Comparison between Hansen-Hurwitz and Horvitz-Thompson estimators for adaptive cluster sampling

Thompson (1990) introduced the adaptive cluster sampling design. This sampling design has been shown to be a useful sampling method for parameter estimation of a clustered and scattered population (Roesch, 1993; Smith et al., 1995; Thompson and Seber, 1996). Two estimators, the modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators, are available to estimate the mean or total of a population. Empirical results from previous researches indicate that the modified HT estimator has smaller variance than the modified HH estimator. We analytically compare the properties of these two estimators. Some results are obtained in favor of the modified HT estimator so that practitioners are strongly recommended to use the HT estimator despite easiness of computations for the HH estimator.     

Adjusted two-stage adaptive cluster sampling

An adjusted two-stage sampling procedure is discussed for adaptive cluster sampling where some networks are large and others are small. A two-stage sample is drawn from the large networks and a single-stage sample is drawn from the rest. The simple random sampling (SRS) procedure without replacement is used at the initial stage. An estimator for the population mean along with its properties is discussed.     

Markov Chain Monte Carlo on optimal adaptive sampling selections

Under a Bayesian population model with a given prior distribution, the optimal sampling strategy with a fixed sample size n is an n-phase adaptive one. That is, the selection of the next sampling units should sequentially depend on the information obtained from the previously selected units, including the observed values of interest. Such an optimal strategy is in general not executable in practice due to its intensive computation. In many survey sampling situations, an important problem is that one would like to select a set of units in addition to a certain number of sampling units which have been observed. If the optimal strategy is an adaptive one, the selection of the additional units should take both the labels and the observed values of the already selected units into account. Hence, a simpler optimal two-phase adaptive sampling strategy under a Bayesian population model is proposed in this article for practical interest. A Markov chain Monte Carlo method is used to approximate the posterior joint distribution of the unobserved population units after the first phase sampling, for the optimal selection of the second phase sample. This approximation method is found to be successful to select the optimal second-phase sample. Finally, this optimal strategy is applied to a set of data from a study of geothermal CO₂ emissions in Yellowstone National Park as a practical illustrative example.     

Performance of adaptive cluster sampling estimators in a multivariate setting

Adaptive cluster sampling has been proven to perform well in a univariate setting, but it may not perform well when there are several parameters of interest. The efficiency of adaptive sampling when there are several variables of interest depends on the relationship of the variables with one another.     

Estimator bias and efficiency for adaptive cluster sampling with order statistics and a stopping rule

Practical problems facing adaptive cluster sampling with order statistics (acsord) are explored using Monte Carlo simulation for three simulated fish populations and two known waterfowl populations. First, properties of an unbiased Hansen-Hurwitz (HH) estimator and a biased alternative Horvitz-Thompson (HT) estimator are evaluated. An increase in the level of population aggregation or the initial sample size increases the efficiencies of the two acsord estimators. For less aggregated fish populations, the efficiencies decrease as the order statistic parameter r (the number of units about which adaptive sampling is carried out) increases; for the highly aggregated fish and waterfowl populations, they increase with r. Acsord is almost always more efficient than simple random sampling for the highly aggregated populations. Positive bias is observed for the HT estimator, with the maximum bias usually occurring at small values of r. Secondly, a stopping rule at the Sth iteration of adaptive sampling beyond the initial sampling unit was applied to the acsord design to limit the otherwise open-ended sampling effort. The stopping rule induces relatively high positive bias to the HH estimator if the level of the population aggregation is high, the stopping level S is small, and r is large. The bias of HT is not very sensitive to the stopping rule and its bias is often reduced by the stopping rule at smaller values of r. For more aggregated populations, the stopping rule often reduces the efficiencies of the estimators compared to the non-stopping-rule scheme, but acsord still remains more efficient than simple random sampling. Despite its bias and lack of theoretical grounding, the HT estimator is usually more efficient than the HH estimator. In the stopping rule case, the HT estimator is preferable, because its bias is less sensitive to the stopping level.     

