Joint versus separate estimation of state and change in category frequencies from repeat stratified two-phase sampling with a fallible classifier

Joint maximum likelihood estimates (JML) of category frequencies and change from repeat stratified two-phase samplingsurveys with a fallible classifier are often seriously biased andhave large root mean square errors when they are obtained for small populations (<5000) with three or more categories and amoderate to small phase II sample size (<1000). JML estimates of state also depend on antecedent or posterior data, a recipe for inconsistency. In these situations, a separate maximum likelihood estimation (SML) of category frequenciesat each survey date appears preferable. SML estimates of net change are obtained as the difference in states. SML standard errors of change are obtained via an estimate of the temporal correlation and variances of state. A bivariate binarylogistic model of change provided the estimate of temporal correlation. SML generally outperformed JMLsignificantly in terms of bias and root mean square errors in eight case studies.     

Towards a sampling strategy for the assessment of forest condition at European level: combining country estimates

A correct characterization of the status and trend of forest condition is essential to support reporting processes at national and international level. An international forest condition monitoring has been implemented in Europe since 1987 under the auspices of the International Co-operative Programme on Assessment and Monitoring of Air Pollution Effects on Forests (ICP Forests). The monitoring is based on harmonized methodologies, with individual countries being responsible for its implementation. Due to inconsistencies and problems in sampling design, however, the ICP Forests network is not able to produce reliable quantitative estimates of forest condition at European and sometimes at country level. This paper proposes (1) a set of requirements for status and change assessment and (2) a harmonized sampling strategy able to provide unbiased and consistent estimators of forest condition parameters and of their changes at both country and European level. Under the assumption that a common definition of forest holds among European countries, monitoring objectives, parameters of concern and accuracy indexes are stated. On the basis of fixed-area plot sampling performed independently in each country, an unbiased and consistent estimator of forest defoliation indexes is obtained at both country and European level, together with conservative estimators of their sampling variance and power in the detection of changes. The strategy adopts a probabilistic sampling scheme based on fixed-area plots selected by means of systematic or stratified schemes. Operative guidelines for its application are provided.     

Monitoring landscape metrics by point sampling: accuracy in estimating Shannon’s diversity and edge density

Environmental monitoring of landscapes is of increasing interest. To quantify landscape patterns, a number of metrics are used, of which Shannon’s diversity, edge length, and density are studied here. As an alternative to complete mapping, point sampling was applied to estimate the metrics for already mapped landscapes selected from the National Inventory of Landscapes in Sweden (NILS). Monte-Carlo simulation was applied to study the performance of different designs. Random and systematic samplings were applied for four sample sizes and five buffer widths. The latter feature was relevant for edge length, since length was estimated through the number of points falling in buffer areas around edges. In addition, two landscape complexities were tested by applying two classification schemes with seven or 20 land cover classes to the NILS data. As expected, the root mean square error (RMSE) of the estimators decreased with increasing sample size. The estimators of both metrics were slightly biased, but the bias of Shannon’s diversity estimator was shown to decrease when sample size increased. In the edge length case, an increasing buffer width resulted in larger bias due to the increased impact of boundary conditions; this effect was shown to be independent of sample size. However, we also developed adjusted estimators that eliminate the bias of the edge length estimator. The rates of decrease of RMSE with increasing sample size and buffer width were quantified by a regression model. Finally, indicative cost–accuracy relationships were derived showing that point sampling could be a competitive alternative to complete wall-to-wall mapping.     

A comparison of two procedures to estimate three basic monitoring landscape metrics for monitoring

An interesting alternative to wall-to-wall mapping approaches for the estimation of landscape metrics is to use sampling. Sample-based approaches are cost-efficient, and measurement errors can be reduced considerably. The previous efforts of sample-based estimation of landscape metrics have mainly been focused on data collection methods, but in this study, we consider two estimation procedures. First, landscape metrics of interest are calculated separately for each sampled image and then the image values are averaged to obtain an estimate of the entire landscape (separated procedure, SP). Second, metric components are calculated in all sampled images and then the aggregated values are inserted into the landscape metric formulas (aggregated procedure, AP). The national land cover map (NLCM) of Sweden, reflecting the status of land cover in the year 2000, was used to provide population information to investigate the statistical performance of the estimation procedures. For this purpose, sampling simulation with a large number of replications was used. For all three landscape metrics, the second procedure (AP) produced a lower relative RMSE and bias than the first one (SP). A smaller sample unit size (50 ha) produced larger bias than a larger one (100 ha), whereas a smaller sample unit size produced a lower variance than a larger sample unit. The efficiency of a metric estimator is highly related to the degree of landscape fragmentation and the selected procedure. Incorporating information from all of the sampled images into a single one (aggregated procedure, AP) is one way to improve the statistical performance of estimators.     

