Finite population properties of individual predictors based on spatial patterns

In this study a conceptual framework for assessing the statistical properties of a non-stochastic spatial interpolator is developed through the use of design-based finite population inference tools. By considering the observed locations as the result of a probabilistic sampling design, we propose a standardized weighted predictor for spatial data starting from a deterministic interpolator that usually does not provide uncertainty measures. The information regarding the coordinates of the spatial locations is known at the population level and is directly used in constructing the weighting system. Our procedure captures the spatial pattern by means of the Euclidean distances between locations, which are fixed and do not require any further assessment after the sample has been drawn. The predictor for any individual value turns in a ratio of design-based random quantities. We illustrate the predictor design-based statistical properties, i.e. asymptotically p-unbiasedness and p-consistency, for simple random sampling without replacement. An application to a couple of environmental datasets is presented, for assessing predictor performances in correspondence of different population characteristics. A comparison with the equivalent non-spatial predictor is presented.     

Detecting Rare Species with Random or Subjective Sampling: a Case Study of Red-Listed Saproxylic Beetles in Boreal Sweden

Abstract:  Efficient sampling design in field studies is important for economical and statistical reasons. We compared two ways to distribute sampling effort over an area, either randomly or subjectively. We searched for red-listed saproxylic (wood-living) beetles in 30 spruce stands in boreal Sweden by sifting wood from dead trees. We randomly selected positions within each stand with a geographic positioning system and sampled the nearest dead tree (random sample). In the same stand we also sampled dead trees that, based on literature, were likely to host such species (subjective sampling). The subjective sampling (two to five samples per stand, depending on stand size) was compared with the higher, random sampling effort (fixed level of 12 samples/stand). Subjective sampling was significantly more efficient. Red-listed species were found in 36% of the subjective samples and in 16% of the random samples. Nevertheless, the larger random effort resulted in a comparable number of red-listed species per stand and in 13 detected species in total (vs. 12 species with subjective sampling). Random sampling was less efficient, but provided an unbiased alternative more suitable for statistical purposes, as needed in, for example, monitoring programs. Moreover, new species-specific knowledge can be gained through random searches.     

Using Niche-Based Models to Improve the Sampling of Rare Species

Abstract:  Because data on rare species usually are sparse, it is important to have efficient ways to sample additional data. Traditional sampling approaches are of limited value for rare species because a very large proportion of randomly chosen sampling sites are unlikely to shelter the species. For these species, spatial predictions from niche-based distribution models can be used to stratify the sampling and increase sampling efficiency. New data sampled are then used to improve the initial model. Applying this approach repeatedly is an adaptive process that may allow increasing the number of new occurrences found. We illustrate the approach with a case study of a rare and endangered plant species in Switzerland and a simulation experiment. Our field survey confirmed that the method helps in the discovery of new populations of the target species in remote areas where the predicted habitat suitability is high. In our simulations the model-based approach provided a significant improvement (by a factor of 1.8 to 4 times, depending on the measure) over simple random sampling. In terms of cost this approach may save up to 70% of the time spent in the field.     

Spatial methods for plot-based sampling of wildlife populations

Classical sampling methods can be used to estimate the mean of a finite or infinite population. Block kriging also estimates the mean, but of an infinite population in a continuous spatial domain. In this paper, I consider a finite population version of block kriging (FPBK) for plot-based sampling. The data are assumed to come from a spatial stochastic process. Minimizing mean-squared-prediction errors yields best linear unbiased predictions that are a finite population version of block kriging. FPBK has versions comparable to simple random sampling and stratified sampling, and includes the general linear model. This method has been tested for several years for moose surveys in Alaska, and an example is given where results are compared to stratified random sampling. In general, assuming a spatial model gives three main advantages over classical sampling: (1) FPBK is usually more precise than simple or stratified random sampling, (2) FPBK allows small area estimation, and (3) FPBK allows nonrandom sampling designs.     

