Triangulation based inclusion probabilities: a design-unbiased sampling approach

A probabilistic sampling approach for design-unbiased estimation of area-related quantitative characteristics of spatially dispersed population units is proposed. The developed field protocol includes a fixed number of 3 units per sampling location and is based on partial triangulations over their natural neighbors to derive the individual inclusion probabilities. The performance of the proposed design is tested in comparison to fixed area sample plots in a simulation with two forest stands. Evaluation is based on a general approach for areal sampling in which all characteristics of the resulting population of possible samples is derived analytically by means of a complete tessellation of the areal sampling frame. The example simulation shows promising results. Expected errors under this design are comparable to sample plots including a much greater number of trees per plot.     

Complex national sampling design for long-term monitoring of protected dry grasslands in Switzerland

We describe a probabilistic sampling design of circular permanent plots for the long-term monitoring of protected dry grasslands in Switzerland. The population under study is defined by the perimeter of a national inventory. The monitoring focus is on the species composition of the protected grassland vegetation and derived conservation values. Efficient trend estimations are required for the whole country and for some predefined target groups (six biogeographical regions and eleven vegetation types). The target groups are equally important regardless of their size. Consequently, intensified sampling of the less frequent groups is essential for sample efficiency. The prior information needed to draw a targeted sample is obtained from the sampling frame and external databases. The logistics and generalized delineation of the target population may pose further problems. Thus, investments in fieldwork and travel time should be well balanced by selecting a cluster sample. Second, any access problems in the field and non-target units in the sample should be compensated for by selecting reserve plots as they otherwise may considerably reduce the effective sample size. Finally, the design has to be flexible as the sampling frame may change over time and sampling intensity might have to be adjusted to redefined budgets or requirements. Likewise, the variables and biological items of interest may change. To fulfil all these constraints and to optimally use the available prior information, we propose a multi-stage self-weighted unequal probability sampling design. The design uses modern techniques such as: balanced sampling, spreading, stratified balancing, calibration, unequal probability sampling and power allocation. This sampling design meets the numerous requirements of this study and provides a very efficient estimator.     

Network inclusion probabilities and Horvitz-Thompson estimation for adaptive simple Latin square sampling

Consider a survey of a plant or animal species in which abundance or presence/absence will be recorded. Further assume that the presence of the plant or animal is rare and tends to cluster. A sampling design will be implemented to determine which units to sample within the study region. Adaptive cluster sampling designs Thompson (1990) are sampling designs that are implemented by first selecting a sample of units according to some conventional probability sampling design. Then, whenever a specified criterion is satisfied upon measuring the variable of interest, additional units are adaptively sampled in neighborhoods of those units satisfying the criterion. The success of these adaptive designs depends on the probabilities of finding the rare clustered events, called networks. This research uses combinatorial generating functions to calculate network inclusion probabilities associated with a simple Latin square sample. It will be shown that, in general, adaptive simple Latin square sampling when compared to adaptive simple random sampling will (i) yield higher network inclusion probabilities and (ii) provide Horvitz-Thompson estimators with smaller variability.     

Halton iterative partitioning master frames

Environmental and Ecological Statistics - A spatial sampling design determines where sample locations are placed in a study area. To achieve reliable estimates of population characteristics, the...     

Spatially-balanced sampling versus unbalanced stratified sampling for assessing forest change: evidences in favour of spatial balance

Large-scale remote sensing-based inventories of forest cover are usually carried out by combining unsupervised classifications of satellite pixels into forest/non forest classes (map data) with subsequent time-consuming visual on-screen imagery classification of a probabilistic sample of pixels taken as the ground truth (reference data). In this paper the estimation of forest change from a sample of reference data is approached by: (i) exploiting map data to construct strata in which changes are occurred, and then adopting the stratified sampling joined with the HT estimator with most sampling effort devoted to strata where changes are occurred irrespective of their size, as suggested in most remote sensing literature regarding land change assessments; (ii) adopting a spatial scheme ensuring spatially balanced samples, as suggested in most recent statistical literature regarding spatial surveys, and exploiting the map data in the difference estimator. The results of a comparison performed on an artificial population of reference data generated from a real population of map data recorded in Sardinia (Italy) discourage the use of unbalanced stratified samples that achieve the worst precision. The best results are obtained by means of spatially balanced samples or stratification with nearly proportional allocation to strata.     