Rao-Blackwell versions of the Horvitz-Thompson and Hansen-Hurwitz in adaptive cluster sampling

Thompson (1990) introduced the adaptive cluster sampling design and developed two unbiased estimators, the modified Horvitz-Thompson (HT) and Hansen-Hurwitz (HH) estimators, for this sampling design and noticed that these estimators are not a function of the minimal sufficient statistics. He applied the Rao-Blackwell theorem to improve them. Despite having smaller variances, these latter estimators have not received attention because a suitable method or algorithm for computing them was not available. In this paper we obtain closed forms of the Rao-Blackwell versions which can easily be computed. We also show that the variance reduction for the HH estimator is greater than that for the HT estimator using Rao-Blackwell versions. When the condition for extra samples is 0$$ " align="middle" border="0"> , one can expect some Rao-Blackwell improvement in the HH estimator but not in the HT estimator. Two examples are given.     

Application of adaptive cluster sampling to low-density populations of freshwater mussels

Freshwater mussels appear to be promising candidates for adaptive cluster sampling because they are benthic macroinvertebrates that cluster spatially and are frequently found at low densities. We applied adaptive cluster sampling to estimate density of freshwater mussels at 24 sites along the Cacapon River, WV, where a preliminary timed search indicated that mussels were present at low density. Adaptive cluster sampling increased yield of individual mussels and detection of uncommon species; however, it did not improve precision of density estimates. Because finding uncommon species, collecting individuals of those species, and estimating their densities are important conservation activities, additional research is warranted on application of adaptive cluster sampling to freshwater mussels. However, at this time we do not recommend routine application of adaptive cluster sampling to freshwater mussel populations. The ultimate, and currently unanswered, question is how to tell when adaptive cluster sampling should be used, i.e., when is a population sufficiently rare and clustered for adaptive cluster sampling to be efficient and practical? A cost-effective procedure needs to be developed to identify biological populations for which adaptive cluster sampling is appropriate.     

Cost analysis of composite sampling for classification

When an environmental sampling objective is to classify all the sample units as contaminated or not, composite sampling with selective retesting can substantially reduce costs by reducing the number of units that require direct analysis. The tradeoff, however, is increased complexity that has its own hidden costs. For this reason, we propose a model for assessing the relative cost, expressed as the ratio of total expected cost with compositing to total expected cost without compositing (initial exhaustive testing). Expressions are derived for the following retesting protocols: (i) exhaustive, (ii) sequential and (iii) binary split. The effects of both false positive and false negative rates are also derived and incorporated. The derived expressions of relative cost are illustrated for a range of values for various cost components that reflect typical costs incurred with hazardous waste site monitoring. Results allow those who are designing sampling plans to evaluate if any of these compositing/retesting protocols will be cost effective for particular applications.     

Stack sampling for organic emissions

Along with increased activity in source sampling for organics, there have been many improvements in the methods of acquiring samples. Much has been learned about how best to proceed, and a number of potentially serious pitfalls have been discovered, characterized, and circumvented. Unfortunately, communication of all of this new technology has not always been effective.

This paper reviews some of the more important fundamental principles involved in stack sampling for organics, briefly describes and discusses recently developed equipment, and points out a few of the more serious pitfalls to be avoided. Extensive references are provided, many of which are often overlooked by newcomers to the field. The conclusion is reached that it is possible to consistently obtain high‐quality samples of organic materials from stationary source stacks, even though knowledge and caution are necessary.     