Improving statistical distinctness in assessing trophic levels: the development of simulated normal distributions

The statistical distinctness in assessing differences of the trophic status between sampling sites was investigated in the present study. Nutrient (phosphate, nitrate, nitrite, ammonia) and phytoplankton (chlorophyll, cell number) variables from nine sampling stations were used for the validation of the statistical procedures. Raw data, transformed data, and simulated data derived on normalized nutrient–phytoplankton frequency distribution were tested. The Kruskal–Wallis (KW) nonparametric statistical procedure was also applied on the raw data as well as the analysis of variance on transformed and simulated data. In all cases, pairwise comparisons for each parameter between stations were performed. The results showed that maximum distinctness between sampling sites for all the six variables was attained using the simulated data. The KW method showed the poorest discrimination between stations. The methodology of producing and using simulated data is described step by step, and the advantages in cases of unequal sampling design or small sample size are discussed.     

Estimating the Number of Tree Species in a Forest Community using Survey Data

Using some large samples as actual populations we study a series of promising estimators for number of species in a population. We recommend a nonparametric estimator, CM2f = cs + fc1/(2c2) where cs is the number of species observed, c1 and c2 are the numbers of species occurring once and twice respectively on the sample plots, and f is the finite population correction for sampling, from such data as collected by Forest Inventory and Analysis (FIA) of the US Forest Service. For FIA samples, there are estimated to be 0, 4, 3, 1 and 6 rare tree species not sampled in Minnesota, Missouri, Pennsylvania, New, York, and Ohio respectively.     

Sampling Scale Effects in Random Fields and Implications for Environmental Monitoring

The concept of a sampling scale triplet of spacing, extent and support is used to define the spatial dimensions of a monitoring network or a field study. The spacing is the average distance between samples, the extent is the size of the domain sampled and the support is the averaging area of one sample. The aim of this paper is to examine what is the bias and the random error (uncertainty) introduced by the sampling scale triplet into estimates of the mean, the spatial variance and the integral scale of a variable in a landscape. The integral scale is a measure of the average distance over which a variable is correlated in space. A large number of two dimensional random fields are generated from which hypothetical samples, conforming to a certain sampling scale triplet, are drawn which in turn are used to estimate the sample mean, spatial variance and integral scale. The results indicate that the biases can be up to two orders of magnitude. The bias of the integral scale is positively related to the magnitude of any of the components of the scale triplet while the bias of the spatial variance is different for different components of the scale triplet. All sampling scale effects are relative to the underlying correlation length of the variable of interest which is closely related to the integral scale. The integral scale can hence be used for sampling design and data interpretation. Suggestions are given on how to adjust a monitoring network to the scales of the variables of interest and how to interpret sampling scale effects in environmental data.     

Sampling The Soil In Long-Term Forest Plots: The Implications Of Spatial Variation

Long-term monitoring of forest soils as part of a pan-European network to detect environmental change depends on an accurate determination of the mean of the soil properties at each monitoring event. Forest soil is known to be very variable spatially, however. A study was undertaken to explore and quantify this variability at three forest monitoring plots in Britain. Detailed soil sampling was carried out, and the data from the chemical analyses were analysed by classical statistics and geostatistics. An analysis of variance showed that there were no consistent effects from the sample sites in relation to the position of the trees. The variogram analysis showed that there was spatial dependence at each site for several variables and some varied in an apparently periodic way. An optimal sampling analysis based on the multivariate variogram for each site suggested that a bulked sample from 36 cores would reduce error to an acceptable level. Future sampling should be designed so that it neither targets nor avoids trees and disturbed ground. This can be achieved best by using a stratified random sampling design.     