Ranked set sampling allocation models for multiple skewed variables: an application to agricultural data

The mean of a balanced ranked set sample is more efficient than the mean of a simple random sample of equal size and the precision of ranked set sampling may be increased by using an unbalanced allocation when the population distribution is highly skewed. The aim of this paper is to show the practical benefits of the unequal allocation in estimating simultaneously the means of more skewed variables through real data. In particular, the allocation rule suggested in the literature for a single skewed distribution may be easily applied when more than one skewed variable are of interest and an auxiliary variable correlated with them is available. This method can lead to substantial gains in precision for all the study variables with respect to the simple random sampling, and to the balanced ranked set sampling too.     

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.     

Optimizing two- and three-stage designs for spatial inventories of natural resources by simulated annealing

This paper reviews design-based estimators for two- and three-stage sampling designs to estimate the mean of finite populations. This theory is then extended to spatial populations with continuous, infinite populations of sampling units at the latter stages. We then assume that the spatial pattern is the result of a spatial stochastic process, so the sampling variance of the estimators can be predicted from the variogram. A realistic cost function is then developed, based on several factors including laboratory analysis, time of fieldwork, and numbers of samples. Simulated annealing is used to find designs with minimum sampling variance for a fixed budget. The theory is illustrated with a real-world problem dealing with the volume of contaminated bed sediments in a network of watercourses. Primary sampling units are watercourses, secondary units are transects perpendicular to the axis of the watercourse, and tertiary units are points. Optimal designs had one point per transect, from one to three transects per watercourse, and the number of watercourses varied depending on the budget. However, if laboratory costs are reduced by grouping all samples within a watercourse into one composite sample, it appeared to be efficient to sample more transects within a watercourse.     

Regression-type estimators for adaptive two-stage sequential sampling

Adaptive two-stage sequential sampling (ATSSS) design was developed to observe more rare units and gain higher efficiency, in the sense of having a smaller variance estimator, than conventional sampling designs with equal effort for rare and spatially cluster populations. For certain rare populations, incorporating auxiliary variables into a sampling design can further improve the observation of rare units and increase efficiency. In this article, we develop regression-type estimators for ATSSS so that auxiliary variables can be incorporated into the ATSSS design when warranted. Simulation studies on two populations show that the regression-type estimators can significantly increase the efficiency of ATSSS and the detection of more rare units as compared to conventional sampling counterparts. Simulation of sampling of desert shrubs in Inner Mongolia (one of the two populations studied) showed that by incorporating a GIS auxiliary variable into ATSSS with the regression estimators resulted in a gain in efficiency over ATSSS without the auxiliary variable. Further, we found that the use of the GIS auxiliary variable in a conventional two-stage design with a regression estimator did not show a gain in efficiency.     

On ranked set sampling for multiple characteristics

We consider the selection of samples in ranked set sampling when several attributes of each sample are of interest. We describe approaches that have appeared previously in the literature and present a novel method that seeks to achieve samples that are nearly balanced with respect to the ranks of all attributes. This method is shown to result in very little loss of precision compared to problems in which only a single sample attribute is of interest.     

Nonparametric mean estimation using partially ordered sets

In ranked-set sampling (RSS), the ranker must give a complete ranking of the units in each set. In this paper, we consider a modification of RSS that allows the ranker to declare ties. Our sampling method is simply to break the ties at random so that we obtain a standard ranked-set sample, but also to record the tie structure for use in estimation. We propose several different nonparametric mean estimators that incorporate the tie information, and we show that the best of these estimators is substantially more efficient than estimators that ignore the ties. As part of our comparison of estimators, we develop new results about models for ties in rankings. We also show that there are settings where, to achieve more efficient estimation, ties should be declared not just when the ranker is actually unsure about how units rank, but also when the ranker is sure about the ranking, but believes that the units are close.     