Review on remediation technologies for arsenic-contaminated soil

• Recent progress of As-contaminated soil remediation technologies is presented. • Phytoextraction and chemical immobilization are the most widely used methods. • Novel remediation technologies for As-contaminated soil are still urgently needed. • Methods for evaluating soil remediation efficiency are lacking. • Future research directions for As-contaminated soil remediation are proposed. Arsenic (As) is a top human carcinogen widely distributed in the environment. As-contaminated soil exists worldwide and poses a threat on human health through water/food consumption, inhalation, or skin contact. More than 200 million people are exposed to excessive As concentration through direct or indirect exposure to contaminated soil. Therefore, affordable and efficient technologies that control risks caused by excess As in soil must be developed. The presently available methods can be classified as chemical, physical, and biological. Combined utilization of multiple technologies is also common to improve remediation efficiency. This review presents the research progress on different remediation technologies for As-contaminated soil. For chemical methods, common soil washing or immobilization agents were summarized. Physical technologies were mainly discussed from the field scale. Phytoextraction, the most widely used technology for As-contaminated soil in China, was the main focus for bioremediation. Method development for evaluating soil remediation efficiency was also summarized. Further research directions were proposed based on literature analysis.     

Variance estimation in the EMAP strategy for sampling discrete ecological resources

The United States Environmental Protection Agency's Environmental Monitoring and Assessment Program (EMAP) is designed to describe status, trends and spatial pattern of indicators of condition of the nation's ecological resources. The proposed sampling design for EMAP is based on a triangular systematic grid and employs both variable probability and double sampling. The Horvitz-Thompson estimator provides the foundation of the design-based estimation strategy used in EMAP. However, special features of EMAP designed to accommodate the complexity of sampling environmental resources on a national scale require modifications of standard variance estimation procedures as well as development of new techniques. An overview of variance estimation methods proposed for application to EMAP's sampling strategy for discrete resources is presented.     

Design-based ranked set sampling using auxiliary variables

A ranked set sampling protocol is proposed when an auxiliary variable is available in addition to the target variable in sample surveys. The protocol may be practically carried out without additional sampling effort or costs. Under the suggested sampling scheme, the estimators usually adopted in surveys with auxiliary information - such as the ratio estimator or the regression estimator - display surprising theoretical properties as well as high performance in practice.     

Adaptive two-stage one-per-stratum sampling

We briefly describe adaptive cluster sampling designs in which the initial sample is taken according to a Markov chain one-per-stratum design (Breidt, 1995) and one or more secondary samples are taken within strata if units in the initial sample satisfy a given condition C. An empirical study of the behavior of the estimation procedure is conducted for three small artificial populations for which adaptive sampling is appropriate. The specific sampling strategy used in the empirical study was a single random-start systematic sample with predefined systematic samples within strata when the initially sampled unit in that stratum satisfies C. The bias of the Horvitz-Thompson estimator for this design is usually very small when adaptive sampling is conducted in a population for which it is suited. In addition, we compare the behavior of several alternative estimators of the standard error of the Horvitz-Thompson estimator of the population total. The best estimator of the standard error is population-dependent but it is not unreasonable to use the Horvitz-Thompson estimator of the variance. Unfortunately, the distribution of the estimator is highly skewed hence the usual approach of constructing confidence intervals assuming normality cannot be used here.     

Comparison of distribution function estimators for forestry applications following variable probability sampling

Practical considerations often motivate employing variable probability sampling designs when estimating characteristics of forest populations. Three distribution function estimators, the Horvitz-Thompson estimator, a difference estimator, and a ratio estimator, are compared following variable probability sampling in which the inclusion probabilities are proportional to an auxiliary variable, X. Relative performance of the estimators is affected by several factors, including the distribution of the inclusion probabilities, the correlation () between X and the response Y, and the position along the distribution function being estimated. Both the ratio and difference estimators are superior to the Horvitz-Thompson estimator. The difference estimator gains better precision than the ratio estimator toward the upper portion of the distribution function, but the ratio estimator is superior toward the lower end of the distribution function. The point along the distribution function at which the difference estimator becomes more precise than the ratio estimator depends on the sampling design, as well as the coefficient of variation of X and . A simple confidence interval procedure provides close to nominal coverage for intervals constructed from both the difference and ratio estimators, with the exception that coverage may be poor for the lower tail of the distribution function when using the ratio estimator.     

Spatial sampling and the environment: some issues and directions

In this paper we summarize research issues for spatial environmental sampling stemming from a NISS/USEPA workshop held on 21-22 September 1994 at Chapel Hill, NC.     