A method to combine non-probability sample data with probability sample data in estimating spatial means of environmental variables

In estimating spatial means of environmental variables of a region from datacollected by convenience or purposive sampling, validity of the results canbe ensured by collecting additional data through probability sampling. Theprecision of the estimator that uses the probability sample can beincreased by interpolating the values at the nonprobability sample points tothe probability sample points, and using these interpolated values as anauxiliary variable in the difference or regression estimator. Theseestimators are (approximately) unbiased, even when the nonprobability sampleis severely biased such as in preferential samples. The gain in precisioncompared to the estimator in combination with Simple Random Samplingis controlled by the correlation between the target variable andinterpolated variable. This correlation is determined by the size (density)and spatial coverage of the nonprobability sample, and the spatialcontinuity of the target variable. In a case study the average ratio of thevariances of the simple regression estimator and estimator was 0.68for preferential samples of size 150 with moderate spatial clustering, and0.80 for preferential samples of similar size with strong spatialclustering. In the latter case the simple regression estimator wassubstantially more precise than the simple difference estimator.     

Sampling Benthic Macroinvertebrates in a Large Flood-Plain River: Considerations of Study Design, Sample Size, and Cost

Estimation of benthic macroinvertebrate populations over large spatial scales is difficult due to the high variability in abundance and the cost of sample processing and taxonomic analysis. To determine a cost-effective, statistically powerful sample design, we conducted an exploratory study of the spatial variation of benthic macroinvertebrates in a 37 km reach of the Upper Mississippi River. We sampled benthos at 36 sites within each of two strata, contiguous backwater and channel border. Three standard ponar (525 cm²) grab samples were obtained at each site ('Original Design'). Analysis of variance and sampling cost of strata-wide estimates for abundance of Oligochaeta, Chironomidae, and total invertebrates showed that only one ponar sample per site ('Reduced Design') yielded essentially the same abundance estimates as the Original Design, while reducing the overall cost by 63%. A posteriori statistical power analysis ( = 0.05, = 0.20) on the Reduced Design estimated that at least 18 sites per stratum were needed to detect differences in mean abundance between contiguous backwater and channel border areas for Oligochaeta, Chironomidae, and total invertebrates. Statistical power was nearly identical for the three taxonomic groups. The abundances of several taxa of concern (e.g., Hexagenia mayflies and Musculium fingernail clams) were too spatially variable to estimate power with our method. Resampling simulations indicated that to achieve adequate sampling precision for Oligochaeta, at least 36 sample sites per stratum would be required, whereas a sampling precision of 0.2 would not be attained with any sample size for Hexagenia in channel border areas, or Chironomidae and Musculium in both strata given the variance structure of the original samples. Community-wide diversity indices (Brillouin and 1-Simpsons) increased as sample area per site increased. The backwater area had higher diversity than the channel border area. The number of sampling sites required to sample benthic macroinvertebrates during our sampling period depended on the study objective and ranged from 18 to more than 40 sites per stratum. No single sampling regime would efficiently and adequately sample all components of the macroinvertebrate community.     

Sampling over time: developing a cost effective and precise exposure assessment program

Studies requiring ambient exposure assessments invariably ask: How often should measurements be taken? Answer to such questions is dictated by budgetary considerations as well as spatial and temporal variability in the data. For example, do we obtain measurements during all seasons, all months within seasons, weeks within months and days within weeks? On one hand, we can obtain a one-time snapshot sample and regard it as representing the "true" mean exposure. On the other hand, we may obtain a large number of measurements over time and then average these in order to represent this "true" mean exposure. The former estimate is the least expensive but may also be the least precise while the latter, may be very precise but prohibitively costly. In this paper, we demonstrate how a pilot study can be undertaken with a potentially promising and feasible sampling plan for the full-scale study. By applying the statistical methodology of variance component analysis (VCA) to the pilot study data and exploiting mathematical relationship between the variance of the overall mean exposure and posited variance components, we can develop a sampling design with decreased sampling costs and/or increased precision of the mean exposure. Our approach was applied to determine sampling design choices for an on-going study that aimed at assessing ambient particulate matter exposure. We conclude that a pilot study followed by the VCA analysis may often lead to sampling design choices that offer considerable cost savings and, at the same time, promise to provide relatively precise estimates of the mean exposure for the subsequent full-scale study.     