Designing an efficient adaptive cluster sample

The choice of neighborhood definition and critical value in adaptive cluster sampling is critical for designing an efficient survey. In designing an efficient adaptive cluster sample one should aim for a small difference between the initial and final sample size, and a small difference between the within-network and population variances. However, the two aims can be at odds with each other because small differences between initial and final sample size usually means small within-network variance. One way to help in designing an efficient survey is to think in terms of small network sizes since the network size is a function of both critical value and neighborhood definition. One should aim for networks that are small enough to ensure the final sample size is not excessively large compared with the initial sample size but large enough to ensure the within-network variance is a reasonable fraction of the population variance. In this study surveys that had networks that were two to four units in size were the most efficient.     

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.     

A hierarchical model for spatial capture-recapture data

Estimating density is a fundamental objective of many animal population studies. Application of methods for estimating population size from ostensibly closed populations is widespread, but ineffective for estimating absolute density because most populations are subject to short-term movements or so-called temporary emigration. This phenomenon invalidates the resulting estimates because the effective sample area is unknown. A number of methods involving the adjustment of estimates based on heuristic considerations are in widespread use. In this paper, a hierarchical model of spatially indexed capture-recapture data is proposed for sampling based on area searches of spatial sample units subject to uniform sampling intensity. The hierarchical model contains explicit models for the distribution of individuals and their movements, in addition to an observation model that is conditional on the location of individuals during sampling. Bayesian analysis of the hierarchical model is achieved by the use of data augmentation, which allows for a straightforward implementation in the freely available software WinBUGS. We present results of a simulation study that was carried out to evaluate the operating characteristics of the Bayesian estimator under variable densities and movement patterns of individuals. An application of the model is presented for survey data on the flat-tailed horned lizard (Phrynosoma mcallii) in Arizona, USA.     

Identifying the largest individual sample value from a two-way composite sampling design

Composite sampling techniques for identifying the largest individual sample value seem to be cost effective when the composite samples are internally homogeneous. However, since it is not always possible to form homogeneous composite samples, these methods can lead to higher costs than expected. In this paper we propose a two-way composite sampling design as a way to improve on the cost effectiveness of the methods available to identify the largest individual sample value.     

Cross-combined composite sampling designs for identification of elevated regions

Analyzing soils for contaminants can be costly. Generally, discrete samples are gathered from within a study area, analyzed by a laboratory and the results are used in a site-specific statistical analysis. Because of the heterogeneities that exist in soil samples within study areas, a large amount of variability and skewness may be present in the sample population. This necessitates collecting a large number of samples to obtain reliable inference on the mean contaminant concentration and to understand the spatial patterns for future remediation. Composite, or Incremental, sampling is a commonly applied method for gathering multiple discrete samples and physically combining them, such that each combination of discrete samples requires a single laboratory analysis, which reduces cost and can improve the estimates of the mean concentration. While incremental sampling can reduce cost and improve mean estimates, current implementations do not readily facilitate the characterization of spatial patterns or the detection of elevated constituent regions within study areas. The methods we present in this work provide efficient estimation and inference for the mean contaminant concentration over the entire spatial area and enable the identification of high contaminant regions within the area of interest. We develop sample design methodologies that explicitly define the characteristics of these designs (such as sample grid layout) and quantify the number of incremental samples that must be obtained under a design criteria to control false positive and false negative (Type I and II) decision errors. We present the sample design theory and specifications as well as results on simulated and real data.     