Spatial structure, sampling design and abundance estimates in sandy beach macroinfauna: some warnings and new perspectives

We discuss methodological aspects directed to quantify the across-shore population structure and abundance of sandy beach macroinfauna. The reliability of estimates derived from design-based (stratified random sampling) and model-based (geostatistics, kriging) approaches is discussed. Our analysis also addresses potential biases arising from environmentally driven designs that consider a priori fixed strata for sampling macroinfauna, as opposed to species-driven sampling designs, in which the entire range of across-shore distribution is covered. Model-based approaches showed, spatially, highly autocorrelated and persistent structures in two intertidal populations of the Uruguayan coast: the isopod Excirolana armata and the yellow clam Mesodesma mactroides. Both populations presented zonation patterns that ranged from the base of the dunes to upper levels of the subtidal. The Gaussian model consistently explained the spatial distribution of species and population components (clam recruits and adults), with a minor contribution (Е%) of unresolved, small-scale variability. The consistent structure of spatial dependence in annual data strongly suggests an across-shore-structured process covering close to 35 m. Kriging predictions through cross-validation corroborated the appropriateness of the models fitted through variographic analysis, and the derived abundance estimates were very similar (maximum difference=7%) to those obtained from linear interpolation. Monthly analysis of E. armata data showed marked variations in its zonation and an unstable spatial structure according to the Gaussian model. The clear spatial structure resulting from species-driven sampling was not observed when data was truncated to simulate an environmentally driven sampling design. In this case, the linear semivariogram indicated a spatial gradient, suggesting that sampling was not performed at the appropriate spatial scale. Further, the cross-validation procedure was not significant, and both density and total abundance were underestimated. We conclude that: (1) geostatistics provides useful additional information about population structure and aids in direct abundance estimation; thus we suggest it as a powerful tool for further applications in the study of sandy beach macroinfauna; and that (2) environmentally driven sampling strategies fail to provide conclusive results about population structure and abundance, and should be avoided in studies of sandy beach populations. This is especially true for microtidal beaches, where unpredictable swash strength precludes a priori stratification through environmental reference points. The need to use adaptive sampling designs and avoid snapshot sampling is also stressed. Methodological implications for the detection of macroecological patterns in sandy beach macroinfauna are also discussed.     

Design based estimation for ranked set sampling in finite populations

In this paper, we consider design-based estimation using ranked set sampling (RSS) in finite populations. We first derive the first and second-order inclusion probabilities for an RSS design and present two Horvitz–Thompson type estimators using these inclusion probabilities. We also develop an alternate Hansen–Hurwitz type estimator and investigate its properties. In particular, we show that this alternate estimator always outperforms the usual Hansen–Hurwitz type estimator in the simple random sampling with replacement design with comparable sample size. We also develop formulae for ratio estimator for all three developed estimators. The theoretical results are augmented by numerical and simulation studies as well as a case study using a well known data set. These show that RSS design can yield a substantial improvement in efficiency over the usual simple random sampling design in finite populations.     

Geographic sampling of urban soils for contaminant mapping: how many samples and from where

Properly sampling soils and mapping soil contamination in urban environments requires that impacts of spatial autocorrelation be taken into account. As spatial autocorrelation increases in an urban landscape, the amount of duplicate information contained in georeferenced data also increases, whether an entire population or some type of random sample drawn from that population is being analyzed, resulting in conventional power and sample size calculation formulae yielding incorrect sample size numbers vis-à-vis model-based inference. Griffith (in Annals, Association of American Geographers, 95, 740–760, 2005) exploits spatial statistical model specifications to formulate equations for estimating the necessary sample size needed to obtain some predetermined level of precision for an analysis of georeferenced data when implementing a tessellation stratified random sampling design, labeling this approach model-informed, since a model of latent spatial autocorrelation is required. This paper addresses issues of efficiency associated with these model-based results. It summarizes findings from a data collection exercise (soil samples collected from across Syracuse, NY), as well as from a set of resampling and from a set of simulation experiments following experimental design principles spelled out by Overton and Stehman (in Communications in Statistics: Theory and Methods, 22, 2641–2660). Guidelines are suggested concerning appropriate sample size (i.e., how many) and sampling network (i.e., where).

