbiased and the standard deviation of the difference between heritability estimates from known and unknown zygosity in the range of 0.02-0.20. Unexpectedly, the estimates of heritability of 10 variables from TEDS using the mixture distribution model were consistently larger than those from the conventional (known zygosity) model. This discrepancy was due to violation of the bivariate normality assumption. A leptokurtic distribution of pair difference was observed for all traits (except non-verbal ability scores of MZ twins), even when the univariate distribution of the trait was close to normality. From an independent sample of Australian twins, the heritability estimates for IQ variables were also larger for the mixture distribution model in six out of eight traits, consistent with the observed kurtosis of pair difference. While the known zygosity model is quite robust to the violation of the bivariate normality assumption, this novel finding of widespread kurtosis of the pair difference may suggest that this assumption for analysis of quantitative trait in twin studies may be incorrect and needs revisiting. A possible explanation of widespread kurtosis within zygosity groups is heterogeneity of variance, which could be caused by genetic or environmental factors. For the mixture distribution model, violation of the bivariate normality assumption will produce biased estimates.

A Finite Mixture Distribution Model for Data Collected from Twins

Most analyses of data collected from a classical twin study of monozygotic (MZ) and dizygotic (DZ) twins assume that zygosity has been diagnosed without error. However, large scale surveys frequently resort to questionnaire-based methods of diagnosis which classify twins as MZ or DZ with less than perfect accuracy. This article describes a mixture distribution approach to the analysis of twin data when zygosity is not perfectly diagnosed. Estimates of diagnostic accuracy are used to weight the likelihood of the data according to the probability that any given pair is either MZ or DZ. The performance of this method is compared to fully accurate diagnosis, and to the analysis of samples that include some misclassified pairs. Conventional analysis of samples containing misclassified pairs yields biased estimates of variance components, such that additive genetic variance (A) is underestimated while common environment (C) and specific environment (E) components are overestimated. The bias is non-trivial; for 10% misclassification, true values of Additive genetic: Common environment: Specific Environment variance components of .6: .2: .2 are estimated as .48: .29: .23, respectively. The mixture distribution yields unbiased estimates, while showing relatively little loss of statistical precision for misclassification rates of 15% or less. The method is shown to perform quite well even when no information on zygosity is available, and may be applied when pair-specific estimates of zygosity probabilities are available.

A finite mixture distribution model for data collected from twins.

Most analyses of data collected from a classical twin study of monozygotic (MZ) and dizygotic (DZ) twins assume that zygosity has been diagnosed without error. However, large scale surveys frequently resort to questionnaire-based methods of diagnosis which classify twins as MZ or DZ with less than perfect accuracy. This article describes a mixture distribution approach to the analysis of twin data when zygosity is not perfectly diagnosed. Estimates of diagnostic accuracy are used to weight the likelihood of the data according to the probability that any given pair is either MZ or DZ. The performance of this method is compared to fully accurate diagnosis, and to the analysis of samples that include some misclassified pairs. Conventional analysis of samples containing misclassified pairs yields biased estimates of variance components, such that additive genetic variance (A) is underestimated while common environment (C) and specific environment (E) components are overestimated. The bias is non-trivial; for 10% misclassification, true values of Additive genetic: Common environment: Specific Environment variance components of.6:.2:.2 are estimated as.48:.29:.23, respectively. The mixture distribution yields unbiased estimates, while showing relatively little loss of statistical precision for misclassification rates of 15% or less. The method is shown to perform quite well even when no information on zygosity is available, and may be applied when pair-specific estimates of zygosity probabilities are available.