3P Sampling Research Articles

Sampling with probability proportional to prediction (3P sampling) using covariates derived from spherical images

Using a two-phase sampling approach with systematic selection of large samples of covariates followed by a sampling with probability proportional to prediction (3P sampling) process to subsample field measures of the parameters of interest can be an efficient design to sample larger forest areas. To assist in obtaining predictions for each sample plot consistently and rapidly, we propose using a 360° spherical camera. In this study, three covariates derived from spherical images were evaluated: (i) basal area (P[BA]); (ii) sum of squared heights per hectare (P[SHT]); and (iii) stem fraction (P[SF]). These covariates were used to estimate volume. Sample simulations showed no biases in volume estimates for any of the three covariates. Overall, P[SF] had the lowest standard error percentages across different simulated sample sizes (10% for five subsamples to 2.5% for 50 subsamples), even though it had the lowest correlations with field volume (correlation = 0.30–0.31). This may be a result of the relatively consistent stand conditions within the study site. Based on our results, standard errors of 5% were obtainable with measurement fractions of about 25% of the number of image-based predictions when using P[SF] or P[BA] and 75% when using P[SHT].

Design-based mapping of tree attributes by 3P sampling.

The estimation of individual values (marks) in a finite population of units (e.g., trees) scattered onto a survey region is considered under 3P sampling. For each unit, the mark is estimated by means of an inverse distance weighting interpolator. Conditions ensuring the design-based consistency of maps are considered under 3P sampling. A computationally simple mean squared error estimator is adopted. Because 3P sampling involves the prediction of marks for each unit in the population, prediction errors rather than marks can be interpolated. Then, marks are estimated by the predictions plus the interpolated errors. If predictions are good, prediction errors are more smoothed than raw marks so that the procedure is likely to better meet consistency requirements. The purpose of this paper is to provide theoretical and empirical evidence on the effectiveness of the interpolation based on prediction errors to prove that the proposed strategy is a tool of general validity for mapping forest stands.

Sampling with probability proportional to prediction: rethinking rapid plant diversity assessment

Rapid biodiversity assessment (RBA) methods are regularly applied to assess plant species richness. One approach is developing sampling designs that integrate expert knowledge. 3P sampling does so by selecting samples with probability proportional to prediction (3P). Higher effort is allocated to areas with high species richness based on predictions made on the ground. 3P sampling for RBA was simulated considering two major factors: knowledge of plant species and types of rapid assessment. Two large census forest plots over 25 ha in size were used. Results showed that sampling error of 3P sampling for RBA was relatively low and could be improved by changing methods of prediction. Sampling was more efficient and accurate when predictions were made with knowledge about abundant species instead of random species. When such prediction was made, knowing only three quarters of the total species richness in a forest performed as well as full knowledge. Randomly walking around in an area and predicting also increased efficiency and accuracy compared to standing stationary at an assessment point. This was counterintuitive to the common practices of establishing ground plots for assessment. Our findings propose that 3P sampling for RBA is workable through engaging local communities in an assessment, which could be cost-effective. Finally, the procedure laid out in this study is the first unequal probability sampling design proposed for RBA.

Potential for wider application of 3P sampling in forest inventory

Sampling with probability proportional to prediction (3P sampling) is useful where the variable of interest to a forest inventory is costly to measure and where there exists a cheaper to measure auxiliary variable, which correlates positively with the variable of interest. Two forms of 3P sampling, termed “classical” and “point-” 3P sampling, have received some use in forest inventory. However, both have limitations that have restricted their use mainly to estimation of tree stem wood volume for timber sales over small forest areas in North America. A more general form of 3P sampling, termed here “ordinary” 3P sampling, has been all but ignored to date. It has potential for use in inventory of a broad range of forest attributes, both floral and faunal and both commercial and environmental, across large or small forest areas. Using a common mathematical approach, the present work derives the estimators of the population mean for these three forms of 3P sampling. Their properties are compared with simple random sampling through Monte Carlo simulations based on two example forest populations. The work lays a basis from which 3P sampling might develop further and enjoy wider application in forest inventory than has been the case previously.

Sampling of Sparse Species with Probability Proportional to Prediction

In this article, a 3P sampling (Probability Proportional to Prediction) approach is presented for surveying sparse species connected to certain types of substrates. The method uses the surveyor's judgement of the probability of finding the species on a substrate as the base for selection of substrates for species inventories. The method is presented together with estimators, variances and variance estimators. The method is first presented for sampling based on all substrates in a study area and then as a subsampling technique in a two-stage design in which plots or strips are selected in a first stage. The presented approach was evaluated as a subsampling technique in a strip survey of calicioid lichen species associated with coarse broadleaf trees. In comparison with a simple random subsampling without replacement, 3P subsampling was in one study area found to be an improvement with 30-55% in terms of standard errors. The improvement was more modest in the other study area, only between 10-16%. The strip survey with 3P subsampling was more cost-efficient than a strip survey without subsampling except in one case. Based on the results in the test, 3P sampling seems to have a potential for sampling sparse species.

Composite and Calibration Estimation Following 3P Sampling

Abstract Two alternatives to the generalized ratio estimator, i.e., Grosenbaugh's 3PSEVENTH, are studied. Both make more complete use of auxiliary information which is available at the conclusion of 3P sampling. Both appear to be more biased but more accurate than 3PSEVENTH. For. Sci. 45(2):179-185.

