Sampling with Partial Replacement Extended To Include Growth Projections

The original theory of Sampling with Partial Replacement (SPR) is modified to incorporate growth model projections of the unmatched plot data on the initial measurement. The Modified Sampling with Partial Replacement (MSPR) theory is illustrated by estimating growth and current volume using simulated plot data. In addition to the assumptions required by SPR, MSPR requires that the growth projection errors be random and have a mean of zero. If these assumptions are met, the modified theory improves the precision of the volume and growth estimators over the original SPR estimator. FOR. SC•. 42(3):328-334. Additional key words: Continuous forest inventory, tree-growth modeling, forest stand projections, volume estimation, growth estimation. are and Cunia (1962) presented a theory of sampling with partial replacement (SPR). The basic aim of the theory was to provide estimators for current stand volume and growth which improved the precision of the estimates by taking advantage of the correlation between repeated measurements. These estimators were based on the theory of weighted means (Brownlee 1965) and provided abasis for combining unmatched temporary sample plot (TSP) data from two occasions with remeasured permanent sample plot (PSP) data. The improvement in precision came first from a direct increase in sample size and second from exploiting the correlation between the matched PSP and the unmatched TSP on both occasions. Since the original paper, several extensions of the original theory have appeared in the literature. Cunia (1965) extended the theory of SPR by using multiple regression estimates. This extension showed that using multiple linear regression to estimate the second occasion parameter was more efficient than using the simple linear estimates, as in the original SPR theory. Cunia and Chevrou (1969) extended the theory of SPR to accept measurements on three or more occasions. Scott (1984) clarified the presentation of SPR theory and resolved the issue of appropriate stimators for variance when samples are small. Van Deusen (1989) formulated the SPR estimators using matrix notation and incorporated addltivity constraints and estimators for components of growth. Newton et al. (1974) developed multivariate estimators for sampling with partial replacement. This extension allowed for the simultaneous estimation in the change of several forest characteristics for two successive forest measurements. The procedure is considered more efficient because it takes advantage of the inherent correlation that exists between different forest characteristics. Han sen and Hahn (1983 ) described the use of growth projection simulators in forest inventory using double sampling for regression estimation Hansen (1990) combined the use of a growth model with several sampling methods including a variation of SPR. This paper, independently of Hansen' s work, extends the mathematical nd statistical presentation of the SPR theory by incorporaUng growth projections for the temporary plots. It is assumed that he growth simulation model provides unbiased estimates of future volume, subject o additive random error so that the initial and projected values are not perfectly correlated. An application of this approach ispresented using simulated plot data. Mike Bokalo is Resource Analyst, Forest Management Division, Land and Forest Services, Alberta Environmental Protection, 8th Floor Bramalea Building, 9920-108 Street, Edmonton, Alberta T5K 2M4; Stephen J. Titus is Professor, Department of Renewable Resources, University of Alberta, T6G 2H1 [author to whom all correspondence should be addressed: (403) 492-2899, Fax (403) 492-4323]; and Douglas P. Wiens is Professor, Department of Mathematical Sciences, University of Alberta, T6G 2H1. Research supported by a grant from the Natural Sciences and Engineenng Research Council of Canada. Acknowledgments: We are very grateful to Mr. Dave J. Morgan, Mr. Tom Lakusta and Mr. Robert Held of the Forest Management Division, Land and Forest Service, Alberta Environmental Protection, for their valuable help and cooperation. This paper is based on work conducted while M. Bokalo was a M.Sc. student and receiving financial assistance from the Alberta Forest Development Research Trust Fund and the Department of Renewable Resources, University of Alberta. Manuscript received December 1, 1994. Accepted October 10, 1995. Copyright ̧ 1996 by the Society of American Foresters 328 Forest Sctence 42(3) 1996 The Modified Sampling with Partial Replacement Procedure (MSPR) Notation for the development of estimators of current volume, growth, and variances follows that of Ware and Cunia (1962). The naming conventions to derive the MSPR estimators are presented in Table 1. The sample volumes from the first occasion are algebraically represented by the letter X. The sample volumes from the second occasion are represented algebraically by the letter Y. Unmatched temporary sample plot volumes from the initial occasion are represented by X u. New TSP on the second occasion are represented by Yn' Remeasured (matched) PSP data are algebraically represented on the first occasion by an Irn and on the second occasion by a Ym. The new source of data for the MSPR procedure is a prediction from a growth model and is represented algebraically by the symbol Zu. The input data for the growth model is the same TSP data X u representing the unmatched data on the initial occasion. Since the TSP data are assumed to be a random sample from the population, and the growth model is assumed to provide unbiased projections, the model predictions are random, unbiased estimates of the population stand volume on the second occasion. It is also assumed that the growth model used to make the predictions is an individual tree growth model composed of relationships derived from independent data. As will be seen in the development of estimators, the magnitude of unbiased random errors in the growth model is reflected in the variance of the estimators. Current Volume Estimator The MSPR volume estimator • MSPR is a linear combination of the five sample averages Xm,Xu, Ym, Yn and •u. The coefficients are to be chosen so that the estimator has a minimum variance, subject to the requirement that it be unbiased. Let gl and 112 be the true mean volumes on the first and second occasions respectively, so that E(.,•m)=e(•,)=111 and e(Fm)=e(•n)=E(•,)=g 2 13xl z = COV[X•,Z•]/O2(Zu) and O•xl z = 111•XIZ112 These are the slope and intercept coefficients of the linear regressions Of Xrn (dependent) on Yrn (independent) and Xu on Zu. The residual variances around the regression lines are: o:xlr = E[ {X m O•xl r [3xlrYm}: ] = o:(Xm)(1 -P:m) (1) O:XlZ = E[ {X•, O•xl z [SxlzZ•, }: ] = 0 • (X•,)(1 p:•,) (2) where Pm and Pu are the coefficients of correlation between Xm and Ym and between Xu and Zu respectively. Note that any linear combination of the five sample averages can be written as an equivalent linear combination containing Ym,Yn,Zu and the differences (residuals) •m --•xIrFm and •'• -•x,z•. We consider these differences, rather than the sample averages •'m and •'• themselves, for two reasons: These differences and the averages Ym, Yn,Z•' are pairwise uncorrelated. This is mathematically convenient, since their variances are then additive. It also allows for straightforward interpretations of the contributions of the individual terms. See the discussion at the end of this section. In the presence of Fm and ju the averages '•m and •'• contain additional information about 112 only through the differences. This explains why the differences are those from regressions of Xm and Xu on Ym and Zu respectively, rather than from Ym and Zu on Xm and Xu. Recall that in Ware and Cunia (1962), the results are expressed in terms of the regression of Ym on Xm. Although expressions analogous to the approach of Ware and Cunia (1962) can be constructed for the MSPR approach, they are much more cumbersome than those presented here.