Rarefaction and nonrandom spatial dispersion patterns

Rarefaction estimates how many species are expected in a random sample of individuals from a larger collection and allows meaningful comparisons among collections of different sizes. It assumes random spatial dispersion. However, two common dispersion patterns, within-species clumping and segregation among species, can cause rarefaction to overestimate the species richness of a smaller continuous area. We use field studies and computer simulations to determine (1) how robust rarefaction is to nonrandom spatial dispersion and (2) whether simple measures of spatial autocorrelation can predict the bias in rarefaction estimates. Rarefaction does not estimate species richness accurately for many communities, especially at small sample sizes. Measures of spatial autocorrelation of the more abundant species do not reliably predict amount of bias. Survey sites should be standardized to equal-sized areas before sampling. When sites are of equal area but differ in number of individuals sampled, rarefaction can standardize collections. When communities are sampled from different-sized areas, the mean and confidence intervals of species accumulation curves allow more meaningful comparisons among sites. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.

A new design for sampling with adaptive sample plots

Adaptive cluster sampling (ACS) is a sampling technique for sampling rare and geographically clustered populations. Aiming to enhance the practicability of ACS while maintaining some of its major characteristics, an adaptive sample plot design is introduced in this study which facilitates field work compared to “standard” ACS. The plot design is based on a conditional plot expansion: a larger plot (by a pre-defined plot size factor) is installed at a sample point instead of the smaller initial plot if a pre-defined condition is fulfilled. This study provides insight to the statistical performance of the proposed adaptive plot design. A design-unbiased estimator is presented and used on six artificial and one real tree position maps to estimate density (number of objects per ha). The performance in terms of coefficient of variation is compared to the non-adaptive alternative without a conditional expansion of plot size. The adaptive plot design was superior in all cases but the improvement depends on (1) the structure of the sampled population, (2) the plot size factor and (3) the critical value (the minimum number of objects triggering an expansion). For some spatial arrangements the improvement is relatively small. The adaptive design may be particularly attractive for sampling in rare and compactly clustered populations with an appropriately chosen plot size factor.     

Randomly selected order statistics in ranked set sampling: A less expensive comparable alternative to simple random sampling

Rank-based sampling designs are powerful alternatives to simple random sampling (SRS) and often provide large improvements in the precision of estimators. In many environmental, ecological, agricultural, industrial and/or medical applications the interest lies in sampling designs that are cheaper than SRS and provide comparable estimates. In this paper, we propose a new variation of ranked set sampling (RSS) for estimating the population mean based on the random selection technique to measure a smaller number of observations than RSS design. We study the properties of the population mean estimator using the proposed design and provide conditions under which the mean estimator performs better than SRS and some existing rank-based sampling designs. Theoretical results are augmented with some numerical studies and a real-life example, where we also study the performance of our proposed design under perfect and imperfect ranking situations.     

Sampling for weed spatial distribution mapping need not be adaptive

Weeds are species of interest for ecologists because they are competitors of the crop for resources but they also play an important role in maintaining biodiversity in agroecosystems. To study their spatial distribution at the field scale, only sampled observations are available due to the cost of sampling. Weeds sampling strategies are static. However, in the domain of spatial sampling, adaptive strategies have also been developed with, for some of them, an important on-line or off-line computational cost. In this article we provide answers to the following question: Are the current adaptive sampling methods efficient enough to motivate a wider use in practice when sampling a weed species at a field scale? We provide a comparison of the behaviour of 8 static strategies and 3 adaptive ones on four criteria: density class estimation, map restoration, spatial aggregation estimation, and sampling duration. From two weeds data sets, we estimated six contrasted Markov Random Field (MRF) models of weed density class spatial distribution and a model for sampling duration. The MRF models were then used to compare the strategies on a large set of simulated maps. Our main finding was that there is no clear gain in using adaptive sampling strategies rather than static ones for the three first criteria, and adaptive strategies were associated to longer sampling duration. This conclusion points out that for weed mapping, it is more important to build a good model of spatial distribution, than to propose complex adaptive sampling strategies.     