基于地统计学的降雨侵蚀力插值方法研究——以江苏省为例

利用江苏省2001-2006年260个站点的自动记录降雨资料,运用卜氏算法计算各站点多年平均降雨侵蚀力,并结合地统计学进行分析与误差对比,结果显示在二阶、球型半变异函数模型参数条件下,普通Kriging插值方法的模拟精度较高,在江苏省具有较好的适用性;基于水土保持工作中雨量站点的选址原则共选取了21个代表站,通过优选出的普通Kriging插值方法对代表站的代表性进行了验证,结果显示选址方法可行.     

A foundation for developing a methodology for social network sampling

Researchers are increasingly turning to network theory to understand the social nature of animal populations. We present a computational framework that is the first step in a series of works that will allow us to develop a quantitative methodology of social network sampling to aid ecologists in their social network data collection. To develop our methodology, we need to be able to generate networks from which to sample. Ideally, we need to perform a systematic study of sampling protocols on different known network structures, as network structure might affect the robustness of any particular sampling methodology. Thus, we present a computational tool for generating network structures that have user-defined distributions for network properties and for key measures of interest to ecologists. The user defines the values of these measures and the tool will generate appropriate network randomizations with those properties. This tool will be used as a framework for developing a sampling methodology, although we do not present a full methodology here. We describe the method used by the tool, demonstrate its effectiveness, and discuss how the tool can now be utilized. We provide a proof-of-concept example (using the assortativity measure) of how such networks can be used, along with a simulated egocentric sampling regime, to test the level of equivalence of the sampled network to the actual network. This contribution is part of the special issue “Social Networks: new perspectives” (Guest Editors: J. Krause, D. Lusseau and R. James).     

基于土壤酶活性变化的铅污染土壤修复基准

近年来,污染土壤修复技术发展很快,而污染土壤修复标准的建立则相对迟缓。为了推进我国该领域的工作,对铅胁迫下土壤酶活性（如过氧化氢酶,磷酸酶,脲酶）随时间的变化进行了研究,以确定棕壤土中铅的土壤修复基准。结果表明：土壤中的铅能够刺激土壤中过氧化氢酶活性的增加,但随着土壤铅浓度的增加,这种刺激作用逐渐减小。土壤过氧化氢酶活性不宜作为铅污染土壤的生物标记物。土壤磷酸酶活性没有一致的变化规律,土壤磷酸酶活性不能作为铅污染土壤的生物标记物。在整个实验时间范围内,土壤脲酶活性与土壤中铅的浓度具有明显的剂量-效应关系（P〈0.01）,土壤脲酶活性可以作为铅污染土壤生物标记物。以土壤脲酶抑制率降低25%为依据,确定棕壤中铅的土壤修复基准为94mg·kg-1,以土壤脲酶抑制率降低45%为依据,确定棕壤中铅的土壤修复基准为178mg·kg-1。     

The use of replicated plot,line and point sampling for estimating species abundance and ecological diversity

The paper deals with the problem of estimating diversity indexes for an ecological community. First the species abundances are unbiasedly and consistently estimated using designs based on n random and independent selections of plots, points or lines over the study area. The problem of sampling elusive populations is also considered. Finally, the diversity index estimates are obtained as functions of the abundance estimates. The resulting estimators turn out to be asymptotically (n large) unbiased, even if a considerable bias may occur for a small n. Accordingly, the method of jackknifing is made use of in order to reduce bias.     

Optimal spatial sampling schemes for environmental surveys

A practical problem in spatial statistics is that of constructing spatial sampling designs for environmental monitoring network. This paper presents a fractal-based criterion for the construction of coverage designs to optimize the location of sampling points. The algorithm does not depend on the covariance structure of the process and provides desirable results for situations in which a poor prior knowledge is available. The statistical characteristics of the method are explored by a simulation study while a design exercise concerning the Pescara area monitoring network is used to demonstrate potential designs under realistic assumptions.