Double sampling with importance sampling to eliminate bias in tree volume estimation of the centroid method

The bias associated with the centroid method for estimating tree volume was removed by double sampling using the centroid method in the first stage of estimation and importance sampling in the second. The bias and efficiency of various sampling methods were investigated in a number of different simulation studies. Three data sets, two of ponderosa pine and the other of mixed species, were used to evaluate the double sampling estimator. We assumed that for small area estimates, such as timber sales, every tree in the population could be visited in the first stage and an estimate of bole volume generated using the centroid method. Both simple random and 3P sampling were studied as methods for selecting the second-stage sample trees. For the estimation of volume over a large area, we chose point-3P sampling for applying the double sampling estimator. The bias associated with using the centroid method alone was reduced in every case. No consistent trends for the three data sets were found when efficiency was measured in terms of the simulation variance. The double sampling estimator was the most efficient in some cases and least efficient in others.

Outlier-resistant estimators for Poisson sampling: a note

Poisson (3P) sampling is a commonly used method for generating estimates of timber volume. The usual estimator employed is the adjusted estimator, Y hata. The efficiency of this estimator can be greatly influenced by the presence of outliers. We formalize such a realistic situation for high-value timber estimation for which Y hata is inefficient. Here, yi approx beta xi for all but a few units in a population for which yi is large and xi very small. This situation can occur when estimating the net volume of high-value standing timber, such as that found in the Pacific Northwest region of the United States. A generalized regression estimator and an approximate Srivastava estimator are not affected by such data points. Simulations on a small population illustrate these ideas.

Composite Estimation for Forest Inventories Using Multiple Value Groups and Strata

Abstract Due to economic and political pressures placed on timber production, information from timber inventories needs to be broken down into information on particular value groups, which are groupings based on individual tree characteristics. Thus estimates of totals and variances need to be computed for individual value groups, all value groups combined, and any collection of value groups. A method for reporting these estimates using a variance-covariance matrix is given. This is followed by two examples of the estimation of totals and variances for value groups applied to a population comprising two strata. Poisson (3P) and sample-tree sampling are used in the first and second strata, respectively. These examples are included to highlight important estimation considerations. A worked example based on an artificial population is also given. The population for the worked example is divided into two strata, with three value groups in each stratum. 3P and sample-tree sampling are used in the first and second strata. South. J. Appl. For. 20(2):103-109.

Error assessment for a provincial timber inventory

In this case study of the Saskatchewan provincial timber inventory, the major sources of errors were identified and suitable procedures presented for their estimation. Applying those procedures on a pilot area, the results showed that sampling error was by far the most important component affecting the accuracy of volume estimates. Variances arising from the use of taper functions to estimate tree volumes were under 0.05%, and variances from 3P subsampling were about 3% of the total variance. A computer simulation experiment showed that a 2% measurement bias in both diameter and height measurements may result in 8% bias in the volume estimate. Key words: 3P sampling, taper functions, error budget

Centroid sampling: A variant of importance sampling for estimating the volume of sample trees of radiata pine

Simulation of importance sampling using a database comprising detailed measurements along the bole of 114 trees of radiata pine (Pinus radiata D. Don) revealed a point on the bole at a relative height of approximately 0.3 which, if selected as the point of importance sampling, gave minimum bias in the estimate of tree volume. A theoretical examination of solids of revolution identified this point as the centroid of the tree. A variant of importance sampling, called centroid sampling, in which the point of sampling is fixed at the height of the centroid rather than being determined at random, was then applied in conjunction with 3P sampling to estimate the volume of a forest plot of 55 trees of 36-year-old radiata pine in Stromlo Forest, Australian Capital Territory. The resulting estimate of stand volume was both precise and accurate.

Adjusted Inclusion Probabilities with 3P Sampling

Abstract The mean square error efficiency of Grosenbaugh's (1964) adjusted 3P estimator is examined relative to four alternative estimators which use regression estimates of the conditional inclusion probabilities and true conditional inclusion probabilities. Actual mean square errors were calculated as the result of exhaustive sampling of populations that ranged in size from 9 elements to 144 elements and that exhibited widely different proportionality between the variable of interest and the auxiliary variable used for probability selection. The adjusted 3P estimator is as efficient or more efficient than all four competitors except when the unconditional probability of inclusion neared unity for some population elements. For. Sci. 33(3):617-631.

Generating the list of random numbers for 3P sampling

Summary This article describes manual and computer based methods for generating random number lists for use in conducting 3P sample surveys. It outlines a procedure for selecting from pages of computer generated output the most satisfactory page of random numbers to use for selecting the 3P sample when the number of units (trees, plots, etc.) to be assessed is small, e.g., <500.

Estimation of bole volume by importance sampling

A sampling methodology to provide an unbiased estimate of bole volume is introduced, with specific regard to its incorporation as an alternative to detailed dendrometry in a 3P sampling framework. In a demonstration of the technique to estimate merchantable volume in a 91-tree population, the average relative sampling error was 7.4%. A two-tool implementation of 3P sampling is proposed and discussed.