Bayesian entropy for spatial sampling design of environmental data

We develop a spatial statistical methodology to design national air pollution monitoring networks with good predictive capabilities while minimizing the cost of monitoring. The underlying complexity of atmospheric processes and the urgent need to give credible assessments of environmental risk create problems requiring new statistical methodologies to meet these challenges. In this work, we present a new method of ranking various subnetworks taking both the environmental cost and the statistical information into account. A Bayesian algorithm is introduced to obtain an optimal subnetwork using an entropy framework. The final network and accuracy of the spatial predictions is heavily dependent on the underlying model of spatial correlation. Usually the simplifying assumption of stationarity, in the sense that the spatial dependency structure does not change location, is made for spatial prediction. However, it is not uncommon to find spatial data that show strong signs of nonstationary behavior. We build upon an existing approach that creates a nonstationary covariance by a mixture of a family of stationary processes, and we propose a Bayesian method of estimating the associated parameters using the technique of Reversible Jump Markov Chain Monte Carlo. We apply these methods for spatial prediction and network design to ambient ozone data from a monitoring network in the eastern US.     

Spatially stratified sampling using auxiliary information for geostatistical mapping

This paper presents a method of spatial sampling based on stratification by Local Moran’s I _i calculated using auxiliary information. The sampling technique is compared to other design-based approaches including simple random sampling, systematic sampling on a regular grid, conditional Latin Hypercube sampling and stratified sampling based on auxiliary information, and is illustrated using two different spatial data sets. Each of the samples for the two data sets is interpolated using regression kriging to form a geostatistical map for their respective areas. The proposed technique is shown to be competitive in reproducing specific areas of interest with high accuracy.     

The effectiveness of marine reserve systems constructed using different surrogates of biodiversity

Biological sampling in marine systems is often limited, and the cost of acquiring new data is high. We sought to assess whether systematic reserves designed using abiotic domains adequately conserve a comprehensive range of species in a tropical marine inter‐reef system. We based our assessment on data from the Great Barrier Reef, Australia. We designed reserve systems aiming to conserve 30% of each species based on 4 abiotic surrogate types (abiotic domains; weighted abiotic domains; pre‐defined bioregions; and random selection of areas). We evaluated each surrogate in scenarios with and without cost (cost to fishery) and clumping (size of conservation area) constraints. To measure the efficacy of each reserve system for conservation purposes, we evaluated how well 842 species collected at 1155 sites across the Great Barrier Reef seabed were represented in each reserve system. When reserve design included both cost and clumping constraints, the mean proportion of species reaching the conservation target was 20–27% higher for reserve systems that were biologically informed than reserves designed using unweighted environmental data. All domains performed substantially better than random, except when there were no spatial or economic constraints placed on the system design. Under the scenario with no constraints, the mean proportion of species reaching the conservation target ranged from 98.5% to 99.99% across all surrogate domains, whereas the range was 90–96% across all domains when both cost and clumping were considered. This proportion did not change considerably between scenarios where one constraint was imposed and scenarios where both cost and clumping constraints were considered. We conclude that representative reserve systems can be designed using abiotic domains; however, there are substantial benefits if some biological information is incorporated.     

Determining association networks in social animals: choosing spatial–temporal criteria and sampling rates

Social Network Analysis has become an important methodological tool for advancing our understanding of human and animal group behaviour. However, researchers tend to rely on arbitrary distance and time measures when defining ‘contacts’ or ‘associations’ between individuals based on preliminary observation. Otherwise, criteria are chosen on the basis of the communication range of sensor devices (e.g. bluetooth communication ranges) or the sampling frequencies of collection devices (e.g. Global Positioning System devices). Thus, researchers lack an established protocol for determining both relevant association distances and minimum sampling rates required to accurately represent the network structure under investigation. In this paper, we demonstrate how researchers can use experimental and statistical methods to establish spatial and temporal association patterns and thus correctly characterise social networks in both time and space. To do this, we first perform a mixing experiment with Merino sheep (Ovis aries) and use a community detection algorithm that allows us to identify the spatial and temporal distance at which we can best identify clusters of previously familiar sheep. This turns out to be within 2–3 m of each other for at least 3 min. We then calculate the network graph entropy rate—a measure of ease of spreading of information (e.g. a disease) in a network—to determine the minimum sampling rate required to capture the variability observed in our sheep networks during distinct activity phases. Our results indicate the need for sampling intervals of less than a minute apart. The tools that we employ are versatile and could be applied to a wide range of species and social network datasets, thus allowing an increase in both the accuracy and efficiency of data collection when exploring spatial association patterns in gregarious species.     