Using Satellite and Other Data to Estimate the Annual Water Demand of an Irrigation District

Locating and forecasting water needs can assist the location of water in dry regions, and improve the management of reservoirs and the canal network. Satellite, ground data, and agrometeorological data were combined to forecast the volume of irrigation water needed during 1993 and 1994 in an irrigation district of 327 km2 located in the Ebro basin, Spain. The main crops were rice, alfalfa plus forage, winter cereals (barley and wheat), sunflower and maize. Their extent was estimated every year by frame area sampling and a regression estimator with satellite data. Initial crop area statistics were obtained by expansion of the sample areas to the entire study area and then a regression estimator with the multitemporal supervised classification of two Landsat-5 TM images was applied. This procedure improved the precision of the estimates by expansion. Net water requiremets (m3 ha-1) of the above mentioned crops were computed from reference evapotranspiration estimates, crop coefficients and effective precipitation. These computations were performed for an average year, i.e. by using long-term averaged meteorological data. Crop hectarage and net crop water requirements were multiplied to obtain, for the entire study area, the volume (hm3 106 m3) of the net crop water requirements. After subtraction of water taken directly from the rivers and non-productive sunflower, the irrigation water volumes were estimated. The comparison of these forecasts with the volumes of water invoiced by the Ebro Basin Water Authority confirmed the feasibility of forecasting the volume of water applied to an individual irrigation district. This is an objective and practical method for estimating the irrigation water volume applied in an irrigated area.     

Using ‘found’ data to augment a probability sample: Procedure and case study

While probability sampling has the advantage of permitting unbiased population estimates, many past and existing monitoring schemes do not employ probability sampling. We describe and demonstrate a general procedure for augmenting an existing probability sample with data from nonprobability-based surveys (found data). The procedure, first proposed by Overton (1990), uses sampling frame attributes to group the probability and found samples into similar subsets. Subsequently, this similarity is assumed to reflect the representativeness of the found sample for the matching subpopulation. Two methods of establishing similarity and producing estimates are described: pseudo-random and calibration. The pseudo-random method is used when the found sample can contribute additional information on variables already measured for the probability sample, thus increasing the effective sample size. The calibration method is used when the found sample contributes information that is unique to the found observations. For either approach, the found sample data yield observations that are treated as a probability sample, and population estimates are made according to a probability estimation protocol. To demonstrate these approaches, we applied them to found and probability samples of stream discharge data for the southeastern US.     

Sampling Considerations for Establishment of Baseline Loadings from Forested Watersheds for TMDL Application

Five methods for estimating maximum daily and annual nitrate (NO3) and suspended sediment loads using periodic sampling of varying intensities were compared to actual loads calculated from intensive stormflow and baseflow sampling from small, forested watersheds in north central West Virginia to determine if the less intensive sampling methods were accurate and could be utilized in TMDL development. There were no significant differences between the annual NO3 load estimates using non-intensive sampling methods and the actual NO3 loads. However, maximum daily NO3 loads were estimated less accurately than annual loads. The ability to estimate baseline NO3 loads fairly accurately with non-intensive concentration data is attributed to the small fluctuation in NO3 concentrations over flow and time, particularly during storms. By contrast, suspended sediment exports determined by any of the non-intensive methods varied significantly and widely from and compared poorly to the actual exports for both daily and annual methods. Weekly sampling better approximated actual annual exports, but there were no significant statistical differences among weekly, monthly, and quarterly estimates. Suspended sediment concentrations varied widely within and among storm events, so that accurate estimates of total annual or maximum daily loads could not be obtained from infrequent sampling.     

Network design factors for assessing temporal variability in ground-water quality

Benchmark major ions and nutrients data were collected biweekly for about two years at 12 wells at two sites in a shallow sand and gravel aquifer in west-central Illinois. The purpose of the study was to explore the time series properties of ground-water quality data collected at a relatively high sampling frequency. A secondary purpose was to determine the relative magnitudes of natural and sampling-related sources of variance in ground-water quality time series. The absence of this kind of information has severely hindered the design of ground-water sampling programs in the past.An autocorrelation analysis showed that the median sampling frequency for which the predicted ratio of effective independent sample size to total sample size was 0.5 (50% sampling redundancy) ranged from 6 to 14 samples per year. For a predicted ratio of effective independent sample size to total sample size of 0.9 (10% sampling redundancy) the sampling frequency ranged from 3 to 6 samples per year. This suggests that, for the wells sampled, sampling frequencies much higher than monthly can result in considerable loss of information, and may not be cost effective. Care was taken in the design of the field and laboratory sampling protocol to minimize the effects of measurement error. The data analysis confirmed that this goal was accomplished. In most cases considerably less than five percent of the total variability could be attributed to sampling and analytical error. Because of the relatively short duration of the study (42 biweekly sampling occasions at most wells) it was not possible to identify the magnitude of seasonal variations reliably.     

