Analyse und praktische Umsetzung unterschiedlicher Methoden des <i>Randomized Branch Sampling</i>

Abstract

Randomized Branch Sampling (RBS) is a multistage sampling procedure, which uses the natural branching of a tree in order to take samples for estimating one or several characteristics of the tree. The three variants of the RBS method that exist at present use unequal probabilities of selection, and selection with replacement (wr) alone or in combination with selection without replacement (wor). In the present work a comparison of the conventional RBS and the variant with selection wor at the first stage is accomplished. In addition, the effects of the choice of the auxiliary variable, the deletion of segments, and the stratification of the tree crown on the variance of the estimate are analysed. A computer program, which allows for the selection of samples both by the conventional and by the modified RBS methods, was developed. Possibilities for the simulation of RBS samples, for the analysis of different auxiliary variables, and the influence of elimination of segments, as well as the stratification on the precision of the estimate are also implemented in the program. In the analysis, data of completely measured trees of three species were used, that is, norway spruce (Picea abies (L.) Karst.), european mountain-ash (Sorbus aucuparia L.) and monterey pine (Pinus radiata D. Don), as well as time estimations from a time study carried out in the field. The results show clearly that the choice of the auxiliary variable affects both the precision of the estimation and the distribution of the samples within the tree crown. Each auxiliary variable produces a set of selection probabilities, which has a certain relationship with the target variable. The more the relationship between the target variable and the selection probabilities approximates a straight line through the origin the more precise the estimate of the target variable and the better the distribution of the samples along the main stem is. The smallest variances were achieved with the diameter of the segments to the power of 2.0 (n orway spruce) up to 2.55 (european mountain-ash) as an auxiliary variable. The stratification of the tree crown as well as the elimination of segments leads to clear improvement of the precision of the estimate depending on species, tree, target variable, and definition and number of strata at the tree. At the trees divided into two strata the decrease of the coefficient of variation of the estimate lies between 10% (european mountain-ash) and 80% (old pine) compared with that from unstratified trees. At the trees divided into three strata, the decrease of the coefficient of variation reaches between 50% (european mountain-ash) and 85% (old pine). Obtaining a certain precision level with the conventional RBS method is mostly more time-consuming than with the variant with selection wor on the first stage. Only with the secondary sample size m=1 the variant with selection wor at the first stage is often more time-consuming than the conventional RBS. In these cases obtaining a certain precision level is up to 5% cheaper with the conventional RBS. When m>1 secondary units are taken with the wor sampling at the first stage, obtaining a certain precision level becomes clearly more time-consuming by the conventional RBS than by the variant with selection wor at the first stage. In general, the larger m the more expensive the conventional RBS becomes compared with the variant with selection wor at the first stage. For example, for the variant with selection wor at the first stage, the sample size m=2 causes a large decrease of the standard error compared with m=1. This, in connection with the short time-spent for an inclusion of the second secondary unit in the sample, implies a large time advantage of this variant compared with the conventional RBS. The smaller the ratio of the variance between the primary units to the entire variance of the estimate the larger is the advantage of the variant with selection wor at the first stage. The advantage mentioned above, observed in unstratified trees without main stem, should be greater in stratified trees where an even higher gain of precision of selection wor compared to the conventional RBS was observed.