Sample based estimation of landscape metrics; accuracy of line intersect sampling for estimating edge density and Shannon’s diversity index

A recent trend is to estimate landscape metrics using sample data and cost-efficiency is one important reason for this development. In this study, line intersect sampling (LIS) was used as an alternative to wall-to-wall mapping for estimating Shannon’s diversity index and edge length and density. Monte Carlo simulation was applied to study the statistical performance of the estimators. All combinations of two sampling designs (random and systematic distribution of transects), four sample sizes, five transect configurations (straight line, L, Y, triangle, and quadrat), two transect orientations (fixed and random), and three configuration lengths were tested, each with a large number of simulations. Reference was 50 photos of size 1 km², already manually delineated in vector format by photo interpreters using GIS environment. The performance was compared by root mean square error (RMSE) and bias. The best combination for all three metrics was found to be the systematic design and as response design the straight line configuration with random orientation of transects, with little difference between the fixed and random orientation of transects. The rate of decrease of RMSE for increasing sample size and line length was studied with a mixed linear model. It was found that the RMSE decreased to a larger degree with the systematic design than the random one, especially with increasing sample size. Due to the nonlinearity in the definition of Shannon diversity estimator its estimator has a small and negative bias, decreasing with sample size and line length. Finally, a time study was conducted, measuring the time for registration of line intersections and their lengths on non-delineated aerial photos. The time study showed that long sampling lines were more cost-efficient than short ones for photo-interpretation.     

The application of adaptive cluster sampling for rare subtidal macroalgae

Adaptive cluster sampling (ACS) is a targeting sampling method that provides unbiased abundance estimators for populations of rare species that may be inadequately sampled with simple random sampling (SRS). ACS has been used successfully to estimate abundances of rockfish and sardine larvae from shipboard surveys. In this study, we describe the application of ACS for subtidal macroalgae. Using SCUBA, we measured abundances of Codium mamillosum, C. pomoides, and Halimeda cuneata at three islands and two levels of wave exposure. The three species were relatively patchy and could be sampled with ACS at one site per dive. Their distributions differed among islands and with exposure to wave energy, with H. cuneata found at only one island. ACS is a useful tool for understanding the spatial distribution and abundance of populations of rare benthic species, but, as was the case in this study, may not be as efficient as sampling with SRS with comparable replication.     

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.     

Structure of turbulent plumes from a momentumless source in a smooth bed

The spatial development of a passive scalar plume is studied within the inhomogeneous turbulence of a boundary layer flow in a recirculating laboratory flume with a smooth bed. The source of the scalar is located flush with the bed, and the low-momentum source design is intended to simulate a diffusive-type scalar release. A weakly diffusive fluorescent dye is used as the scalar. Planar laser-induced fluorescence (PLIF) techniques were used to record the structure of the plume at a spatial resolution of 150 μm. The measured structure of the mean concentration field is compared to an analytical solution for shear-free, homogeneous turbulence. The laboratory plume exhibits spatial development in the mean concentration field that deviates from the self-similar behavior predicted by the analytical solution; this deviation is due to the mean shear and inhomogeneity of the turbulence. In particular, the influence of the viscous sublayer on the plume development is seen to be significant. Nonetheless, the analytical solution replicates some of the features seen in the laboratory plume, and the solution suggests methods of reducing the laboratory data even for cases where the results deviate from the analysis. We also examine the spatial development of the root-mean-square (rms) fluctuating concentration field, and use scalar probability density functions to examine the relationship between the mean and fluctuating concentrations.