A cost-precision model for marine environmental monitoring,based on time-integrated averages

Ongoing marine monitoring programs are seldom designed to detect changes in the environment between different years, mainly due to the high number of samples required for a sufficient statistical precision. We here show that pooling over time (time integration) of seasonal measurements provides an efficient method of reducing variability, thereby improving the precision and power in detecting inter-annual differences. Such data from weekly environmental sensor profiles at 21 stations in the northern Bothnian Sea was used in a cost-precision spatio-temporal allocation model. Time-integrated averages for six different variables over 6 months from a rather heterogeneous area showed low variability between stations (coefficient of variation, CV, range of 0.6–12.4%) compared to variability between stations in a single day (CV range 2.4–88.6%), or variability over time for a single station (CV range 0.4–110.7%). Reduced sampling frequency from weekly to approximately monthly sampling did not change the results markedly, whereas lower frequency differed more from results with weekly sampling. With monthly sampling, high precision and power of estimates could therefore be achieved with a low number of stations. With input of cost factors like ship time, labor, and analyses, the model can predict the cost for a given required precision in the time-integrated average of each variable by optimizing sampling allocation. A following power analysis can provide information on minimum sample size to detect differences between years with a required power. Alternatively, the model can predict the precision of annual means for the included variables when the program has a pre-defined budget. Use of time-integrated results from sampling stations with different areal coverage and environmental heterogeneity can thus be an efficient strategy to detect environmental differences between single years, as well as a long-term temporal trend. Use of the presented allocation model will then help to minimize the cost and effort of a monitoring program.     

Ordinal time series model for forecasting air quality index for ozone in Southern California

Air quality index (AQI) for ozone is currently divided into six states depending on the level of public health concern. Generalized linear type modeling is a convenient and effective way to handle the AQI state, which can be characterized as non-stationary ordinal-valued time series. Various link functions which include cumulative logit, cumulative probit, and complimentary log-log are considered, and the partial maximum likelihood method is used for estimation. For a comparison purpose, the identity link, which yields a multiple regression model on the cumulative probabilities, is also considered. Random time-varying covariates include past AQI states, various meteorological processes, and periodic components. For model selection and comparison, the partial likelihood ratio tests, AIC and SIC are used. The proposed models are applied to 3 years of daily AQI ozone data from a station in San Bernardino County, CA. An independent year-long data from the same station are used to evaluate the performance of day-ahead forecasts of AQI state. The results show that the logit and probit models remove the non-stationarity in residuals, and both models successfully forecast day-ahead AQI states with almost 90 % of the chance.     

M-估计在水质监测中的应用

Ｍ估计是极大似然估计的一种简称。以水质蒸馏后４氨基安替比林分光光度法测定挥发酚和纳氏试剂比色法测定氨氮为例,分别考察了Ｍ估计与传统的最小二乘法（ＬＳ）所获取的校准曲线方程。回收试验结果表明个别异常值（＜３０％）不影响Ｍ估计,而ＬＳ估计则不然,前者在水质监测中的稳健性亦得以充分证明     

Improving the precision of sample-based forest damage inventories through two-phase sampling and post-stratification using remotely sensed auxiliary information

Many countries have a national forest inventory (NFI) designed to produce statistically sound estimates of forest parameters. However, this type of inventory may not provide reliable results for forest damage which usually affects only small parts of the forest in a country. For this reason, specially designed forest damage inventories are performed in many countries, sometimes in coordination with the NFIs. In this study, we evaluated a new approach for damage inventory where existing NFI data form the basis for two-phase sampling for stratification and remotely sensed auxiliary data are applied for further improvement of precision through post-stratification. We applied Monte Carlo sampling simulation to evaluate different sampling strategies linked to different damage scenarios. The use of existing NFI data in a two-phase sampling for stratification design resulted in a relative efficiency of 50 % or lower, i.e., the variance was at least halved compared to a simple random sample of the same size. With post-stratification based on simulated remotely sensed auxiliary data, there was additional improvement, which depended on the accuracy of the auxiliary data and the properties of the forest damage. In many cases, the relative efficiency was further reduced by as much as one-half. In conclusion, the results show that substantial gains in precision can be obtained by utilizing auxiliary information in forest damage surveys, through two-phase sampling, through post-stratification, and through the combination of these two approaches, i.e., post-stratified two-phase sampling for stratification.