Obtaining species: sample size considerations

Suppose fish are to be sampled from a stream. A fisheries biologist might ask one of the following three questions: ‘How many fish do I need to catch in order to see all of the species?’, ‘How many fish do I need to catch in order to see all species whose relative frequency is more than 5%?’, or ‘How many fish do I need to catch in order to see a member from each of the species A, B, and C?’. This paper offers a practical solution to such questions by setting a target sample size designed to achieve desired results with known probability. We present three sample size methods, one we call ‘exact’ and the others approximate. Each method is derived under assumed multinomial sampling, and requires (at least approximate) independence of draws and (usually) a large population. The minimum information needed to compute one of the approximate methods is the estimated relative frequency of the rarest species of interest. Total number of species is not needed. Choice of a sample size method depends largely on available computer resources. One approximation (called the ‘Monte Carlo approximation’) gets within ±6 units of exact sample size, but usually requires 20–30 minutes of computer time to compute. The second approximation (called the ‘ratio approximation’) can be computed manually and has relative error under 5% when all species are desired, but can be as much as 50% or more too high when exact sample size is small. Statistically, this problem is an application of the ‘sequential occupancy problem’. Three examples are given which illustrate the calculations so that a reader not interested in technical details can apply our results.     

Bridging the gap between mechanistic and adaptive explanations of territory formation

How animals divide space can have fundamental implications for the population dynamics of territorial species. It has recently been proposed that space can be divided if animals tend to avoid fight locations, rather than the winner of fights gaining access to exclusive resources, behaviour that generates exclusive territories in two-dimensional space. A game-theory model has shown that this avoidance behaviour can be adaptive, but the adaptiveness has not been investigated in a spatially realistic context. We present a model that investigates potential strategies for the acquisition of territories when two-dimensional space must be divided between individuals. We examine whether exclusive territories form when animals avoid all encounters with others, or only those encounters that have led to losing fights, under different fighting costs and population densities. Our model suggests that when fighting costs are high, and the population density is low, the most adaptive behaviour is to avoid fight locations, which generates well-defined, exclusive territories in a population that is able to resist invasion by more aggressive strategies. Low fighting costs and high population densities lead to the break-down of territoriality and the formation of large, overlapping home ranges. We also provide a novel reason as to why so-called paradoxical strategies do not exist in nature: if we define a paradoxical strategy as an exact mirror-image of a common-sense one, it must respond in the opposite way to a draw as well as to wins and losses. When this is the case, and draws are common (fight outcomes are often not clear-cut in nature), the common-sense strategy is more often adaptive than a paradoxical alternative.Communicated by P. Bednekoff     

Spatially-balanced adaptive web sampling

The paper deals with sampling from a finite population that is distributed over space and has a highly uneven spatial distribution. It suggests a sampling design that allocates a portion of the sample units that are well spread over the population and sequentially selects the remaining units in sub-areas that appear to be of more interest according to the study variable values observed during the survey. In order to estimate the population mean while using this sampling design, a computationally intense estimator, obtained via the Rao–Blackwell approach, is proposed and a resampling method is used that makes the inference computationally feasible. The whole sampling strategy is evaluated through several Monte Carlo experiments.     

Efficiency and robustness of ant colony transportation networks

Efficient and robust transportation networks are key to the effectiveness of many natural systems. In polydomous ant colonies, which consist of two or more spatially separated but socially connected nests, resources must be transported between nests. In this study, we analyse the network structure of the inter-nest trails formed by natural polydomous ant colonies. In contrast to previous laboratory studies, the natural colonies in our study do not form minimum spanning tree networks. Instead the networks contain extra connections, suggesting that in natural colonies, robustness may be an important factor in network construction. Spatial analysis shows that nests are randomly distributed within the colony boundary and we find nests are most likely to connect to their nearest neighbours. However, the network structure is not entirely determined by spatial associations. By showing that the networks do not minimise total trail length and are not determined only by spatial associations, the results suggest that the inter-nest networks produced by ant colonies are influenced by previously unconsidered factors. We show that the transportation networks of polydomous ant colonies balance trail costs with the construction of networks that enable efficient transportation of resources. These networks therefore provide excellent examples of effective biological transport networks which may provide insight into the design and management of transportation systems.